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2004 Archive

  • 01/05/04
    P. Gill - UCSD
    Organizational Meeting

  • 01/07/04
    Thalia Jeffres - University of Michigan/MSRI
    Regularity of heat operator on cone

  • 01/08/04
    Justin Roberts - UCSD
    Organizational Meeting

  • 01/08/04
    Audrey Terras - UCSD
    Organizational Meeting

  • 01/09/04
    Jeff Rabin - UCSD
    Super Riemann surface II

  • 01/12/04
    Michael Roitman - University of Michigan
    Vertex Operator Algebras

  • 01/13/04
    Olvi Mangasarian - UCSD
    Knowledge-Based Kernel Approximation

    Prior knowledge, in the form of linear inequalities that
    need to be satisfied over multiple polyhedral sets, is
    incorporated into a function approximation generated by a
    linear combination of linear or nonlinear kernels. In
    addition, the approximation needs to satisfy conventional
    conditions such as having given exact or inexact function
    values at certain points. Determining such an approximation
    leads to a linear programming formulation. By using a
    nonlinear kernel and imposing the prior knowledge in the
    feature space rather than the input space, the nonlinear
    prior knowledge translates into nonlinear inequalities in
    the original input space. Through a number of computational
    examples, it is shown that prior knowledge can significantly
    improve function approximation.

  • 01/13/04
    Wee Teck Gan - UCSD
    Uniqueness of Joseph ideal

    Given a complex simple Lie algebra, Joseph has constructed a
    completely prime ideal in the universal enveloping algebra whose
    associated variety is the minimal non-zero nilpotent orbit. He also
    claimed that J is characterized by this property when the Lie algebra is
    not of type A. Unfortunately there is a gap in the proof. We shall
    describe a simple proof along different lines. (Joint work with G. Savin).

  • 01/13/04
    Salah Baouendi - UCSD
    Local and global groups of diffeomorphisms of CR manifolds

    Joint Seminar with UCI

  • 01/13/04

  • 01/15/04
    Justin Roberts - UCSD
    Hyperbolic Space

  • 01/15/04
    Kevin O'Bryant - UCSD Visitor
    The spectra of a floor sequence

  • 01/15/04

  • 01/20/04
    Philip Gill - UCSD
    On Unconstrained Optimization

  • 01/20/04
    Bertram Kostant - MIT
    Minimal coadjoint orbits and symplectic induction

    Let $(X,w)$ be an integral symplectic manifold and let $(L,Delta)$ be a quantum line bundle, with connection, over X having w as curvature. With this data, one can define an induced symplectic manifold Y with $/dim(Y) = dim(X)+2$. This is applied to show that the 5 split exceptional Lie groups arise symplectically from the symplectic induction of coadjoint orbits of certain classical groups.

  • 01/22/04
    Henry Tuckwell - UCSD Visitor
    Stochastic Nonlinear Neural and Epidemic Networks

    Mathematical models of neurons are formulated as nonlinear
    stochastic ordinary or partial differential equations.
    Progress in determining the activity of single neurons for
    models of varying degrees of mathematical complexity by analytical
    and simulation methods will be described. Exact results for the global
    activity in a network of such elements will also be
    obtained and in particular their approximations by means of
    diffusion processes. Analogous models for
    viral dynamics and epidemic networks may also be considered.

  • 01/22/04
    Audrey Terras - UCSD
    Fun with Zeta and L-Functions of Graphs

    This talk will be a survey of joint work with Harold Stark on
    zeta and L-functions of finite connected graphs, emphasizing parallels
    with the zeta and L-functions of number theory.

  • 01/23/04

  • 01/26/04

  • 01/27/04
    Jianliang Qian - UCLA
    A Local Level Set Eulerian Method for Paraxial Geometrical Optics

    Geometrical optics and its ingredients, eikonals and amplitudes, have
    wide applications, such as optimal control, robotic navigation and
    computer vision. We propose a local level set method for constructing
    the geometrical optics term in the paraxial formulation for the high
    frequency asymptotics of 2-D acoustic wave equations. The geometrical
    optics term consists of two multivalued functions: a traveltime function
    satisfying the eikonal equation locally and an amplitude function
    solving a transport equation locally. The multivalued traveltimes are
    obtained by solving a level set equation and a traveltime equation with a
    forcing term. The multivalued amplitudes are computed by a new
    Eulerian formula based on the gradients of traveltimes and takeoff
    angles. As a byproduct the method is also able to capture the caustic
    locations. The proposed Eulerian method is second-order accurate and
    has complexity of $O(N^2 Log N)$. Several examples including the well
    known Marmousi synthetic model illustrate the accuracy and efficiency of
    the Eulerian method. We will also discuss the extension of the method
    to anisotropic elastic wave equations and other possible future

  • 01/27/04
    Karin Baur - UCSD Visitor
    Higher secant varieties of the minimal adjoint orbit

    The adjoint group of a simple complex Lie algebra Lie(G) has
    a unique minimal orbit which we denote by C. We describe for
    classical Lie algebras, for any natural number k,
    the Zariski closure of the union kC of all spaces spanned
    by k points on C. The image of this set in the projective
    space {Bbb P}(Lie(G)) is usually called the (k-1)-st
    secant variety of {Bbb P}(C), and its dimension and
    defect are easily determined from our explicit description.
    We give the smallest k for which the closure of kC is
    equal to the Lie algebra and compare these results with
    the upper bound on secants of general varieties given
    in a theorem of F. Zak (eg 1993).

    This talk describes recent joint work with Jan Draisma.

  • 01/29/04
    Kazuhiro Kuwae - Yokohama City University
    On Calabi`s strong maximum principle via local Dirichlt forms

    I will talk about a stochastic proof of an extension of the strong maximum principle by E. Calabi in the framework of local Dirichlet forms associated with strong Feller diffusions.

  • 01/29/04
    Graham Hazel - UCSD Graduate Student
    Teichmuller space

  • 01/29/04
    Stefan Erickson - UCSD Graduate Student
    Variations of a Theme of Stark

  • 01/29/04
    Jason Colwell - Univ. of Southern California
    The Conjecture of Birch and Swinnerton-Dyer for elliptic curves with complex multiplication by a nonmaximal order

    Gross has refined the Birch Swinnerton Dyer Conjecture in the case of
    an elliptic curve with complex multiplication by a nonmaximal order.
    Gross Conjecture has been reformulated in the language of derived
    categories and determinants of perfect complexes. Burns and Flach have
    realized that this immediately leads to a refinement of Gross
    Conjecture. The conjecture is now expressed as a statement concerning a generator of the image of a map of 1-dimensional modules. This
    conjecture is proved by a construction which shows it to follow from
    the Explicit Reciprocity Law and Rubin`s Main Conjecture.

  • 02/02/04
    John Wilson - UCSD Visitor
    Aspects of Growth of Groups

  • 02/03/04
    Xiaochun Li - UCLA
    The Hilbert transform along $C^{1+epsilon}$ vector fields

    Let $v$ be a vector field from ${mathbb R}^2$ to the unit circle
    ${mathbb S}^1$. We study the operator $$ H_vf(x)= p.v. int_{-1}^{1}f(x-tv(x))frac{dt}{t},.$$ We prove that if the vector field $v$ has $1+epsilon$ derivatives, then $H_v$ extends to a bounded map from $L^2$ into itself.

  • 02/03/04
    John Etnyre - Pennsylvania State University
    Contact Geometry Topology and Dynamics

    Contact geometry is a venerable subject that arose out of the study of Geometric Optics in the1800s. Though the years it has repeatedly cropped up in many areas of mathematics, but only in thepast 30 years or so has it received serious attention. Recently there has been great progress inunderstanding contact structures. Depending on ones perspective contact structures sometimes seemlike topological objects, sometimes geometric objects and sometimes dynamical objects. In this talkI will begin by discussing how contact structures arise out of natural problems and how they have deep connections with topology and dynamics. Then after surveying a few topics about contact
    structures in low dimensions I will define contact homology in certain situations. Contact homologyis a new invariant of contact structures (and/or certain submanifolds of them) that is similar, in
    spirit, to Gromov-Witten invariants of symplectic manifolds or Floer homology of Lagrangiansubmanifolds in symplectic manifolds. I then will proceed to discuss applications of contact homology, in particular, I will describe how it yields potentially new invariants of submanifolds of
    Euclidean space.

  • 02/03/04
    Hung Yean Loke - National Univ of Singapore
    Analytic continuations of Gelfand-Zetlin bases

  • 02/03/04
    Jason Bell - University of Michigan
    Hilbert Series of Prime $PI$ rings

  • 02/03/04
    Lincoln Lu - UCSD Visitor
    Spectra of random power law graphs

    Many graphs arising in various information networks exhibit the power
    law behavior the number of vertices of degree $k$ is proportional
    to $k^{-eta}$ for some positive $eta$. Such graphs are called
    power law graphs. In the study of the spectra of power law graphs,
    there are basically two competing approaches. One is to prove
    analogues of Wigner`s semi-circle law while the other predicts that
    the eigenvalues follow a power law distributions. We will show that
    both approaches are essentially correct if one considers the
    appropriate matrices. We will prove that (under certain mild
    conditions) the eigenvalues of the (normalized) Laplacian of a random
    power law graph follow the semi-circle law while the spectrum of the
    adjacency matrix of a power law graph obeys the power law. Our
    results are based on the analysis of random graphs with given expected
    degrees and their relations to several key invariants. These results
    have implications on the usage of spectral techniques in many areas
    related to pattern detection and information retrieval.

    This is a joint work with Prof. Fan Chung and Prof. Van Vu.

  • 02/05/04
    Tom Fleming - UCSD Graduate Student
    Hyperbolic simplexes and the figure-eight knot

  • 02/05/04
    Joachim Rosenthal - University of Notre Dame
    Algebraic Methods in Cryptography

    Modern cryptography is very algebraic by nature. In this overview
    talk we will explain the major secret and public key
    cryptographic protocols. For secret key systems, Rijndael has
    become the new standard and we will describe this protocol
    through a sequence of algebraic operations in a finite ring $R$.

    In the area of public key cryptography the major protocols are
    the RSA protocol, the traditional Diffie-Hellman and the ElGamal
    protocol. The last two protocols are based on the hardness of
    the discrete logarithm problem in a finite group.

    The discrete logarithm problem can be viewed as a semi-group
    action on a set. This leads naturally to a generalized
    Diffie-Hellman key exchange and a generalized ElGamal one-way
    trapdoor function.

    Using this point of view we will provide several interesting
    semi-group actions on finite sets. Our main focus will be
    examples of semi-ring actions on a semi-module. These examples
    may lead to practical new one-way trapdoor functions.

    The presented results constitute joint work with Gerard Maze and
    Chris Monico.

  • 02/06/04
    Damla Sentürk - UC Davis
    Covariate Adjusted Regression

    We introduce covariate adjusted regression (CAR) for situations where both predictors and response in a regression model are not directly observable, but are contaminated with a multiplicative factor that is
    determined by the value of an unknown function of an observable covariate. We demonstrate how the regression coefficients can be
    estimated by establishing a connection to varying coefficient regression. The proposed covariate adjustment method is illustrated with an analysis of the regression of plasma fibrinogen concentration as response on serum transferrin level as predictor for 69 hemodialysis patients. In this example, both response and predictor are thought to be influenced in a multiplicative fashion by body mass index. A bootstrap hypothesis test enables us to test the significance of the regression parameters. We establish the asymptotic distribution of the parameter estimates for this new covariate adjusted regression model.

  • 02/06/04

  • 02/06/04
    Bjorn Dundas - Norwegian University of Science and Technology
    Elliptic cohomology through 2-bundles?

  • 02/09/04
    Aurore Delaigle - UC Davis and Catholic University of Louvain
    Density Estimation and Deconvolution Problems

    R E C R U I T M E N T

    We consider estimation of a density from a sample that contains
    measurement errors. This problem, known as a deconvolution problem, has
    applications in many different fields such as astronomy, chemistry or
    public health, since in real data applications, it happens quite often
    that the observations are made with error. The contaminating density, or
    error density, is often assumed to be known. In this context, a so-called
    deconvolution kernel density estimator has been proposed in the literature
    (see for example Carroll and Hall (1988) or Stefanski and Carroll (1990)).

    The behavior of the deconvolution kernel density estimator depends
    strongly on a smoothing parameter called the bandwidth. We discuss several
    possible ways of choosing an appropriate bandwidth in practice.

    We next consider deconvolution kernel estimation of a density with left
    and/or right unknown finite endpoints. From the contaminated sample, we
    estimate the boundary of the support by the value which maximizes a
    certain diagnostic function. This function can for example be based on the
    derivative of a deconvolution kernel estimator of the density. We
    establish asymptotic properties of the proposed estimator and study the
    practical aspects of the method via a simulation study.

