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2007 Archive

  • 01/08/07
    Ualbai Umirbaev - Eurasian National University \\ Astana, Kazakhstan
    Automorphisms of polynomial and free algebras

  • 01/09/07
    Nitu Kitchloo - UCSD
    Formal groups laws and their applications in topology

    This is the organizational meeting for our topology learning semiar. The topic this quarter is formal group laws and their applications in topology. Nitu will give an introduction to the topic and we will distriubute the talks.

  • 01/09/07
    Jiawang Nie - Institute for Mathematics and its Applications (IMA) \\ University of Minnesota
    Semidefinite and polynomial optimization

    This lecture will talk about semidefinite programming (SDP) and
    its applications in global polynomial optimization. Firstly, after
    introducing SDP, we will how to represent k-elliptic curves by SDP.
    Secondly, after an overview of the sum of squares (SOS) relaxation, which
    can be reduced to SDP, we will present gradient SOS relaxation. While the
    general SOS relaxation has a gap in finding the global minimum, the gradient
    SOS relaxation can find the global minimum whenever a global minimizer
    exists. Lastly, we will show how to exploit sparsity in SOS and its
    applications in sensor network localization.

  • 01/09/07
    Daniel R. Reynolds - Assistant Project Scientist \\ Dept. of Mathematics UCSD
    Mathematical modeling and simulation in fusion energy research

    Fusion energy holds the promise of a clean, sustainable and safe
    energy source for the future. While research in this field
    has been ongoing for the last half century, much work
    remains before it may prove a viable source of energy. In
    this talk, I discuss some of the scientific and engineering
    challenges remaining in fusion energy, and the role of
    mathematics research in overcoming these obstacles. In
    particular, I will discuss some of the mathematical models
    used in studying fusion stability and refueling, how
    solutions to those models may be approximated, and introduce
    some model improvements to better simulate fusion processes.

  • 01/10/07
    Mohammed Abouzaid - University of Chicago
    Homological mirror symmetry for toric varieties

    I will begin by explaining the statement of the Homological
    Mirror Symmetry conjecture for Fano toric varieties and outline how
    Lefschetz fibrations have been used to prove the conjecture in some cases.
    I will then show how Mikhalkin's flavour of tropical geometry can be used
    to prove half of the homological mirror conjecture for all smooth
    projective toric varieties (dropping the Fano condition!).

  • 01/10/07
    Bakhadyr Khoussainov - Department of Computer Science \\ University of Auckland
    Automatic structures

    We introduce the concept of automatic structure. Informally, these
    are structures that can be defined in terms of automata. By automata
    we mean any of the following machines: finite automata, tree automata,
    Buchi automata, and Rabin automata.

    Nerode and the speaker initiated the systematic study of automatic
    structures in 94. An important property of automatic structures is
    that these structures are closed under the first order interpretations
    and have effective semantics. In particular, the first order theory
    of any automatic structure is decidable.

    The theory of automatic structures has become an active research
    area in the last decade with new and exciting results. In this talk
    we survey recent results in the area and outline some of the interesting
    proofs. The talk will provide many examples.

    Some of the results of the talk are published in LICS 01-04 and
    STACS04 conference proceedings. Results are joint with Nerode,
    Rubin, Stephan, and Nies.

  • 01/11/07
    Laura DeMarco - University of Chicago
    Complex dynamics and potential theory

    I will begin with the basics of the two subjects, with the goal of
    explaining how each has been used as a tool in the other. The first half
    will be devoted to foundational results, dating to the 1930s for potential
    theory and the 1980s for complex dynamics. The second half will be
    devoted to more recent developments.

  • 01/11/07
    Maia Averett - Graduate Student, UCSD
    That group's a real klass act: A little introduction to K-theory

    K-theory is a valuable tool with applications in various areas of mathematics, including topology, geometry, algebraic geometry, and number theory. In this talk I will attempt to give a rough outline of what K-theory is about, in the algebraic as well as the geometric context. Given a ring R and a space X, I will define the groups K(R) and K(X), and explain the relation between the two. I also hope to outline some applications in various fields. This talk should be accessible and I hope interesting to students of both algebra and geometry alike.

  • 01/11/07
    Allen Knutson - UCSD
    Moduli spaces and quotients by groups

    Many mathematical objects come in continuous families, prompting the
    desire to define a ``universal family'' that contains each such object
    exactly once up to isomorphism. When this isn't possible (because the
    family would be too bad to be worthwhile -- I'll talk about this
    we can try to come close, by including only ``stable'' objects.

    Frequently the universal family is constructed by starting with a
    bigger family that includes each object many times, then dividing
    by a group action that implements the isomorphisms. There are two ways to
    do this, one algebro-geometric (complex) and one symplecto-geometric
    (real), and I'll give some idea of why they agree.

    The main example will be the space of $N$ ordered points on the
    Riemann sphere, modulo M\"obius transformations. These are unstable if two

  • 01/11/07
    Mihnea Popa - University of Chicago
    Vanishing theorems and Fourier-Mukai transforms

    Vanishing theorems for cohomology groups are one of the
    essential tools of modern algebraic geometry, and have particularly
    important applications in higher dimensional geometry. Under strong
    positivity assumptions on line bundles, for example ampleness, there are
    well-known "standard" vanishing theorems, like those of Kodaira, Nakano
    and Kawamata-Viehweg. They have very useful partial analogues, called
    Generic Vanishing Theorems - first discovered by Green and Lazarsfeld -
    when the positivity hypotheses are weakened. I will describe all of the
    above and their importance, and then explain that recent techniques based
    on Fourier-Mukai functors and homological algebra can be used to widely
    extend the context of generic vanishing, and relate it to standard
    vanishing. As an application, I will explain how to generalize the results
    of Green-Lazarsfeld to a version of Kodaira vanishing under weak
    positivity hypotheses.

  • 01/11/07
    Nigel Boston - University of South Carolina
    Arboreal Galois representations

    We describe various results on Galois groups of iterates of a given quadratic polynomial. Joint work with Rafe Jones (University of Wisconsin).

  • 01/11/07
    Igor Kukavica - Mathematics, USC
    Conditional regularity for solutions of the Navier-Stokes equations

    We consider sufficient conditions for regularity
    of Leray-Hopf solutions of the Navier-Stokes equation. By a
    result of Neustupa and Panel, a Leray-Hopf weak solution is
    regular provided a single component of the velocity is
    bounded. In this talk we will survey existing and present
    new results on one component and one direction
    regularity. We will also show global regularity for a class
    of solutions of the Navier-Stokes equation in thin
    domains. This is a joint work with M. Ziane.

  • 01/16/07
    Kenley Jung - UCLA
    Applications of geometric measure theory to von neumann algebras

    One of the central problems in the study of von Neumann algebras
    is to find computable invariants which can distinguish nonisomorphic
    algebras. In the 1980s, Dan Voiculescu developed a noncommutative probability
    theory in order to understand a particular class of such von Neumann algebras.
    Specifically, he defined subsets of $R^n$ called microstate spaces which
    model the behavior of a generating set of a given von Neumann algebra.
    Since these spaces are subsets of $R^n$, classical analytic tools such as
    volume can be applied to them.

    I will discuss how the application of ideas from geometric measure theory
    to microstate spaces has provided insight into the general problem of
    invariants and answered some longstanding questions in von Neumann

  • 01/18/07
    Ery Arias-Castro - UCSD
    Searching for a trail of evidence in a maze

    Consider the complete regular binary tree of depth M oriented from the root to the leaves. To each node we associate a random variable and those variables are assumed to be independent. Under the null hypothesis, these random variables have the standard normal distribution while under the alternative, there is a path from the root to a leaf along which the nodes have the normal distribution with mean A and variance 1, and the standard normal distribution away from the path. We show that, as M increases, the hypotheses become separable if, and only if, A is larger than the square root of 2 ln 2. We obtain corresponding results for other graphs and other distributions. The concept of predictability profile plays a crucial role in our analysis.

    Joint work with Emmanuel Candes, Hannes Helgason and Ofer Zeitouni.

  • 01/18/07
    Andrew Linshaw - UCSD
    An introduction to vertex algebras

    Vertex algebras arose out of conformal field theory, and were
    first defined mathematically by Borcherds in 1986. Since then, they have
    found applications in many areas of mathematics, including representation
    theory, number theory, finite group theory, and geometry. Vertex algebras
    are vector spaces (generally infinite-dimensional) which are equipped with
    a family of bilinear products (indexed by the integers) which in general
    are neither commutative nor associative. In many ways they behave like
    ordinary associative algebras, and the usual categorical and formal
    algebraic notions like homomorphisms, ideals, quotients, and modules over
    vertex algebras are easy to define.

    In this talk, I'll define vertex algebras, give some basic examples,
    indicate how to do computations, and hopefully state some interesting open

  • 01/18/07

  • 01/18/07
    Todd Kemp - CLE Moore Instructor \\ Department of Mathematics, MIT
    Dimension in global analysis and free probability

    How can you tell the dimension of a manifold? One answer lies in studying
    the flow of heat on the manifold. {\em Heat flow} is a smoothing process on
    Riemannian manifolds, whose long-term behaviour is intimately linked to
    global geometry. However, the {\em short-time} smoothing behaviour is
    universal: it depends only upon the dimension of the manifold,
    and determines the dimension uniquely.


    In non-commutative geometry, the over-arching principal is to study a
    non-commutative algebra, pretend it is an algebra of smooth functions
    or differential operators on a {\em non-commutative manifold}, and import
    analytic and algebraic tools from global analysis to discover geometric
    facts about this manifold.


    While using heat flow is an excessively difficult way to determine the
    dimension of a manifold, it yields one approach to define dimension for
    non-commutative manifolds. In the context of {\em free probability} (one
    branch of non-commutative geometry concentrating on analytic properties
    of free groups), this leads, inexorably, to the somewhat comical-sounding
    conclusion that {\em all free groups have dimension $6$}.


    In this talk, I will outline those aspects of free probability which relate to
    heat kernel analysis, and make the connection between dimension and
    heat flow clear. I will also discuss recent joint work with Roland Speicher,
    showing that {\em all free semigroups have dimension $4$}.

  • 01/22/07
    John B. Greer - Courant Institute of Mathematical Sciences \\ New York University
    Upper bounds on the coarsening rates of discrete ill-posed nonlinear diffusions

    I will discuss a recent proof of a weak upper bound on the
    coarsening rate of the discrete-in-space version of an
    ill-posed, nonlinear diffusion equation. The continuum
    version of the equation violates parabolicity and lacks
    a complete well-posedness theory. In particular,
    numerical simulations indicate very sensitive
    dependence on initial data. Nevertheless, models based
    on its discrete-in-space version, which I will discuss,
    are widely used in a number of applications, including
    population dynamics (chemotactic movement of bacteria),
    granular flow (formation of shear bands), and computer
    vision (image denoising and segmentation). The bounds
    have implications for all three applications. This is
    joint work with Selim Esedoglu (U. of Michigan

  • 01/22/07
    Daniel Krashen - Yale University
    Geometry and division algebras

    In this talk I will introduce the study of finite
    dimensional division algebras and the Brauer group, and I will discuss
    the fundamental problem of computing the index of a Brauer class. It
    turns out that algebraic geometry can play an important role in this
    problem. In particular I will describe the theory of twisted sheaves
    and how it allows one to solve this problem in certain cases.

  • 01/23/07
    Daniel Krashen - Yale University
    Zero-dimensional cycles on homogeneous varieties

    The study of cycles on homogeneous varietes has seen a great deal of
    activity in the past few years. In particular, new results about
    cycles on quadric hypersurfaces has resulted in fundamental
    breakthroughs in the theory of quadratic forms.

    The goal of this talk will be to answer a basic question about cycles
    on homogeneous of more general types. In particular, we will address
    the problem of calculating the group of zero dimensional cycles on
    such varieties. In the case of quadrics, this was first done by Swan
    in 1989 (and also independently by Karpenko). Similar computations
    were made later for certain classes of other homogeneous varieties by
    Merkurjev and Panin.

    By using the geometry of Hilbert schemes of points on homogeneous
    varieties, we will describe how to extend the previous results and to
    compute the group of zero cycles for some homogeneous varieties of
    each of the classical types.

  • 01/23/07
    Julien Dubedat - New York University \\ Courant Institute of Mathematical Sciences
    Schramm-Loewner Evolutions on Riemann surfaces

  • 01/23/07
    Jason Bandlow - UCSD Graduate Student
    A new proof of the hook formula

    A new proof of the hook formula
    Abstract: The hook-length formula is a well known result expressing the
    number of standard tableaux of shape $\lambda$ in terms of the lengths
    of the hooks in the diagram of $\lambda$. Many proofs of this fact have
    been given, of varying complexity. I'll give a new and simple proof
    which uses only some power series and partial fractions expansions.
    Other versions of the hook formula will also be discussed.

  • 01/24/07
    John Erik Fornaess - University of Michigan, Ann Arbor
    Plurisubharmonic defining functions

    Suppose a domain has a defining function which is plurisubharmonic at each boundary point.
    Does it have a plurisubharmonic defining function?
    I will report on some recent progress on this question. This is joint work with Anne Katrin Herbig.

  • 01/25/07
    David Clark - UCSD Graduate Student
    Exotic 7-Spheres Are Hot

    Exotic spheres, manifolds homeomorphic but <italic>not diffeomorphic</italic> to the standard sphere, had eluded mathematicians since the dawn of differential topology. In 1956, John Milnor found one in seven dimensions. His short paper on the result stunned the math world and won him the Field's medal. In this talk, we'll survey the ideas of smooth structures on manifolds and smooth vector bundles, and with a few smooth moves we'll construct Milnor's exotic sphere.

    Note: This is meant to be a highly accessible talk; some familiarity with algebraic topology will enhance the experience, but is not strictly necessary!

  • 01/25/07
    Nora Ganter - University of Illinois Urbana-Champaign
    The relationship between elliptic cohomology and string theory - orbifold genera, product formulas and cohomology operations

    Elliptic cohomology is a field at the intersection of number theory,
    algebraic geometry and algebraic topology. Its definition is very
    technical and highly homotopy theoretic. While its geometric definition
    is still an open question, elliptic cohomology exhibits striking formal
    similarities to string theory, and it is strongly expected that a
    geometric interpretation will come from there.

    To illustrate the interaction between the two fields, I will speak about
    my work on orbifold genera and product formulas:
    After a very informal introduction to elliptic cohomology, I will
    discuss string theory on orbifolds and explain how a formula by
    Dijkgraaf, Moore, Verlinde and Verlinde on the orbifold elliptic genus
    of symmetric powers of a manifold motivated my work in elliptic
    cohomology. I will proceed to explain why elliptic cohomology provides a
    good framework for the study of orbifold genera.

  • 01/25/07
    Eyal Goren - McGill University
    Superspecial abelian varieties

    Superspecial abelian varieties are positive characteristic
    phenomenon. These are abelian varieties enjoying many special, even
    super special l;-) properties, which I'll attempt to explain. After some
    background about stratifications of moduli spaces and how, in that
    context, the superspecial locus is the smallest, I will describe some
    more recent work on superspecial abelian varieties. In particular, work
    of M.-H. Nicole and A. Ghitza that makes use of the superspecial locus
    to construct modular forms and mention very briefly some beautiful
    graphs that can be constructed from the superspecial locus.

  • 01/25/07
    Alex Ghitza - Colby College
    Galois representations (mod p) and modular forms

    The relation between modular forms and Galois representations
    is more popular than ever, thanks to recent progress on the
    Fontaine-Mazur conjecture by Kisin, and on Serre's conjecture by Khare
    and Wintenberger. After a short introduction to this circle of ideas, I
    will discuss ongoing work on higher-dimensional generalizations, in the
    context of Gross' philosophy of modular Galois representations.
    I will attempt to make the talk as self-contained and accessible as

  • 01/29/07
    Vladimir Kirichenko - Kiev University, Ukraine
    Tiled orders and Frobenius rings

    By a tiled order we mean a right Noetherian prime semiperfect and semidistributive ring with the nonzero Jacobson radical. For example, a serial tiled order $A$ is a Noetherian (but non-Artinian) serial indecomposable ring. A ring $A$ is decomposable if $A = A_1 \times A_2$, otherwise $A$ is indecomposable. Every serial tiled order $A$ is hereditary and Gorenstein, i.e., $inj.dim_A A_a = inj.dim_A {_A} A = 1$.

    Let $R(A)$ be the Jacobson radical of a tiled order $A$. For any tiled order $A$ there exists a countable set of two sided ideals $I_1 \supset I_2 \supset \dots$, where $R^2 (A) \supset I_1, I_{k+1} \neq I_k$ and all quotient rings $A/I_k$ are Frobenius.

    For any permutation $\sigma \in S_n$ there exists a Frobenius ring $B$ with the Nakayama permutaion $\sigma$. We consider the exponent matrices of tiled orders, in particular, Gorenstein matrices. We discuss the relations between exponent matrices and quivers of tiled orders, cyclic Gorenstein orders and doubly stochastic matrices, Gorentstein matrices and tiled orders of injective dimension one.

  • 01/30/07
    Gordan Savin - University of Utah
    Bernstein's center for real groups

  • 01/30/07
    Sarah Mason - University of California, Berkeley
    Decomposing the Schur functions and their crystal graphs

    The Schur functions can be decomposed into "nonsymmetric Schur
    functions" obtained through a certain specialization of Macdonald
    polynomials. We explore several combinatorial properties of these
    polynomials and a connection to crystal graphs.

  • 02/01/07
    Wee Teck Gan - UCSD
    The unreasonable effectiveness of modular forms in arithmetic

    How many times can a prime number be expressed as the sum of n squares? What is the asymptotic distribution of integer points on a family of ellipsoids $ax^2 + by^2 + cz^2 = n$ as $n$ tends to infinity?
    I will explain how modular forms can be used to address these questions and others.

