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#### 2010 Archive

##### Geometric superrigidity and harmonic maps

We discuss rank one and higher rank superrigidity for the isometry groups of a class of complexes which includes hyperbolic buildings as a special case. Our method uses harmonic maps to singular spaces.

##### Random Walker Rankings for BCS Football Teams

Division I (BCS) college football is the only major sport in the United States where a champion is named without a playoff system. Much of a team's fate in the BCS rests on its ranking, which is generated through a combination of computer and human polls. In this talk, a computer method of ranking teams using random walkers will be explored. A basis procedure will be presented, and then more complicated examples in which different factors (such as score differential) will be considered. Original work by Thomas Callaghan, Peter J. Mucha, and Mason A. Porter.

##### Counting boxes

I will give an introduction to 3-dimensional partitions
and their role in modern counting problems: ideal sheaves,
stable pairs, and holomorphic curves. Emphasis will be placed
on the elementary examples that led to the discovery of the
full structure.

##### Flexible Penalty Methods for Nonlinear Optimization

Penalty functions and filter methods are two commonly-used algorithms for solving nonlinear constrained problems. In practice they suffer from a number of disadvantages that slow their convergence. Flexible penalty functions are an attempt to utilize the best characteristics of both and have been demonstrated to exhibit good global convergence properties.

##### Finding patterns that do not contain many monochromatic constellations

Fix some set of points. Then a constellation in [n] is a scaled translated copy of the points. Given a 2-coloring of [n] there must be some monochromatic copies of any fixed constellation (assuming n is sufficiently large). In this paper we outline a method to experimentally find a block coloring of [n] that avoids many monochromatic copies of the constellation. We also show that for constellations with three points we can always beat random coloring.
(Joint work with Kevin Costello (Georgia Tech.) and Ron Graham
(UCSD).)

##### Pattern avoidance on partial permutations

Motivated by the concept of partial words, we introduce an analogous concept of partial permutations. A partial permutation of length $n$ with $k$ holes is a sequence of symbols $\pi=\pi_1\pi_2\dotsb\pi_n$ in which each of the symbols from the set $\{1,2,\dotsc,n-k\}$ appears exactly once, while the remaining $k$ symbols of $\pi$ are holes''.\\

We introduce pattern-avoidance in partial permutations and prove that most of the previous results on Wilf equivalence of permutation patterns can be extended to partial permutations with an arbitrary number of holes. We also show that Baxter permutations of a given length $k$ correspond to a Wilf-type equivalence class with respect to partial permutations with $(k-2)$ holes. Lastly, we enumerate the partial permutations of length $n$ with $k$ holes avoiding a given pattern of length at most four, for each $n\ge k\ge 1$. \\

This is joint work with Anders Claesson, V$\mathrm{\acute{i}}$t Jel$\mathrm{\acute{i}}$nek, and Eva Jel$\mathrm{\acute{i}}$nkov$\mathrm{\acute{a}}$.

##### Blending in mathematics

Mathematics is one of the richest, if more abstruse, areas of higher human cognition. It is a formal system, founded on a minimum of primitive concepts, but involving cognitive mechanisms, such as blending and framing, in an iterative manner, which lead to the rich structure of "higher" mathematics. The use of such cognitive mechanisms is done in a very controlled way, so as to maintain the rigor of the discipline. It is suggested that in mathematics blending and other such mechanisms are incorporated into the formal structure of the discipline. This thesis is examined via a number of examples. This has the effect that blends are easy to make in mathematics. On the other hand, before blends and other processes were incorporated into mathematics, some blends that are obvious, even necessary, in hindsight, have taken long times — sometimes centuries — to be realized. We hypothesize there is a cognitive cost to actualizing blends, which must be overcome. This phenomenon is investigated via the historical record. \\

\noindent Co-Sponsored by the Embodied Cognition Lab, Dept. of Cognitive Science, and the UCSD/SDSU Mathematics and Science Education joint doctoral program

##### A Characterization of The Standard Polynomial

The standard polynomial can be characterized by
its degree and its exponent
(the exponent of a polynomial is the exponent of the
T-ideal it generates).

##### The formation of singularities in the curve shortening flow.

I will talk about a self-contained proof of curve shortening flow in the plane which does not use the monotonicity formula or the classification of singularities based on the work of Andrews and Bryan.

##### The Hodge Conjecture

The Hodge conjecture claims that on a compact complex manifold X, the
cohomology class of every harmonic (p,p) form can represented as a
rational sum of the cohomology classes of subvarieties of X. This very
dense statement suggests a deep connection between the analytic tool of
forms and the geometric data of subvarieties on X. In this talk, we will
focus primarily on unraveling the full statement of the Hodge conjecture,
and discuss a few of the more accessible cases proven to date, including
the Lefschetz (1,1) theorem.