    This is joint work with Irène Gijbels.

  • 02/10/04

  • 02/10/04
    Martin Kimball - Caltech
    Modularity in four dimensions

    Langlands conjectured that a continuous complex Galois
    representation can
    be associated to an automorphic ''representation'' such that their $L$-functions
    agree. We reduce the problem for representations into $GL(4)$ with solvable
    image into several cases. We prove that in certain cases, Langlands
    modularity conjecture holds. In particular, we obtain a new case of
    Artin s $L$-function conjecture.

  • 02/11/04
    Vin de Silva - Stanford University
    Low-dimensional structure in high-dimensional data

    T T O P $quad$ R E C R U I T M E N T

    In recent years, vast quantities of
    scientific data have been
    generated in various disciplines,
    thanks to the almost universal use of
    computer technology. What are we to do with this data?
    Can we find simple
    ways of understanding complex data?
    I discuss two areas of recent research
    into this problem: ($1$) nonlinear dimensionality reduction;
    ($2$) topological estimation.

  • 02/12/04
    de Silva - Stanford University
    Algebraic Topology and Sparse Matrix Computations

  • 02/12/04
    Wee Teck Gan - UCSD

  • 02/12/04
    Jonathan Goodman - Courant Institute, visiting Standford University
    Probability laws for time stepping methods for stochastic differential equations

    In Monte Carlo estimation of expected values of functionals of solutions of SDEs, we have to generate approximate sample paths, probably by time stepping. Typically, the bias is roughly proportional to a power of the time step, h. The power depends on the time stepper and on the functional. For a functional F(X(T)), so called weak error esimates give first order convergence for the forward Euler and Milstein methods. To bound the bias for more complicated functionals, we study the joint distribution of the path observed at the time step times: (X(h), x(2h), ... X(nh)), where nh=T. We compute the difference between the joint density for the time stepper and the exact joint density. For forward Euler this does not go to zero with h, but for Milstein it goes as sqrt(h). Our analysis leads to Runge Kutta method that is simpler than Milstein but has similar statistical properties. The talk will include background material including the definition of the Milstein method. This is joint work with Peter Glynn and Jose Antonio Perez.

  • 02/12/04
    Nitya Kitchloo - Johns Hopkins University
    Topology of Infinite dimensional Groups

    I will give a general framework to study the topology of
    infinite dimensional groups via their actions on contractible spaces known
    as Buildings. The groups of interest to us will be loop groups, Kac-Moody
    groups, symplectomorphism groups and similar transformation groups. I will
    give examples of natural buildings associated to such groups and use them
    to derive some consequences.

  • 02/13/04
    Hao Fang - Courant Institute, NYU
    Spectral invariants of torsion type and applications

    R E C R U I T M E N T

    In this talk I will discuss some recent results on spectral invariants of
    torsion type.

    Spectral invariants of elliptic operators on compact manifolds are
    important global geometric quantities. The Ray-Singer torsion is a
    spectral invariant with significant topological implications. We discuss a
    new spectral invariant which generalizes Ray-Singer`s construction. This
    new invariant behaves nicely under holonomy restrictions. In particular,
    it coincides with the BCOV torsion (first constructed in Mirror Symmetry
    theory) when restricted to Calabi-Yau manifolds.

    As an application in algebraic geometry, we prove a Shafarevich type
    theorem for holomorphic moduli of polarized Calabi-Yau manifolds. We also
    show some links between our new invariant and automorphic forms,
    generalizing classical results in the two dimensional case.

    Parts of the results are joint work with Lu and Yoshikawa.

  • 02/13/04
    Ron Douglas - Texas A&M
    Hilbert modules and complex geometry

  • 02/13/04
    Yijun Zuo - Michigan State University
    Data depth and some applications

    Order related procedures (such as median, quantiles, and
    nonparametric procedures) in one-dimensional data analysis and inference
    have played such important roles that their analogues in high dimensions
    have been sought for years (but without many satisfactory results). The
    task is non-trivial because there is no natural and clear order principle
    in high dimensions. On the other hand, data depth turns out to be a quite
    promising tool for a center-outward ordering of multi-dimensional
    observations. In this talk motivations of data depth are discussed.
    Examples illustrating notions of data depth including half-space,
    simplicial and projection depth are provided. Applications of data depth
    in location, in regression, and in other settings are discussed. Depth
    based procedures can outperform their competitors by maintaining a good
    balance between efficiency and robustness. Computing issues and some
    future research directions of data depth are briefly addressed.

  • 02/13/04
    Nitya Kitchloo - Johns Hopkins University
    Topology of Symplectomorphism Groups

    I describe the topology of (the classifying space of) the
    symplectomorphism groups of a family of symplectic 4-manifolds. In
    particular, we calculate the integral cohomology of these classifying
    spaces. We also study the space of compatible complex structures on these
    symplectic manifolds and outline a proof showing that this space is

  • 02/17/04
    Fengbo Hang - Princeton University
    Strong and weak density of smooth maps for the Dirichlet energy

    The Sobolev space $W^{1,2}(M,N)$ between two Riemannian manifolds
    $M$ and $N$ appears naturally in the calculus of variations. We will
    discuss necessary and sufficient (topological) conditions for smooth maps
    to be strongly or weakly dense in this space. These problems are of
    analytical interest and closely related to the theory of harmonic maps.

  • 02/17/04

  • 02/17/04
    Hans Wenzl - UCSD
    Spinor representations and reconstruction

    It is shown that the commutant of the Pin-group action on
    tensor powers of the spinor representation is already generated by
    the elements which appear in the second tensor power, and a complete
    set of relations is given. This can be used to completely classify
    all tensor categories whose Grothendieck semiring is the one of a
    classical group.

  • 02/19/04
    Sean Raleigh - UCSD Graduate Student
    The Gromov norm

  • 02/19/04
    Joe Buhler - CCR
    Factorization of Polynomials over Local Fields

    Motivated by a ``mass formula" due to Serre for local fields,
    this talk will investigate the probability that a uniformly random
    monic polynomial with coefficients in the p-adic integers factors
    completely into linear factors.

  • 02/19/04
    M. Fukushima - Kansai University, Osaka, Japan
    Poisson point processes attached to symmetric diffusions

  • 02/19/04
    Gunnar Carlsson - Stanford University
    Algebraic topology as a tool in data analysis

    I will discuss some attempts to recover topological information about geometric objects from ``point cloud data" sampled from the object. Examples will include data sets obtained from image data, and the problem of recognizing shapes, i.e. subcomplexes of Euclidean $3$-space.

  • 02/19/04
    J.M. Lee - University of Washington
    Foliations of CR manifolds and estimates for tangential Cauchy-Riemann

    The $overlinepartial_b$-Neumann problem is the analog for CR manifolds of
    the $overlinepartial$-Neumann problem. All positive results about this
    problem so far have applied to domains with a defining function depending
    only on the real and imaginary parts of a single CR function. The key
    feature of such domains is that they are foliated (away from a
    characteristic curve) by compact, strictly pseudoconvex CR submanifolds of
    real codimension 2. I will describe a new approach to finding estimates
    based on decomposing the operator into its tangential and transverse parts
    with respect to this foliation. Estimates for the tangential parts follow
    from known results about the tangential Cauchy-Riemann complex on compact CR
    manifolds, while estimates for the transverse part reduce to elliptic
    estimates in the plane. For certain domains in the Heisenberg group, my
    student Robert Hladky has used this method to obtain sharp boundary
    regularity, even near characteristic points.

  • 02/20/04
    Ben Weinkove - Columbia University
    The J-flow and the Mabuchi energy

    The J-flow is a parabolic flow on compact Kahler manifolds with two Kahler metrics. It was discovered by S. Donaldson and X. X. Chen independently. Donaldson defined it in the setting of moment maps and symplectic geometry. Chen described the flow as the gradient flow of the J-functional appearing in his formula for the Mabuchi energy. The Mabuchi energy is an important functional on the space of Kahler potentials. Its critical points give constant scalar curvature metrics, and its lower boundedness is related to stability in the sense of geometric invariant theory. I will show that under a condition on the initial data, the J-flow converges to a critical metric. I will then explain how this implies the lower boundedness of the Mabuchi energy for an open set of Kahler classes on manifolds with negative first Chern class.

  • 02/20/04

  • 02/20/04

  • 02/20/04
    Michel Grueneburg - Standford University
    Yamabe flow on three manifolds

  • 02/20/04
    Gunnar Carlsson - Stanford University
    Representations of Galois groups and algebraic K-theory of fields

    This talk will discuss some conjectures about the relationship between the algebraic K-theory of a field on the one hand and the so-called derived completed complex representation theory of the absolute Galois group of the field. I will also discuss the connection of this derived completion with the study of the space of deformations of a given representation.

  • 02/23/04
    Tullio Ceccherini-Silberstein - Universita delgi Studi di Roma
    On amenable algebras

  • 02/23/04
    Agata Smoktunowicz - Institute of Mathematics Polish Academy of Sciences
    On Graded Domains

  • 02/23/04
    Serge Guillas - University of Chicago
    Time series of functional data and environmental statistics

  • 02/24/04
    Jennifer Erway - UCSD Graduate Student
    An Optimization Problem in Biomechanics

  • 02/24/04
    Peter Trapa - University of Utah
    Unipotent representations and the theta correspondence.

    Fix a real reductive group $G$. Suppose $mathcal{O}'$ is a
    nilpotent orbit in $mathfrak{g}'$, the dual of the complexified Lie
    algebra of $G$. To each $X' in mathcal{O}'$, one may associate an sl(2)
    triple, say $X', Y'$, and $H'$. Since $(1/2)H'$ lives in a Cartan
    subalgebra of $mathfrak{g}'$, it defines an infinitesimal character for
    $G$. One piece of the Arthur conjectures predicts that the smallest
    representations of $G$ with infinitesimal character $(1/2)H'$ should
    appear as local components of automorphic forms; in particular, they
    should be unitary. (Two good examples to keep in mind are the trivial
    representation and limits of discrete series with zero infinitesimal
    character; the former corresponds to the principal orbit $mathcal{O'}$
    and the latter to the zero orbit.) In this talk, we explain how to prove
    a large part of this conjecture for certain classical groups using the
    theta correspondence.

  • 02/24/04
    Van Vu - UCSD
    New bound on Erdos distinct distances problem

    One of the most well known questions of Erdos in discrete geometry is
    the following: Given n points in $R^d$, what is the smallest number of
    distinct distances among them ? Here d is fixed and n tends to inifnity.
    We denote by $f_d(n)$ is smallest number of distince distances.

    The problem of determining $f_d(n)$ has been attacked by many
    researchers (including Erdos, Beck, Chung,
    Trotter, Szemeredi, Beck, Ssekely, Solymosi-Toth, Sharir, etc) for
    decades. In this talk, I will give a brief overview and also present a new result
    (joint with Solymosi). This result gives an almost sharp estimate for
    $f_d(n)$ for relatively large dimension $d$. The main tool is what we
    call "decomposition technique", which appears to be useful in other problems as well.

  • 02/26/04
    Sean Raleigh - UCSD Graduate Student
    Mostow rigidity

  • 02/26/04
    David Siegmund - Stanford University
    An Urn Model of Diaconis

    In attempting to understand the "meat ax" of finite group theory,
    Diaconis has formulated an urn model. In the simplest case, balls
    numbered 0 and 1 are placed in an urn. At times
    $n = 1,2,...,$ two balls are drawn with
    replacement. Those balls are replaced in the
    urn, and a new ball that contains the sum mod $2$ of the
    numbers on the drawn balls is added to the urn. A conjecture
    is that the fraction of balls numbered $1$ converges to $1/2$.
    This conjecture and some generalizations are proved
    as a two-fold application of the almost supermartingale
    convergence theorem of Robbins and Siegmund $(1972)$.

    This is joint research with Benny Yakir.

  • 02/26/04
    Noureddine El Karoui - Stanford University
    The Tracy-Widom law holds when $n, p, p/n
    ightarrow infty$, with application to PCA

    Principal Component Analysis (PCA) is a tool used across the spectrum of scientific applications. In modern practice, it is often applied to $n imes p$ data matrices with $n$ and $p$ both large. Classical theory (Anderson
    1963) fails to apply in this setting. Using random matrix theory, Johnstone
    (2000) recently shed light on some
    theoretical aspects of PCA in this setup. Specifically, when the entries of the
    $n imes p$ matrix
    $X$ are iid ${cal N}(0,1)$ and $n/p
    ho in (0,infty)$, he showed
    $lambda_{n,p}$ , the largest eigenvalue of the empirical covariance matrix
    $X'X$, converges to the so-called Tracy-Widom distribution (after proper
    recentering and rescaling).
    We will show that the result holds when $n,p
    ightarrow infty$ and
    ightarrow 0$ or $
    infty$, in effect removing the need to worry about the limiting behavior of
    We will also present preliminary results for rates of convergence. Finally, we
    will illustrate how these and
    related theoretical insights might be used in practice.