  • 02/01/07
    Jim Haglund - University of Pennsylvania
    The combinatorics of nonsymmetric Macdonald polynomials

  • 02/01/07
    Shih-Hsien Yu - Mathematics, Hong Kong University
    On recent developments on the Green's function for the Boltzmann equation and its application to nonlinear problems

    In this talk we will survey the development of the Green's
    function for the Boltzmann equation. The talk will include
    the motivation from the field of hyperbolic conservation
    laws, the connection between the Boltzmann equation and the
    hyperbolic conservation laws, and the particle-like and the
    wave-like duality in the Boltzmann equation. With all these
    components one can realize a clear layout of the Green's
    function for the Boltzmann equation. Finally we will present
    the application of the Green's function to an
    initial-boundary value problem in the half space domain.

  • 02/05/07
    Frank Kelly - University of Cambridge
    Flow level models of Internet congestion control

    Variability in the number of simultaneous flows present can have a substantial impact on the perceived performance of packet networks such as the Internet. While the packet level behaviour of a given set of flows is by now well understood, less is known about the stochastic behaviour of the number of flows in progress on different routes through the network. In this talk we describe recent work on Brownian models of networks in heavy traffic. Joint work with Ruth Williams.

  • 02/05/07
    Adrian Wadsworth - UCSD
    Valuations on central simple algebras

  • 02/06/07
    Christian Haesemeyer - University of Illinois Urbana Champaign
    On the algebraic K-theory of singularities

    Algebraic K-theory is a highly complicated invariant of
    algebraic varieties that encodes arithmetic, geometric and algebraic
    In this talk, I will try to make this distinction somewhat less vague and
    explain how to isolate some of the algebraic and geometric information
    K-theory provides about singularities, leading to proofs of various
    longstanding conjectures.

  • 02/06/07
    Mauricio de Oliveira - Department of MAE, UCSD
    Numerical optimization assisted by noncommutative symbolic algebra

    This talk describes how a symbolic computer algebra
    tool (NCAlgebra) that handles symbolic matrix
    (noncommutative) products can be used to assist the
    numerical solution of semidefinite programs where the
    variables are matrices. The idea is to keep matrix
    variables aggregated at all steps of a primal-dual
    interior-point algorithm in which symbolic expressions
    are automatically generated and used iteratively.

  • 02/06/07
    Gregg Musiker - UCSD, Graduate Student
    Combinatorics of elliptic curves and chip-firing games

    For a given elliptic curve $E$ over a finite field $F_q$, we let $N_k =
    \#E(F_{q^k})$, where $F_{q^k}$ is a $k$th degree extension of the finite
    field $F_q$. Because the Zeta Function for $E$ only depends on $q$ and
    $N_1$, the sequence $\{N_k\}$ only depends on those numbers as well.
    More specifically, we observe that these bivariate expressions for $N_k$
    are in fact polynomials with integer coefficients, which alternate in sign
    with respect to the power of $N_1$.

    This motivated a search for a combinatorial interpretation of these
    coefficients, and one such interpretation involves spanning trees of a
    certain family of graphs. In this talk, I will describe this
    combinatorial interpretation, as well as applications and directions for
    future research. This will include determinantal formulas for $N_k$,
    factorizations of $N_k$, and the definition of a new sequence of
    polynomials, which we call elliptic cyclotomic polynomials.

    One of the important features of elliptic curves which makes them the
    focus of contemporary research is that they admit a group structure.
    During the remainder of this talk I will describe chip-firing games, how
    they provide a group structure on the set of spanning trees, and numerous
    ways that these groups are analogous to those of elliptic curves.

  • 02/06/07
    Jian Song - Johns Hopkins University
    Canonical K\"ahler metrics and the K\""ahler-Ricci flow"

    The existence of K\"ahler-Einstein metrics on a compact K\""ahler manifold

  • 02/08/07
    Allen Knutson - UCSD
    Moduli spaces and quotients by groups

    Many mathematical objects come in continuous families, prompting the
    desire to define a ``universal family'' that contains each such object
    exactly once up to isomorphism. When this isn't possible (because the
    family would be too bad to be worthwhile -- I'll talk about this behavior),
    we can try to come close, by including only ``stable'' objects.

    Frequently the universal family is constructed by starting with a
    bigger family that includes each object many times, then dividing
    by a group action that implements the isomorphisms. There are two ways to
    do this, one algebro-geometric (complex) and one symplecto-geometric
    (real), and I'll give some idea of why they agree.

    The main example will be the space of $N$ ordered points on the
    Riemann sphere, modulo M\"obius transformations. These are unstable if

  • 02/08/07
    David Meyer - UCSD
    Quantum correlated equilibria in games

    Many quantum games can be understood as protocols for the communication and processing of quantum information, and should be compared to classical games with communication. After introducing two of the standard quantum game protocols, I'll explain how this comparison works, and its consequences.

  • 02/08/07
    David Whitehouse - Institute for Advanced Studies
    Central L-values and toric periods for GL(2)

    Waldspurger established a connection between the vanishing of
    certain L-values and the vanishing of period integrals over tori. Subsequent
    work of Gross, Zhang and others has, in certain cases, made this connection
    more precise. I will describe a different approach, via the relative trace
    formula, to obtain such refinements in general. This is joint work with
    Kimball Martin.

  • 02/08/07
    Toby Colding - Courant Institute and MIT
    Embedded minimal surfaces

    The study of minimal surfaces is a central problem in geometry and analysis that dates back to the 1700's when the catenoid and helicoid were discovered. I will survey recent advances, focusing on joint work with Bill Minicozzi that describes the structure of a general embedded minimal surface in terms of the catenoid and helicoid. I hope to give an overview of how these results have played a role in the solution of some old problems.

  • 02/08/07
    Jim Lin
    Finding a Thesis Advisor

    One of the most important choices a graduate student will make in their graduate career will be choosing a thesis advisor. It is never too early for students to begin thinking about choosing an area of specialty and choosing among the 55 UCSD math faculty who might supervise them.
    How did other students find a thesis advisor? What are the key factors to consider when choosing an advisor? What do professors look for before they accept a student as their thesis student? How does finding a thesis advisor lead to find a problem for a thesis?
    The goal of this seminar is to share information about the process of finding a thesis advisor. We will have three graduate students-Manda Riehl, Jonathan Armel and Kristin Jehring describe their experiences finding a thesis advisor and what happens after a thesis advisor is found.
    We will also have two faculty, Lance Small and Michael Holst describe what they look for in a graduate student before they accept him or her as their thesis student. Jim Lin will serve as moderator. All students, especially first, second and third year students, are cordially invited to attend.

  • 02/13/07
    Gabriel Nagy - UCSD
    Initial data for numerical relativity

    We consider the conformal decomposition of Einstein's
    constraint equations introduced by Lichnerowicz and York, on a
    compact manifold with boundary. We show that there exists a solution
    to the coupled Hamiltonian and momentum constraint equations when the
    derivative of the mean extrinsic curvature is small enough, and
    assuming that the Ricci scalar of the background metric is bounded,
    though it can change sign on the manifold. The solutions are in
    general not uniquely determined by the source functions and boundary
    data. The proof technique is based on finding barriers for the
    Hamiltonian constraint equation which are independent of the
    solutions of the momentum constraint equation, and then using
    standard fixed-point methods for increasing operators in Banach
    spaces. This work generalizes a previous work by Isenberg and
    Moncrief on closed manifolds.

  • 02/13/07

  • 02/15/07
    Ross Richardson - UCSD, Graduate Student
    Randomness and Regularity \`a la Szemeredi.

    A major advance in combinatorics occured in 1972,
    when E. Szemeredi provided a complicated and ingeneous combinatorial
    proof of the Erd\"os-Turan conjecture: every subset

  • 02/15/07

  • 02/15/07
    David Meyer - UCSD
    Quantum correlated equilibria in games (part 2)

    Many quantum games can be understood as protocols for the communication and processing of quantum information, and should be compared to classical games with communication. After introducing two of the standard quantum game protocols, I'll explain how this comparison works, and its consequences.

  • 02/15/07
    Shing-Tung Yau - Harvard University and UCI
    Canonical metrics on complex manifolds

  • 02/16/07
    Matthew Hedden - Massachusetts Institute of Technology
    On knot Floer homology and complex curves

    \noindent Suppose we view the three-dimensional sphere as: $S^3 = \{(z,w) \subset \mathbb{C}^2|\ |z|^2 + |w|^2 = 1\}. $ If we are given a complex curve $V_f = \{(z,w)|0 = f(z,w) \in \mathbb{C} [z,w]\},$ we can then examine the intersection $K = V_f \cap S^3.$
    In the transverse case, this intersection $K$ will be a link i.e. an embedded
    one-manifold in the three-sphere. This talk will be interested in the question:
    Question: Which links can arise from complex curves in the above manner?
    I will discuss the history of this problem, focusing first on the case where
    $f(z,w)$ has an isolated singularity at the origin where the question is completely
    answered. I’ll then discuss how a powerful set of knot invariants
    defined by Ozsvath and Szabo and independently by Rasmussen using the
    theory of pseudo-holomorphic curves can provide information on the above
    question. More precisely, Ozsvath and Szabo and Rasmussen defined a numerical
    invariant of knots, denoted $\tau(K)$, which we show provides an obstruction
    to knots arising in the above manner. More surprisingly, suppose
    we focus on knots whose exteriors, $S^3 - K$, admit the structure of a fiber
    bundle over the circle, the so-called $fibered$ knots. In this case we show
    that $\tau(K)$ detects exactly when a fibered knot arises as the intersection of
    the three-sphere with a complex curve satisfying a certain genus constraint.
    Our proof relies on connections between Ozsvath-Szabo theory and certain
    geometric structures on three-manifolds called contact structures.

  • 02/20/07
    Cheng Yeaw Ku - Math Department, Caltech
    Intersecting families of permutations and partial permutations.

    A set of permutations on $n$ points is intersecting if, for
    any two of its elements, there is some point which is sent to the same
    point by both of them. How large can such a set be? Similarly, a set
    of partial permutations (meaning injections defined on some $r$ points
    of the $n$-set, for some fixed $r$) is intersecting if, for any two of its
    elements, there is some point on which they are both defined and is
    sent to the same point by both of them. Again, how large can such a
    set be?

    We shall survey and discuss some results on these problems. We will
    also mention some fascinating conjectures in this area.

    This talk includes joint work with Peter Cameron and Imre Leader.

  • 02/22/07
    Keh-Shin Lii - University of California, Riverside
    Modeling marked point processes

    New probability models are proposed for the analysis of marked point
    processes. These models deal with the type of data that arrive
    or are observed in possibly unequal time intervals such as
    financial transactions, earthquakes among others. The models
    treat both the time between event arrivals and the observed marks
    as stochastic processes. We adopt a class of bivariate
    distributions to form the bivariate mixture transition
    distribution(BMTD). In these models the
    conditional bivariate distribution of the next
    observation given the past is a mixture of conditional
    distributions given each one of the last p observations or a
    selection of past p events. The identifiability of the model is
    investigated, and EM algorithm is developed to obtain estimates
    of the model parameters. Simulation and real data examples are
    used to demonstrate the utility of these models.

  • 02/22/07
    Allen Knutson - UCSD
    Moduli spaces and quotients by groups: Part ][

    Last time I described the (na\"ive) algebraic geometry way to define

  • 02/22/07
    James Barrett - King's College London
    Dirichlet L-functions at strictly positive integers and Fitting invariants of K-groups

    For K a C.M. abelian extension of a totally real base-field k with
    Galois group G, Solomon has recently constructed for each prime p a Z p[G] ideal of Q p[G] related to values of Dirichlet L-functions at s=1 and conjectured that this ideal is contained within Z p[G]. Jones has subsequently shown that for each odd p the Equivariant Tamagawa Number Conjecture (or ETNC) implies that Solomon's ideal should actually be contained within the Fitting ideal of the class-group of O K. I shall explain how to define analogous ideals related to values of Dirichlet L-functions at integers r strictly greater than 1 and provide a sketch of the techniques used to show that the ETNC relates these `higher Solomon ideals' to the Fitting ideals of certain natural cohomology groups (and thus, when the Quillen-Lichtenbaum conjecture is valid, to Fitting ideals of Quillen K-groups of O K). In particular, for certain choices of K/k and r these results are
    unconditional as the relevant cases of the ETNC and Quillen-Lichtenbaum conjecture are known to be valid.

  • 02/23/07
    David R. Morrison - UCSB/KITP
    Ricci flow and membrane theory

  • 02/24/07
    Bernd Siebert - Freiburg
    Tropical geometry and mirror symmetry

  • 02/24/07

  • 02/24/07

  • 02/26/07
    Jason Bell - Simon Fraser University
    Subfields of Division Rings

  • 02/27/07

  • 02/28/07

  • 03/01/07
    Rafael De Santiago - Graduate Student, University of California, Irvine
    Interest Rate Markets with Stochastic Volatility

    We analyze stochastic volatility effects in the context of the bond market. The short rate model is of Vasicek type and the focus of our analysis is the effect of multiple scale variations in the volatility of this model. Using a singular perturbation approach we can identify a parsimonious representation of multiscale stochastic volatility effects. The results are illustrated with numerical simulations. We also present a framework for model calibration and look at applications to bond option pricing.

  • 03/01/07

  • 03/01/07
    Kristin Lauter - Microsoft Research
    Class invariants for genus two

    Evaluating certain Siegel modular functions at CM points on the moduli space of principally polarized abelian surfaces give algebraic
    numbers which we call class invariants. The construction of class
    invariants is motivated by explicit class field theory, specifically,
    the construction of units with possible applications to Stark conjectures.
    Class invariants can also be viewed as invariants of the binary sextic
    defining a genus 2 curve whose Jacobian corresponds to the CM point
    on the moduli space. The explicit construction of genus two curves
    with CM is motivated by cryptographic applications.

    When evaluating certain Siegel modular functions at CM points,
    the coefficients of the minimal polynomials have striking factorizations.
    In joint work with Eyal Goren, we studied primes that appear in the
    factorization of the denominators, and proved a bound on such primes
    closely related to the discriminant of the CM field. In more recent work,
    we study the primes appearing simultaneously in the numerators of
    CM values of certain Siegel modular functions in dimension 2.
    This work generalizes the work of Gross and Zagier for the modular
    j-function and is related to a conjecture of Bruinier and Yang on
    intersection numbers.

  • 03/05/07
    A.Yu.Olshanskii - Vanderbilt University (USA) \\ Moscow University (Russia)
    Hyperbolic groups: homomorphisms and direct limits

    All groups under our consideration are finitely generated. Asymptotic cones (AC) of groups were
    introduced by M.Gromov in 1981. He used them for the description of groups with polynomial
    growth. AC of groups are homogeneous geodesic, metric spaces. There exists a group having
    non-homeomorphic cones. All AC of G are R-trees iff the group G is hyperbolic. In a recent joint
    paper with D.Osin and M.Sapir, we called a group G lacunary hyperbolic (LH) if at least one AC
    of G is an R-tree. We characterize LH groups as direct limits of hyperbolic groups satisfying
    certain restrictions on the hyperbolicity constants and injectivity radii. We show that the class
    of LH groups is very large. Many group-theoretical couner-examples (E.G., some Tarski
    monsters) are LH groups. Among new examples, we construct a group having an AC with
    a non-trivial countable fundamental group. This solves Gromov's problem of 1993.

  • 03/06/07
    Henning Hohnhold - UCSD
    Universal deformations in algebraic topology: the Hopkins-Miller theorem

    I'm going to explain the theorem of Hopkins and Miller (and partly Goerss) that gives a version of Lubin-Tate deformation theory in the context of algebraic topology. More concretely, the theorem says that there is a functor $(k,\Gamma) \mapsto E_{(k,\Gamma)}$ from formal groups laws over perfect fields of characteristic $p>0$ to a very nice category of commutative ring spectra, namely $E_{\infty}$-ring spectra. It has the property that the formal group law of the cohomology theory associated with the ring spectrum $E_{(k,\Gamma)}$ is the universal deformation of $(k,\Gamma)$. By functoriality, we obtain an action of the (extended) Morava stabilizer group on the spectrum $E_{(\mathbb{F}_{p^n},H_n)}$, where $H_n$ denotes the Honda formal group law of height $n$. One application is the construction of the higher real $K$-theories $EO_n$ as the homotopy fixed point spectra obtained from the action of finite index subgroups of the Morava stabilizer subgroup.

  • 03/06/07
    Yuri Bakhturin - Memorial University, Canada
    Large Lie Algebras

  • 03/08/07
    Sebastien Roch - University of California, Berkeley
    Markov Models on Trees: Reconstruction and Applications

    Markov models on trees arise naturally in many fields, notably in molecular
    biology - as models of evolution; in statistical physics - as models of
    spin systems; and in networking - as models of broadcasting. In this talk,
    I will discuss various inference problems motivated especially by
    applications in statistical phylogenetics, i.e. the reconstruction of
    evolutionary histories of organisms from their molecular sequences. In
    particular, I will consider the "root reconstruction" problem: how
    accurately can one guess the value at the root of the tree, given the state
    at the leaves? I will focus on recent work establishing new conditions for
    the impossibility of such reconstruction. I will also discuss the related
    "phylogenetic reconstruction" problem: given enough samples at the leaves,
    can one reconstruct the tree that generated this data and, if so, how
    efficiently? I will present a recent result on a sharp transition in the
    number of samples required to recover the tree topology, using a connection
    to the root reconstruction problem above. Time permitting, I will describe
    briefly connections to computational learning theory and network tomography
    as well. This is joint work with S. Bhamidi, C. Borgs, J. Chayes, C.
    Daskalakis, E. Mossel, and R. Rajagopal.

  • 03/08/07
    Jiri Lebl - Graduate Student, UCSD
    Of all the pseudoconvex domains, she had to walk into mine

    In this talk, I will talk about an incredibly wonderful theorem of Charles
    Fefferman about biholomorphisms of strictly pseudoconvex domains. I will
    also talk about a certain kernel named after a famous movie actress. You
    will find out what is a biholomorphic mapping, strictly pseudoconvex domain,
    kernel, etc... I might also tell you some historical background, where I
    will make up the bits that I don't actually know.

  • 03/08/07
    Mohammed Ziane - Mathematics, USC
    Regularity results for the Navier-Stokes equations and the primitive equations of the ocean

    I will present some recent results on the Serrin-type conditional regularity of the Navier-Stokes equations. Basically, if one component of the weak solution of the Navier-Stokes equation belongs to a Serrin type space of regularity then the weak solution is regular and is unique. The second part of the talk is devoted to the primitive equations of the ocean with the Dirichlet boundary condition for which we prove the global regularity. This is a joint work with I. Kukavica.