##### Proximal Support Vector Machine Classification

Instead of a standard support vector machine (SVM) that classifies points by assigning them to one of two disjoint halfspaces, points are classified by assigning them to the closest of two parallel planes (in input or feature space) that are pushed apart as far as possible. This formulation, which can also be interpreted as regularized least squares, leads to an extremely fast and simple algorithm for generating a linear or nonlinear classifier that merely requires the solution of a single system of linear equations. In contrast, standard SVMs solve a quadratic or a linear program that require considerably longer computational time. Computational results on publicly available datasets indicate that the proposed proximal SVM classifier has comparable test-set correctness to that of standard SVM classifiers, but with considerably faster computational time that can be an order of magnitude faster. The linear proximal SVM can easily handle large datasets as indicated by the classification of a 2-million point 10-attribute dataset in 20.8 seconds. All computational results are based on 6 lines of a MATLAB code.

##### Hall Littlewood Operators in Theory of Parking Functions and Diagonal Harmonics

This is the first of a series of lectures covering recent progress on
Haglund-Morse-Zabrocki leading to the compositional refinement of the
Shuffle conjecture. Then give a detailed presentation of the joint work
with Guoce Xin and Mike Zabrocki leading to the compositional
refinement of the q,t-Catalan and Schroeder results. These lectures will
cover the symmetric function side of the result. The combinatorial side
is work of Angela Hicks. Her results consist of two surprising
Parking function bijections refining the q,t-Catalan and Schroeder
recursion that will be presented by her sometime later this year to complete
the proof of these two special cases of the Haglund-Morse-Zabrocki
conjectures.

##### Thinking Inside the Box: Line Search Algorithms for Constrained Optimization

We discuss line search algorithms - algorithms used to figure out how far in a given direction to travel to best minimize a function - and give two major examples. From there, we look at how several current optimization routines adapt line searches to handle simple constraints. Then, we introduce a new method that combines the advantages of two different strategies and show its benefits using numerical results.

##### Frame Based Image Restoration

Efficient algorithms of image restoration and data recovery are derived by exploring sparse approximations of the underlying solutions by redundant systems such as wavelet frames and Gabor frames. Several algorithms and numerical simulation results for image restoration, compressed sensing, and matrix completion will be presented in this talk.

##### Can Complexity Theory Ratify the Invisible Hand of the Market''?

It is not from the benevolence of the butcher, the brewer, or the baker, that we expect our dinner, but from their regard for their own interest.'' Each participant in a competitive economy is led by an invisible hand to promote an end which was no part of his intention'' -- Adam Smith, 1776.\\

With his treatise, The Wealth of Nations, 1776, Adam Smith initiated the
field of economics, and his famous quote provided this field with its
central guiding principle. The pioneering work of Walras (1874) gave a
mathematical formulation for this statement, using his notion of market
equilibrium, and opened up the possibility of a formal ratification. \\

Mathematical ratification came with the celebrated Arrow-Debreu Theorem
(1954), which established existence of equilibrium in a very general model
of the economy; however, an efficient mechanism for finding an equilibrium
has remained elusive.
The question of algorithmic ratification was taken up in the earnest within
theoretical computer science a decade ago, and attention soon gravitated on
markets under piecewise-linear, concave utility functions. As it turned out,
the recent resolution of this open problem did not yield the hoped-for
mechanism; however, it did mark the end of the road for the current
approach. It is now time to step back and plan a fresh attack, using the
powerful tools of modern complexity theory and algorithms. After providing a summary of key developments through the ages and a gist of
the recent results, we will discuss some ways of moving forward.\\

(Based in part on recent work with Mihalis Yannakakis.)

##### On the Question of Global Regularity for Three-dimensional Navier-Stokes Equations and Relevant Geophysical Models

\noindent The basic problem faced in geophysical fluid dynamics is
that a mathematical description based only on fundamental physical
principles, the so-called the Primitive Equations'', is often
prohibitively expensive computationally, and hard to study
analytically. In this talk I will survey the main obstacles in
proving the global regularity for the three-dimensional
Navier-Stokes equations and their geophysical counterparts. Even
though the Primitive Equations look as if they are more difficult to
study analytically than the three-dimensional Navier-Stokes
equations I will show in this talk that they have a unique global
(in time) regular solution for all initial data.

\vskip .2in

\noindent Inspired by this work I will also provide a new global
regularity criterion for the three-dimensional Navier-Stokes
equations involving the pressure.

\vskip .2in

\noindent This is a joint work with Chongsheng Cao.