  • 02/26/04
    Kristin Lauter - Microsoft
    Generating curves over finite fields with a known number of points

    It is often useful in cryptography to be able to generate
    an elliptic curve over a finite field with a given number of points.
    This talk will explain the complex multiplication (CM) method for
    constructing suitable elliptic curves and explain a variant which is
    joint work with A. Agashe and R. Venkatesan. I will also explain the CM
    method for generating genus 2 curves and some interesting problems which

  • 03/01/04
    Johan Hoffman - Courant Institute of Mathematical Sciences/New York University
    Adaptive DNS/LES: A New Approach to Computational Turbulence Modeling

    We present a new approach to CFD and Computational Turbulence Modeling
    using adaptive stabilized Galerkin finite element methods with duality
    based a posteriori error control for chosen output quantities of interest,
    with the output based on the exact solution to the Navier-Stokes
    equations, thus circumventing introducing and modeling Reynolds stresses
    in averaged Navier-Stokes equations. We refer to our methodology as
    Adaptive DNS/LES, where automatically by adaptivity certain features of
    the flow are resolved in a Direct Numerical Simulation DNS, while certain
    other small scale turbulent features are left unresolved in a Large Eddy
    Simulation LES. The stabilization of the Galerkin method giving a weighted
    least square control of the residual acts as the subgrid model in the LES.
    The a posteriori error estimate takes into account both the error from
    discretization and the error from the subgrid model. A crucial observation
    from computational examples is that the contribution from subgrid modeling
    in the a posteriori error estimation can be small, making it possible to
    simulate aspects of turbulent flow without accurate modeling of Reynolds
    stresses. Using the a posteriori error estimates we further consider the
    question of uniqueness of weak solutions to the Navier-Stokes equations,
    where we give computational evidence of both uniqueness and non-uniqueness
    in outputs of weak solutions.

  • 03/02/04
    Lihe Wang - University of Iowa (talk held at UCI)
    Regularity theory for general curvature flow


  • 03/02/04
    Xianzhe Dai - UCSB (talk held at UCI)
    Positive Mass Theorem and Stability of Manifolds with Parallel Spinors


  • 03/02/04
    Chung Graham - UCSD
    Random walks on directed graphs

  • 03/03/04
    Michael Donohue - UCSD Graduate Student

  • 03/03/04

  • 03/03/04
    Bo Li - University of Maryland
    Continuum Modeling and Analysis of Epitaxial Growth of Thin Films

    Epitaxial growth is a modern technology to grow thin solid films by depositing atoms or molecules onto an existing layer of material. Microscopic processes in epitaxial growth include the deposition of atoms onto a surface, desorption of adsorbed atoms (adatoms) into the gas phase, surface diffusion of adatoms, attachment and detachment of adatoms to and from atomic steps, adatom island nucleation, and island coalescence. These processes are characterized by fluctuation, non-equilibrium, and multiple spatial and temporal scales. In this talk, I will first present a derivation of an island dynamics model for epitaxial growth that includes step kinetics. This is an improvement of the classical Burton-Cabrera-Frank model that assumes
    the adatom equilibrium along steps. An adaptive finite element method for the new model will be described. I will then focus on a class of fourth-order diffusion equations that model the coarsening in the surface dynamics of growing films after the roughening transition. Such growth equations are gradient flows of certain free energies. I will
    show bounds for coarsening rates and the decay of energy, derive the energy asymptotics in the large-system-limit, and predict the exact scaling laws for the coarsening under the hypothesis of realization.

  • 03/04/04
    Jana Comstock - UCSD Graduate Student

  • 03/05/04
    Sergey Kataev - University of Kentucky
    Non-overlapping patterns in permutations and words

    A descent in a permutation a(1)a(2)...a(n) is an i such that a(i) >
    a(i+1). The distribution of the number of descents is given by the well
    known Eulerian numbers.

    We define a new statistic, namely the maximum number of non-overlapping
    descents in a permutation. Two descents i and j overlap if |i-j| = 1. We
    find the distribution of this new statistic using partially ordered
    generalized patterns (POGPs). The POGPs are generalizations of the
    Babson-Steingrimsson patterns, which in turn generalize the classical
    permutation patterns.
    A segment pattern is a pattern whose occurrence in a permutation is
    required to consist of consecutive letters of the permutation. As an
    example, the number of descents corresponds to the segment pattern 21.

    Let P be an arbitrary segment pattern. Using POGPs, we give the
    exponential generating function (e.g.f.) for the entire distribution of
    the maximum number of non-overlapping occurrences of P, provided we know
    the e.g.f. for the number of permutations that avoid P.

    We also discuss POGPs in words (with repeated letters), where we get results
    similar to those for permutations.

  • 03/05/04
    Stephen Watson - Northwestern University
    Coarsening Dynamics of Faceted Crystal Surfaces: The Annealing-to-Growth Transition

    The current renaissance in the study of evolving faceted
    crystal surfaces
    was prompted by the discovery of nano-scale faceted pyramidal islands
    (quantum dots) on Si films, as well as the appearance of novel in situ
    imaging techniques.
    We consider the coarsening dynamics of
    faceted crystal surfaces that pertain to two distinct continuum models;
    so-called thermal annealing (A) and net-growth (G) regimes.
    We present a novel theoretical framework which
    unites the two problems by recognizing
    their common (leading-order) kinematic framework;
    namely, piecewise-affine surfaces evolving on a suitably slow time scale.
    Dynamic evolution laws, intrinsic to such surfaces, are then found
    for each problem through novel matched asymptotic analysis.
    Our theory resolves the long-standing annealing-to-growth enigma,
    whereby the scaling law for the increase in time, t, of the
    characteristic facet size, L, is observed to undergo
    a transition when switching from (A) to (G).
    In addition, the nature of the transition in surface morphology
    between (A) and (G) follows readily from the stability
    properties of the associated dynamical systems.
    Large scale simulations, which are rich in both topological and
    statistical terms, will also be presented.

  • 03/05/04

  • 03/05/04

  • 03/08/04
    Jing Wang - IMA, Department of Mathematics, University of Minnesota
    Analysis of a variational approach to multi-focal lens design

    In this talk, we will consider the problem of designing multi-focal
    optical lenses. This problem has been actively studied over the past
    20 years and the results obtained along the way of research have been
    used directly in the lens industry.

    We study a variational approach for the multi-focal lens design problem
    and find out that the Euler-Lagrange equation associated with the
    variational problem is a fourth-order nonlinear elliptic partial
    differential equation. We consider two linearizations of the equation
    along with three types of boundary conditions, and analyze the existence,
    uniqueness as well as regularity of the solutions under these types of
    boundary conditions. Finally, we propose a numerical method with a special
    type of spline functions for solving the linearized partial differential
    equations, and the method is shown to be very efficient.

  • 03/08/04
    Mark Sapir - Vanderbilt University
    Isoperimetric functions of groups: connections between group theory, topology, logic and computer science

    We give a survey of a series of papers by Birget, Olshanskii, Rips and myself about asymptotic invariants of groups and connections between them and other areas of mathematics. In particular we give an algebraic characterization of groups with word problem in NP, construct manifolds with arbitrary isoperimetric functions $(say f(x)=x^{pi+e})$, and finitely presented counterexamples to von Neumann conjecture (a finitely presented non-amenable group without free subgroups).

  • 03/09/04
    T. Sadykov - University of Western Ontario, London
    Algebraic functions and holonomic systems of partial differential equations

    I will present a joint work with Alicia Dickenstein and Laura
    Matusevich. Algebraic functions (defined as solutions to algebraic
    equations with general symbolic coefficients) are classically known
    to satisfy certain systems of linear partial differential equations
    with polynomial coefficients. In the talk I will consider a more
    general class of systems of differential equations. We prove that
    such systems are holonomic and that their complex holomorphic solutions
    have moderate growth. We also provide an explicit formula for the
    holonomic rank of these systems as well as bases in their spaces of
    complex holomorphic and Puiseux polynomial solutions.

  • 03/09/04
    Liz Fenwick - UCSD Graduate Student
    A Morphing Algorithm for Generating Near Optimal Grids

  • 03/09/04

  • 03/10/04
    R. Garcia - UC Santa Barbara
    Complex symmetry with applications to analytic function theory

    We study a few classes of Hilbert space operators which are complex
    symmetric with respect to a preferred orthonormal basis. The existence of
    this additional symmetry has notable implications and in particular it
    explains from a unifying point of view some classical results.

  • 03/10/04
    Anthony Shaheen - UCSD Graduate Student

  • 03/11/04

  • 03/11/04
    Frank Quinn - Virginia Tech
    History of manifolds

    Tracing the use of the term over $150$ years gives insight into the way mathematicians name things, and the things mathematicians name.

  • 03/11/04
    Allen Knutson - UC Berkeley
    Why do matrices commute?

    Put another way: is every polynomial in $2 n^2$ variables that vanishes
    on a pair of commuting matrices, in the ideal generated by the obvious
    $n^2$ quadratic relations?

    Alas, we do not know (and I cannot answer the question either).
    I will introduce several other related schemes that seem easier to study,
    like the space of pairs of matrices whose commutator is diagonal,
    which I will prove is a reduced complete intersection, one of whose
    components is the commuting variety. Conjecturally, it has only one
    other component, and I will explain where that one comes from.
    Along the way we will also see a rather curious invariant of permutations,
    and much simple linear algebra.

  • 03/11/04
    B. Choi - Yonsei University (Korea)
    ARMA model identification

  • 03/12/04

  • 03/12/04
    Stefan Friedl - Munich, Germany
    Examples of topologically slice knots

  • 03/15/04
    Ronghui Xu - Harvard School of Public Health and Dana-Farber Cancer Institute
    Proportional Hazards Model with Mixed Effects

    In this talk we describe our work on proportional hazards model with mixed effects (PHMM) for right-censored data. Our motivation came from a
    multi-center clinical trial in lung cancer, where treatment effects were
    found to vary substantially among the centers. We provide a general
    framework for handling random effects in proportional hazards
    regression, in a way similar to the linear, non-linear and generalized
    linear mixed effects models that allow random effects of arbitrary
    covariates. This general framework includes the frailty models as a
    special case. Semi parametric maximum likelihood estimates of the
    regression parameters, the variance components and the baseline hazard,
    and empirical Bayes estimates of the random effects can be obtained via
    an MCEM algorithm. Variances of the parameter estimates are approximated
    using Louis formula. The model found interesting applications in
    recurrent events, twin data and genetic epidemiology. Following the
    introduction of the model, our recent work has included topics on model
    diagnostics and model selection. We will elaborate on one of these
    topics during the talk.

  • 03/16/04
    Frank Chang - UCSD Graduate Student
    Division Algebras over Generalized Local Fields

  • 03/18/04
    Tonghai Yang - University of Wisconsin at Madison
    The CM-Values of Hilbert Modular Functions

  • 03/30/04

  • 03/30/04
    Ivan Cherednik - U of North Carolina Chapel Hill
    From Hankel trasform to Verlinde Algebras I

    Lie groups provide a formalization of the concept of symmetry in the classical theory of special functions, combinatorics, and physics. From this viewpoint, DAHA describes abstract Fourier transforms, especially those making the Gaussian Fourier-invariant. The classical Fourier transform, the Hankel transform, and the one from the theory of Gaussian sums are well known examples. Thus DAHA formalizes an important part of the classical Fourier analysis.

    The Verlinde algebras are the key finite-dimensional examples. There exist three different Verlinde algebras of type A 1:
    1) the one connected with the Hankel transform, and, hopefully,
    with the massless conformal field theory,
    2) the major Verlinde algebra associated with the Kac-Moody
    fusion and the massive CFT and,
    3) the algebra presumably describing the fusion of the
    (1,p)-Virasoro model.
    They will be discussed in the lectures.

  • 04/01/04

  • 04/01/04
    Peter Teichner - UCSD

    We will read a paper by Princeton graduate student Jacob Rasmussen, on Khovanov homology and the slice genus. It is a breakthrough paper, proving the so called Milnor conjectures by purely combinatorial means. These conjectures predict the 4-dimensional genus of torus knots, and their only known proof involves Gauge theory, or more precisely, Seiberg-Witten theory.

    Rasmussen gives a proof using only methods from combinatorial skein theory and a certain spectral sequence. He obtains a concordance invariant for knots, which is absolutely mind blowing. It can show that certain knots with trivial Alexander polynomial are not smoothly slice, even though by Freedmans theorem they are topologically slice. Hence the invariant sees the difference between smooth and topological phenomena in dimension four.