  • 03/12/07
    Larissa Horn

  • 03/12/07
    Kristin Jehring

  • 03/13/07
    Hailiang Liu - Department of Mathematics, Iowa State University
    Computing Multi-valued Solutions for Euler-Poisson Equations

    In this talk we first review the critical threshold phenomena
    for Euler-Poisson equations, which arise in the semiclassical
    approximation of Schrodinger-Poisson equations and plasma
    dynamics. We then present a phased space-based level set
    method for the computation of multi-valued velocity and
    electric fields of one-dimensional Euler-Poisson equations.
    This method uses an implicit Eulerian formulation in an
    extended space, which incorporates both velocity and electric
    fields into the configuration space. Multi-valued velocity
    and electric fields are captured through common zeros of two
    level set functions, which solve a linear homogeneous
    transport equation in the field space. The superposition
    principle for multi-valued solutions is established.

  • 03/13/07
    Bertram Kostant - Massachusetts Institute of Technology
    On maximal Poisson commutative subalgebras of S(g), complete integrability, and corresponding Darboux coordinates on any reductive Lie algebra $\frak g$

    Recently, using Gelfand-Zeitlin and the space of Hessenberg matrices, Wallach and I found natural Darboux coordinates (as a classical mechanical solution of the Gelfand-Zeitlin question) on $\frak g$ for the case where $\frak g$ is the space of all matrices. Now, at least locally, I do the same for any reductive $\frak g$ using a beautiful result of A. A. Tarasov on Fomenko-Miscenko theory and old results of mine on a generalization of the Hessenberg matrices.

  • 03/13/07
    Allen Knutson - UCSD
    Shifting, matroids, and Littlewood-Richardson

    To prove the Erd\H os-Ko-Rado theorem about extremal collections of
    subsets of $1,\ldots,n$, they invented the {\em shifting} technique,
    which preserves the number of subsets in a collection but simplifies
    (in some senses) the collection. After a finite number of shifts,
    one's collection becomes invariant under shifting, and easily studied.

    Given a finite set of $n$ vectors in a $k$-dimensional vector space,
    the collection of subsets that form bases of the vector space
    satisfies some combinatorial properties. Abstracting them, Whitney
    defined {\em matroids}. The matroids that are invariant under shifting
    have been classified, and correspond to partitions inside a
    $k \times (n-k)$ rectangle. The shift of a matroid usually is not a matroid.

    I'll present a new version of the Littlewood-Richardson rule,
    that starts with a certain matroid, and alternately shifts it
    (breaking matroidness) and decomposes as a union of maximal submatroids.
    The leaves of the tree so constructed are labeled with fully
    shifted matroids, hence partitions. To actually carry out
    such a calculation in practice requires some new algorithms.

    Unlike all other known Littlewood-Richardson rules, this matroid
    shifting rule has an easy generalization to multiplication of Schubert
    (not just Schur) polynomials, where it is still a conjecture.

    This work is joint with Ravi Vakil.

  • 03/15/07
    Denis Bell - University of North Florida
    Quasi-invariant measures on path space

    Let $N$ denote a manifold equipped with a finite Borel measure
    $\gamma$. A vector field $Z$ on $N$ is said to
    be admissible with respect
    to $\gamma$
    if $Z$ admits an integration by parts formula.
    The measure $\gamma$ is
    said to be quasi-invariant under $Z$
    if the class of null sets of $\gamma$
    is preserved by the flow generated
    by $Z$. In this talk we study the law $\gamma$ of an elliptic
    diffusion process with values in a closed compact manifold.
    We construct a
    class of admissible vector fields for $\gamma$, show that $\gamma$
    is quasi-invariant under these vector fields,
    and give a formula for the
    associated family of Radon-Nikodym
    derivatives $d\gamma_s\over d\gamma$.

  • 03/15/07

  • 03/15/07
    Eric Lauga - Department of Mathematics, MIT
    Some Modeling Problems Inspired by Swimming Microorganisms

    In this talk, we use biological locomotion on small scales as an
    inspiration (and an excuse) to solve a number of modeling
    problems in small-scale fluid mechanics. We first solve for the
    swimming kinematics of elastic swimmers, devices which exploit
    flow-induced deformation of elastic filaments for
    propulsion. More generally, we then show how soft surfaces can
    be exploited for propulsion without inertia. Finally, we
    describe how the viscoelastic nature of the surrounding fluid
    can affect the kinematics and energetics of simple swimmers.

  • 03/15/07
    Efim Zelmanov - UCSD
    An overview of Abstract Algebra in the 20th century

    I will try to discuss Abstract Algebra from its emergence to
    the present day and its place among other mathematical areas.

  • 03/15/07
    Jonathan Sands - University of Vermont and UCSD
    Dedekind Zeta functions at s=-1 and the Fitting ideal of the tame kernel in a relative quadratic extension

    Abstract: Brumer's conjecture states that Stickelberger elements
    combining values of L-functions at s=0 for an abelian extension of
    number fields E/F should annihilate the ideal class group of E when it
    is considered as module over the appropriate group ring.
    In some cases, an ideal obtained from these
    Stickelberger elements has been shown to equal
    a Fitting ideal connected with the ideal class group.
    We consider the analog of this at s=-1, in which the class group
    is replaced by the tame kernel, which we will define.
    For a field extension of degree 2, we show that there is an exact equality
    between the Fitting ideal of the tame kernel and the most natural
    higher Stickelberger ideal; the 2-part of this equality is conditional on
    the Birch-Tate conjecture.

  • 03/15/07
    Jinchao Xu - Mathematics Department, The Pennsylvania State University
    Design, Analysis and Application of Optimal PDE Solvers

    A number of recent results, including special discretization
    schemes, adaptive methods and multilevel iterative methods for
    the resulting algebraic systems, will be presented in this talk
    for various partial differential equations (PDEs). With a
    careful and combined use of qualitative properties of PDEs, the
    underlying functional spaces and their discretizations, many
    different kinds of equations will be treated with similar
    techniques. After an introduction to some practically efficient
    methods such as the algebraic multigrid method for the Poisson
    equations, it will be shown how more complicated systems such as
    linear elasticity equations, electro- magnetic equations, porous
    media, Stokes equations and more general Newtonian/non-Newtonian
    models can be reduced to the solution of a sequence of Poisson
    equation and its simple variants. The efficiency of these
    algorithms will be illustrated by theoretical analysis, numerical
    examples and engineering applications.

  • 03/16/07
    Robert Blair Angle - UCSD

  • 03/21/07
    Albert Chau - University of Waterloo
    Pseudolocality in Ricci flow and applications

  • 03/22/07
    Piotr Kokoszka - Utah State University
    Discriminating between long memory and change-point models

    Over the last two decades long memory time series have become an
    established modeling tool in many areas of science and technology,
    including geosciences, medical sciences, telecommunication networks
    and to some extend financial econometrics. It has however been
    recently realized that practically all statistical procedures
    intended to detect and estimate long memory give spurious results if
    a time series without long memory is perturbed by nonstationarities,
    like trends or breaks (change-points). For example, if a mean of a
    short memory time series changes, a test for the presence of long
    memory will incorrectly indicate that the time series has long
    memory. Similarly, a test for the presence of change point, will
    incorectly show that that a change point is present if the time
    series is stationary with long memory. A growing body of research
    which has accumulated over the last decade is concerned with finding
    and illustrating cases of such spurious inference, without
    addressing the issue how to choose between the two modeling
    approaches. In this talk we will discuss two new statistical tests
    aimed at distinguishing between the two approaches and apply them to
    a financial and a hydrological time series. The talk will focus on
    the ideas rather than technicalities and will be broadly accessible.

  • 04/04/07
    Kelly McKinnie - Emory University
    Noncyclic and indecomposable p-algebras

    In this talk we will discuss questions about finite dimensional central simple p-algebras. In particular we will
    discuss the first construction of a noncyclic p-algebra due to
    Amitsur and Saltman. From there we will talk about the structure
    of these algebras under prime to p extensions and the conditions
    under which they are indecomposable.

  • 04/05/07
    Jason Bandlow - UCSD Graduate Student
    Permutations and the Plights of Prisoners

    200 prisoners, scheduled for execution, are given one chance for
    survival. Their 200 names are put in a row of 200 boxes, one name per
    box. Each prisoner will enter this room, one at a time, and open 100
    boxes with the goal of finding his or her own name. If every
    prisoner does this successfully, all will go free. If any one
    fails to find his or her own name, all will be executed. The
    boxes will be closed after each prisoner and once a prisoner has
    entered the room, any attempt at communication with the others
    will be punished by execution. However, the prisoners are
    allowed to strategize beforehand. In fact, a strategy exists which
    gives, roughly, a $\frac{1}{3}$ probability of survival. Can you find
    it? No Googling!

    We'll consider this problem and the fate of other unfortunate
    hypothetical prisoners, mostly as an excuse to discuss some
    combinatorics of permutation enumeration.

  • 04/05/07
    Vladimir Sverak - University of Minnesota
    PDE aspects of Navier-Stokes Equations

    Solutions of incompressible Navier-Stokes equations can
    exhibit a wide spectrum of different types of behavior.
    In various regimes,
    the equations contain as special limiting cases
    for example the classical
    heat equation, the non-linear Schroedinger
    equation, various other dispersive equations
    with strange dispersion relations,
    various non-trivial finite-dimensional dynamical systems,
    some classical geometric semilinear elliptic equations, etc.
    In addition, when thinking about
    realistic fluid flows and applications,
    ideas from statistical mechanics enter the picture. In
    the lecture I will explain
    (a limited number of) some PDE aspects of these equations.

  • 04/09/07
    Rafal Synowiecki - AGH University of Science and Technology \\ Krakow, Poland
    Resampling nonstationary time series with periodic and almost periodic structure

  • 04/10/07
    Justin Roberts - UCSD
    Ozsvath--Szabo homology

    This term's topology learning seminar will be on Ozsvath-Szabo homology.

    About 7 years ago, Ozsvath and Szabo invented this construction (which they call "Heegaard Floer homology") in an attempt to give a different definition of Seiberg-Witten theory. Their theory has been incredibly successful in applications to low-dimensional topology.

    In brief, they show how to associate a family of homology groups to a 3--manifold by choosing a Heegaard splitting and computing a suitable Lagrangian intersection Floer homology. Some of the most important features of the construction are:

    1. 4--dimensional cobordisms induce maps between homology groups; the invariants of closed 4-manifolds are conjecturally equal to the Seiberg-Witten invariants.

    2. There is a version of the homology for knots in $S^3$, which leads to an exact formula (not just a bound!) for the genus of knots. Consequently their homology distinguishes the unknot, and can be used to prove many old conjectures about surgery on knots.

    3. The theory gives rise to powerful invariants of contact structures on 3--manifolds and can distinguish tight from overtwisted.

    4. The homology for knots, unlike all earlier gauge-theoretic invariants, can actually be calculated by purely combinatorial means. There is a strong hope that this will eventually lead to a complete combinatorial calculation of the Ozsvath-Szabo/Seiberg-Witten/Donaldson invariants of 4-manifolds.

    The first meeting will be Tuesday 10th April, in room 7218, at 10.30am.
    I will give an introductory talk and then we will arrange the schedule of speakers for the rest of the term. Anyone is welcome to attend - attendance does not necessarily lead to being volunteered for a talk!

  • 04/10/07
    Jean-Paul Berrut - University of Fribourg \\ Switzerland
    A formula for the error of finite sinc--interpolation over a fixed finite interval

    Sinc--interpolation is an infinitely smooth interpolation on the whole real
    line based on a series of shifted and dilated sinus--cardinalis functions used as
    Lagrange basis. It often converges very rapidly, so for example for functions
    analytic in an open strip containing the real line and which decay fast enough at
    infinity. This decay does not need to be very rapid, however, as in Runge's function
    $1/(1+x^2)$. Then one must truncate the series, and this truncation error is much
    larger than the discretisation error (it decreases algebraically while the latter
    does it exponentially).

    In our talk we will give a formula for the error commited when merely using
    function values from a finite interval symmetric about the origin.
    The main part of the formula is a polynomial in the distance between the nodes
    whose coefficients contain derivatives of the function at the extremities.

  • 04/10/07
    Fan Chung Graham - UCSD
    Open problems for large sparse graphs

  • 04/12/07
    Jacek Leskow - Polish-American Graduate School of Business \\ Nowy Sacz, Poland
    Relative measurability and time series analysis \\ A non-stochastic perspective

    The concept of relative measure was fairly popular among Polish
    mathematician of 1930 in Lvov. Steinhaus and Urbanik were working
    on introducing a relative measure and relative measurability into
    the area of random variables.
    Recent work on signal processing and time series has led to
    re-discovery of the 'old-school' theorems and application to
    data generated by signals or time-series. Some fundamental work
    was done by Garnder and continuation of this work was done
    by Leskow and Napolitano.

    A short informal introduction to nonstochastic approach
    to time series inference via relative measurability will
    be presented. Applications to signal forecasting will be

  • 04/12/07
    Nolan Wallach
    Invariant Theory

    Let G be a group acting``nicely" on a space, $X$, with some
    structure(topological, differentiable, algebraic, combinatorial.). A
    basic problem is to find an effective way of determining if two points
    $x,y$ are in the same orbit (or at least in an appropriate closure of an
    orbit). In this lecture I will look at methods that can be used in
    concert with computers to
    approach such problems through the determination of "enough" computable
    invariant functions. There will be several examples including measures of
    quantum entanglement.

  • 04/12/07
    Wee Teck Gan - UCSD
    Organizational Meeting and The Local Langlands Conjecture for GSp(4)

    There will be a 10 minute organizational meeting for the schedule of talks this quarter, after which I will discuss a recent proof of the local Langlands conjecture for GSp(4). Joint with with Shuichiro Takeda.

  • 04/12/07
    Sergey Yekhanin - Massachusetts Institute of Technology
    New Locally Decodable Codes and Private Information Retrieval Schemes

    A q-query Locally Decodable Code (LDC) is an error-correcting
    code that encodes an n-bit message x as a codeword $C(x)$, such that one can
    probabilistically recover any bit $x_i$ of the message by querying only $q$
    of the codeword $C(x)$, even after some constant fraction of codeword bits
    been corrupted. The goal of LDC related research is to minimize the length
    such codes.

    A q-server private information retrieval (PIR) scheme is a cryptographic
    protocol that allows a user to retrieve the $i-th$ bit of an $n-bit$ string $x$
    replicated between $q$ servers while each server individually learns no
    information about $i$. The goal of PIR related research is to minimize the
    communication complexity of such schemes.

    We present a novel algebraic approach to LDCs and PIRs and obtain vast
    improvements upon the earlier work. Specifically, given any Mersenne prime
    $p=2^t - 1$, we design three query LDCs of length $Exp(n^{1/t})$, for every
    $n$. Based
    on the largest known Mersenne prime, this translates to a length of less
    than $Exp(n^{10^{-7}})$, compared to $Exp(n^{1/2})$ in the previous
    We also design 3-server PIR schemes with communication complexity of
    $O(n^{10^{-7}})$ to access an n-bit database, compared to the previous best
    scheme with complexity $O(n^{1/5.25})$.

    It has often been conjectured that there are infinitely many Mersenne primes. Under this conjecture, our constructions yield three query locally
    decodable codes of subexponential length and three server private
    retrieval schemes with subpolynomial communication complexity.

  • 04/16/07
    Michael Kinally - UCSD

  • 04/16/07
    Rafal Synowiecki - AGH University of Science and Technology \\ Krakow, Poland
    Resampling nonstationary time series with periodic and almost periodic structure; part II

  • 04/17/07

  • 04/17/07
    Benjamin Schlein - Department of Mathematics, UC Davis
    Derivation of the Gross-Pitaevskii equation

    In this talk, I am going to discuss some recent results concerning the derivation, from many body quantum mechanics, of a cubic nonlinear Schroedinger equation, known as the Gross-Pitaevskii equation, for the dynamics of Bose-Einstein condensation. This is a joint work with L. Erdos and H.-T. Yau.

  • 04/17/07
    Sebastian Cioaba - UCSD
    The spectral radius and the diameter of connected graphs

    Recently, Wang, Chakrabarti, Wang and Faloutsos have shown that the spectral radius of a graph plays an important role in modeling virus propagation in networks. This led Van Dam and Kooij to consider the following problem: which connected graph on n nodes and diameter D has minimal spectral radius ? Van Dam and Kooij answered this question for $D=n-1,n-2,n-3,n/2,2,1$ and provided a conjecture for the case $D=n-e$, when e is fixed. In this talk, I give an overview of their work and I will outline a proof of their conjecture for $e=4$ and
    possible extensions for $e>4$.

    This is joint work in progress with Edwin Van Dam (University of Tilburg,
    The Netherlands).

  • 04/18/07
    Ana Dudek - AGH University of Science and Technology \\ Krakow, Poland
    A resampling method for point processes

  • 04/19/07
    Ron Getoor - UCSD
    Walsh's interior reduite

    This will be an expository talk. I'll begin by introducing the concepts of reduite (reduced function) and balayage (swept measure) in classical potential theory and their interpretations in terms of Brownian motion. I'll then discuss the extension of these ideas to Markov processes as in Hunt's fundamental memoir. After introducing h-transforms I'll be able to define the interior reduite and discuss some of its properties following Walsh. If time permits I'll give some indications of recent work in this area by Fitzsimmons and myself.

  • 04/19/07
    Larissa Horn - UCSD Graduate Student
    We live but a fraction of our lives

    As any good TA knows, adding fractions can be tricky. But being able to work with fractions can enrich your mathematical life in many ways. Unfortunately, if you happen to be a ring, this luxury is not inherent. However, you could still hope to embed into some larger ring containing inverses--even to the extent that you become a field or division ring.

    We will look first at the theory of localization for commutative rings where everything is fairly straightforward. Then, we will see what happens when we try to generalize to noncommutative rings and look at some examples of good and bad cases.

    And the quote... Thoreau.