##### Hall Littlewood Operators in Theory of Parking Functions and Diagonal Harmonics II

This is the second of a series of lectures covering recent progress on
Haglund-Morse-Zabrocki leading to the compositional refinement of the
Shuffle conjecture. Then give a detailed presentation of the joint work
with Guoce Xin and Mike Zabrocki leading to the compositional refinement
of the q,t-Catalan and Schroeder results. These lectures will
cover the symmetric function side of the result. The combinatorial side
is work of Angela Hicks. Her results consist of two surprising
Parking function bijections refining the q,t-Catalan and Schroeder
recursion that will be presented by her to complete
the proof of these two special cases of the Haglund-Morse-Zabrocki
conjectures.

##### Millennium Problem Series: P vs NP

be easily verified are always easy to solve to begin with. We'll find out
how to ask this question formally, and discuss what makes this problem so
hard. We'll look at some of the problem's history, its present, and what
may lie ahead for P and NP.

##### Character sums for primitive root densities

It follows from the work of Artin (1927, 1958) and Hooley (1967) that,
under the assumption of the generalized Riemann hypothesis, every
non-square rational number r different from -1 is a primitive root
modulo infinitely many primes.
Moreover, the set of these primes has a natural density that can
be written as the product of a naive density' and a somewhat
complicated correction factor reflecting the entanglement of the
number fields that underly the density statement.

We show how the correction factors arising in Artin's original primitive
root problem and some of its generalizations can be interpreted as
character sums describing the nature of the entanglement.
The resulting description in terms of local contributions
is so transparent that it greatly facilitates explicit
computations, and naturally leads to non-vanishing criteria
for the correction factors.

##### Nonsmooth, Nonconvex Optimization

There are many algorithms for minimization when the objective function
is differentiable, convex, or has some other known structure,
but few options when none of the above hold, particularly when
the objective function is nonsmooth at minimizers, as is often
the case in applications. We describe two simple algorithms
for minimization of nonsmooth, nonconvex functions.
Gradient Sampling is a relatively new method that, although
computationally intensive, has a nice convergence theory.
The method is robust and the convergence theory has recently
been extended to constrained problems.
BFGS is an old method, developed for smooth problems, for which we have
very limited theoretical results, but some remarkable empirical observations,
extensive success in applications, and a rather bold conjecture.
Limited Memory BFGS is a popular extension for large problems,
and it too is applicable to the nonsmooth case, although our experience
with it is more mixed.

##### Laminations by Riemann surfaces

I will talk about recent joint work with Nessim Sibony and Erlend Wold on Rimann surface laminations.
A compact set is laminated by Riemann surfaces if locally the set is a disjoint union of complex discs which vary continuously.

##### Quantum integrability and Combinatorics: qKZ, ASM, TSSCPP

We shall review the recent progress, based on the use of the quantum
Knizhnik-Zamolodchikov equation, in the study of certain statistical
models of loops with remarkable combinatorial properties and which are
the subject of the Razumov-Stroganov conjecture. Then we shall present
some applications to enumerative combinatorics, settling in particular
some old conjectures on Alternating Sign Matrices and Plane Partitions.

##### Millennium Problem Series: Poincare Conjecture

If an n-manifold has the same homotopy type as the n-sphere, is it homeomorphic to the n-sphere? The Poincare Conjecture and its generalizations tell us yes! Come learn about a Millennium Problem that has been solved. We will discuss what the conjecture is actually saying, its rich history, and the idea behind Perelman's proof.

##### Some Applications of Geometry to Computer Graphics---Payback

For more than a dozen years I have been interested in Mathematical Visualization. This is the process of creating computer realizations of mathematical objects and then displaying them on a computer screen. For the most part, creating the computer avatar" of the mathematical object is the hard" and interesting part, and then displaying it comes relatively easily. But this is only because the field of Computer Graphics has developed many powerful and efficient algorithms that we can borrow and adapt for the display process. For the most part I have been a consumer", but along the way I have noticed several places where quite sophisticated concepts from geometry can markedly improve the algorithms currently used in computer graphics, and in this talk I will discuss two of these. The first is how most efficiently to use a mouse to rotate a three-dimensional object on a computer screen. The second is how to sprinkle a large number of points on a surface embedded in three-space. Here, sprinkle" means that the number of points in any region of the surface should be proportional to its area.