    As usual, the talks in this seminar will be given by the participants, with two survey lectures at the beginning given by Justin and Peter. Rasmussens paper, supplemented by some survey articles, will be used as reference for later talks.

  • 04/01/04
    Ivan Cherednik - U Of North Carolina
    Double affine Hecke Algebras

    Introduced 13 years ago, DAHA play now a solid role in modern representation theory with wide spectrum of applications from Harmonic Analysis and Combinatorics to Algebraic Geometry and Mathematical Physics.
    I will begin with general motivation in the theory of spherical functions and the definitions in the rank one case. The talk will be mainly about the so-called nonsymmetric Verlinde algebras, which are in the focus of the new theory, including applications to the ''non-cyclotomic'' Gaussian sums and recent results on the diagonal coinvariants (the combinatorics of two sets of variables).

  • 04/02/04

  • 04/06/04
    Gabriel Nagy - UCSD
    Finite elements in space-time

  • 04/06/04
    Jenia Tevelev - University of Texas at Austin
    Higher-dimensional versions of stable rational curves

    The space of ordered n-tuples of points on a projective line has a compactification, due to Grothendieck and Knudsen, with many remarkable properties: it has a natural moduli interpretation, namely it is the moduli space of stable n-pointed rational curves. It has a natural Mori theoretic meaning, namely it is the log canonical model of the interior.
    For a curve in the interior, there is a description of the limiting stable n-pointed rational curve, due to Kapranov, in terms of the Tits tree of $PGL_2$. We study these properties for the higher-dimensional versions of the Grothendieck-Knudsen space, the Chow quotients of Grassmannians.

  • 04/06/04
    Ambar Sengupta - Louisiana State University
    A YM2 sampler: Results from low dimensional Gauge theories

    A discussion of results arising from Yang-Mills gauge theory on compact surfaces and the relationship between YM2 and Chern-Simons.

  • 04/06/04
    Ivan Cherednik - U. of North Carolina Chapel Hill
    From Hankel transform to Verlinde Algebras II

    Lie groups provide a formalization of the concept of symmetry in the classical theory of special functions, combinatorics, and physics. From this viewpoint, DAHA describes abstract Fourier transforms, especially those making the Gaussian Fourier-invariant. The classical Fourier transform, the Hankel transform, and the one from the theory of Gaussian sums are well known examples. Thus DAHA formalizes an important part of the classical Fourier analysis.

    The Verlinde algebras are the key finite-dimensional examples. There exist three different Verlinde algebras of type A 1:
    1) the one connected with the Hankel transform, and, hopefully,
    with the massless conformal field theory,
    2) the major Verlinde algebra associated with the Kac-Moody
    fusion and the massive CFT and,
    3) the algebra presumably describing the fusion of the
    (1,p)-Virasoro model.
    They will be discussed in the lectures.

  • 04/06/04

  • 04/08/04
    Kristin Lauter - Microsoft
    Class invariants for quartic CM-Fields

  • 04/08/04
    Frank Sottile - Clay Mathematical Institute & MSRI
    Certificates of algebraic positivity

    Positivity is a distinguishing property of the field of real numbers. Writing a polynomial as a sum of squares gives a certificate that it is positive. Hilbert showed that a positive homogeneous quartic polynomial in three variables (ternary quartic) is a sum of three squares of quadratic polynomials. He also showed that there are positive polynomials of every higher degree or greater number of variables with no such sum of squares representations. This led to his 17th problem - to determine whether a positive polynomial is a sum of squares of rational functions. This was answered in the affirmative by Artin in 1926.

    Recently, positive polynomials have have undergone a revival. In the 1990s Lasserre realized that recent theoretical results from real algebraic geometry and semi-definite programming could be combined to give effective algorithms for solving a class of relaxations of hard optimization problems. The relaxation replaces positivity by sum of squares representation.

    I will briefly survey the history of positive polynomials and these modern applications, and then discuss a recent strengthening of Hilberts Theorem on ternary quartics: a positive ternary quartic is a sum of squares in exactly 8 inequivalent ways.

  • 04/09/04

  • 04/09/04
    Cristian Popescu - UCSD
    L-functions at s=0 - Stark s Conjecture

  • 04/13/04
    Ryan Szypowski - UCSD Graduate Student
    Mixed finite element formulations and differential complexes

    The choice of a proper finite element space is essential to guarantee the
    stability of a discretised system of partial differential equations. In
    many cases, the differential geometric structure of the system can be
    captured by a differential complex, and finding an appropriate discrete
    differential complex may lead to a finite element space which yields a
    stable discretisation. Two problems, one from electromagnetism and
    the other from elasticity, will be discussed in this context.

  • 04/13/04
    Nolan Wallach - UCSD
    Exceptional spherical triples

    Let G be a connected, simply connected, simple group over
    the complex numbers. Let K be the fixed point group of an involutive
    automorphism of G and let P be a parabolic subgroup of G such that KP has
    interior in G. Then we say that (G,K,P) is a spherical triple if there
    is a Borel subgroup of K, B, such that B has an open orbit in G/P. This
    condition is equivalent with having (degenerate) principal series for the
    non-compact real form corresponding to K be multiplicity free. In this
    talk we will give a classification of these triples for the exceptional

  • 04/13/04

  • 04/15/04
    Ben Cooper - UCSD Graduate Student
    Khovanov homology for knots

  • 04/15/04
    Dinesh Thakur - University of Arizona
    Zeta values for function fields

    We will describe the arithmetic of zeta values in the
    function field context. It is a curious mix of many analogies,
    some strong theorems where corresponding number field statements
    are conjectures, and some questions where the number field situation
    is understood, but the function field situation is not even
    conjecturally understood

  • 04/16/04
    Finbarr Sloane - National Science Foundation
    Quality criteria for design research

    The goal of this presentation is to explore the multiple definitions of design research in education, and particularly in mathematics education. In doing so, I revisit components of Brown, A. L. (1992)*, which highlight the issues for warrant. I argue that as yet her concerns have gone relatively unheeded by the education research community. I point to the need for mixed methodologies in support of design based insights and develop a validity framework that is inclusive of design adaptation issues.

    * Design experiments: Theoretical and methodological challenges in creating complex interventions in classroom settings. Journal of the Learning Sciences, 2, 141-178.Brown, A. L. (1992)

  • 04/16/04
    Cristian Popescu - UCSD
    The Brumer-Stark Conjecture

  • 04/19/04
    Alex Lubotzky - Hebrew University of Jerusalem
    Subgroup growth of Lattices in Semisimple Lie groups

    The subgroup growth of a group is the study of the number of
    its index n subgroups as a function of n. We will present some sharp
    estimates on this growth for lattices L in higher rank semi-simple Lie
    groups G. A surprising phenomenom is that this growth depends only on the
    root system of G and not on L. In the most general case, the results
    depend on the generalized Riemann Hyphothesis but a number of results can
    be also proved unconditionally.

  • 04/20/04
    Avy Soffer - Rutgers University
    N-Soliton scattering of nonintegrable Systems

    The scattering of solitons off each other and with radiation is studied for NLS in three or more dimensions. The general problem of finding the large time behavior of such equations, including the proof of Asymptotic Completeness, and stability is discussed.

    Recent solution of this problem for NLS with small radiation initial data will be described

  • 04/20/04
    Yean Loke
    Exceptional Lie algebras

    In this talk, I will describe a construction of the exceptional
    Lie algebras of type F and E using the octonions and the principle of triality.

  • 04/20/04
    Career Seminars

    Featuring the following speakers:

    Robert Hecht-Nielsen - Founder, HNC Software; Professor, Electrical and Computer Engineering UCSD

    Beth Smith - Mathematics Assistant Professor, Grossmont

    Saul Molina - Assistant Systems Administrator, Math Dept UCSD

    Space is limited - reserve your place - email

  • 04/20/04

  • 04/20/04
    Alex Lubotzky - Hebrew University of Jerusalem
    From Ramanujan graphs to Ramanujan complexe

    Ramanujan graphs are k-regular graphs with optimal bounds on their eigenvalues. They play a central role in various questions in combinatorics and computer secience. Their construction is based on the work of Deligne and Drinfeld on the Ramanujan conjecture for GL(2). The
    recent work of Lafforgue which settles the Ramanujan conjecture for GL(n) over function fields opens the door to study of Ramanujan complexes: these are higher dimensional analogues which are obtained as quotients of the Bruhat-Tits building of PGL(n) over local fields.

  • 04/21/04
    Xiaofeng Sun - UC Irvine
    The geometry of the moduli spaces of Riemann surfaces

    We introduce and study new complete Kahler metrics on the moduli and the Teichmuller spaces of Riemann surfaces, the Ricci and the perturbed Ricci metric. They are asymptotically equivalent to the Poincare metric. The perturbed Ricci metric has bounded negative sectional and Ricci curvatures. As corollaries we prove the equivalence of these new metrics to several classical metrics such as the Kahler-Einstein metrics, proving a conjecture of Yau in the early 80s. Other consequences will also be discussed. This is joint work with K. Liu and S.-T. Yau.

  • 04/22/04

  • 04/22/04
    Tom Fleming - UCSD Graduate Student
    Lee homology for knots

  • 04/22/04
    Herber Goins - California Institute of Technology
    On the modularity of wildly Ramified Galois Representations

    There has been great interest in two-dimensional representations of Galois
    groups, from conjectures of Artin concerning complex projective
    representations of the symmetries of the Platonic solids, to conjectures
    of Shimura and Taniyama concerning p-adic representations associated to
    elliptic curves. Many of these conjectures were recently answered in the
    affirmative by Wiles and Taylor using techniques from arithmetic algebraic
    geometry. In this talk, we explain how these results can be extended even
    further, and give some applications.

  • 04/23/04

  • 04/24/04
    Spring 2004 Southern California Analysis and Partial Differential Equations Seminar

    Saturday Schedule

    10:30 - 11:00 AM Coffee and assorted danish

    11:00 - 12:00 PM Gunther Uhlmann, University of Washington

    Boundary rigidity and the Dirichlet-to-Neumann map

    1:30 - 2:30 PM Chuu-Lian Terng, UCI & Northeastern University

    Periodic and homoclinic orbits of the modified 2+1 chiral model

    2:45 - 3:45 PM Oded Schramm, Microsoft Research

    Conformally invariant random processes in two dimensions

    3:45 - 4:15 PM Coffee break

    4:15 - 5:15 PM Michael Lacey, Georgia Tech

    Hilbert transforms and smooth families of lines

  • 04/25/04
    Spring 2004 Southern California Analysis and Partial Differential Equations Seminar

    Sunday Schedule

    9:30 - 10:00 AM Coffee and assorted danish

    10:00 - 11:00 AM Jean-Pierre Rosay, University of Wisconsin-Madison

    Pseudo-holomorphic discs and rough almost complex structures. (Isolated solutions of non-linear PDE)

    11:15-12:15 PM Christoph Thiele, UCLA

    Basic questions for the Nonlinear Fourier transform

  • 04/27/04
    Jean-Pierre Rosay - University of Wisconsin, Madison
    The Kobayashi metric on almost complex manifolds

  • 04/27/04

  • 04/27/04

  • 04/27/04
    Hershy Kisilevski - Concordia University - Montreal
    Vanishing and non-Vanishing Dirichlet twists of Elliptic L-functions

  • 04/27/04
    K. Williams - MIT Graduate Student
    Enumeration of totally positive Grassmann cells

    Alex Postnikov recently gave a combinatorially explicit cell
    decomposition of the totally nonnegative part of a Grassmannian, denoted $Gr_{kn}+$, and showed that this set of cells is isomorphic as a graded poset to many other interesting graded posets. The main result of our
    work is an explicit generating function which enumerates the cells in $Gr_{kn}+$ according to their dimension. As a corollary, we give a new proof that the Euler characteristic of $Gr_{kn}+$ is 1. Additionally, we use our result to produce a new q-analog of the Eulerian numbers, which interpolates between the Eulerian numbers, the Narayana numbers, and the binomial coefficients.

  • 04/28/04
    Imre Patyi - UCSD
    Complex submanifolds in Hilbert space

    We show that any complex submanifold in Hilbert space defined by global equations has a basis of pseudoconvex open neighborhoods, that holomorphic vector bundles are acyclic over smooth complete intersections in Hilbert space, and that such complete intersections are holomorphic retracts of some pseudoconvex open neighborhoods.