  • 04/19/07
    Wee Teck Gan - UCSD
    The Local Langlands Conjecture for GSp(4)

    We will discuss a recent proof of the local Langlands conjecture for GSp(4). This is joint work with Shuichiro Takeda.

  • 04/20/07

  • 04/24/07

  • 04/24/07
    Olvi Mangasarian - UCSD
    Nonlinear knowledge in kernel machines

    Prior knowledge over arbitrary general sets is incorporated into
    nonlinear support vector machine approximation and classification problems as linear constraints of a linear program. The key tool in this incorporation is a theorem of the alternative for convex functions that converts nonlinear prior knowledge implications into linear inequalities
    without the need to kernelize these implications. Effectiveness of the
    proposed formulation is demonstrated on synthetic examples and on
    important breast cancer prognosis problems. All these problems
    exhibit marked improvements upon the introduction of prior knowledge
    over nonlinear kernel approaches that do not utilize
    such knowledge.

  • 04/24/07
    Adriano M. Garsia - UCSD
    Hilbert series of invariants, constant term identities and Kostka-Foulkes polynomials

    We seek for the Hilbert series of the ring of invariant polynomials
    in the $2n+n^2$ variables $\{u_i,v_j,x_{i,j}\}_{i,j=1}^n$ under the
    action of $GL_n[C]$ by right multiplication on the row vector
    $u=(u_1,u_2,\ldots ,u_n)$, left multiplication on the column vector
    $v=(v_1,v_2,\ldots ,v_n)$ and by conjugation on the matrix $\|x_{i,j}\|_{i,j=1}^n$.
    We reduce the computation of this Hilbert series to the evaluation of
    the constant term of a certain rational function. Remarkably, the
    final result hinges on the explicit evaluation of certain
    Kostka-Foulkes polynomials.

  • 04/24/07
    Christine Guenther - Pacific University
    Stability of Ricci flow at homogeneous solitons

  • 04/26/07
    Henning Hohnhold - UCSD
    What is a stack?

    The goal is to explain what a stack is and why people think they are interesting objects to study. I'll start by saying what people (who want to classify a certain class of mathematical objects, say triangles or elliptic curves or...) expect from a moduli space and why it can easily happen that such a moduli `space' doesn't exist. This is the situation in which stacks get their chance. They are some kind of a generalization of a the notion of a `space', where the meaning of space depends on what you are interested in (for example, in the context of algebraic geometry one might say that stacks are a generalization of the notion of a scheme). I'll then use an example to motivate the axioms appearing in the definition of a stack. Depending on how much time I have I will say what an algebraic stack is and maybe how Deligne and Mumford employed the notion to prove something interesting.

  • 04/30/07
    Tonghai Yang - University of Wisconsin at Madison
    An arithmetic intersection formula on a Hilbert modular surface

    On a Hilbert modular surface over $\mathbb Z$, there are two
    families of arithmetic cycles. One family consists of the
    Hirzebruch-Zagier divisors
    $\mathcal T_m$ of codimension $1$, indexed by positive integers $m$, and
    another consists of the CM cycles $CM(K)$ of codimension 2, indexed by
    quartic CM number fields $K$. When $K$ is not biquadratic, $\mathcal T_m$
    and $CM(K)$ intersect properly, and a natural
    question is, what is the intersection number? In this talk, we present a
    conjectural formula for the intersection number of
    Bruinier and myself. We give two partial results in this talk. If time
    permits, I will also briefly describe two applications:
    one of the consequences is
    a generalization of the Chowla-Selberg formula, and another is a
    conjecture of Lauter on Igusa invariants.

  • 05/01/07
    Dave Clark - UCSD, Graduate Student
    Ozsvath-Szabo invariants of knots

  • 05/01/07
    Michael Yampolsky - University of Toronto
    Computability and complexity of Julia sets

    Informally speaking, a compact set in the plane is computable if
    there exists an algorithm to draw it on a screen with an arbitrarily
    high magnification. We investigate the question of computability
    of a Julia set of a quadratic polynomial and obtain some surprising
    (joint work with Mark Braverman)

  • 05/01/07
    Hieu Nguyen - UCSD Graduate Student
    Remarks on the Local Behavior of the Finite Element Method

    Our main goal is to study the behavior of the error in finite element approximations of partial differential equations. The error typically has two components -- local error and global (pollution) error. We also will discuss the very interesting phenomena of superconvergence, and in particular, how to determine superconvergence points, and what advantages can be derived from them.

  • 05/01/07
    Gyula Y. Katona - Alfr'ed R'enyi Institute of Mathematics \\ Hungarian Academy of Sciences
    Hamiltoninan Chains in Hypergraphs

    An r-uniform hypergraph is a generalization of graphs, we consider a
    subset of all r-element subsets of a given vertex set. A Hamiltionian
    chain is a generalization of hamiltonian cycles for hypergraphs, it is a
    "cycle" that contains all vertices. Among the several possible ways of
    generalizations this is probably the most strong one, it requires the
    strongest structure. Since there are many interesting questions about
    hamiltonian cycles in graphs, we can try to answer these questions
    for hypergraphs, too. I give a survey on
    results about such questions.

  • 05/03/07
    Vladimir Rotar - San Diego State University
    On asymptotic proximity of probability distributions and the non-classical invariance principle

    Usually, a limit theorem of Probability Theory is a theorem that concerns convergence of a sequence of distributions $P_n$ to a distribution $P$. However, there is a number of works where the traditional setup is modified, and the object of study is two sequences of distributions, $P_n$ and $Q_n$, and the goal consists in establishing conditions implying the convergence
    $P_n - Q_n ->0 (1)$
    In particular problems,$P_n$ and $Q_n$ are, as a rule, the distributions of the r.v.'s $f(X_1,...,X_n)$ and $f(Y_1,...,Y_n)$, where $f(.)$ is a function, and $X_1,X_2$,... and $Y_1,Y_2$,... are two sequences of r.v.'s. The aim here is rather to show that different random arguments $X_1,...,X_n$ may generate close distributions of $f(X_1,...,X_n)$ , than to prove that the distribution of $f(X_1,...,X_n)$ is close to some fixed distribution (which, above else, may be not true). Clearly, such a framework is more general than the traditional one. First, as was mentioned, the distributions $P_n$ and $Q_n$, themselves do not have to converge. Secondly, the sequences $P_n$ and $Q_n$ are not assumed to be tight, and the convergence in $(1)$ covers situations when a part of the probability mass or the whole distributions "move away to infinity'", while the distributions $P_n$ and $Q_n$, are approaching each other.
    We consider a theory on this point, including the very definition of convergence $(1)$, and a particular example of the invariance principle in the general non-classical setup.

  • 05/03/07
    Kristin Jehring - UCSD, Graduate Student
    A Brief Introduction to Mathematical Finance

    In this talk we will develop the concept of arbitrage and discuss the theory of options pricing. Time permitting, we will present the famous Black-Scholes option pricing formula.

  • 05/03/07
    Eric Wambach - California Institute of Technology
    On automorphic representations of unitary groups

    We will present an integral representation for automorphic
    representations on $U(3) x GL(2)$. It involves parabolic induction to $U(4)$.
    The resulting formula can be applied to determine the cuspidal
    automorphic representations of $U(3)$ that occur in the restriction of
    the Siegel induced residual spectrum of $U(4)$.

  • 05/03/07

  • 05/08/07

  • 05/08/07
    Haomin Zhou - Department of Mathematics \\ Georgia Institute of Technology
    Variational PDE Models in Wavelet Inpainting

    We propose variational models for image inpainting in wavelet domain,
    which aims to filling in missing or damaged wavelet coefficients
    in image reconstruction. The problem is motaviated by error concealment
    in image processing and communications. It is closely related to
    classical image inpainting, with the difference being that the
    inpainting regions are in the wavelet domain. This
    brings new challenges to the reconstructions. The new variational
    models, especially total variation minimization in conjunction
    with wavelets lead to PDE's,in the wavelet domain and can be solved
    numerically. The proposed models have effective and automatic control
    over geometric features of the inpainted images including sharp edges, even
    in the presence of substantial loss of wavelet coefficients, including in
    the low frequencies. This work is joint with Tony Chan (UCLA) and
    Jackie Shen (Minnesota).

  • 05/08/07
    Alon Regev - UCSD, Graduate Student

  • 05/10/07
    Ben Cooper - UCSD Graduate Student
    Manifolds with unsolvable $\pi_1$

    What does a compact Riemannian manifold with unsolvable word problem look like from within? I will discuss Nabutovsky's work on the subject.

  • 05/10/07
    Patrick Guidotti - Mathematics \\ University of California, Irvine
    Maximal Regularity and Free Boundary Problems

    After introducing the concept of Maximal Regularity for parabolic problems and illustrating its usefulness for dealing with (fully) nonlinear problems, a brief introduction to Free Boundary Problems will be given. The focus will then shift to a class of Free Boundary Problems. Maximal regularity results as well as elliptic regularity results will be presented which are needed in the analysis of the Free Boundary Problems of interest.

  • 05/10/07
    Eknath Ghate - Tata Institute \\ University of California, Los Angeles
    The local behaviour of ordinary Galois representations

    Ordinary forms of weight at least 2 give rise to locally
    reducible Galois representations. Greenberg has asked
    whether these representations are semi-simple. One
    expects this to be the case exactly when the underlying
    form has CM. We shall speak about various results towards
    this expectation that use p-adic families of forms
    and deformation theory. This is joint work with Vatsal.

  • 05/10/07
    Lee Lindblom - Physics \\ California Institute of Technology
    New insights into gauge freedom and constraints in numerical relativity

    Numerical relativity has undergone a revolution during the past two years, with several groups now routinely performing accurate simulations of binary black hole systems with multiple orbits, mergers, and ringdown of the holes to a final single hole equilibrium state. This talk will discuss some of the mathematical developments that made this revolution possible. In particular new insights will be discussed into how the gauge degrees of freedom may be specified in the Einstein equations, and how this changes the behavior of the constraints of the theory.

  • 05/11/07
    Allen Knutson - UCSD
    The Steinberg scheme and the Robinson-Schensted correspondence

    The Robinson-Schensted correspondence is one between the
    set of permutations and pairs of same-shape standard Young tableaux.
    I'll recall a few of the combinatorial aspects of this.

    The Steinberg scheme (for $GL_n$) is a set of triples, one nilpotent matrix
    and two flags invariant under the nilpotent, whose components correspond
    to permutations. I'll recall why this is (for those who haven't been
    coming to the seminar), and show that they also correspond to pairs
    of standard Young tableaux. The basic linkage between the linear algebra
    and the combinatorics is that Jordan canonical forms of nilpotent matrices
    correspond to partitions.

    This talk will only require linear algebra, and a willingness to talk
    about the ``components'' of an algebraic set.

  • 05/14/07

  • 05/15/07
    Ben Cooper - UCSD, Graduate Student
    Ozsvath-Szabo invariants of 4-manifolds

  • 05/15/07

  • 05/15/07

  • 05/15/07
    Reimundo Heluani - University of California, Berkeley
    Supersymmetry of the Chiral de Rham Complex

  • 05/15/07
    A. Wong \\ M. Colarusso \\ D. Rogalski
    Finding Jobs in Academia

    We will have three panelists who have recently found jobs: Aaron Wong, Assistant Professor, tenure track at Nevada State College, Henderson, Nevada, Mark Colarusso, Visiting Assistant Professor, University of Notre Dame, Indiana, and Dan Rogalski, Assistant Professor, UCSD. They will describe their experiences applying for an academic job. Some of the questions they will answer are: How many applications should I send out? How do I prepare for an interview? What should I write in my cover letter and resume? What are important qualifications for a teaching job, postdoc job, tenure track research job?
    The discussion will be followed by a question and answer period.

  • 05/15/07
    Po-Shen Loh - Princeton University
    Constrained Ramsey Numbers

    For two graphs $S$ and $T$, the constrained Ramsey number $f(S, T)$ is the minimum $n$ such that every edge coloring of the complete graph on
    $n$ vertices (with any number of colors) has a monochromatic subgraph isomorphic to $S$ or a rainbow subgraph isomorphic to $T$. Here, a
    subgraph is said to be rainbow if all of its edges have different
    colors. It is an immediate consequence of the Erd\H{o}s-Rado
    Canonical Ramsey Theorem that $f(S, T)$ exists if and only if $S$ is a
    star or $T$ is acyclic. Much work has been done to determine the rate
    of growth of $f(S, T)$ for various types of parameters. When $S$ and
    $T$ are both trees having $s$ and $t$ edges respectively, Jamison,
    Jiang, and Ling showed that $f(S, T) \leq O(st^2)$ and conjectured
    that it is always at most $O(st)$. They also mentioned that one of
    the most interesting open special cases is when $T$ is a path. We
    study this case and show that $f(S, P_t) = O(st\log t)$, which differs
    only by a logarithmic factor from the conjecture. This substantially
    improves the previous bounds for most values of $s$ and $t$.

  • 05/17/07
    Guillaume Bonnet - University of California, Santa Barbara
    Non-linear SPDEs for Highway Traffic Flows: Theory, and Calibration to Traffic Data

    Highway traffic flows are generally modeled by partial differential equations
    (PDEs). These models are used by traffic engineers for
    road design, planning or management. However, they
    often fail to capture important features of
    empirical traffic flow studies, particularly at small
    scales. In this talk, I will propose a fairly simple stochastic model for
    highway traffic flows in the form of a nonlinear stochastic partial differential
    equation (SPDE) with random
    coefficients driven by a Poisson random measure. I will discuss the
    well posedness of the proposed equation as well as the
    corresponding inverse problem that I will illustrate by its
    calibration to high resolution traffic data from highway
    101 in Los Angeles. I will also present a more sophisticated spde
    in the form of a system of coupled hyperbolic-parabolic equations.

  • 05/17/07
    David Eisenbud - Mathematical Research Sciences Institute \\ University of California, Berkeley
    Fibers of a Generic Projection and Asymptotic Regularity

    Varieties were originally studied by comparing them with
    the hypersurfaces that are their generic projections--curves
    in the plane and surfaces in three-space, for example. In
    low dimensions, the fibers of these generic projections are
    pretty well understood, but there are serious obstructions
    to extending this understanding to all dimensions.
    I'll survey what's known, explain some examples, and present
    a new conjecture about these fibers. A connection with the
    regularity of powers of an ideal (asymptotic regularity)
    plays an interesting role.

  • 05/17/07
    Nolan Wallach - UCSD
    Quantum wavelet transforms

  • 05/17/07
    Jeff Achter - Colorado State University
    Monodromy of hyperelliptic curves

  • 05/17/07
    Max Gunzburger - Mathematics and School of Computational Science \\ Florida State University
    Reduced-order modeling for complex systems

    The computational approximation of solutions of complex systems such as the Navier-Stokes equations is often a formidable task. For example, in feedback control settings where one often needs solutions of the complex systems in real time, it would be impossible to use large-scale finite element or finite-volume or spectral codes. For this reason, there has been much interest in the development of low- dimensional models that can accurately be used to simulate and control complex systems. We review some of the existing reduced-order modeling approaches, including reduced-basis methods and especially methods based on proper orthogonal decompositions techniques. We also discuss a new approach based on centroidal Voronoi tessellations. We discuss the relative merits and deficiencies of the different approaches and also the inherent limitations of reduced-order modeling in general.

  • 05/18/07
    Oded Yacobi - UCSD, Graduate Student
    The Jacobson-Morosov theorem

  • 05/22/07
    Sean Raleigh - UCSD, Graduate Student
    Contact 3-manifolds

  • 05/22/07
    Steve Butler - UCSD
    Anti-coverings of graphs

    Spectral graph theory has enjoyed much success in using eigenvalues of
    matrices associated with a graph to understand some structural property or
    bound various kinds of behavior of the graph. When two graphs share many
    eigenvalues in common it can often be traced to some sort of common
    structure that they share. Well known examples of this are common
    coverings or equitable partitions.

    We will consider another variation of this where (for the normalized
    Laplacian) two graphs do not project to a common graph but share a common
    ``anti-covering'' (which we will define). We will also consider
    anti-covers for the adjacency matrix and use it to establish the following
    linear algebra result (among others): {\it Let $M$ be an $n{\times}n$
    real symmetric matrix and $|M|$ be the $n{\times}n$ matrix found by taking
    (entrywise) the absolute values of $M$; then there exists a nonnegative
    symmetric $2n{\times}2n$ matrix ${\cal N}$ such that the spectrum of
    ${\cal N}$ is the union of the spectrums of $M$ and $|M|$.}

  • 05/24/07
    Karl Frederickson - UCSD, Graduate Student
    Fun with Singularities

    Singularities offer the opportunity to study some interesting mathematics
    from a relatively simple point of view. First I'll describe what an
    affine variety is and what it means for one to have a singularity. Then,
    I'll discuss the work of Milnor, Brieskorn and others on the topology of a
    complex hypersurface near one of its singular points, which has
    connections to knot theory and exotic spheres. If I have time, I'll talk
    about a special class of singularities coming from invariant theory that
    were discovered by Klein, and the McKay correspondence, which gives an
    unexpected connection between resolutions of singularities and the Dynkin
    diagrams used to classify compact Lie groups.

  • 05/24/07
    Everett Howe - The Center for Communications Research
    Even sharper upper bounds for the number of points on curves

  • 05/29/07
    Sean Raleigh - UCSD, Graduate Student
    Contact 3-manifolds, II

  • 05/29/07
    Simon Tavener - Department of Mathematics \\ Colorado State University
    A posteriori error estimation and adaptivity for an operator decomposition approach to conjugate heat transfer

    Operator decomposition methods are an attractive solution strategy
    for computing complex phenomena involving multiple physical processes,
    multiple scales or multiple domains. The general strategy is to
    decompose the problem into components involving simpler physics
    over a relatively limited range of scales, and then to seek the
    solution of the entire system through an iterative procedure
    involving solutions of the individual components.

    We construct an operator decomposition finite element method for a
    conjugate heat transfer problem consisting of a fluid and a
    solid coupled through a common boundary. Accurate a posteriori
    error estimates are then developed to account for both local
    discretization errors and the transfer of error between fluid and
    solid domains. These estimates can be used to guide adaptive mesh
    refinement. We show that the order of convergence of the operator
    decomposition finite element method is limited by the accuracy of
    the transferred gradient information, and demonstrate how a simple
    boundary flux recovery method can be used to regain the optimal
    order of accuracy in an efficient manner.