##### Enumeration in the rook monoid.

The rook monoid $R_n$ is the semigroup of 0/1 matrices of size
n with at most one 1 in each row and column. The subgroup of invertible
elements of $R_n$ is the symmetric group, and almost all questions about
permutations make sense for the rooks. In this talk, without assuming
any background in the subject, we
1. review some semigroup theoretic properties of $R_n$,
2. briefly explain the role of $R_n$ in the theory of algebraic
monoids,
3. present some recent combinatorial results on $R_n$.
In particular, we show that the celebrated numbers of mathematics
such as Eulerian numbers, Catalan numbers, Stirling numbers, etc.,
all appear rather naturally in enumeration in $R_n$.

##### Staircase tableaux and the asymmetric exclusion process

The ASEP is a model from statistical mechanics which describes a system of interacting particles hopping left and right on a one-dimensional lattice of n sites with open boundaries. In the bulk, the rate of hopping left is q times the rate of hopping right, and particles may enter and exit at both sides of the lattice at rates alpha, beta, gamma, and delta. We introduce some new tableaux (staircase tableaux) and use them to describe the stationary distribution of the ASEP with all parameters general. These tableaux seem to have very interesting combinatorial properties. For example, the staircase tableaux of size n have cardinality $4^n$ n!, and distinguished subsets of them have cardinality (2n-1)!!, (n+1)!, and $C_n$ (Catalan numbers). I'll close with applications to Askey-Wilson polynomials, and several open problems. This is joint work with Sylvie Corteel (and part of it is additionally joint with Dennis Stanton).

##### Landau-Ginzburg/Calabi-Yau correspondence

A far reaching correspondence from physics suggests that
the Gromov-Witten theory of a Calabi-Yau hypersurface of weight
projective space (more generally a toric variety) can be computed
by the singularity theory of its defining polynomial. In this talk,
I will present some of works (jointly with Alessandro Chiodo)
towards establishing this correspondence mathematically as well as
some of surprises and speculations.\\

Talk time runs to 10:30 AM.

##### SYZ for blowups and mirror symmetry for hypersurfaces in toric varieties

We will present joint work with Mohammed Abouzaid and Ludmil
Katzarkov, investigating mirror symmetry for blowups from the
perspective of the Strominger-Yau-Zaslow conjecture. Namely, we first
describe how to construct a Lagrangian torus fibration on the blowup of
a toric variety along a codimension 2 subvariety contained in a toric
hypersurface. Then we discuss the SYZ mirror and its instanton
corrections, to provide an explicit description of the mirror
Landau-Ginzburg model (up to higher order corrections to the
superpotential). This construction allows one to geometrically construct
mirrors of essentially arbitrary hypersurfaces in toric varieties. We
will focus on examples such as pairs of pants and curves of arbitrary genus.

##### Toward HMS for quotient surface singularities

Talk time runs to 4:30 PM.

##### Amoeba, coamoeba and non-commutative Donaldson-Thomas invariants

I this talk I will discuss non-commutative Donaldson-Thomas
invariants for a class of toric Calabi-Yau manifolds. The generating
function of these invariants is computed by a crystal melting model,
which is determined by the tropical limit'' of the coamoeba of the
Newton polynomial of the toric diagram. In the thermodynamic limit, the
limit shape of the crystal gives amoeba of the mirror Calabi-Yau
manifold. If time allows, I will also discuss the wall crossing of
Donaldson-Thomas invariants.\\

Talk time runs to 10:30 AM.

##### Mirror symmetry in three-dimensional topological field theory

Talk time runs to 4:30 PM.

##### Black holes, Donaldson-Thomas invariants and tropical geometry

Talk times runs until 10:30 AM.

##### Applications of Microlocalization

I will review several applications of
the microlocalization theorem relating the
Fukaya category of a cotangent
bundle to constructible sheaves on the base.
In combination with T-duality, this produces
equivalences between coherent sheaves and
constructible sheaves in a variety of settings.
To give a feel for this perspective, I
will focus on a few simple examples. \\

Talk time runs to 4:30 PM.

##### Lifting tropical curves in subvarieties

Talk time runs until 10:30 AM.

##### Non-generic tropical enumerative problems

Tropical geometry is useful not only in computing coarse
(symplectically invariant) enumerative invariants (such as
those coming from the Gromov-Witten theory), but also in
more general enumerative problems, in particular, those
involving special position of the constraints.\\

In the talk we'll look at those more general enumerative
problems and show how they may be solved tropically.

##### From $A_{\infty}$-pre-categories to $A_{\infty}$-categories

It is well known that "Fukaya category" is in fact an
$A_{\infty}$-pre-category in sense of Kontsevich and Soibelman.
The reason is that in general the morphism spaces are defined
only for transversal pairs of Lagrangians, and higher products
are defined only for transversal sequences of Lagrangians. It has been
conjectured by Kontsevich and Soibelman that for any graded commutative ring
$k,$ quasi-equivalence classes of $A_{\infty}$-pre-categories over $k$ are in
bijection with quasi-equivalence classes of $A_{\infty}$-categories over $k$ with
strict (or weak) identity morphisms.
In this talk I will sketch a proof of this conjecture for essentially small
$A_{\infty}$-(pre-)categories, in the case when $k$ is a field. In particular, it
follows that we can replace Fukaya $A_{\infty}$-pre-category with a quasi-equivalent
actual $A_{\infty}$-category.