  • 04/29/04
    Gregg Musiker - UCSD Graduate Student
    Khovanov homology and Reidemeister moves

  • 04/29/04
    Stefan Erickson - UCSD Graduate Student
    Prime divisibility in the Fibonacci numbers

  • 04/30/04
    Barry Smith - UCSD Graduate Student
    The Brumer-Stark Conjecture for function fields

  • 04/30/04
    Dimitris Politis - UCSD
    Bootstrap methods for time series: a selective overview

    A tutorial overview of time-domain Bootstrap Methods for time series will be given with emphasis on block-bootstrap and subsampling. Comparisons of the methods
    will be discussed, as well as the crucial issue of practical block size choice.

  • 05/04/04
    Eugene Izhikevich - Neurosciences Institute, La Jolla
    Brain models as systems of delay differential equations

  • 05/04/04

  • 05/04/04
    John Lott - Michigan and MSRI
    Notions of generalized Ricci curvature



  • 05/04/04
    Ben Chow - UC San Diego
    Ricci flow and Fukaya theory in dimension 3



  • 05/04/04
    John Walsh - University of British Columbia
    Probabilistic interpretation of the interior reduite

  • 05/06/04
    Xaiojun Huang - Harvard University

  • 05/06/04
    Henning Hohnhold - UCSD Graduate Student
    Spectral sequence

  • 05/06/04
    S. Fraenkel
    The structure of complementary sets of integers: a 3-shift theorem

    Let $0 < alpha < beta$ be two irrational numbers satisfying
    $1/alpha + 1/beta = 1$. Then the sequences $a'_n = lfloor
    {nalpha}rfloor$, $b'_n = lfloor{nbeta}rfloor$, $nge 1$, are
    complementary over $IZ_{ge 1}$. Thus $a'_n = {rm mex_1}
    {a'_i,b'_i : 1 le i< n}$, $n geq 1$ (${rm mex_1}(S)$, the
    smallest positive integer not in the set $S$). Suppose that $c =
    beta-alpha$ is an integer. Then $b'_n = a'_n+cn$ for all $n ge

    We define the following generalization of the sequences $a'_n$,
    $b'_n$: Let $c,;n_0inIZ_{ge 1}$, and let $XsubsetIZ_{ge 1}$
    be an arbitrary finite set. Let $a_n = {rm mex_1}(Xcup{a_i,b_i
    : 1 leq i< n})$, $b_n = a_n+cn$, $nge n_0$. Let $s_n =
    a'_n-a_n$. We show that no matter how we pick $c,;n_0$ and $X$,
    from some point on the {it shift sequence/} $s_n$ assumes either
    one constant value or three successive values; and if the second
    case holds, it assumes these values in a very distinct
    fractal-like pattern, which we describe.

    This work was motivated by a generalization of Wythoff's game to
    $Nge 3$ piles.

  • 05/11/04
    Adriano Garsia - UCSD
    The $n!/k$ conjecture and other amenities

    This talk covers a variety of conjectures not resolved by Mark Haiman's
    proof of the $n!$-conjecture. We also review what came to be called
    "Science Fiction" which taken literally is blatantly false at the
    representation theoretical level, yet it beautifully explains many of the
    experimentally observed properties of the Macdonald $q,t$-Kostka

  • 05/11/04
    Dimitri Gioev - University of Pennsylvania and Courant Institute
    Universality in random matrix theory for orthogonal and symplectic ensembles (joint with P.Deift)

    We give a proof of the Universality Conjecture
    in Random Matrix Theory for orthogonal $(beta=1)$ and
    symplectic $(beta=4)$ ensembles in the scaling limit
    for a class of weights $w(x)=exp(-V(x))$
    where V is a polynomial. For such weights
    the associated equilibrium measure is supported on
    a single interval.
    Our starting point is Widom's representation
    of the correlation kernels for the beta=1,4 cases
    in terms of the unitary $(beta=2)$ correlation kernel
    plus a correction term which involves orthogonal
    polynomials $(OP's)$ with respect to the weight w introduced above.
    We do not use skew orthogonal polynomials.
    In the asymptotic analysis of the correction terms
    we use amongst other things
    differential equations for the derivatives
    of OP's due to Tracy-Widom,
    and uniform Plancherel-Rotach type asymptotics for OP's
    due to Deift-Kriecherbauer-McLaughlin-Venakides-Zhou.
    The problem reduces to a small norm problem
    for a certain matrix of a fixed size
    that is equal to the degree of the polynomial potential.

  • 05/11/04

  • 05/12/04
    Christopher K. King - Northeastern University
    Matrix inequalities and quantum information theory

    Quantum information theory has generated interesting conjectures concerning products of completely positive maps on matrix algebras. This talk will describe the background to these conjectures and also some recent proofs in special cases. Matrix inequalities have played an important role in these results, and the talk will describe applications of the Lieb-Thirring inequality and an extension of Hanner's inequality to this problem.

  • 05/13/04
    Li Yu - UCSD Graduate Student
    The Khovanov-to-Lee spectral sequence

  • 05/13/04
    Al Hales - Center for Communications Research
    Jordan decomposition in integral group rings

    Let A be a square matrix with rational entries. Then A can be written as the
    sum S + N where S and N also have rational entries, S is semisimple, N is nilpotent, and
    S and N commute. This representation is unique, and is called the Jordan decomposition
    of A. It can be considered as a coordinate-free, and ambient-field-free, version of the
    usual Jordan canonical form for matrices. This decomposition (in its multiplicative
    version) is particularly useful in the study of algebraic groups.

    If G is a finite group and a is an element of the rational group ring Q[G],
    i.e. a is a linear combination of group elements with rational coefficients, then there
    is an analogous decomposition: a = s + n where s and n lie in Q[G], s is semisimple, n
    is nilpotent, and s and n commute (this representation is also unique).

    Consider the integral versions of these decompositions: if the matrix A has
    integer entries, need S and N have integer entries? if the element a in Q[G] has integer
    coefficients, need s and n have integer coefficients? We give complete answers to these
    questions. The multiplicative version of the integral group ring question is much more
    subtle, however, and we only have partial results on this problem.

  • 05/14/04

  • 05/14/04
    Caleb Emmons - UCSD Graduate Student
    Advancement talk

  • 05/18/04
    Tai Alexis Melcher - UCSD Graduate Student
    Hypoelliptic heat kernel inequalities on Lie groups

  • 05/18/04
    Christopher Sogge - Johns Hopkins University
    Nonlinear wave equations in waveguides

    In joint work with J. Metcalfe and A. Stewart, we prove global and almost
    global existence theorems for nonlinear wave equations with quadratic
    nonlinearities in infinite homogeneous waveguides. We can handle both the
    case of Dirichlet boundary conditions and Neumann boundary conditions. In
    the case of Neumann boundary conditions we need to assume a natural
    nonlinear Neumann condition on the quasilinear terms. The results that we
    obtain are sharp in terms of the assumptions on the dimensions for the
    global existence results and in terms of the lifespan for the almost global
    results. For nonlinear wave equations, in the case where the infinite part
    of wave guide has spatial dimension three, the hypotheses in the theorem
    concern whether or not the Laplacian for the compact base of the wave guide
    has a zero mode or not.

  • 05/19/04
    Jiaping Wang - University of Minnesota
    Green's form estimates and application

  • 05/20/04
    Anthony Mendes - UCSD Graduate Student
    Building generating functions brick by brick

  • 05/20/04
    Jana Comstock - UCSD - Graduate Student
    Rasmussen's invariant

  • 05/20/04
    Thomas R. Fleming - UCSD Graduate Student
    Advancement to Candidacy

  • 05/20/04
    Edward Odell - University of Texas, Austin
    Ramsey theory and Banach spaces

    Ramsey's original theorem states that if one finitely colors the $k$
    element subsets of ${\Bbb N}$ then there exists an infinite subsequence $M$
    of ${\Bbb N}$ all of whose $k$ elements subsets have the same color.
    This theorem and stronger versions entered into Banach space theory in the
    They were ideal for studying subsequences of a given sequence
    $(x_i)\subseteq X$ (infinite dimensional separable Banach space).
    We survey some of these applications and the following problem.
    $X$ is said to satisfy the ultimate Ramsey theorem if for every finite
    coloring $(C_i)_{i=1}^n$ of its unit sphere $S_X$ and $\varepsilon>0$ there exists
    an infinite dimensional subspace $Y$ and $i_0$ so that
    $S_Y\subseteq (C_{i_0})_\varepsilon =\{x:|x-z|<\varepsilon$ for some $z\in C_{i_0}\}$.
    What spaces $X$ (if any) have this property?
    We survey other results including Gowers' block Ramsey theorem for Banach

  • 05/21/04
    Jim Lin - UCSD

  • 05/21/04
    Stefan Erickson - UCSD Graduate Student
    The Brumer-Stark conjecture in characteristic p, III

  • 05/24/04
    Kevin O'Bryant - UCSD
    There are arbitrarily long arithmetic progressions of primes

    Abstract: On of the most famous problems in number theory is to show that there are arbitrarily long arithmetic progressions, all of whose terms are prime numbers. At the start of the year, this was known to be true only for 3-term progressions. Recently, Ben Green and Terrence Tao have released a manuscript which purports to prove this conjecture. I will discuss some of the history of this problem, and outline their argument. I intend for this talk to be accessible to a broad audience, and hope that this will lead to a series of talks (not all by me) verifying the Green/Tao proof.

  • 05/24/04
    Michele Ann Schuman - UCSD Graduate Student
    A new look at problems of Herstein and Kaplansky

  • 05/25/04
    Jeffrey S. Ovall - UCSD Graduate Student
    Duality-based adaptive refinement for elliptic PDEs

  • 05/25/04
    Jim Lin - UCSD
    Secondary operations

  • 05/27/04
    Sean Raleigh - UCSD Graduate Student
    Estimates for the 4-ball genus

  • 05/27/04
    Kevin O'Bryant - UCSD
    Extended constructions of Sidon-type sets

    Abstract: An $(h,g)$ Sidon set is a set $S$ of integers with the property that the coefficients of
    $(\sum_{s \in S} z^s)^h$
    are bounded by $g$. These arose naturally is Simon Sidon's study of Fourier Series, and have become a standard topic in combinatorial number theory. I will present joint work with Greg Martin giving constructions of such sets, and discuss numerous open problems.

  • 07/01/04
    Professors Shangyou Zhang, Huo-yuan Duan, - University of Delaware and National University of Singapore
    A generalized BPX framework covering the V-cycle nonnested multigrid method

    Fifteen years ago, Bramble, Pasciak and Xu developed a framework
    for analyzing the multigrid method (The analysis of multigrid
    algorithms with nonnested spaces or noninherited quadratic forms,
    Mathematics of Computation, 1991.) It was, known as BPX Framework,
    widely used in the analysis of mutligrid and domain decomposition
    methods. The framework was extended several times, covering less
    regularity or non-symmetric, and other cases. However an apparent
    limit of the framework is that it could not incorporate the number
    of smoothings in the V-cycle analysis. Therefore, the framework
    is limited in nonnested multigrid methods (as one-smoothing multigrid
    methods won\'t converge for almost all nonnested cases), and
    it produces variable V-cycle, relaxed coarse-level correction,
    or non-uniform convergence rate V-cycle methods, or other non-optimal
    results in analysis thus far. Therefore, most non-nested finite
    element problems were show to converge only for W-cycles based
    BPX framework, or an even earlier frame work of Bank and Dupont.

    This paper completes a long time effort in extending the BPX Framework
    so that the number of smoothings is included in the V-cycle analysis.
    We will apply the extended BPX Framework to the analysis of many V-cycle
    nonnested multigrid methods. Some of them were previously proven
    for two-level and W-cycle iterations only.

  • 07/06/04
    Jacob Sterbenz - Princeton University
    Non-linear wave equations

  • 07/08/04
    Karen Ball - University of Indiana
    Deterministic thinnings of poisson point processes

  • 08/04/04
    Melvin Leok - Caltech, Control and Dynamical Systems
    Foundations of Computational Geometric Mechanics

    Structure preserving numerical integrators aim to preserve as many of the physical invariants of a dynamical system as possible, since this typically results in a more qualitatively accurate simulation. Computational geometric mechanics is concerned with a class of structured integrators based on discrete analogues of Lagrangian and Hamiltonian mechanics.

    Discrete theories of exterior calculus and connections on principal bundles provide some of the mathematical foundations of computational geometric mechanics, and address the question of how to obtain canonical discretizations that preserve, at a discrete level, the important properties of the continuous system.

    Some recent progress on the construction of a combinatorial formulation of discrete exterior calculus based on primal simplicial complexes, and circumcentric dual cell complexes will be presented. These techniques have been used to systematically recover discrete vector differential operators such as the Laplace--Beltrami operator.