    This is joint work with Don Estep and Tim Wildey.

  • 05/30/07
    D. Jacob Wildstrom - UCSD, Graduate Student
    Dynamic Resource Location on Generalized Distance Metrics

  • 05/30/07
    Mark Colarusso - UCSD, Graduate Student
    The Gelfand-Zeitlin algebra and polarizations of generic adjoint orbits for classical groups

    We will discuss our current research which describes and constructs polarizations of regular adjoint orbits for certain classical groups. This research generalizes recent work of Bertram Kostant and Nolan Wallach. Kostant and Wallach construct polarizations of regular adjoint orbits in the space of $n\times n$ complex matrices $M(n)$. They accomplish this by defining an $\frac{n(n-1)}{2}$ dimensional abelian complex Lie group $A$ that acts on $M(n)$ and stabilizes adjoint orbits. Note that the dimension of this group is exactly half the dimension of a regular adjoint orbit in $M(n)$. This fact allows $A$ orbits of dimension $\frac{n(n-1)}{2}$ contained in a given regular adjoint orbit to form the leaves of a polarization of an open submanifold of that orbit. We study the $A$ orbit structure on $M(n)$ and generalize the construction to complex orthogonal Lie algebras $\mathfrak{so}(n)$. In the case of $M(n)$, we obtain complete descriptions of $A$ orbits of dimension $\frac{n(n-1)}{2}$ and thus of leaves of polarizations of all regular adjoint orbits. For $\mathfrak{so}(n)$, we construct polarizations of certain regular semi-simple adjoint orbits.

  • 05/31/07
    David Solomon - UCSD
    Beyond Stickelberger

    In 1890 Stickelberger published his eponymous theorem in Math. Annalen giving an explicit annihilator for the `minus' (or imaginary) part of the class group of a cyclotomic field as a Galois module. However, Stickelberger's wonderful theorem raises more questions than it answers. And strangely, many obvious ones have only begun to receive serious attention - let alone answers - in the late 20th and early 21st centuries. For instance:

    Is there a similar result for an arbitrary (abelian) extension of number fields?

    Is the `Stickelberger ideal' the full annihilator of minus part the class group?

    What about the `plus' (or real) part?

    The first question leads to Brumer's Conjecture. The answer to the second question is certainly negative, for several different reasons which we shall try to disentangle. This leads to comparisons with the Fitting ideal of the class group and of its Pontrjagin dual, and so to very recent work by Greither, the speaker and others, which we shall survey.

    If time allows we should like to report on some recent approaches to the third question.

  • 05/31/07
    John Sullivan - University of Illinois at Urbana-Champaign \\ Technischen Universitaet Berlin
    Two connections between combinatorial and differential geometry

    There is a rich interplay between combinatorial and differential
    We will give first a geometric proof of a combinatorial result, and
    a combinatorial analysis of a geometric moduli space. The first is
    work with Ivan Izmestiev, Rob Kusner, Guenter Rote, and Boris
    the second with Karsten Grosse-Brauckmann, Nick Korevaar and Rob

    In any triangulation of the torus, the average vertex valence is 6.
    Can there be a triangulation where all vertices are regular (of valence 6) except for one of valence 5 and one of valence 7? The answer is no.
    To prove this, we give the torus the metric where each triangle is
    equilateral and then explicitly analyze its holonomy. Indeed,
    from Riemann surfaces can characterize exactly which euclidean cone
    metrics have full holonomy group no bigger than their restricted
    holonomy group (at least when the latter is finite).

    Next we consider the moduli space $M_k$ of Alexandrov-embedded surfaces
    of constant mean curvature which have k ends and genus 0 and are contained in a slab. We showed earlier that $M_k$ is homeomorphic to an open manifold
    $D_k$ of dimension $2k-3$, defined as the moduli space of spherical metrics
    on an open disk with exactly k completion points. In fact, $D_k$ is the
    ball $B^{2k-3}$; to show this we use the Voronoi diagram or Delaunay
    triangulation of the k completion points to get a tree, labeled by
    logarithms of cross-ratios. The combinatorics of the tree are tracked
    by the associahedron, and the labels give us a complexification of the
    cone over its dual. We note similarities to the spaces of labeled
    used in phylogenetic analysis.

  • 05/31/07
    Dave Levermore - Mathematics \\ University of Maryland
    From Boltzmann Equations to Gas Dynamics

    Maxwell (1866) and Boltzmann (1872) developed a recipe to go from certain Newtonian laws of molecular dynamics to the Navier-Stokes system of gas dynamics. This recipe was controversial at the time. Mathematicians such as Hilbert, Klein, Poincare, and Zermelo were drawn into the debate. Hilbert featured it at the 1900 ICM in the articulation of his sixth problem, and made important contributions towards its resolution. The problem however remains largely open. Recent significant advances start with the DiPerna-Lions (1990) theory of global solutions to Boltzmann equations and lead to the Golse-Saint Raymond (2004) proof of the incompressible Navier-Stokes limit. This lecture will introduce the Boltzmann equation and survey some "new" connections to linear and weakly nonlinear gas dynamics that are the focus of recent research.

  • 06/05/07

  • 06/05/07
    Emre Mengi - UCSD
    A Backward Approach for Model Reduction

    The differential equation $\dot{x}(t) = Ax(t) + Bu(t)$ coupled with the
    algebraic equation $y(t) = Cx(t) + Du(t)$ where $A\in\mathbb{C}^{n\times n}$,
    $B\in\mathbb{C}^{n\times m}$, $C\in\mathbb{C}^{p\times n}$ is
    called a state space system and commonly employed to represent
    a linear operator from an input space to an output space in control
    theory. One major challenge with such a representation is that
    typically $n$, the dimension of the intermediate state function $x(t)$,
    is much larger than $m$ and $p$, the dimensions of the input
    function $u(t)$ and the output function $y(t)$. To reduce the order of
    such a system (dimension of the state space) the traditional
    approaches are based on minimizing the $H_{\infty}$ norm of the
    difference between the transfer functions of the original system and
    the reduced-order system. We pose a backward error minimization
    problem for model reduction in terms of the norms of the
    perturbations to the coefficients $A$, $B$ and $C$ such that the
    perturbed systems are equivalent to systems of order $r<n$. It follows
    from the fact that singular values are insensitive to perturbations that
    a system with a small backward error has a small forward error, that
    is the difference between the transfer functions is small in $H_{\infty}$
    norm. We derive a singular value characterization for a simplified
    version of the backward error minimization problem. The singular
    value characterization is a generalization of a formula recently
    derived for the Wilkinson distance problem, the norm of the smallest
    perturbation to a matrix so that the perturbed matrix has a multiple
    eigenvalue. We suggest methods to estimate the Wilkinson distance
    and minimize the backward error for model reduction.

  • 06/05/07
    Ben Weinkove - Harvard University
    The Calabi-Yau equation and symplectic geometry

    The Riemannian Penrose inequality in dimensions less than 8
    Abstract: The Positive Mass Theorem states that a complete asymptotically flat manifold of nonnegative scalar curvature has nonnegative mass. The Riemannian Penrose inequality provides a sharp lower bound for the mass when black holes are present. More precisely, this lower bound is given in terms of the area of an outermost minimal surface, and equality is achieved only for Schwarzschild metrics. The Riemannian Penrose inequality was first proved in three dimensions in 1997 by G. Huisken and T. Ilmanen for the case of a single black hole. In 1999, H. Bray extended this result to the general case of multiple black holes using a different technique. In this talk we extend Bray's technique to dimensions less that 8. This is joint work with H. Bray.

  • 06/05/07
    Dan Lee - Duke University
    The Riemannian Penrose inequality in dimensions less than 8

    The Positive Mass Theorem states that a complete asymptotically flat manifold of nonnegative scalar curvature has nonnegative mass. The Riemannian Penrose inequality provides a sharp lower bound for the mass when black holes are present. More precisely, this lower bound is given in terms of the area of an outermost minimal surface, and equality is achieved only for Schwarzschild metrics. The Riemannian Penrose inequality was first proved in three dimensions in 1997 by G. Huisken and T. Ilmanen for the case of a single black hole. In 1999, H. Bray extended this result to the general case of multiple black holes using a different technique. In this talk we extend Bray's technique to dimensions less that 8. This is joint work with H. Bray.

  • 06/06/07

  • 06/07/07
    Bruce K. Driver - UCSD
    Path Integrals and Quantization

    I will discuss Feynman's path integral interpretation of quantum mechanics over curved configurations spaces, i.e. Riemannian manifolds. We will see how curvature of the configuration space enters in the interpretation (and the ambiguity) of Feynman's path integral prescription.

  • 06/07/07
    John D'Angelo - University of Illinois at Urbana - Champaign
    Positivity conditions in complex geometry

    I will begin with some elementary remarks about Hermitian
    symmetric functions on complex manifolds. I will introduce
    various positivity condtions for such functions and discuss
    the relationships among them. Examples include globalizable metrics on Hermitian
    line bundles. A version of the Cauchy-Schwarz inequality for
    Hermitian symmetric functions and how it relates to complex geometry will
    be a major part of the talk. To conclude I will pose several accessible problems

  • 06/07/07
    Michael Friedlander - Computer Science \\ University of British Columbia
    Exact regularization of convex programs

    An optimization problem is ill-posed if its solution is not unique or is acutely sensitive to data perturbations. A common approach to such problems is to construct a related problem with a well-behaved solution that deviates only slightly from the original solution set. The strategy is often used in data fitting applications, and also within optimization algorithms as a means for stabilizing the solution process.

    This approach is known as regularization, and deviations from solutions of the original problem are generally accepted as a trade-off for obtaining solutions with other desirable properties.

    In fact, however, there exist necessary and sufficient conditions such that solutions of the regularized problem continue to be exact solutions of the original problem. We present these conditions for general convex programs, and give some applications of exact regularization.

    (Joint work with Paul Tseng.)

  • 06/07/07
    Harold Stark - UCSD

  • 06/08/07
    Jason Bandlow - UCSD, Graduate Student
    Combinatorics of Macdonald Polynomials and Extensions

  • 06/11/07
    Mor Harchol-Balter - Computer Science Department \\ Carnegie Mellon University
    Analysis of Join-the-Shortest-Queue Routing in Web Server Farms

    We present the first analysis of the Join-the-Shortest-Queue (JSQ) routing policy for Web server farms. Web server farms involve a collection of Processor-Sharing (PS) servers, whereas prior analyses of JSQ have always assumed First-Come-First-Serve (FCFS) servers. This work introduces a new technique: Single-Queue-Approximation (SQA), and uses the technique to prove some interesting insensitivity properties for Web server farms.
    Based on joint work with: Varun Gupta, Karl Sigman, and Ward Whitt.

  • 06/12/07
    Rachel Pries - Colorado State Univ
    Boundary methods for the the p-rank strata of curves

  • 06/12/07
    Rob Ellis - Illinois Institute of Technology
    Two-batch liar games on a general bounded channel

    We consider a 2-person perfect information ``liar'' game, often called a R\'enyi-Ulam game. The basic game is that of ``twenty questions'' played between questioner Paul and responder Carole; Paul searches for a distinguished element $x$ in a search space $[n]$ by asking Yes-No questions of the form ``is $x\in A$'', where $A\subseteq [n]$. Carole responds `Yes' or `No', lying in up to $k$ responses. The fully off-line game is equivalent to $k$-error-correcting codes.

    We extend this game to a general channel $\mathcal{C}$ which governs the manner in which Carole may lie. Specifically, given the alphabet $[t]:=\{1,\ldots,t\}$, Paul searches for $x\in[n]$ by partitioning $[n]=A_1\cup \cdots \cup A_t$ and asking for $a$ such that $x\in A_a$. A lie is a tuple $(a,b)\in[t]\times [t]$ with $a\neq b$. The channel $C$ specifies an arbitrary set of lie strings of bounded length $\leq k$ from which Carole may choose a string and intersperse its lies, in order, among her responses. For example, when $t=2$, Carole lies with $(1,2)$ when she responds with 2 (``No'') when the correct response is 1 (``Yes''). We further restrict Paul to ask his questions in two off-line batches. We show that the maximum size of the search space $[n]$ for which Paul can guarantee finding the distinguished element is $t^{q+k}/(E_k(C){q \choose k})$ as $q\rightarrow\infty$, where $E_k(C)$ is the number of lie strings in $\mathcal{C}$ of maximum length $k$, generalizing previous work of Dumitriu and Spencer, and of Cicalese, Deppe, and Mundici. We similarly solve the pathological liar variant. This is joint work with Kathryn Nyman (Loyola University-Chicago).

  • 06/13/07
    Andrejs Treibergs - University of Utah
    An eigenvalue estimate and a capture problem

    Suppose n pursuers starting at the origin chase a single prey
    starting at 1, all doing standard independent Brownian motions on the real line. Bramson and Griffeath (1991) showed that the expected capture time is infinite for three or fewer pursuers and, after simulations, conjectured that it is finite for four or more. Li and Shao (2001) proved it for five or more pursuers. In recent work with Ratzkin, we show that it finite for four, completing the proof. We use the idea of Li and Shao to reduce the problem to an estimate of the first Dirichlet eigenvalue of a domain in the sphere.

    I'll discuss eigenvalues, describe the reduction, the eigenvalue estimates, and some related numerics.

  • 06/15/07
    Jon Grice - UCSD, Graduate Student

  • 06/15/07
    Bruno Pelletier - Université Montpellier 2
    Nonparametric set estimation

    We consider the problem of estimating a set S from a random sample of
    points of S, which amounts at estimating the support of the
    underlying probability density. Set estimation has applications in
    various situations, including medical diagnosises, image analysis,
    and quality control for example. We focus on the simple set estimator
    defined as the union of balls centered at the random points. Using
    tools from Riemannian geometry, and under mild analytic conditions on
    the underlying density of the data, we derive the exact rate of
    convergence of this set estimator.
    In closed connection with the problem of set estimation, we study the
    estimation of the number of connected components of a level set of a
    multivariate probability density. This allows one to assess the
    number of clusters of a statistical population, which is an essential
    problem of unsupervised learning. We introduce an estimator based on
    a graph, and using similar geometrical tools, we establish the
    asymptotic consistency of the methodology.

  • 09/07/07
    Andre Kundgen - California State University, San Marcos
    Graphs with many maximum independent sets

    A graph with independence number alpha is called (alpha,k)-balanced
    if every induced subgraph on k vertices has independence number alpha
    as well. We will discuss the maximum number of vertices in an
    (alpha,k)-balanced graph for fixed k and alpha, a problem with obvious
    connections to Ramsey Theory. We focus specifically
    on the case k=2alpha which is motivated from polyhedral combinatorics.

    (Joint work with A. Brieden, Z. Furedi and R. Ramamurthi)

  • 09/20/07
    Michele D'Adderio - UCSD Graduate Student
    Towards a geometric theory of algebras

  • 09/25/07
    Bo Li - UCSD
    Electrostatic free energy and its variations in implicit solvent models

    n this talk, I will first recall a mean field approximation of
    electrostatic free energy for an ionic solution, and discuss two
    issues: (1) Rigorous mathematical justification of the existence
    of equilibrium concentrations and their Boltzmann relations; (2)
    The effect of inhomogeneous Dirichlet boundary condition to the
    solution of the related Poisson-Boltzmann equation for the
    electrostatic potential. I will then consider a class of variational
    implicit solvent models for the solvation of biomolecules, and
    present a formal derivation of the first variation of the
    electrostatic free energy with respect to the location change of the
    dielectric boundary. This result is needed for level-set relaxation
    and force calculations of biomolecular structures and dynamics.

  • 09/25/07

  • 10/02/07
    Benzhuo Lu - Howard Hughes Medical Institute
    Electrodiffusion: A continuum modeling framework for biomolecular systems with realistic spatiotempo

    A computational framework is presented for the continuum modeling of
    cellular biomolecular diffusion influenced by electrostatic driving
    forces. This framework is developed from a combination of numerical
    methods, geometric meshing and computer visualization tools. In
    particular, a hybrid of (adaptive) finite element and boundary element
    methods is adopted to solve the Smoluchowski equation (SE), the Poisson
    equation (PE), and the Poisson-Nernst-Planck equation (PNPE) in order
    to describe electrodiffusion processes. The finite element method is
    used because of its flexibility in modeling irregular geometries and
    complex boundary conditions. The boundary element method is used due
    to the convenience of treating the singularities in the source charge
    distribution and its accurate solution to electrostatic problems on
    molecular boundaries. Nonsteady-state diffusion can be studied using
    this framework, with the electric field computed using the densities
    of charged small molecules and mobile ions in the solvent. A solution
    for mesh generation for biomolecular systems is supplied, which is an
    essential component for the finite element and boundary element computations.
    The partially coupled Smoluchowski equation and Poisson-Boltzmann equation
    (PBE) are considered as special cases of the PNPE in numerical algorithm,
    and therefore can be solved in this framework as well. The possible
    extensions of the physical model in this frame are also discussed. Some
    example computations are reported for: reaction-diffusion rate coefficient,
    ion density distribution, time-dependent diffusion process of the
    neurotransmitter consumption.

  • 10/04/07
    Dipendra Prasad - Tata Institute \\ University of California, San Diego
    On p-adic modular forms

  • 10/07/07
    Vyacheslav Yasilev - University of Tomsk, Russia
    On an adaptive choice of bandwidth for non-parametric kernel density estimators

    The properties of adaptive non-parametric kernel estimators for the multivariate probability density $f(x)$ (and its derivatives) of identically distributed random vectors $\varepsilon_n,\ n\geq 1$ at a given point are studied. It is supposed that the vectors $\varepsilon_n,\ n\geq 1$ form a martingale-difference process $(\varepsilon_n)_{n\ge 1}$ and the function to be estimated belongs to a class of densities slightly narrower than the class of densities with the following condition on the highest derivatives of the order $\nu:$

    |f^{(\nu)}(y)-f^{(\nu)}(x)|\leq \Delta(\|x-y\|),\qquad x,y\in {R^m},

    where $\Delta(t),\ t\geq 0,$ is some positive, bounded from above, monotonously decreasing for $t,$ small enough unknown function.