##### A proof of Aldous' spectral gap conjecture

Aldous' spectral gap conjecture asserts that on any finite graph the random walk process and the random transposition process (which is also known as stirring process or interchange process) have the same spectral gap. I will give a proof of this conjecture using a recursive strategy. The approach is a natural extension of the method already used to prove the validity of the conjecture on trees. The novelty is an idea based on electric network reduction, which reduces the problem to the proof of an explicit inequality for a random transposition operator involving both positive and negative rates. The proof of the latter inequality uses suitable coset decompositions of the associated matrices on permutations. (Joint work with Tom Liggett and Pietro Caputo.)

##### Characterization of Solenoidal Groups

A topological group $G$ is said to be solenoidal if it contains a
dense one-parameter subgroup. That is, there exists a continuous
homomorphism from the real numbers into $G$ such that the image is
dense. We can obtain information about a topological group by
studying its character group, the set of all continuous homomorphisms
from $G$ into the circle group $T$. This is analogous to studying the
dual of a vector space in functional analysis. We show that a compact
group is solenoidal if and only if its character group is
topologically isomorphic with a subgroup of the real line with the
discrete topology. Along the way, we encounter the$a$-adic numbers,
the $a$-adic solenoid, and ultimately a corollary which tells us that
all compact solenoids can be expressed in terms of an uncountable
product of $a$-adic solenoids.

##### The Virtual Lung Project at UNC

This lecture will provide an overview of the Virtual Lung Project at UNC, which has been active for several years. The goal is to build predictive tools for medical applications, with a major focus on mucus transport. The Cystic Fibrosis Center at UNC is the main driver of the project, which grounds our basic science projects in chemistry, mathematics and physics to clinical practice. I will give an overview of basic facts about lung function and dysfunction, and the approaches we have undertaken, including experimental, theoretical and numerical. I will give some detail in the second half of the talk about projects in my research group related to characterization of mucus viscoelastic properties at biologically relevant length and force scales, and to diffusive transport of foreign particles in mucus layers.

##### Random regular graphs: from random walks to geometry and back.

The class of random regular graphs has been the focus of extensive study highlighting its excellent expansion properties. For instance, it is well known that almost every regular graph of fixed degree is essentially Ramanujan, and understanding this class of graphs sheds light on the general behavior of expanders. In this talk we will present several recent results on random regular graphs, focusing on the interplay between the spectrum, geometry and behavior of the simple random walk in these graphs.
We will first discuss the relation between spectral properties and the abrupt convergence of the random walk to equilibrium, derived from precise asymptotics of the number of paths between vertices. Following the study of the graph geometry we proceed to its random perturbation via exponential weights on the edges (first-passage-percolation). We then show how this allows the derivation of various key features of the classical Erd\H{o}s-R\'enyi random graph near criticality, such as the asymptotics of the diameter of the largest component and the mixing time of the random walk on it.
Based on joint works with Jian Ding, Jeong Han Kim, Yuval Peres and Allan Sly.

##### Ocean color remote sensing

The objective of ocean color remote sensing is to retrieve the values of certain geophysical parameters, either atmospheric or oceanic, from satelite measurements of the upwelling radiation field. These include the phytoplankton type and concentration, sediments, yellow substances, aerosol type and vertical distribution, for instance. The subject therefore refers to a variety of inverse problems. In this talk, we present some statistical methodologies adapted to these problems. \\

This is a joint work with Robert Frouin (Scripps Institution of Oceanography, UC San Diego).

##### The detectability lemma: making sense of the notion of quantum constraint satisfaction

The quantum analogue of a constraint satisfaction problem is a sum of local Hamiltonians---each local Hamiltonian specifies a local constraint whose violation contributes to the energy of the given quantum state. Formalizing the intuitive connection between the ground (minimal) energy of the Hamiltonian and the minimum number of violated constraints is problematic, since the number of constraints being violated is not well defined when the terms in the Hamiltonian do not commute. This presentation will indicate how to make this connection and explain the relationship to the quest for a quantum PCP result.

##### Real analytic domains in $\mathbb C^3$

We will present some new results on pseudoconvex domains with real analytic boundaries in $\mathbb C^3$. A first step towards solving d-bar and the peak point problem is to find a pseudoconvex bumping of the same order as the type. We will give some new results, discuss work in progress and also talk about some open problems.