    For discrete connections, the discrete analogue of the Atiyah sequence of a principal bundle is considered, and a splitting of the discrete Atiyah sequence is related to discrete horizontal lifts and discrete connection
    forms. Continuous connections can be obtained by taking the limit of discrete connections in a natural way.

    Examples spanning the work on exterior calculus and connections include the discrete Levi-Civita connection for a semidiscrete Riemannian manifold, and the curvature of an abstract simplicial complex endowed with a metric on the vertices.

  • 08/12/04
    Eric Brussel - Emory University
    Computing the Brauer group with Cup products

  • 09/14/04
    Kagba Suaray - UCSD Graduate Student
    On Kernel density estimation for censored data

  • 09/16/04
    Aaron Lauda - Cambridge University
    Frobenius categories

  • 09/28/04
    Hanspeter Kraft - University of Basel, Switzerland
    Equations for field extensions and covariants of the symmetric

    For a given finite separable field extension $L/K$ we would like to find a generator $x \in L$ whose equation is as simple as possible. For example, one would like to have as many vanishing coefficients as possible. More generally, the transcendence degree of the subfield $k$ of $K$ generated by the coefficients of the equation should be as small as possible. It was shown by Buhler and Reichstein that the minimal transcendence degree one can reach has an interpretation in terms of rational covariants of the symmetric group $S_n$ where $n := [L:K]$. This number is called the essential dimension of $L/K$ or of $S_n$ it can be defined for every finite group. We will describe the relation between covariants and equations for field extensions and will explain the main results in this context.

  • 09/28/04
    Hanspeter Kraft - University of Basel, Switzerland
    A result of Hermite and equations of degree 5 and 6

    A classical result from 1861 due to Hermite says that every separable equation of degree 5 can be transformed into an equation of the form $x^5 + b x^3 + c x + d = 0$. Later, in 1867, this was generalized to equations of degree 6 by Joubert. We show that both results can be understood as an explicit analysis of certain covariants of the symmetric groups $S_5$ and $S_6$. In case of degree 5, the classical invariant theory of binary forms of degree 5 comes into play whereas in degree 6 the existence of an outer automorphism of $S_6$ plays an essential r\^ole. Although these consequences for equations of degree 5 and 6 have been cited and used many times in the literature, it seems unclear if the methods and ideas of Hermite and Joubert have really been understood.

  • 09/29/04
    Ning Zhang - U.C. Riverside
    Dolbeault groups of the loop space of the Riemann sphere

    The loop space of a complex manifold M, consisting of all maps from the circle $S^1$ to M with some fixed $C^k$ or Sobolev regularity, is an infinite dimensional complex manifold. We identify an infinite dimensional subgroup of the Picard group of holomorphic line bundles on the loop space of the Riemann sphere, and show that the space of holomorphic sections of any such line bundle is finite dimensional. We also compute the (0,1) Dolbeault group of the loop space of the Riemann sphere (which is infinite dimensional).

  • 09/30/04
    J. Roberts - UCSD
    Fibrations in homotopy theory

  • 09/30/04
    Hanspeter Kraft - University of Basel, Switzerland
    Covariant dimension for finite groups

    We define another invariant for a finite group $G$, the so-called {\it covariant dimension of $G$} which is closely related to the essential dimension of $G$ which we introduced in the previous talk. It is the minimal possible dimension of the image of a faithful (regular) covariant of $G$. It turns out that many results for the essential dimension of a finite group $G$ can be carried over to the covariant dimension of $G$ where they admit surprisingly simple proofs. In this way we have been able to reconstruct the main results of Buhler and Reichstein for the essential dimension and their applications to field extensions. But still there remain a number of interesting open questions which are quite elementary and easy to understand. Some of them will be discussed at the end of the talk.

  • 09/30/04
    Richard Stong - Rice University
    Decomposing Cartesian products into Hamiltonian cycles

    Decompositions of graphs have a number of applications, for example, programming for multiple processor computers. In particular, this makes decompositions of high dimensional cubes into hamiltonian cycles of interest. These decompositions are most easily approached by looking at decompositions of more general cartesian products. For the undirected case, these decompositions are fairly well understood, but I want to present a new way of looking at these constructions. This new outlook allows one to solve the directed case as well.

  • 10/05/04

  • 10/05/04
    Andrey Todorov - UC Santa Cruz
    CY manifolds whose moduli space is a locally symmetric space

    In this talk we will discuss CY manifolds that are double covers of $CP^3$ ramified over 8 planes. We will show that the moduli space of those CY manifolds is a locally symmetric space. The proof of this fact is based on the proof that the Yukawa coupling does not admit quantum corrections. The meaning of this statement will be explain.

  • 10/05/04
    K. Baur - UCSD
    Nice parabolic subalgebras and a normal form for admissible

    Lynch's vanishing theorem for generalized Whittaker modules
    applies to generic homomorphisms of the nilradical of a class of parabolic subalgebras which we have named nice. In this talk we describe the classification of nice parabolic subalgebras of simple Lie algebras. Our results give rise to a new characterisation of even nilpotents. We will also give a normal form of Richardson elements for nice parabolic subalgebras in the classical case.

  • 10/06/04
    J. Eggers - UCSD
    An elementary introduction to juggling and juggling Mathematics

    Have you ever seen someone juggle and wonder how he or she does
    it? Or, are you able to juggle but have wondered how the process might be described mathematically? In this talk, I will introduce the concept of a juggling sequence and explain how juggling sequences can be used to describe simple juggling patterns. I will discuss some of the mathematical questions related to juggling sequences. Finally, I will
    illustrate some simple juggling patterns by juggling them (when I'm not
    picking up the balls off the floor, that is) and exhibit a pattern that
    was unknown prior to the discovery of juggling sequences.

    Refreshments will be provided.

  • 10/07/04
    L. Yu - UCSD Graduate Student
    Spectral sequence of a fibration

  • 10/07/04
    J. Schweinsberg - UCSD
    Family size distributions for multitype Yule processes

    Qian, Luscombe, and Gerstein (2001) introduced a model of gene
    duplication that we may formulate as follows. Consider a multitype
    Yule process starting with one individual in which there are no deaths
    and each individual gives birth to a new individual at rate one.
    When a new individual is born, it has the same type as its parent
    with probability $1 - r$ and is a new type, different from all previously
    observed types, with probability r. We refer to individuals with
    the same type as families and provide an approximation to the joint
    distribution of family sizes when the population size reaches $N$.
    We also show that if $1 << S << N^{1-r}$, then the number of families
    of size at least S is approximately $CNS^{-1/(1-r)}$, while if
    $N^{1-r} << S$ the distribution decays more rapidly than any power.

  • 10/07/04
    Peter Stevenhagen - Leiden University, Netherlands
    Efficient construction of elliptic curves

    Over the last 20 years, efficient algorithms have been developed to count the number of points of a given elliptic curve over a finite field. We discuss the inverse problem of constructing elliptic curves with a given number of points over a finite field. The difficulty of the problem depends on its exact wording. We present a solution to the problem that easily handles curves of the size occurring in cryptographic practice, and explain why it should be expected to do so.

  • 10/11/04

  • 10/12/04
    Josip Globevnik - University of Ljubljana
    Analyticity on circles

    Let $F$ be an open family of circles in the complex plane and let $P$ be the union of all the circles from $F$. Suppose that $f$ is a continuous function on $P$ which extends holomorphically from each circle $C$ belonging to $F$ (that is, its restriction to $C$ extends holomorphically through the disc bounded by $C$). Is $f$ holomorphic on $P$? We show that this question is naturally related to a question in $C^2$ and then, for certain families $F$, we show how some standard facts from several complex variables can be used to deal with this question.

  • 10/12/04

  • 10/12/04
    R. Andersen - UCSD Graduate Student
    Modeling the small world phenomenon using a hybrid model with local network flow

    The 'small world phenomenon' as observed by the psychologist Stanley Milgram in the 1960s is that most pairs of people are connected by a short chain of acquaintances. More generally we say a graph exhibits the small world phenomenon if the average distance between vertices is small due to a few random-like edges combined with a highly regular local structure. Randomly generated graphs with a power law degree distribution are often used to model large real-world networks. While these graphs have small average distance, they do not have the local structure associated with the small world phenomenon. We define a hybrid model which combines a global graph (a random power law graph) with a local graph (a graph with high local connectivity defined by network flow). We also present an efficient algorithm to extract a highly connected local graph from a given realistic network. We show that the hybrid model is robust in the following sense. Given a graph generated by the hybrid model, we can recover a good approximation to the original local graph in polynomial time.

  • 10/13/04
    A. Terras - UCSD
    Primes in graphs

    In this talk we will compare the primes (2,3,5,7,11,...) with primes in a
    graph which are classes of closed paths in the graph. In the process, we will look at the Riemann zeta function from number theory and the Ihara zeta function from graph theory and meditate on the Riemann hypothesis.

  • 10/14/04
    M. Gurvich - UCSD Graduate Sudent
    Classifying spaces of groups

  • 10/14/04
    Dr. Kristin Lauter - Microsoft
    Computing modular polynomials and applications to cryptography

    (Joint work with Denis Charles)

    The nth modular polynomial parameterizes pairs of elliptic curves with
    an isogeny of degree n between them. Modular polynomials provide the
    defining equations for modular curves, and are useful in many different
    aspects of computational number theory and cryptography. For example,
    computations with modular polynomials have been used to speed elliptic
    curve point-counting algorithms.

    The standard method for computing modular polynomials consists of
    computing the Fourier expansion of the modular j-function and solving a
    linear system of equations to obtain the integral coefficients. The
    computer algebra package MAGMA incorporates a database of modular
    polynomials for n up to 59.

    The idea of the talk is to compute the modular polynomial directly
    modulo a prime p, without first computing the coefficients as integers.
    Once the modular polynomial has been computed for enough small primes,
    our approach can also be combined with the Chinese Remainder Theorem
    (CRT) approach to obtain the modular polynomial with integral
    coefficients or with coefficients modulo a much larger prime using
    Explicit CRT. Our algorithm does not involve computing Fourier
    coefficients of modular functions.

  • 10/18/04

  • 10/19/04
    Zhilin Li - North Carolina State University
    Removing source singularity for certain interface problems

    In many two-phase flows and moving interface problems,
    the flux and the solution of the governing PDE often
    have finite jumps across the interface. Such jumps often result from source distributions along the interface. One famous application is Peskin's immersed boundary model/method which has been widely used. Using a level set function to represent the interface, we have proposed a new method that can transfer an interface with discontinuity in the flux and/or in the solution to an interface problem with a smooth solution if the coefficient PDE is continuous. The new formulation is based on extensions of the jumps along the normal line of the interface. If the coefficient of the PDE is also discontinuous, then the transformation leads to a new interface problem with homogeneous, also called natural, jump conditions. This will greatly simplify the immersed interface method because no surface derivatives of the jump conditions is needed anymore. Theoretical and numerical analysis including implementation details and numerical example will also
    be presented.

  • 10/19/04

  • 10/19/04
    Sergi Elizalde - MSRI
    Refined enumeration of pattern-avoiding permutations

    The talk will begin with a short survey of some of the main enumerative
    results in the subject of restricted (or pattern-avoiding) permutations.

    Next I will discuss some recent work concerning statistics on restricted
    permutations, motivated in part by the surprising result of Zeilberger et
    al. which states that the number of fixed points has the same distribution
    in 321-avoiding permutations as in 132-avoiding permutations. I will give two combinatorial proofs of this result, and generalize it to other
    statistics. Bijections to Dyck paths play an important role in the proofs.

    Finally I will consider permutations avoiding several patterns
    simultaneously, as well as generalized patterns (i.e., with the
    requirement that some elements occur in adjacent positions). Using similar bijections we obtain generating functions enumerating these restricted permutations with respect to several statistics.

  • 10/20/04
    Cristian Popescu - UCSD
    Right triangles and related Diophantine stories

    I will discuss links between a certain class of Pythagorean triples and
    certain aspects of modern number theory, such as the theory of elliptic
    curves, their L-functions, and the various still unsolved conjectures
    predicting their behavior.

  • 10/21/04
    David Clark - UCSD Graduate Student
    Classifying spaces of vector bundles

  • 10/21/04
    Natella O'Bryant - U. C. Irvine
    Weak convergence of stochastic stratified processes

    During the last decade several stochastic stratified processes were introduced in the probability literature. They include the fiber Brownian motion introduced by Bass and Burdzy as a process that switches between two-dimensional and one-dimensional evolution, a Markov process on a `whiskered sphere' described by Sowers, processes on trees like the Walsh process, the Evans process, spider martingales, and Brownian snakes. We study the rate of weak convergence to one such process. Using a Wasserstein-type metric as the distance that metrizes weak convergence, we show that the rate of this convergence can be controlled by the rates of convergence of other related processes.