    An asymptotic mean square criterion is proposed. The optimality, in asymptotically minimax sense of adaptive estimators of density derivatives, is proved for a class of the Bartlett kernel estimators with a random data-driven bandwidth.

    It's well-known that the optimization of the asymptotic value of
    the mean squared error for the Bartlett kernel density estimators leads to the optimal bandwidth depending on unknown functions. Therefore it is not quite simple to apply these estimators to practice.

    The paper proposes an adaptive approach to this problem, which
    is based on the idea of changing the unknown functions in optimal bandwidth by a sequence of estimators converging to the unknown values of these functions. It is shown, that the constructed adaptive kernel estimators keep all the asymptotic
    properties of the sharp-optimal non-adaptive Bartlett estimators.

    An example of the adaptive estimator, optimal in the sense of the introduced criterion is considered. This estimator has simple structure and may be easily of practical usage in real statistical
    problems. The proposed estimators possess the property of uniform asymptotic normality and almost sure convergence.

  • 10/08/07
    Benedict Gross - Harvard University
    Parameters of Discrete Series Representations

  • 10/09/07
    Justin Roberts - UCSD
    Introduction to topological conformal field theories

    A (1+1)-dimensional Topological Quantum Field Theory is a tensor functor from the category of 2-dimensional cobordisms to the category of vector spaces. It is easy to give a characterisation of such functors: they are determined by the vector space associated to a single circle together with the structure maps it inherits from the disc and the pair of pants, which make it into a finite-dimensional Frobenius algebra.

    A (1+1)-dimensional Conformal Field Theory is a much more subtle thing, being a functor from the category of Riemann surfaces (2-dimensional cobordisms equipped with complex structures or "moduli") to a category of Hilbert spaces.

    Somewhere between lies the idea of Topological Conformal Field Theory, which is a "chain level" version of a CFT. It is determined by a chain complex on which the spaces of chains of the (morphism spaces of the) category of Riemann surfaces act. Such a structure arises in several places in modern topology, most notably in the theory of Gromov-Witten invariants of symplectic manifolds and in the Sullivan-Chas string topology of a loop space.

    This term we aim to read Kevin Costello's paper "Topological Conformal Field Theories and Calabi-Yau categories" (math.QA/0412149), which gives an algebraic characterisation of TCFTs analogous to the "Frobenius algebra" classification of TQFTs. In the first talk in the series I will try to give an overview of what the paper says, and we will organise talks for the rest of the term.

  • 10/09/07
    Randy Bank - UCSD
    Convergence Analysis of a Domain Decomposition Paradigm

    We describe a domain decomposition algorithm
    for use in several variants of the parallel
    adaptive meshing paradigm of Bank and Holst.
    This algorithm has low communication, makes
    extensive use of existing sequential solvers,
    and exploits in several important ways data
    generated as part of the adaptive meshing paradigm.
    We show that for an idealized version of the
    algorithm, the rate of convergence is independent
    of both the global problem size N and the number
    of subdomains p used in the domain decomposition
    partition. Numerical examples illustrate the
    effectiveness of the procedure.

  • 10/09/07
    Sebastian Cioaba - UCSD
    Eigenvalues of Graphs

    In this talk, I will present some new and old connections
    between the eigenvalues of a graph and its structure.

  • 10/09/07
    Yu Ding - California State University, Long Beach
    Degenerate singularity in 3-dimensional Ricci flow

  • 10/11/07
    Jacob Sterbenz - UCSD
    An introduction to non-linear dispersive equations I

    We'll give a broad introduction to the subject of non-linear dispersive equations by starting with an overview of some of the most basic examples of non-linear systems, and then discussing specific linear and non-linear estimates which have been used to study these equations.

  • 10/11/07
    Eric Tressler - UCSD, Graduate Student
    The Devil's Strategy is to Give Up

    The Angel lives on an infinite chessboard, and plays a game with the Devil. Each turn, the Devil removes a square; the Angel then flies up to 1000 king moves away, to any square still remaining on the board. Can the Devil trap the Angel?

    Berlekamp, Conway, and Guy introduced this game in 1982, and the Angel's fate remained conspicuously unknown until recent papers by four independent authors all showed that the Devil's cause is hopeless. One of the papers, by Andr\'{a}s M\'{a}th\'{e}, achieves this by considering a ``Nice Devil.'' This paper will be presented after some background is explored.

  • 10/11/07
    Jason Schweinsberg - UCSD
    Modeling the effect of beneficial mutations on the genealogy of a population

    The solutions of many important problems in population genetics require
    tools from probability theory because the evolution of a population over
    time is modeled as a random process. In this talk, we focus on the question
    of how to describe the effect that beneficial mutations have on the
    genealogy of a population. We present two approximations. The first, which
    involves coin tossing, is simple but not very accurate. The second, which
    is based on a stick-breaking construction, leads to much more accurate

  • 10/11/07
    Audrey Terras - UCSD
    A new kind of zeta function: When number theory meets graph theory

    I will present an introduction to zeta and L-functions of graphs by
    comparison with the zeta and L-functions of number theory. Basic
    properties will be discussed, including: the Ihara formula saying that the
    zeta function is the reciprocal of a polynomial. I will then explore graph
    analogs of the Riemann hypothesis, the prime number theorem, Chebotarev's
    density theorem, zero (pole) spacings, and connections with expander
    graphs and quantum chaos. References include my joint papers with Harold
    Stark in Advances in Mathematics. There is also a book I am writing on my

  • 10/15/07
    Uzy Hadad - Hebrew University, Israel
    Uniform Kazhdan Constant for some families of linear groups

    Let $R$ be a ring generated by $l$ elements with stable
    range $r$. Assume that the group $EL_d(R)$ has Kazhdan constant
    $\epsilon_0>0$ for some $d \geq r+1$. We prove that there exist
    $\epsilon(\epsilon_0,l) >0$ and $k \in \mathbb{N}$, s.t. for every
    $n \geq d$, $EL_n(R)$ has a generating set of order $k$ and a
    Kazhdan constant larger than $\epsilon$. As a consequence, we
    obtain for $SL_n(\mathbb{Z})$ where $n \geq 3$, a Kazhdan constant
    which is independent of $n$ w.r.t generating set of a fixed size.

  • 10/16/07
    Jacob Sterbenz - UCSD
    An introduction to non-linear dispersive equations II

    In this talk we'll continue with linear and non-linear equations, focusing on several "monotonicity principles". These include the so called Morawetz and local smoothing estimates.

  • 10/16/07
    Maia Averett - UCSD, Graduate Student
    Open-closed topological field theories

  • 10/16/07
    Philip E. Gill - UCSD
    Numerical Linear Algebra and Optimization

    In the formulation of practical optimization methods,
    it is often the case that the choice of numerical
    linear algebra method used in some inherent calculation
    can determine the choice of the whole optimization
    algorithm. The numerical linear algebra is particularly
    relevant in large-scale optimization, where the linear
    equation solver has a dramatic effect on both the
    robustness and the efficiency of the optimization.

    We review some of the principal linear algebraic issues
    associated with the design of modern optimization
    algorithms. Much of the discussion will concern the use
    of direct and iterative linear solvers for large-scale
    optimization. Particular emphasis will be given to some
    recent developments in the use of regularization.

  • 10/16/07
    Ray Luo - Department of Molecular Biology and Biochemistry \\ University of California, Irvine
    Implicit Solvent Modeling, Poisson-Boltzmann equations, and related topics

  • 10/16/07
    Blair Sullivan - Princeton University
    Feedback arc sets and girth in digraphs

    Given a directed graph $G$ with girth at least $m+1$ (and no parallel edges),
    let $\beta(G)$ denote the size of the smallest subset $X \subseteq E(G)$
    so that $G \setminus X$ has no directed cycles, and let $\gamma(G)$ be the
    number of non-edges. Prior joint work with Maria Chudnovsky and Paul Seymour
    showed that when $m = 3$, $\beta(G) \leq \gamma(G)$, and we conjectured
    $\beta(G) \leq \frac{1}{2}\gamma(G)$.
    Can one say anything stronger if $m > 3$? In this talk, I will discuss
    a new conjecture giving a ratio between $\beta(G)$ and $\gamma(G)$, namely
    $\beta(G) \leq \frac{2}{m^2-m-1}\gamma(G)$, for $m \geq 3$. The talk will
    also cover two new results in this direction: the bound
    $\beta(G) \leq \frac{1}{3}\gamma(G)$ when $m=4$, and for
    circular interval graphs, a generalization of previous methods which
    gives a new bound for all $m$.

  • 10/18/07
    Nitu Kitchloo - UCSD

    I'll try to explain how one may enumerate manifolds endowed with
    some structure. For example, we may want to list all almost complex or
    parallelizable manifolds. This problem is too general as posed, so we will
    work up to cobordism (to be defined). The result will be a very elegant
    framework which was first explored by R. Thom, and now belongs to the
    toolbox of any self-respecting algebraic topologist.

  • 10/18/07
    Harold Stark - UCSD

  • 10/18/07
    Peter Topping - University of Warwick
    Posing Ricci flow on Riemann surfaces

  • 10/18/07
    Pan Peng - Harvard University
    On a proof of the Labastida-Marino-Ooguri-Vafa conjecture

    Based on large N Chern-Simons/topological string duality, in a series of papers, J.M.F. Labastida, M. Marino, H. Ooguri and C. Vafa conjectured certain remarkable new algebraic structure of link invariants and the existence of infinite series of new integer invariants. In this lecture, I will describe a proof of this conjecture. Moreover, I will show that these new integer invariants vanish at large genera. In the end of the talk, some application in the knot theory and related problems (e.g., the famous volume conjecture), will also be discussed.

  • 10/23/07
    Mohammad Ali Ebrahimi-Fardooe - UCSD, Graduate Student
    Image inpainting

    Image inpainting involves filling in part of an image (or video) using
    information from the surrounding area. In this talk I will discuss
    connection of Navier-Stokes equations (NSE) in image inpainting. This
    important connection suggests the possibility of other hybrid methods
    turbulence models in image inpainting.
    Recently, the three-dimensional (3d) Navier-Stokes-Voight (NSV)
    were suggested as a regularizing model for the 3d NSE. We would like
    investigate how we can tune the relevant parameters of this model to
    optimize the end result in image inpainting.

  • 10/23/07
    Yongcheng Zhou - Biochemistry and Mathematics, UCSD
    Modeling, computation and applications of electrostatic stress of bimolecules

    Electrostatic forces play a crucial role in biomolecular interactions such as
    protein-ligand association, protein induced membrane deformation or DNA curvature
    and deformation in protein-DNA complexes. Although the electrostatic forces
    derived from the potential solution of the Poisson-Boltzmann equation (PBE)
    have been widely used in studying these interactions with Brownian dynamics
    simulations, the modeling and computation of these forces in the continuum
    framework are not well established. In this talk I will propose the models of
    the electrostatic body force and surface force compatible with the PBE, and
    discuss the stable computation methods of these forces using a new stable
    regularization scheme of the PBE. Numerical experiments will be presented to
    verify these models and their usefulness of in studying the rigid body motion
    of biomolecules.

  • 10/23/07
    Ben Weinkove - Harvard
    Canonical metrics and Kahler geometry

    I will discuss canonical metrics on Kahler manifolds,
    including Kahler-Einstein and constant scalar curvature Kahler
    metrics. By conjectures of Yau, Tian and Donaldson, the existence of
    such a metric should be equivalent to the properness of the relevant
    energy functional and to the algebraic `stability' of the manifold.
    I will describe some work on energy functionals in Kahler geometry
    and methods for finding canonical metrics.

  • 10/25/07
    Ben Wilson - UCSD, Graduate Student
    Symmetric Bilinear Forms and 4-Manifolds

    A large part of this talk will be devoted to symmetric bilinear forms and inner product spaces, which should be of interest to geometers and algebraists alike. I'll outline some of the theory over general (commutative) rings and then turn to the classification of non-degenerate symmetric bilinear forms over the integers. The case of indefinite forms is completely understood and not too hard. On the contrary, the case of positive definite forms is quite difficult and, surprisingly, turns out to be related to the question of packing oranges in Euclidean boxes.

    In the second part of the talk, we will see that symmetric bilinear forms come up as invariants in the theory of $4$-dimensional manifolds (their intersection forms). Freedman showed that a (topological) $4$-manifold is (almost) classified by its associated bilinear form and that indeed every possible form arises in this way. On the other hand, Donaldson showed that the intersection form of a \emph{smooth} $4$-manifold has a very special structure. Combining the two results led to examples of many topological $4$-manifolds that do not admit a smooth structure, a revolution of the (previously almost non-existent) theory of $4$-manifolds, two fields medals, lots of other interesting research, and this talk.

  • 10/25/07
    Ery Arias-Castro - UCSD
    Searching for a Trail of Evidence in a Maze

    Suppose we observe a security network composed of
    sensors with each sensor returning a value indicating whether the
    sensor is at risk (high value) or not (low value). A typical attack
    leaves a trail where the sensors return higher-than-normal values.
    The goal is to detect a possible attack. Within a simplified
    framework, we will see that if the sensor do not return high-enough
    values (we will quantify that), then detection is impossible.

    Formal abstract: Consider the complete regular binary tree of depth M
    oriented from the root to the leaves. To each node we associate a
    random variable and those variables are assumed to be independent.
    Under the null hypothesis, these random variables have the standard
    normal distribution while under the alternative, there is a path from
    the root to a leaf along which the nodes have the normal distribution
    with mean A and variance 1, and the standard normal distribution away
    from the path. We show that, as M increases, the hypotheses become
    separable if, and only if, A is larger than the square root of 2 ln 2. We obtain corresponding results for other graphs and other
    distributions. The concept of predictability profile plays a crucial
    role in our analysis.

    Joint work with Emmanuel Candes, Hannes Helgason and Ofer Zeitouni.

  • 10/25/07
    Karen Acquista - Boston University
    The Weil group of a 2-local field

    I will explain the cohomological approach to local class
    field theory, and how it generalizes to higher local fields. For
    example, a 2-local field $K$ is a complete discrete valuation field with
    residue field a local field; class field theory for K states, in part,
    that there is an Artin map from $K_2(K)$ to the abelianized absolute
    Galois group of $K$.

    I will show how the cohomological approach can be used to construct a
    Weil group for any finite extension of 2-local fields.

  • 10/29/07
    Murray Schacher - University of California, Los Angeles \\ Center for Communications Research
    Distinguishing quaternions by splitting fields

  • 10/30/07
    Dan Budreau - UCSD
    A-infinity algebras

  • 10/30/07
    Zeyun Yu - Math and Biochemistry, UCSD
    Molecular Mesh Generation and Processing

    Molecular surface and volumetric meshes are important for molecular
    modeling and simulation. I'll talk about mesh generation using a
    so-called "soft-model" approach, meaning that each atom is treated as
    a smoothly decaying Gaussian function instead of a hard sphere, and
    the molecular surfaces are given as level sets of the summation of
    such functions from all atoms in a molecule. The mesh generation
    toolchain consists of several steps: (1) from molecules to 3D volumes;
    (2) from volumes to initial surface meshes; (3) surface mesh
    post-processing; and (4) volumetric mesh generation and smoothing. A
    number of examples on molecules taken from Protein Data Bank (PDB)
    will be demonstrated.

  • 10/30/07
    Sami Assaf - University of Pennsylvania
    A combinatorial proof of Macdonald positivity

    Since they were introduced by Macdonald 20 years ago,
    Macdonald polynomials have been widely studied and have been found to
    have applications in such areas as representation theory, algebraic
    geometry, group theory, statistics, and quantum mechanics. The
    Macdonald integral form may be defined as the unique function
    satisfying certain triangularity and orthogonality conditions, from
    which symmetry follows. The Macdonald Positivity Conjecture (now
    Theorem) states that the coefficients of the Macdonald integral form
    expanded into Schur functions are non-negative integers. The original
    proof, due to Haiman in 2001 building on joint work with Garsia, uses
    difficult machinery in algebraic geometry and does not provide a
    combinatorial understanding of the coefficients. In this talk we
    present a purely combinatorial proof of Macdonald positivity and give
    a combinatorial interpretation for the Schur coefficients. The proof
    utilizes an elegant monomial expansion for Macdonald polynomials
    discovered by Haglund in 2004 and a new combinatorial tool called a
    dual equivalence graph.

  • 11/01/07
    Sam Buss - UCSD
    Large Numbers, Busy Beavers, Noncomputability, and Incompleteness

    This talk surveys fundamental concepts in computability theory, including undecidability (noncomputability) and the G\"odel incompleteness theorems. The motivation and proof sketches are based on the Berry paradox."

  • 11/01/07
    Karen Acquista - Boston University
    The Weil group of a 2-local field

    I will explain the cohomological approach to local class
    field theory, and how it generalizes to higher local fields. For
    example, a 2-local field K is a complete discrete valuation field with
    residue field a local field; class field theory for K states, in part,
    that there is an Artin map from $K_2(K)$ to the abelianized absolute
    Galois group of K.

    I will show how the cohomological approach can be used to construct a
    Weil group for any finite extension of 2-local fields.

  • 11/05/07
    Tom Dorsey - Center for Communications Research
    Minimal Ring Extensions of Prime Rings

  • 11/06/07
    Jacob Sterbenz - UCSD
    \Huge \bf Some simple $L^p$ estimates for oscillatory integrals Name

    This week we'll talk about proving $L^p$ estimates for solutions to constant coefficient dispersive equations. This is really a (quite specialized) problem in a much larger field known as "geometric harmonic analysis". We'll discuss some of the simplest tools for proving estimates of this type, and also a bit about the structure of the oscillatory integrals themselves. If time permits, we'll give some applications of these estimates to non-linear problems.