##### Experimental evidence for the occurrence of E(8) in nature and the radii of the Gossett circles

A recent experimental discovery involving the spin structure of electrons in
a cold one dimensional magnet points to a model involving the exceptional
Lie group E(8). The model predicts 8 particles the ratio of whose masses are
the same as the ratios of the radii of the circles in the famous Gossett
diagram (going back to 1900) of what is now understood to be a 2 dimensional
projection of the 240 roots of E(8) arranged in 8 concentric circles. The
ratio of the radii of the two smallest circles (read 2 smallest masses) is
the golden number. This beautifully has been found experimentally. The ratio
of the radii of the other masses has been written down conjecturally by
Zamolodchikov. This again agrees with the analogous statement for the radii
of the Gossett circles.

Some time ago we found an operator A (very easily defined and reexpressed
by Vogan as an element of the group algebra of the Weyl group) on 8-space
whose spectrum is exactly the squares of the radii of the Gossett circles.

The operator A is written in terms of the coefficients $n_i$ of the highest
root. In McKay theory the $n_i$ are the dimensions of the irreducible
representations of the binary icosahedral group. Our result works for any
simple Lie group not just E(8).

##### $Sl_2$-Tilings

Call SL2-tiling a filling of the discrete plane by elements of a
ring (the coefficients) in such a way that each connected 2 by 2 submatrix has
determinant 1. Similar objects have been studied by Coxeter and Conway; they
call them frieze-patterns. Given a bi-infinite word on {x, y}, interpreted as a path in the discrete plane, called the frontier, put 1s at its vertices. Then one may uniquely complete
this picture to an SL2-tiling; it turns out that the coefficients of the tiling are
all positive integers; we prove this by giving explicit matrix product formulas for these
coefficients. Our constructions are motivated by the so-called "frises",
associated to acyclic digraphs. In a joint work with I. Assem and D. Smith, we showed that the sequences of the frise all satisfy a linear recursion if and only if the digraph is a Dynkin diagram, or an affine diagram, with an acyclic orientation.

##### Some early developments around the Dirichlet problem.

The Dirichlet problem states: Given a domain and a
continuous function f on its boundary, find a harmonic function u in
the domain and continuous on the closure of the domain such that u=f
on the boundary. I will give some highlights in the early
developments around it. In one direction, motivation from physics and
electrostatics, the work by Green, Neumann and Fredholm and the
general solution by Poincare and Perron will be talked about. In
another direction, we will mention the Dirichlet principle and its
relation with Dirichlet problem. It will be a very informal talk from
a historical perspective and no background will be assumed.

##### Water: Active or Passive Player in Cavity-Ligand Recognition & Binding?

We use a straightforward approach based on potential of mean force calculations from explicit
solvent molecular dynamics simulations to derive complete thermodynamic signatures of
cavity-ligand binding. Using extensive sampling, free energy, enthalpy, and
entropy estimates are obtained along the binding coordinate for a set of
systems with varying cavity and ligand physicochemical properties, thus revealing an
unprecedented, nanoscale picture of hydration thermodynamics.
Despite the simplicity of the model systems considered, a broad range of thermodynamic signatures
is found in which water (rather then cavity or ligand) contributions appear to
largely determine the overall system enthalpy-entropy compensation.

##### "Explicit Formulae for Zeta Functions and L-Functions of Graphs

We will briefly introduce graph theory Zeta and
L-functions. We will then proceed to develop an explicit formula for
both, similar to those appearing classically. Along the way, there
will be a discussion of applications and the Riemann Hypothesis (or
lack thereof) for graphs.

##### A self-normalized approach to statistical inference for time series

In the inference of time series (e.g. hypothesis testing and confidence interval construction), one often needs to obtain a consistent estimate for the asymptotic covariance matrix of a statistic. Or the inference can be conducted by using resampling (e.g. moving block bootstrap) and subsampling techniques. What is common for almost all the existing methods is that they involve the selection of a smoothing parameter. Some rules have been proposed to choose the smoothing parameter, but they may involve another user-chosen number, or assume a parametric model. In this talk, we introduce the so-called self-normalized (SN) approach in the context of confidence interval construction and change point detection. The self-normalized statistic does not involve any smoothing parameter and its limiting distribution is nuisance parameter free. The finite sample performance of the SN approach is evaluated in simulated and real data examples.