    For a related classical example of a Hamiltonian system on a cylinder, the corresponding estimates on the rate of convergence are obtained using two different methods. One of these methods reveals an explicit expression for the Wasserstein distance in the classical case.

  • 10/25/04
    Daniel Goldstein - CCR
    Inequalities for finite group permutation modules

    If $f$ is a nonzero complex-valued function defined on a finite abelian group $A$ and $\hat f$ is its Fourier transform, then $|f|| \hat f| \ge |A|$, where $f$ and $\hat f$ are the supports of f and $\hat f$. In this paper we generalize this known result in several directions. In particular, we prove an analogous inequality where the abelian group $A$ is replaced by a transitive right $G$-set, where $G$ is an arbitrary finite group. We obtain stronger inequalities when the $G$-set is primitive and we determine the primitive groups for which equality holds. We also explore connections between inequalities of this type and a result of Chebotarev on complex roots of unity, and we thereby obtain a new proof of Chebotarev’s theorem.

  • 10/26/04
    Tom Bewley - UCSD (Flow Control Lab, Dept. of MAE)
    Mathematical challenges in the control and optimization of fluid systems

    This talk will discuss the mathematical challenges of the control-oriented
    analysis of Navier-Stokes systems in the following three areas:

  • 10/26/04

  • 10/27/04
    Cristian Popescu - UCSD
    Right triangles and related Diophantine stories, Part II

    I will discuss links between a certain class of Pythagorean triples and
    certain aspects of modern number theory, such as the theory of elliptic
    curves, their L-functions, and the various still unsolved conjectures
    predicting their behavior.

    Refreshments will be provided.

  • 10/27/04
    Andre Neves - Stanford University
    The Yamabe invariant of $RP^3\#(S^1\times S^2)$

  • 10/28/04
    Van Vu - UCSD
    On an urn model of Diaconis

    We consider the following urn model proposed by Diaconis. There is an urn with $n$ balls, each ball has value 0 or 1. Pick two random balls and add a new ball with their sum (modulo 2) to the urn. (Thus the number of balls increases by one each time). What is the limiting distribution?

    In general, let $G$ be a finite additive group. We begin with an $a$ set
    $g_1, \dots, g_n$ of (not necessarily different) elements of $G$. (In fact, typically $n$ is much larger than $|G|$). At time $i=0,1,2,\dots,$ we choose two random elements $a$ and $b$ from the set $g_1, \dots, g_{n+i}$ and add the sum $g_{n+i+1} = a+b$ to the set.

    Siegmund and Yakir studied the distribution of the set $g_1,g_2\dots$ and showed that it converges to the uniform distribution. This result leaves open the question: How quickly does this convergence occur? In this talk we address this issue.

  • 10/28/04
    Maia Averett - UCSD Graduate Student
    Cohomology of classifying spaces

  • 10/28/04
    Kevin Costello - UCSD Graduate Student
    Singularity of random matrices

  • 10/28/04
    Ramin Takloo-Bighash - Princeton University
    Rational points and automorphic forms

    In this talk, I will report on a recent joint work with Joseph A. Shalika and Yuri Tschinkel on the distribution of rational points of bounded height on certain group compactifications.

  • 10/28/04
    Jiming Jiang - U. C. Davis
    Partially observed information and inference about non-gaussian mixed linear models

    In mixed linear models with nonnormal data, the Gaussian Fisher-information matrix is called quasi information matrix (QUIM). The QUIM plays an important role in evaluating the asymptotic covariance matrix of the estimators of the model parameters, including the variance components. Traditionally, there are two ways to estimate the information matrix: the estimated information matrix and the observed one. Because the analytic form of the QUIM involves parameters other than the variance components, for example, the third and fourth
    moments of the random effects, the estimated QUIM is not available. On the other hand, because of the dependence and nonnormality of the data, the observed QUIM is inconsistent. We propose an estimator of the QUIM consisting partially of an observed form and partially of an estimated one. We show that this estimator is consistent and computationally very easy to operate. The method is used to derive large sample tests of statistical hypotheses that involve the variance
    components in a non-Gaussian mixed linear model. Finite sample
    performance of the test is studied by simulations and compared with
    the delete-group jackknife method that applies to a special case of non-Gaussian mixed linear models.

  • 10/29/04
    Gilad Gour - UCSD
    Concurrence monotones and remote entanglement distributions

    I extend the definition of concurrence into a family of entanglement
    monotones, which I call concurrence monotones. I will discuss their
    properties and advantages as computational manageable measures of entanglement. I will then show that the concurrence monotones provide bounds on quantum information tasks. As an example, I will discuss their applications to remote entanglement distributions (RED) such as entanglement swapping and Remote Preparation of Bipartite Entangled States (RPBES). I will present a powerful theorem which states what kind of (possibly mixed) bipartite states or distributions of bipartite states can not be remotely prepared. The theorem establishes an upper bound on the amount of G-concurrence (one member in the concurrence family) that can be created between two single-qudit nodes of quantum networks by means of tripartite RED. For pure bipartite states the bound on the G-concurrence can always be saturated by RPBES.

  • 11/02/04
    A. Henke - University of Leicester
    A generalization of James' column removal formula

    Let $K$ be a field and $GL_n(K)$ the group of invertible n by n matrices over $K$. About a century ago, I. Schur described the irreducible polynomial representations of $GL_n(K)$ for the field $K$ of complex numbers. The same problem over a field $K$ of prime characteristic is still widely open. In this context, G. James proved in 1981 a so-called column removal formula. It relates polynomial representations in different homogeneous degrees. Combinatorially, it is based on removing the first column of an associated Young diagram. We generalize this operation of column removal to considering ``complements" of Young diagrams. Algebraically this establishes equivalences between categories of polynomial representations in different degrees. As by-product it establishes and explains certain repeating patterns in decomposition numbers of general linear and symmetric groups. This work is related to work by Beilinson-Lusztig and MacPherson. It is joint work with Fang and Koenig.

  • 11/03/04

  • 11/04/04

  • 11/04/04
    Tony Shaheen - UCSD Graduate Student
    The trace formula for finite upper half planes

  • 11/04/04
    Zinovy Reichstein - University of British Columbia
    Cayley maps for algebraic groups

    The exponential map plays an important role in Lie theory; it allows one to linearize a Lie group in the neighborhood of the identity element, thus reducing many questions about Lie groups to (more tractable) questions about Lie algebras. Unfortunately (at least for an algebraic geometer),
    the exponential map is not algebraic; it is given by an infinite series and thus cannot be defined in the setting of algebraic groups.
    \noindent The next best thing is to linearize the conjugation action of $G$ on itself in a Zariski neighborhood of the identity element. For special orthogonal groups $SO_n$ this is done by the classical Cayley map, which has been used in place of the exponential map in some applications. In the 1980s D. Luna asked which other simple algebraic groups admit a ``Cayley map". In this talk, I will discuss the background of this problem and a recent solution, obtained jointly with N. Lemire and V. L. Popov.

  • 11/05/04
    Zinovy Reichstein - University of British Columbia
    What can be solved in radicals?

    Galois theory tells us that that some polynomials \vskip .1in

    $f(x) = x^n + a_1 x^{n-1} + ... + a_{n-1} x + a_n$ \vskip .1in

    \noindent of degree $n > 4$ cannot be solved in radicals. Equivalently, some $S_n$-covers cannot be split by a solvable base extension. J. Tits asked whether an analogous assertion remains true if $S_n$ is replaced by a connected group $G$. In this talk I will discuss the background of this problem and recent results (obtained jointly with V. Chernousov and $P$. Gille) which indicate that solvability in radical may, indeed be possible in this setting. In particular, I will explain a connection we found between Tits' question and a variant Hilbert's 13th problem.

  • 11/06/04

  • 11/06/04

  • 11/09/04
    Bo Li - UCSD
    Variational properties of unbounded order parameters

    Order parameters such as the surface gradient in thin film growth can be unbounded as the size of an underlying system increases. Such unbounded order parameters can be modeled by a variational problem in which the effective free energy consists of a negative logarithmic function of the order parameter and a usual regularizing term. In this talk, we will first describe such a model for unbounded order parameters and compare it with a usual Ginzberg-Landau type model for domain walls. We will also give heuristic arguments to show why low energy configurations should have large value of the order parameter. Rigorous results will then be presented, and proved using the direct method in the calculus of variations. We will conclude the talk by a discussion on the related dynamics.

  • 11/09/04
    Paul Aspinwall - Duke University
    Superpotentials for D-branes and A-infinity algebras

    Worldvolumes of $D$-branes on Calabi-Yau threefolds give rise to
    supersymmetric gauge theories. If the $D$-brane is marginally stable
    these theories are quiver gauge theories. Using the derived category
    of coherent sheaves to analyze such $D$-branes, the superpotential of
    these gauge theories may be determined systematically.

  • 11/09/04
    Wee Teck Gan - UCSD
    Exceptional theta correspondences over finite fields

    If $G_1 x G_2$ is a subgroup of a finite group $H$, one can restrict a representation of $H$ to $G_1 x G_2$ and examine how it decomposes. One obtains in this way a function from irreducible representations of $G_1$ to (possibly reducible, possibly zero) representations of $G_2$. In certain cases, this function turns out to be very nice. We examine a particular case of this, when the groups involved are finite groups of Lie type. A large portion of the talk will be on recalling the classification of irreducible representations of such groups due to Deligne-Lusztig and Lusztig.

  • 11/09/04
    Sam Hsiao - University of Michigan
    Canonical characters on quasisymmetric functions and bivariate Catalan numbers

    In a recent preprint ``Combinatorial Hopf algebras and generalized
    Dehn-Sommerville relations"

    \noindent(math.CO/0310016), Aguiar, Bergeron, and Sottile set up a framework for studying combinatorial invariants encoded by quasisymmetric functions. They show that every character (i.e., multiplicative linear functional) on a graded connected Hopf algebra decomposes uniquely as a product of an even character and an odd character, both of which can in theory be computed from the even and odd parts of the ``universal character" on the Hopf algebra of quasisymmetric functions. \vskip .1in

    \noindent In my talk I will introduce some of these ideas and then go on to give explicit formulas for the even and odd parts of the universal
    character. They can be described in terms of Legendre's beta function evaluated at half-integers, or in terms of bivariate Catalan numbers:
    $$C(m,n)=(2m)!(2n)!/(m!(m+n)!n!).$$ I will explain how properties of characters and of quasisymmetric functions can be used to derive several interesting identities among bivariate Catalan numbers and in particular among Catalan numbers and central binomial coefficients. \vskip .1in

    \noindent This work is joint with Marcelo Aguiar.

  • 11/10/04
    Orest Bucicovschi - UCSD Graduate Student
    Prime Time

    I will talk about factorization of integers into primes, public key
    cryptography, and more. \vskip .1in

    \noindent Refreshments will be served!

  • 11/10/04
    Guofang Wei - UC Santa Barbara
    The covering spectrum of a compact length space

    We define a new spectrum for compact length spaces and Riemannian manifolds called the ``covering spectrum" which roughly measures the size of the one dimensional holes in the space. More specifically, the covering spectrum is a set of real numbers $\delta>0$ which identify the distinct $\delta$ covers of the space. We investigate the relationship between this covering spectrum, the length spectrum, the marked length spectrum and the Laplace spectrum. We analyze the behavior of the covering spectrum under Gromov-Hausdorff convergence. This is a joint work with C. Sormani.

  • 11/12/04
    Laurent Baratchart - INRIA
    Bounded extremal problems on the circle and Toeplitz operators

    We consider the problem of best approximating a given function in
    $L^2$ of a subset of the unit circle by the trace of an $H^2$ function whose norm on the complementary set is bounded by a prescribed constant. It is known that the solution can be obtained by solving a spectral equation for a certain Toeplitz operator. We show how diagonalization of such an operator ‡ la Rosenblum-Rovnyak allows to estimate the rate of convergence. We also consider the same problem in $L^\infty$ rather than $L^2$ norm, and present its solution that involves some unbounded Toeplitz operator.

  • 11/12/04
    Dr. Carsten Gundlach - University of Southampton, U.K.
    Towards well-posed initial-boundary value problems for numerical relativity

    I'll review how the stability of simulations in numerical relativity is related to having a well-posed continuum problem, and why well-posedness is not a property of the Einstein equations as such, but of the way in which they are formulated as an initial-boundary value (time evolution) problem. After reviewing the related concepts of well-posedness, strong hyperbolicity and symmetric hyperbolicity, I discuss applications to the Einstein equations, and in particular recent work. I conclude with an outlook on what strategies seem most promising to achieve well-posedness.