  • 11/06/07
    Will Wylie - University of California, Los Angeles
    Rigidty of Gradient Ricci solitons

  • 11/06/07
    John Shopple - UCSD Graduate Student
    The Level Set Method in a Finite Element Setting

    The level set method is a way of tracking moving interfaces
    Topological changes (splitting or joining) of several moving
    interfaces are
    captured naturally by the level set method. I will briefly introduce
    level set method, then discuss some of my current research: the level
    method in a finite element setting, body fitting a mesh to the
    reinitialization and application to finding equilibrium solute-solvent

  • 11/06/07
    Yongcheng Zhou - Biochemistry and Mathematics, UCSD
    Modeling, Computation and Applications of Electrostatic stress of Bimolecule

    Electrostatic forces play a crucial role in biomolecular interactions
    such as protein-ligand association, protein induced membrane deformation
    or DNA curvature and deformation in protein-DNA complexes. Although the
    electrostatic forces derived from the potential solution of the
    Poisson-Boltzmann equation (PBE) have been widely used in studying these
    interactions with Brownian dynamics simulations, the modeling and
    computation of these forces in the continuum framework are not well
    established. In this talk I will propose the models of the electrostatic
    body force and surface force compatible with the PBE, and discuss the stable
    computation methods of these forces using a new stable regularization scheme
    of the PBE. Numerical experiments will be presented to verify these models
    and their usefulness of in studying the rigid body motion of biomolecules.

  • 11/08/07
    Daniel Vallieres - UCSD
    The Euler Identity

    In this food for thought seminar, we will reflect upon the beautiful Euler identity
    $$e^{i \pi} = -1 $$
    Apparently, Euler viewed in this equation all the body of mathematics. The number $e$ is related to analysis, the number $\pi$ to geometry, the number $i$ to algebra, and the number $-1$ to arithmetic or synonymously number theory.
    \\ \\
    The following identities are well known:
    \item The $x$ and $y$ coordinates of the third roots of unity on the unit circle
    \item $\sin(\frac{2 \pi}{3}) = \frac{\sqrt{3}}{2}$ and $\cos(\frac{2 \pi}{3}) = -\frac{1}{2}$
    \item $\sin(\frac{4 \pi}{3}) = -\frac{\sqrt{3}}{2}$ and $\cos(\frac{4 \pi}{3})= -\frac{1}{2}$
    \item The $x$ and $y$ coordinates of the fourth roots of unity on the unit circle
    \item $\sin(\frac{2 \pi}{4}) = 1$ and $\cos(\frac{2 \pi}{4}) = 0$
    \item $\sin(\frac{4 \pi}{4}) = 0$ and $\cos(\frac{4 \pi }{4}) = -1$
    \item $\sin(\frac{6 \pi}{4}) = -1$ and $\cos(\frac{6 \pi}{4}) = 0$
    The following ones are less well known:
    \item The $x$ and $y$ coordinates of the fifth roots of unity on the unit circle
    \item $\sin(\frac{2 \pi}{5})= \frac{\sqrt{5 + 2 \sqrt{5}}}{\sqrt[5]{176 + 80\sqrt{5}}}$ and $\cos(\frac{2 \pi}{5}) = \frac{1}{\sqrt[5]{176 + 80\sqrt{5}}}$
    \item $\sin(\frac{4 \pi}{5}) = {\frac{\sqrt{5 - 2\sqrt{5}}}{\sqrt[5]{-176 + 80\sqrt{5}}}}$ and $\cos(\frac{4 \pi}{5}) = -\frac{1}{\sqrt[5]{-176 + 80\sqrt{5}}}$
    \item $\sin(\frac{6 \pi}{5}) = - \frac{\sqrt{5 - 2\sqrt{5}}}{\sqrt[5]{-176 + 80 \sqrt{5}}}$ and $\cos(\frac{6 \pi}{5}) = - \frac{1}{\sqrt[5]{-176 + 80\sqrt{5}}}$
    \item $\sin(\frac{8 \pi}{5}) = -\frac{\sqrt{5 + 2\sqrt{5}}}{\sqrt[5]{176 + 80\sqrt{5}}}$ and $\cos(\frac{8\pi}{5}) = \frac{1}{\sqrt[5]{176 + 80\sqrt{5}}}$
    We shall explain a painless way of deriving these formulas. For that, we will do a little bit of analysis, geometry, algebra and arithmetic.

  • 11/08/07
    Jacob Sterbenz - UCSD
    Waves in elastic media, and the Huygens' principle.

    In this talk we'll discuss the phenomena of wave propagation in
    elastic membranes and solids. Using a few simplifying assumptions, one can
    use Newton's equations to write down a partial differential equation for how
    such a medium attempts to return to equilibrium from an initial displacement
    or initial impulse.

    There are several striking consequences one can derive from these partial
    differential equations. The first is that even in an isotropic solid there
    are two different kinds of waves, each of which propagates with a distinct
    speed! The second is that there is a marked difference between how waves
    propagate through a solid as opposed to how they propagate across a
    membrane. In the former case the disturbance completely leaves any bounded
    region and heads outward unless it is reflected back by some sort of
    boundary. But in the latter case there is always some lingering "residue" of
    a wave in any bounded region, no matter how far the bulk portion of the wave
    has progressed. There are still open research questions concerning this
    curious phenomena, and we'll try to discuss these by the end of the talk.

  • 11/08/07
    Ping-Shun Chan - UCSD
    Odd Degree Cyclic base change for U(3)

    I will present the base-change lifting of automorphic, or local admissible, representations of the unitary group U(3), with respect to an odd degree cyclic field extension.

  • 11/08/07
    Shengli Kong - University of California, Irvine
    Some remarks on the Ricci flow

  • 11/13/07

  • 11/13/07
    Natalia Berloff - Department of Applied Mathematics and Theoretical Physics \\ University of Cambridge
    Mathematical Models of Superfluids

    In the past decade, superfluid ultracold gases and liquids
    have become the physicists' preferred domain for exploring
    novel phenomena in condensed matter, and for testing many-body
    theories of how atoms interact with each other. There is an
    explosion of the new quantum fluids discovered, that challenge
    both mathematicians and physicists by their fascinating, yet
    enigmatic behaviors. The most famous of them is the ability
    to flow through narrow gaps without any friction and energy
    dissipation and the existence of quantized vortices. Different
    problems and approaches in mathematical modeling of the
    macroscopic superfluid dynamics will be discussed. Particular
    emphasis will be given to the formation of coherent structures
    and the quantized vortex dynamics.

  • 11/13/07
    Gaurav Arya - Department of Nanoengineering, UCSD
    Mesoscale modeling and simulation of chromatin

    Eukaryotic double-stranded DNA achieves cellular compaction through several
    hierarchical levels of organization. First, DNA wraps around nucleosomes that
    comprise of two copies each of the positively charged core histones H2A, H2B,
    H3 and H4. The resulting "bead-on-a-string" nucleoprotein complex folds further
    at physiological salt, and in the presence the linker histone, into the 30-nm
    chromatin fiber. The thermodynamic and structural details of how histone tails
    (N-termini of core histones) and linker histones critically compact and modulate
    chromatin structure as well as regulate gene transcription are not well understood.
    I present a new mesoscopic model of chromatin that represents nucleosome cores
    as rigid bodies with an electrostatic surface, linker DNA as a discrete worm-like
    chain model, and histone tails as protein bead chains, to elucidate the physical
    role of each histone tail and the linker histone in chromatin folding. An
    end-transfer configurational-bias Monte Carlo approach provides the positional
    distribution of histone tail and their physical interactions at different salt
    milieus. Analyses indicate that the H4 tails mediate the strongest internucleosomal
    interactions; the H3 tails crucially screen electrostatic repulsion between the
    linker DNAs; and the H2A and H2B tails mediate fiber/fiber interactions. The
    primary function of the linker histones is to decrease the nucleosome triple angles,
    resulting in highly compact chromatin with a different internucleosomal interaction
    pattern than that obtained in linker-histone deficient chromatin. The development
    of this model also opens new avenues for studying higher-order structures of
    chromatin and the role of posttranslational modifications and variants of histone
    tails in gene regulation.

  • 11/13/07
    Mike Hansen - UCSD
    SAGE and Symmetric Functions

    SAGE is a free, open-source computer algebra that aims to
    be a replacement for Mathematica, Maple, Matlab, and Magma and was
    started by William Stein, a number theorist at the University of
    Washington. Recently, a lot of support has been added to SAGE for
    working with combinatorial objects such as partitions, permutations,
    tableaux, etc. Additionally, SAGE now has fairly good support for
    symmetric functions, which are "polynomials" in infinitely many
    variables which are invariant under any permutation of those
    variables. In this talk, I will give a brief introduction to SAGE and
    give an overview of symmetric functions and how to work with them in
    SAGE. I will also talk briefly about some generalizations of
    symmetric functions including Hall-Littlewood polynomials and
    Macdonald polynomials.

  • 11/15/07
    David Scheinker - Graduate Student, UCSD
    The Math Behind the Magic

    Do you ever find yourself disappointed and confused when an
    interesting result follows from an algebraic identity that seems to
    have come out of nowhere? Do arguments that are "so elegant that they
    conceal what is really going on" leave you longing for more? WHAT IF
    I'm going to present a short proof of a classical result that
    follows from an unmotivated algebraic identity. I will then walk
    through an intuitive, Functional Analytic, derivation of the theorem
    introducing and explaining definitions as I go. Ideally the long proof
    of the theorem will completely illustrate the meaning of the short
    one. The emphasis of the talk will be on introducing interesting ideas
    in functional analysis in the context of an extended example.

  • 11/15/07
    Bill Helton - UCSD
    Positive Thinking and Other Inequalities

    A polynomial $p$ is bigger than $q$ if $p(x)> q(x)$ for all $x$ and a classical area of mathematics (called semialgebraic geometry) focuses on providing algebraic ways to test if such inequalities hold.

    Semialgebraic geometry can be generalized to polynomials with matrix variables which is fortunate, since many problems in systems engineering have matrix unknowns. The difficulty here is that the polynomials are in variables which do not commute.
    Particularly important (for numerical solutions) is convexity and there has been a recent quest to classify convex noncommutative polynomials. The conclusion turns out to be simple. The talk will be on the mathematics and the motivation.

  • 11/15/07
    Henry Cohn - Senior Researcher \\ Microsoft Research
    Mysteries of Euclidean sphere packing bounds

    The best bounds known for the Euclidean sphere packing problem (at least in dimensions 4 through 36) are proved using linear programming bounds due to Cohn and Elkies. These bounds are derived using a simple argument based on Poisson summation, but optimizing the bounds has proved difficult. It is conjectured that the bounds are in fact sharp in 8 and 24 dimensions, and this has been verified to 30 decimal places, but it has not been proved. In this talk, I'll discuss the challenges that must be overcome to prove it as well as some new mysteries that have arisen recently in this area.

  • 11/15/07
    Herbert Heyer - University of T\"ubingen"
    \bf \Huge Bi-invariant L$\bf \acute{e}$vy processes

    For a Gelfand pair $(G,H)$ consisting of a locally compact group $G$ and a compact subgroup $H$
    of $G$, the $H$-bi-invariant measures on $G$ can be identified with measures on the double coset space $G/\!/H$. The canonical projection from $G$ onto $G/\!/H$ induces a convolution of measures on $G/\!/H$ such that $G/\!/H$ becomes a commutative hypergroup. Central to the present talk will be the discussion of
    L\'evy processes with values in the hypergroup $G/\!/H$ and their characterization in terms of complex-valued martingales.

    In order to achieve this goal some harmonic analysis on hypergroups has to be developed: the convolution hemigroups associated with L\'evy processes will be studied with the help of a Fourier transform.

    On Sturm-Liouville hypergroups the characterization of Gaussian hemigroups is obtained via martingales involving moments. For convolution semigroups, i.e.\ stationary L\'evy processess, the results yield characterizations of radial Brownian motions on Euclidean and hyperbolic spaces.

  • 11/19/07
    Zoran Grujic - University of Virginia
    Regularity of Koch-Tataru solutions to the 3D Navier-Stokes equations revisited

    Koch-Tataru solutions are global-in-time small data mild
    solutions to the 3D NSE emanating from small data in
    $BMO^{-1}$. There have been several recent works in which
    the regularizing-decay rate estimates for Koch-Tataru
    solutions have been obtained. Spatial analyticity of solutions
    then follows as a consequence. I will present a different
    approach in which the spatial analyticity is obtained
    directly together with explicit estimates on the time-evolution
    of the domain of analyticity. The regularizing-decay rate
    estimates will then follow at once.

  • 11/19/07
    Greg Knese - University of California, Irvine
    Bernstein-Szeg\H{o} measures on the two dimensional torus

    This talk is about how reproducing kernels and certain
    rearrangements of finite dimensional subspaces of two variable
    polynomials can reprove several interesting results. These include
    recent results of Geronimo and Woerdeman on orthogonal polynomials on
    the two dimensional torus as well as Ando's inequality (and its close
    relative, Pick interpolation theorem on the bidisk). Considerable
    time will be spent on one variable in order to make the ideas clear.

  • 11/19/07
    Daniel Wulbert - UCSD
    The Locator Problem

    The locator problem models the following physical situation.
    Suppose one lands an unmanned space craft on an unobservable terrain (e.g., under the clouds of Venus or on the backside of the moon) and wishes to determine the location of the landing site. Suppose that the craft can sample the altitude at the landing site and at several other spots (say at 100 meters east and at 100 meters west). However, the
    craft does not have a map of the altitudes of the terrain (i.e.,
    $a(x,y)$), but only a single function, $p(x,y)$ that approximates the altitude. The locator problem is to find a function p from a family of functions $P$ that minimizes the error between the actual location of the craft and the computed location of the craft using the approximation $p$.
    The error is to be minimized over all possible locations that is we seek the $p$ in $P$ to minimize

    $$\Arrowvert (x,y)- p^{(-1)}(a(x,y))\Arrowvert$$

    This is equivalent to classical approximation questions about existence and uniqueness of best approximations from this (non-linear) family of inverse functions. The question is most interesting when the elements in the setting are the most fundamental and basic: for example, when $P$ is the polynomials of degree n and the norm is the uniform norm or the
    $L_1$ norm.

    Although this is a rich theoretical setting with five fundamental
    elements to define (various metrics, data collections and families of approximating functions) and potentially has useful applications, we
    have the only known solutions. These are for $P$ the increasing
    polynomials of degree $n$; the domain and range being the unit interval
    and the norm being either the uniform norm or the $L_1$ norm. In this
    setting there exist best locator functions and they are unique.

  • 11/20/07
    Nitu Kitchloo - UCSD
    Generators and relations for TCFTs

  • 11/20/07
    Becca Thomases - University of California, Davis
    Analysis and Computations for Viscoelastic Fluids

    Viscoelastic flow modeled by the Oldroyd-B equations will be discussed from an analytical and computational perspective. First I will present a local energy decay theorem which applies to a large class of hyperbolic systems including the Oldyoryd-B model. This decay theorem is used to prove that global smooth solutions exist for small initial data. While small solutions are global, the problem for large data is much more complicated. I will present recent computational work on the Oldroyd-B equations which indicates that the system develops singularities exponentially in time at hyperbolic stagnation points in the flow. The singularities arise in the stress field of the flow and the algebraic structure of these singularities depends critically on an important elasticity parameter, the Weissenberg number. A local approximation to the solution at the hyperbolic stagnation point is constructed and there is excellent agreement between the local solution and the simulations. In addition, past a critical Weissenberg number the flow pattern becomes quite sensitive to time periodic perturbations of the background forcing (or changes in initial data) and there is a transition from small scale local mixing around the stable and unstable manifolds to global mixing in the fluid.

  • 11/20/07
    Hieu Nguyen - UCSD Graduate Student
    Adaptive Finite Element Methods For Solving PDEs

    In this talk, we study three versions of
    adaptive finite element method: r, h and p-adaptive. While
    the first two are well developed and widely used, much less
    is known about the third one. By looking at some ideas and
    techniques used in r-version and h-version, we propose a
    tentative plan to construct a p-version of adaptive finite
    element method. The key thing makes it possible is the
    recently result of Bank, Xu and Zheng on generalizing
    gradient recovery technique for linear elements to
    derivative recovery for Lagarange elements of order p

  • 11/20/07
    Luis Silvestre - Courant Institute \\ New York University
    Fully nonlinear integro-differential equations

    We study nonlinear integro-differential equations. Typical examples
    are the ones that arise from stochastic control problems with
    discontinuous Levy processes. We can think of these as nonlinear
    equations of fractional order. Indeed, second order elliptic PDEs are
    limit cases for integro-differential equations. Our aim is to extend
    the theory of fully nonlinear elliptic equations to this class of
    equations. We are able to obtain a result analogous to the Alexandroff
    estimate, Harnack inequality and $C^{1,\alpha}$ regularity. As the
    order of the equation approaches two, in the limit our estimates
    become the usual regularity estimates for second order elliptic pdes.
    This is a joint work with Luis Caffarelli.

  • 11/20/07
    Katherine Stange - Brown University
    Elliptic Nets

    Elliptic divisibility sequences are integer recurrence sequences, each
    of which is associated to an elliptic curve over the rationals
    together with a rational point on that curve. I'll give the background
    on these and present a higher-dimensional analogue over arbitrary base
    fields. Suppose $E$ is an elliptic curve over a field $K$, and $P_1$, ...,
    $P_n$ are points on E defined over $K$. To this information we associate
    an $n$-dimensional array of values of $K$ satisfying a complicated
    nonlinear recurrence relation. These are called elliptic nets. All
    elliptic nets arise from elliptic curves in this manner. I'll explore
    some of the properties of elliptic nets and the information they
    contain, relating them to generalised Jacobians and to the Tate
    and Weil pairings.

  • 11/20/07
    Olga K. Dudko - Department of Physics and NSF Center for Theoretical Biological Physics \\ UCSD
    Single-Molecule Pulling Experiments: Theory, Analysis and Interpretation

    Mechanical forces are generated during nearly every facet of the
    cell cycle. Recent advances in experimental techniques enable
    experimentalists to exert forces on individual molecules and observe
    their response in real time. Thus, the single-molecule approach
    has changed the way many physical, chemical and biological problems
    are addressed. We present a theory for extracting kinetic information
    from single-molecule pulling experiments at constant force or
    constant loading rate. Our procedure provides estimates of not
    only i) the intrinsic rate coefficient and ii) the location of the
    transition state, as in the widely used phenomenological approach
    based on Bell's formula, but also iii) the free energy of activation.
    We illustrate the use of our approach by applying it to sets of data
    obtained from nanopore unzipping of individual DNA hairpins and from
    unfolding of single protein molecules with the atomic force microscope.