##### Towards universal semantic communication

\small Is it possible for two intelligent players to communicate
meaningfully with each other, without any prior common background? What
does it even mean for the two players to understand each other? In
addition to being an intriguing question in its own right, we argue that
this question also goes to the heart of modern communication
infrastructures, where misunderstandings (mismatches in protocols)
between communicating players are a major source of errors. We believe
that questions like this need to be answered to set the foundations for a
robust theory of (meaningful) communication.\\
In this talk, I will describe what computational complexity has to say
about such interactions. Most of the talk will focus on how some of the
nebulous notions, such as intelligence and understanding, should be
defined in concrete settings. We assert that in order to communicate
"successfully," the communicating players should be explicit about their
goals - what the communication should achieve. We show examples that
illustrate that when goals are explicit the communicating players can
achieve meaningful communication.\\

Based on joint works with Oded Goldreich (Weizmann) and Brendan Juba (MIT). \\

Lunch will be served following the seminar. Your RSVP must be received
by Mon., Feb. 22nd (for catering purposes), aallen@ece.ucsd.edu

##### Modifying Finite Elements to use the Computational Efficiency of Diagonally Implicit Runge-Kutta Methods

First I will show how full space time finite elements is
similar to solving via fully implicit Runge Kutta methods. Next I will
propose a modified finite element method that uses diagonally implicit
Runge Kutta methods. I would like to prove that this new method retains
the advantages of space time finite elements while reducing the
computational complexity. Simulation results with error estimates will
be shown that support this claim.

##### Stanley Depth and Interval Partitions

In 1982 Stanley developed a geometric/combinatorial invariant
for a finitely generated $\mathbb{Z}^n$-graded module $M$ over a
polynomial ring in $n$ variables and conjectured that this quantity was
an upper bound on the depth of $M$. Since then relatively little
progress has been made in resolving Stanley's conjecture, in part
because of the difficulty of calculating the Stanley depth. Recently,
Herzog, Vladoiu, and Zheng provided an algorithm for calculating the
Stanley depth of the quotient of monomial ideas over a polynomial ring.
We use their result to resolve the Stanley depth of the maximal
square-free monomial ideal and end up with a previously unknown property
of the boolean lattice. We extend these ideas to partially resolve the
case of square-free Veronese ideals. \\

This is joint work with Csaba Bir$\mathrm{\acute{o}}$, Dave Howard, Mitch Keller, Noah
Streib, Yi-Huang Shen, and Tom Trotter.

##### *Honk* The Riemann Hypothesis

I will attempt to partially answer some of the following questions: What is the Riemann zeta function and why is it important? What do zeros of zeta have to do with prime numbers? What is the Riemann hypothesis? Why would we like it to be true? I will give some history along the way, including a discussion of Riemann's epoch-making memoir \emph{On the number of primes less than a given magnitude}. No background is necessary.

##### Bootstrap for goodness of fit statistics

Motivation for this topic comes from a problem in
X-ray astronomy. Nonparametric goodness of fit statistics
are generally based on the empirical distribution function.
The distribution free property of these test statistics do
not hold in the multivariate case or when some parameters
are estimated. Bootstrap methods to estimate the null
distributions in such cases will be presented. The results
hold not only in the univariate case but also under the
multivariate setting. These ideas are taken a step
further to develop resampling methods for inference,
when the data is from an unknown distribution which may or
may not belong to a specified family of distributions.

##### State and Parameter Estimation in Models of Nonlinear Systems

The problem of using observed information in
estimating unobserved state variables and fixed
parameters in a model of a nonlinear dynamical system
when the measurements are noisy, the model has errors,
and the state of the system as measurements begin is
uncertain is cast as an exact path integral. In this
formulation the measurements are seen as a guiding
"potential" directing a chaotic system to the correct
region of phase space. The path integral is evaluated
directly for small systems such as the Lorenz 1996
model. In the measurement window, there are indications
that the distribution of state variables is nearly
Gaussian. The properties of the path integral can yield
an approximation to the number of required
measurements.

A saddle point evaluation of the path integral is seen
to be 4DVAR, and implementing this in the case where
the dynamics is error free?--i.e. deterministic?--is
shown to require regularization to achieve a smooth
surface on which the estimation search can proceed. The
number of observations required to regularize the
search can be estimated from this requirement.

The regularized 4DVAR method is applied to experiments
on a small nonlinear circuit as well as to the Lorenz
1996 model.

##### On nonlinear wave equations

We will begin discussing the paper Strichartz estimates on schwarzschild black hole backgrounds'' by Tataru, et. al.