  • 11/12/04
    Ludmil Katzarkov - UC Irvine
    Uniformization theorem for smooth projective varieties

    Building on the work of Gromov and Schoen, Kollar and Simpson we
    will show that the universal covering of a smooth projective varieties
    with a linear fundamental groups is holomorphically convex. At the end we will connect this result with some conjectures of Zelmanov's on
    Burnside type of groups.

  • 11/15/04
    Daniel S. Rogalski - MIT
    Projectively simple rings

    Projectively simple rings are those positively graded rings which are as close to being simple as possible. We show that under suitable hypotheses, such rings can be described using constructions from algebraic geometry. This naturally leads to an interesting geometric question: which varieties have automorphisms which fix no proper subvarieties? This is joint work with James Zhang and Zinovy Reichstein.

  • 11/16/04

  • 11/16/04
    Peter Petersen - UCLA
    Eigenvalue pinching

    Joint Seminar with UCI

  • 11/16/04
    Hans Wenzl - UCSD
    On restriction rules from $Gl(N)$ to $O(N)$

    We show that the Grothendieck semiring of $O(N)$ is a quotient of the one of $O(\infty)$. Using this, we can express restriction multiplicities from $GL(N)$ to $O(N)$ via an alternating sum formula over the elements of a reflection group in terms of classical multiplicites due to Littlewood.

  • 11/16/04
    Rick Schoen - Stanford University
    On the isoperimetric inequality for minimal surfaces

    Joint Seminar with UCI

  • 11/17/04
    Mark Gross - UCSD
    Amoebas and tropical geometry

    Suppose one has a polynomial equation in two variables, $f(x,y)=0$. This defines a curve, but if we work over the complex numbers, it in fact
    defines a surface in $C^2$, which is difficult to visualize. I will discuss
    ways of visualizing these surfaces by introducing the concepts of amoebas, and their relationship to a new notion known as tropical geometry. \vskip .2in

    \noindent Refreshments will be served!

  • 11/18/04
    Nitu Kitchloo - UCSD
    Bott periodicity

  • 11/18/04
    Herbert Heyer - Tübingen University
    Transient convolution semigroups

    The existence of recurrent stochastic processes with stationary independent increments in an arbitrary locally compact group has remained a challenging research topic over a period of four decades. On the other hand, there exists an elaborate theory of transient processes in groups leading to interesting problems in potential theory. In the present expository talk, which is designed to be more or less self contained, the speaker will revisit the well-known Kesten-Spitzer transience criterion for processes in {\it abelian} locally compact groups, and will describe a purely measure theoretic approach to it.

  • 11/18/04
    Leanne Robertson - Smith College
    Class numbers of real cyclotomic fields

    The class numbers of the real cyclotomic fields ${\bf Q}(\cos(2\pi/{p^n}))$ are very difficult to compute. Indeed, they are not known for a single prime $p>67$. We analyze these class numbers using the Cohen-Lenstra heuristics on class groups and are led to make the following conjecture: For all but finitely many primes $p$, the class number of ${\bf Q}(\cos(2\pi/ {p^n}))$ is equal to the class number of ${\bf Q}(\cos(2\pi/ {p}))$ for all positive integers $n$. It is possible that there are no exceptional primes $p$ at all. Work in progress to test this conjecture empirically will also be discussed. This is joint work with Joe
    Buhler and Carl Pomerance.

  • 11/18/04
    Rick Schoen - Stanford University
    Current directions in mathematical general relativity

  • 11/22/04
    A. Volberg - Michigan State University
    Story of analytic capacity and problems of Painleve, Ahlfors and Vitushkin

    We will tell the story of the solution of several famous problems of Painlev´e, Ahlfors and Vitushkin. \vskip .1in

    \noindent Essentially, the theory of nonhomogeneous Calder´on-Zygmund (CZ) operators is the topic of the lecture. The main cornerstone of the theory of CZ operators turns out not to be a cornerstone at all. Namely, one can completely get rid of
    homogeneity of the underlying measure. The striking application of this theory is the solution of the series of problems of Painlev´e, Ahlfors and Vitushkin on the borderline of Harmonic Analysis and Geometric Measure Theory. \vskip .1in

    \noindent We will show how the ideas from nonhomogeneous CZ theory interplay with Tolsa’s ideas of capacity theory with Calder´on-Zygmund kernels to give Tolsa’s solution of the famous Vitushkin conjecture of semiadditivity of analytic capacity. We also show what changes should be maid if we want to grow the dimension and to prove the semiadditivity of Lipschitz harmonic capacity in $Rn, n > 2$, where the wonderful tool of Menger-Melnikov’s curvature extensively used in $2D$ is “cruelly missing”.

  • 11/24/04
    Jason Lee - UCSD Graduate Student
    Combinatorial Game Theory: Arithmetic with Nimbers

    A combinatorial game is a two-player game with perfect information and no chance elements. The players take turns making moves under clearly defined rules, and the last player able to make a move wins the game. NIM is one of the most well-known combinatorial games, and many people know that optimal play in Nim involves an addition operation which is just an XOR operation in disguise. Nim has a richer structure than this, however, as there is also a multiplication operation, and some unexpected and delightful things pop up such as Fermat powers (numbers of the form $2^{2^n}$). We'll explore nim fields -- finite fields of objects that aren't numbers, but NIMBERS, and really bizarre things happen -- even if I could convince you that 4 times 4 is 6, could you ever believe that the cube root of the nimber 2 is infinity? \vskip .1in

    \noindent Refreshments will be served!

  • 11/29/04
    Sergei Krutelevich - University of Ottawa
    Jordan algebras, exceptional groups and higher composition laws

    Higher composition laws in number theory were discovered by M.
    Bhargava several years ago. They may be viewed as a generalization of
    Gauss's law of composition of binary quadratic forms. M. Bhargava also
    discovered a mysterious connection between higher composition laws and exceptional Lie groups. \vskip .1in

    \noindent In our talk we will describe an unexpected relation between higher composition laws and the Freudenthal construction in the theory of Jordan algebras. We will show how this construction can be used to
    shed additional light on existing composition laws, as well as provide
    new examples of spaces with similar properties.

  • 11/30/04
    Allen Knutson - University of California at Berkeley
    Gluing Young tableaux into a ball

    Young tableaux, beloved of combinatorialists, tolerated by representation theorists and geometers, seem at first glance to be an
    unruly combinatorial set. I'll define a simplicial complex in which
    they index the facets, and slightly more general objects (Buch's
    ``set-valued tableaux'') label the other interior faces. The theorem that says we're on a right track: This simplicial complex is homeomorphic to a ball. I'll explain why this is surprising, useful, and shows why Buch didn't discover the exterior faces too. Finally, I'll explain how algebraic geometry forced these definitions on us (or, ``How I made my peace with Young tableaux''). This work is joint with Ezra Miller and Alex Yong.

  • 12/01/04
    Ronghui 'Lily' Xu - UCSD
    An Introduction to Biostatistics

    We first give an introduction to biostatistics, starting with what statistics is all about. We will elaborate on three areas of biostatistics: clinical trials, survival analysis and computational biology (bioinformatics). Time permitting, we will also briefly describe the biostatistics graduate programs around the country, and the excellent career opportunities open to those with a biostatistics degree.

  • 12/02/04
    Magdalena Musat - UCSD
    The condenser problem

    In classical potential theory on $R^n (n >= 3)$, the condenser theorem states the following. Let $U$ and $V$ be open subsets of $R^n$ with disjoint closures, where $U$ is relatively compact. There exists a Newtonian potential $p$ of a signed measure $mu = mu^+ - mu^-$ such that: \vskip .1in

    \noindent 1. $0 <= p <= 1$ \vskip .1 in

    \noindent 2. $p = 1 on U, p = 0 on V$ \vskip .1in

    \noindent 3. $mu^+$ is supported in the closure of $U$, $mu^-$ is supported in the closure of $V$ \vskip .1in

    \noindent The fundamental proof of this theorem was given by A. Beurling and J. Deny in the framework of Dirichlet spaces. Later K. L. Chung and R. K. Getoor studied the condenser problem in the probabilistic context of Markov processes. They showed that the condenser potential at a point x is simply the probability that Brownian motion starting at $x$ hits $U$ before it hits $V$. In this talk we discuss the condenser problem in the potential-theoretic framework of balayage spaces. We introduce the notion of a fine condenser potential, for which existence and uniqueness are proved for arbitrary superharmonic functions on sets $U, V$ which are not necessarily open. The probabilistic interpretation carries over to this setting by replacing the hitting times by the penetration times. These results are joint work with Jürgen Bliedtner.

  • 12/02/04
    Nitu Kitchloo - UCSD

  • 12/02/04
    Dr. Federico Ardila - Microsoft
    Bergman complexes and the space of phylogenetic trees

    Motivated by studying the amoeba of a system of polynomial equations, we associate to each matroid $M$ a polyhedral complex $B(M)$, called the ``Bergman complex". I will describe the topology and combinatorics of this complex. Somewhat surprisingly, the space of phylogenetic trees is (essentially) a Bergman complex, and we obtain some new results about it as a consequence. If $M$ is oriented, the Bergman complex $B(M)$ has a ``positive part" $B+(M)$, which I will also describe. \vskip .1in

    \noindent If time allows, I will show that for a Coxeter arrangement $A$, $B(A)$ is closely related to de Concini and Procesi's compactification of the complement of $A$, and $B+(A)$ is dual to a known Coxeter generalization of the associahedron. \vskip .1in

    \noindent My talk will assume no previous knowledge of matroids and arrangements. \vskip .1in

    \noindent Parts of this work are joint with Carly Klivans, Lauren Williams, and Vic Reiner.

  • 12/02/04
    Caleb Emmons - UCSD Graduate Student
    Rubin's conjecture in multiquadratic extensions

  • 12/02/04
    Dr. Melvin Leok - The University of Michigan
    Computational geometric mechanics and its applications to geometric control theory

    The geometric approach to mechanics serves as the theoretical underpinning of innovative control methodologies in geometric control theory. These techniques allow the attitude of satellites to be controlled using changes in its shape, as opposed to chemical propulsion, and are the basis for understanding the ability of a falling cat to always land on its feet, even when released in an inverted orientation. \vskip .1in

    \noindent Curiously, while the geometric structure of mechanical systems plays a critical role in the construction of geometric control algorithms, these algorithms have typically been implemented using numerical schemes that ignore the underlying geometry. \vskip .1in

    \noindent Geometric integration is the field of numerical analysis that focuses on developing geometric structure-preserving integrators, and computational geometric mechanics focuses on developing geometric integrators for dynamical systems arising from mechanics. \vskip .1in

    \noindent This talk will introduce some of the discrete differential geometric tools necessary to implement control algorithms in a manner that respects and preserves the geometry of the problem. These tools include discrete analogues of Lagrangian mechanics, the exterior calculus of differential forms, and connections on principal bundles. \vskip .1in

    \noindent This is joint work with Mathieu Desbrun (CS, Caltech), Anil Hirani (JPL), Jerrold Marsden (CDS, Caltech), Alan Weinstein (Math, Berkeley).

  • 12/07/04
    Dr. Balazs Szegedy - Microsoft
    Limits of dense graph sequences and reflection positivity

    We say that a sequence of dense graphs $G_n$ is convergent if for every fixed graph $F$ the density of copies of $F in G_n$ tends to a limit $f(F)$. Many theorems and conjectures in extremal graph theory can be formulated as inequalities for the possible values of the function $f$. We prove that every such inequality follows from the positive definiteness of the so-called connection matrices. Moreover we construct a natural limit
    object for the sequence $G_n$ namely a symmetric measurable function on the unit square. Along the line we introduce a rather general model of random graphs which seems to be interesting on its own right. \vskip .1in

    \noindent Joint work with L. Lovasz (Microsoft Research).

  • 12/09/04
    Jim Lin - UCSD
    Cohomology of loop spaces

  • 12/09/04
    Roland W. Freund - University of California, Davis
    Pade-type reduced-order modeling of higher-order systems

    A standard approach to reduced-order modeling of higher-order
    linear dynamical systems is to rewrite the system as an equivalent first-order system and then employ Krylov-subspace techniques for reduced-order modeling of first-order systems. While this approach results in reduced-order models that are optimal in a Pade sense, in general, these models do not preserve the form of the original higher-order system. \vskip .1in

    \noindent In this talk, we present a new approach to reduced-order modeling of higher-order systems based on projections onto suitably
    partitioned Krylov basis matrices that are obtained by applying Krylov-subspace techniques to an equivalent first-order system. We show that the resulting reduced-order models preserve the form of the original higher-order system. Moreover, possible additional properties such as passivity or reciprocity are also preserved. While the resulting reduced-order models are no longer optimal in the Pade sense, we show that they still satisfy a Pade-type approximation property. We also discuss some implementation details and present some numerical examples.