  • 11/20/07
    Vsevolod (Seva) Lev - The University of Haifa \\ University of California, San Diego
    Projecting difference sets onto the positive orthant

    A combinatorial geometry problem, related (in a surprising way)
    to the Graham's g.c.d. conjecture, is as follows. Let $n\ge 1$ be
    an integer. Given a vector $(a_1 , ... , a_n)\in R^n$, write
    $$ a^+ := ( \max(a_1,0) , ... , \max(a_n,0) ) $$
    (the "projection of $a$ onto the positive orthant"), and for a set
    $A\subset R^n$ put
    $$ A^+ := \{ a^+ : a\in A \}. $$
    How small $|(A-A)^+|$ can be for a set $A\subset R^n$ of given
    cardinality $|A|$? We discuss previously known results and report
    on recent developments due to Ron Holzman, Rom Pinchasi, and
    the presenter.

  • 11/27/07
    Justin Roberts - UCSD
    TCFTs and A-infinity categories

  • 11/27/07
    Hakan Nordgren - UCSD Graduate student
    Well-posedness for the equations of motion of an incompressible, inviscid, self-gravitating fluid with free boundary

    Building on work by Hans Lindblad, Daniel Coutand and Steve Shkoller, we prove that the equations of motion of an incompressible, inviscid, self-gravitating fluid with free boundary are well-posed. The methodology consists of the use of a smoothing operator which smoothes along the boundary of the fluid. This then allows the use of a fixed-point-type argument, which provides a solution to a smoothed version of the problem. Estimates subsequently show that these solutions converge to a solution of the full problem.

  • 11/27/07
    Alex Bilik and Bo Li - Department of Mathematics, UCSD
    Fluid density modeling and simulations of solvation of nonpolar molecules

  • 11/27/07
    Fan Chung Graham - UCSD
    The pagerank and heat kernel of a graph

    We will give four proofs of the Cheeger inequality which relates
    the eigenvalues of a graph with various isoperimetric variations of the Cheeger constant. The first is a simplified proof of the classical Cheeger inequality using eigenvectors. The second is based on a rapid mixing result for random walks by Lov\\'asz and Simonovits. The third uses PageRank, a quantitative ranking of the vertices introduced by Brin and Page. The fourth proof is by an improved notion of the heat kernel pagerank. The four proofs lead to further improvements of graph partition algorithms and in particular the local partition algorithms with cost proportional to its output instead of in terms of the total size of the graph.

  • 11/29/07
    Dipendra Prasad - Tata Institute, India
    Introduction to Rigid Analytic Geometry

  • 11/29/07
    Chris Tiee - UCSD Graduate Student
    Goin' with the Ricci Flow

    We first introduce the honored guest, Ricci Flow and present some simple examples. We'll then get excited about why it is used: it is essentially a nonlinear heat equation which is used for smoothin' out the rough edges around some baddie metrics. However there are some real rogues who are a hopeless case, and develop singularities when goin' with the flow. These can ``excised'' under certain conditions, and we can continue on our merry way.

    Next, we describe why it is called a ``flow,'' and chat about how flows fit into the big picture in solving PDEs in general. We really hope that this will not only (partially) demystify the now famous Ricci Flow, but also give newbie analysts and PDE-ers a big picture of what kinds of totally awesome modern tools and methods they'll be encountering.

  • 11/29/07
    Barbara Neuhierl - Siemens Corporate Research and Technologies, Munich
    The Lattice-Boltzmann-Method for Computational Aeroacoustics

    The so-called Lattice-Boltzmann-Method is a relatively young numerical
    method for the computation of fluid mechanics. In contrary to
    'classical' techniques of computational fluid dynamics where the
    Navier-Stokes-Equations - a system of partial differential equations
    describing the macroscopic behavior of a flow - are discretized and
    solved, the Lattice-Boltzmann-Method starts from the microscopic
    description of a flow, the Boltzmann-equation. It is a time-explicit
    procedure based on a strongly simplified kinetic theory, yet it is
    approximating the Navier-Stokes equations.

    The presentation gives a short overview about some basic principles of
    the Lattice-Boltzmann-Method. Typical engineering applications, in
    particular the feasibility of describing aeroacoustic phenomena (i.e.
    sound generated by flows which typically requires a time-dependent
    calculation of pressure and density within a fluid), are pointed out,
    and a partitioned approach for the coupled simulation of flows and
    structures is presented.

  • 11/29/07
    Francesco Montrone - Siemens Corporate Research and Technologies, Munich
    Design of High Voltage Devices based on Sensitivity Analysis

    The design of electrodes' and insulators' shapes with respect to
    electric field stresses in High Voltage Devices is formulated as a
    nonlinear optimization problem under constraints.
    We apply a sensitivity equation method in order to achieve gradient
    information for the optimal design problem. This enables us to use the
    SQP-Algorithm for the optimization task while a FEM software package
    accomplishes the electric field computation.
    Results are presented for gas-insulated transmission lines and
    circuit-breakers in highly integrated switch-gears.

  • 11/29/07
    Ery Arias-Castro - UCSD
    Searching for a Trail of Evidence in a Maze

    Suppose we observe a security network composed of
    sensors with each sensor returning a value indicating whether the
    sensor is at risk (high value) or not (low value). ?A typical attack
    leaves a trail where the sensors return higher-than-normal values.?
    The goal is to detect a possible attack.? Within a simplified
    framework, we will see that if the sensor do not return high-enough
    values (we will quantify that), then detection is impossible.

    Formal abstract: Consider the complete regular binary tree of depth M
    oriented from the root to the leaves.? To each node we associate a
    random variable and those variables are assumed to be independent.?
    Under the null hypothesis, these random variables have the standard
    normal distribution while under the alternative, there is a path from
    the root to a leaf along which the nodes have the normal distribution
    with mean A and variance 1, and the standard normal distribution away
    from the path. We show that, as M increases, the hypotheses become
    separable if, and only if, A is larger than the square root of? 2 ln
    2.? We obtain corresponding results for other graphs and other
    distributions.? The concept of predictability profile plays a crucial
    role in our analysis.

    Joint work with Emmanuel Candes, Hannes Helgason and Ofer Zeitouni.

  • 11/29/07
    Dipendra Prasad - Tata Institute, India
    Period integrals and central critical L-values

    Period integrals of automorphic forms, such as Fourier coefficients of modular forms, is a subject of much contemporary interest.
    In this talk we survey some of the literature on the subject, and
    include some of my own works which relate these global concepts to
    analogous local questions.

  • 11/29/07
    Harm Derksen - University of Michigan
    Algorithms for Invariant Rings

    The main goal in Invariant Theory is to study the ring of all
    polynomials that are invariant under an group action. Invariant
    rings are not always finitely generated, because of Nagata's
    counterexample to Hilbert fourteenth problem.
    By results of Hilbert, Nagata and Haboush, invariant rings of
    reductive groups are finitely generated. Unfortunately, most
    finite generation proofs are not constructive. In particular,
    they do not provide algorithms for finding a set of generators
    for the invariant ring. In this talk I will discuss
    various algorithms for generators of invariant rings. I will also
    present recent results of Gregor Kemper and myself:
    We found the first algorithm for generators of invariant rings
    of reductive groups actions on affine varieties
    in arbitrary characteristic.
    We also found an algorithm for generators of invariant
    rings for unipotent group actions on the polynomial ring.
    In that case, the ring of invariants may not be finitely generated,
    but the output of the algorithm presents the ring of invariants as the
    ring of regular functions on some explicitly given quasi-affine variety.

  • 12/01/07
    Jack Xin - University of California, Irvine
    Asymptotic front speeds in random flows

  • 12/01/07
    Alexei Borodin - California Institute of Technology
    Large time fluctuations of the totally asymmetric simple exclusion process

  • 12/01/07
    Jean-Pierre Fouque - University of California, Santa Barbara
    On Volatilities

  • 12/01/07
    Ken Alexander - University of Southern California
    The effect of disorder on polymer depinning transitions

  • 12/01/07
    Amber Puha - University of California, Los Angeles
    The Fluid Limit of a Shortest Remaining Processing Time Queue

  • 12/03/07
    Jan Vondrak - Princeton University
    Approximation algorithms for combinatorial allocation problems

    Combinatorial allocation problems have been a subject of recent interest
    due to their role in on-line auctions and electronic commerce. An
    allocation problem entails a finite set of "items" that should be
    distributed among participating "players" in order to maximize a certain
    "social utility" function. A particular case of interest is the Submodular
    Welfare Problem, where the utility functions are assumed to be submodular.

    Our recent result is that a $(1-1/e)$-approximation can be achieved for
    the Submodular Welfare Problem, which is known to be optimal. The
    $(1-1/e)$-approximation can be extended to a more general problem for which
    a $1/2$-approximation was known since 1978 [Fisher,Nemhauser,Wolsey]. I will
    discuss our improvements and the techniques that we use - randomization,
    replacing the discrete problem by a continuous one, and approximately
    solving a non-linear optimization problem using a continuous greedy

    Partly joint work with G. Calinescu, C. Chekuri and M. Pal.

  • 12/04/07
    Jared Wunsch - Northwestern University
    Diffraction of Waves on Singular Spaces

  • 12/04/07
    Mark Gross - UCSD
    TCFT and mirror symmetry

  • 12/04/07
    Jeff Ovall - California Institute of Technology
    Efficient and reliable error estimation for elliptic eigenvalue problems

  • 12/04/07
    Peter Keevash - School of Mathematical Sciences \\ Queen Mary, University of London
    A hypergraph regularity method for generalised Turan problems

    We introduce a new method for analysing certain problems in
    extremal combinatorics that involve small forbidden configurations. The
    cornerstone of our approach is a quasirandom counting lemma for
    quasirandom hypergraphs, which extends the standard counting lemma by
    not only counting copies of any particular configuration but also
    showing that these copies are evenly distributed. We demonstrate the
    power of the method by proving a conjecture of Mubayi on the codegree
    threshold of the Fano plane, that any 3-graph on n vertices for which
    every pair of vertices is contained in more than $n/2$ edges must contain
    a Fano plane, for n sufficiently large. For projective planes over
    fields of odd size we show that the codegree threshold is between $n/2-q+1$ and $n/2$, but for $PG_2(4)$ we find the somewhat surprising phenomenon that the threshold is less than $(1/2-c)n$ for some absolute $c>0$.

  • 12/05/07
    Hans Wenzl - UCSD
    Constructions of Subfactors

    We review basics of subfactor theory in von Neumann algebras.
    We then show how one can construct examples using the notion of
    commuting squares. This will be illustrated using actions of finite
    groups. These are models for more general examples using tensor categories.

  • 12/05/07
    Sue Sierra - University of Michigan
    Rings graded equivalent to the Weyl algebra

    Let $A$ be the first Weyl algebra, in the Euler gradation. We classify graded rings $B$ such that $gr-A$ and $gr-B$ are equivalent (we say that $A$ and $B$ are graded equivalent), and produce some surprising examples. In particular, we show that $A$ is graded equivalent to an idealizer in a localization of $A$.

    In the process, we derive a concise new characterization of equivalences of graded module categories that generalizes the classical Morita theorems.

  • 12/06/07
    Ben Wilson - UCSD Graduate Student
    Symmetric Bilinear Forms and 4-Manifolds

    A large part of this talk will be devoted to symmetric bilinear forms and inner product spaces, which should be of interest to geometers and algebraists alike. I'll outline some of the theory over general (commutative) rings and then turn to the classification of non-degenerate symmetric bilinear forms over the integers. The case of indefinite forms is completely understood and not too hard. On the contrary, the case of positive definite forms is quite difficult and, surprisingly, turns out to be related to the question of packing oranges in Euclidean boxes.

    In the second part of the talk, we will see that symmetric bilinear forms come up as invariants in the theory of $4$-dimensional manifolds (their intersection forms). Freedman showed that a (topological) $4$-manifold is (almost) classified by its associated bilinear form and that indeed every possible form arises in this way. On the other hand, Donaldson showed that the intersection form of a \emph{smooth} $4$-manifold has a very special structure. Combining the two results led to examples of many topological $4$-manifolds that do not admit a smooth structure, a revolution of the (previously almost non-existent) theory of $4$-manifolds, two fields medals, lots of other interesting research, and this talk.

  • 12/06/07
    Daniel Wulbert - UCSD
    Cake Cutting

    A region, $X$, (called a cake) is to be "sliced" so that each of a panel of m judges assess that the division as fair. Each judge has his or her own measure, $ui(S)$ of the value of each part, $S$ of the cake and $ui(X)=1$.

    There are two settings. In the first the cake is to be distributed to two people so that every judge believes that the portions given to each recipients $(U and X-U)$ is worth exactly $½$ (i.e. $ui(U)= ½ = ui(X-U) for all i=1,2, … , m)$.

    In the second setting, the m $(m > 2)$ judges are taking a portion of the cake (i.e., $Ui$) for themselves. They want a division of the cake (i.e., $Uj \cap Uk$ is empty for each $j\neq k$ and $U1 +U2+ … +Um = X$) so that each believes they received more than their fair share of the cake (i.e. $ui(Ui) > 1/m for each i)$.

    Both settings have solutions.

    The solutions give an introduction to measure theory and a fixed point theorem.

  • 12/06/07
    Cristian D. Popescu - UCSD
    On the Coates-Sinnott-Lichtenbaum Conjectures -- Quillen $K$-theory and special values of $L$-functions

    The conjectures in the title were formulated in
    the late 1970s as vast generalizations of the classical theorem of
    Stickelberger. They make a very subtle connection between the
    $\Bbb Z[G(F/k)]$--module structure of the Quillen K-groups K${_\ast}(O_F)$ in an abelian extension $F/k$ of
    number fields and the values at negative integers of the
    associated $G(F/k)$--equivariant $L$--functions $\Theta_{F/k}(s)$.

    These conjectures are known to hold true if the base field $k$ is
    $\Bbb Q$, due to work of Coates-Sinnott and Kurihara. In this
    talk, we will provide evidence in support of these conjectures
    over arbitrary totally real number fields $k$.

  • 12/06/07
    George Kyriazis - University of Cyprus \\ University of South Carolina
    \bf \Huge Weighted spaces of Distributions on the interval $[-1,1]$ and the unit ball

    The Littlewood-Paley theory is extended to weighted spaces of
    distributions on $[-1,1]$ with Jacobi weights
    w(t)=(1-t)^\alpha(1+t)^\beta, <i>italic</i>
    and to the unit ball $B^d$
    in $R^d$ with weights $W_\mu(x)= (1-|x|^2)^{\mu-1/2}$, $\mu \ge 0$.
    Almost exponentially localized polynomial elements (needlets)
    $\{\varphi_\xi\}$, $\{\psi_\xi\}$ are constructed
    in complete analogy with the classical case on $R^d$,
    it is shown that weighted Triebel-Lizorkin and Besov spaces
    can be characterized by the size of the needlet coefficients
    $\{\langle f,\varphi_\xi\rangle\}$
    in respective sequence spaces.

  • 12/11/07

  • 12/11/07
    Dan Wulbert - UCSD
    A Unified Liapanouv Theorem

    Let $\mu_1, \mu_2, ... , \mu_m$ be non-atomic probability measures on a measurable space $(X, \Sigma)$.

    Theorem (Liapanouv 1940) ${\mu(\cup) = (\mu_1(\cup) + \mu_2(\cup) + ... + \mu_m(\cup) ): \cup \ {\text{in}}\ \Sigma }$ is a compact convex set.

    If in addition there is a topology on X and ‚$\Sigma$ is the Borel sets (or the Baire sets, respectively) we can ask when the range of the vector valued measure $\mu$ is obtained even when the measure is restricted to the sets $\cup$ which are open (or the support of a non-negative continuous function, resp.).

    We will give a couple applications of the Classical Theorem. We will then cast the Liapanouv Theorem in an equivalent form about the range of a vector of integrals on X.

    In that form we will give a single theorem that, in addition to proving the Classical Liapanouv Theorem, also characterizes when the open sets (or the supports of continuous functions, resp.) suffice. That is let L be a cone of functions. Let S be the supports of functions in L, and let ‚$\Sigma$ be the sigma-algebra generated by S. The three cases above result when $L = L\infty$, the upper-semi-continuous functions on X, and C(X) respectively.

  • 12/13/07
    Philippe Rigollet - Georgia Institute of Technology
    Model selection, aggregation and stochastic convex optimization using mirror averaging algorithms

    Several statistical problems where the goal is to minimize an unknown convex risk function, can be formulated in the general framework of stochastic convex optimization. For example the problem of model selection and more generally of aggregation can be treated using the machinery of stochastic optimization in several frameworks including density estimation, regression and convex classification. We describe a family of general algorithms called "mirror averaging algorithms" that yield and estimator (or a classifier) which attains optimal rates of model selection in several interesting cases. The theoretical results are presented in the form of exact oracle inequalities similar to those employed in optimization theory. The practical performance of the algorithms is illustrated on several real and artificial examples and compared to standard estimators or classifiers.

  • 12/18/07
    Yi Zhao - Department of Mathematics and Statistics \\ Georgia State University
    An exact result and its application on hypergraph Tur\\'an numbers

    We first prove an exact result for hypergraphs: given $r\ge 2$, let
    $p$ be the
    smallest prime factor of $r-1$. If $n> (p-1)r$ and $G$ is an $r$-
    uniform hypergraph on
    $[n]$ such that every $r+1$ vertices contain $0$ or $r$ edges, then
    $G$ is either
    empty or a star, $\{E\subset [n]: |E|=r, E\ni x\}$ for some $x\in
    Then we use it to slightly improve best known bounds for hypergraph
    Tur\'an numbers. We show that $\pi(K^r_{r+1})\leq 1- \frac{1}{r} -
    \left(1- \frac{1}{r^{p-1}}\right)\frac{(r-1)^2}{2r^p({r+p\choose
    p-1}+{r+1\choose 2})}$
    when $r\ge 4$ is even.

    This is joint work with Linyuan Lu.