##### Linear and Non-Linear Extremal Problems in Analytic Function Spaces

In this talk, I will give a survey of some extremal problems in various analytic function spaces. I will start by discussing the duality relationships that were first applied to such problems by S. Ya. Khavinson in the east and simultaneously by Rogosinski and Shapiro in the west in the late 1940s and early 1950s. I will then show how one can use this duality to establish the generic form of extremal solutions in both Hardy and Bergman spaces. I will discuss some explicit solutions in these spaces as well as in Fisher spaces and will show why the search for explicit solutions naturally leads to non-linear problems, for non-vanishing functions. Many open problems still exist in this area, the most famous of which is the Krzyz conjecture. Finally, I will discuss some current joint ongoing work with D. Khavinson on a specific problem in Bergman spaces.

##### Malmhedens theorem revisited

In 1934 Harry Malmheden discovered an elegant geometric algorithm for solving the Dirichlet
problem in a ball. Although his result was rediscovered independently by Duffin 23 years later,
it still does not seem to be widely known. In this talk we return to Malmheden’s theorem, give
an alternative proof of the result that allows generalization to polyharmonic functions and, also,
discuss applications of his theorem to geometric properties of harmonic measures in balls in $R^n$. \\

Joint work with M. Agranovsky and H. S. Shapiro.

##### Cohomology reveals when helicity is a diffeomorphism invariant

Using knot invariants as our guide, we seek to understand vector field invariants. One vector field invariant is helicity, which calculates the average linking number of the field's flowlines. Computed analogously to Gauss' linking integral, it is widely useful in the physics of fluids. Helicity is invariant under certain diffeomorphisms of its domain - we seek to understand which ones.\\

By integrating over the configuration space of 2n points on a circle, Bott and Taubes computed finite-type invariants of knots. By analogy, we realize helicity as an integral over the configuration space of 2 points on a domain in Euclidean space. We extend this framework to differential $(k+1)$-forms on domains $R^{2k+1}$ and express helicity as a cohomology class. This topological approach allows us to find a general formula for how much helicity changes when the form is pushed forward by a diffeomorphism of the domain. We classify the helicity-preserving diffeomorphisms on a given domain, finding new ones on the two-holed solid torus and proving that there are no new ones on the standard solid torus. \\

This is joint work with Jason Cantarella.

##### Millennium Problem Series: Navier-Stokes Equations

I will talk about fluids. Personally, I've never seen a
fluid whose velocity blows up in finite time, and I'm quite certain I
never will. However, somebody is offering a million dollars to show
this (or prove me wrong) for fluids which move according to the
Navier-Stokes Equations. What's even worse is such fluids most likely
don't exist! Why? You'll have to come to the talk for that.

##### $n$-groups

In this talk, we give a brief introduction to a natural generalization of groups, called n-groups. Just as discrete groups represent the homotopy types of acyclic spaces, n-groups realize homotopy types of connected topological spaces X such that $\pi_i(X)=0$ for $i>n$. In this talk, we adopt the formalism of simplicial sets, and define n-groups as simplicial sets satisfying certain a filling condition (introduced by Duskin).\\

In the first part of the talk, we explain what a 2-group look like: this material is contained in any textbook on simplicial sets. We indicate how 2-groups arise in topological quantum field theory.\\

In the second part of the talk, we explain a generalization of Lie theory to n-groups, in which the role of Lie algebras is taken by differential graded Lie algebras, and the role of the ordinary differential equations underlying Lie theory is taken by the Maurer-Cartan equation for flat superconnections on simplices.

##### K-theory, topological insulators, and quantum computation

This informal blackboard talk will describe some recent connections between geometric K-theory and condensed matter physics. The intended end point is the a design" for a topological quantum computer posted on the physics arXiv last week: arXiv:1003.2856.

##### On a conjecture of Brouwer regarding the connectivity of strongly regular graphs

A $(v,k,\lambda,\mu)$-strongly regular graph (SRG for short) is a finite undirected graph without loops or multiple edges such that (i) it has $v$ vertices, (ii) it is regular of degree $k$, (iii) each edge is in $\lambda$ triangles, (iv) any two nonadjacent points are joined by $\mu$ paths of length 2. The connectivity of a graph is the minimum number of vertices one has to remove in order to make it disconnected (or empty).
In 1985, Brouwer and Mesner used Seidel's characterization of strongly regular graphs with eigenvalues at least $-2$ to prove that the vertex-connectivity of any $(v,k,\lambda,\mu)$-SRG equals its degree $k$. Also, they proved that the only disconnecting sets of size $k$ are the neighborhoods $N(x)$ of a vertex $x$ of the graph.
A natural question is what the minimum number of vertices whose removal will disconnect a $(v,k,\lambda,\mu)$-SRG into non-singleton components. In 1996, Brouwer conjectured that this number is $2k-\lambda-2$. In this talk, I will report some progress on this problem.
This is joint work with Kijung Kim and Jack Koolen (POSTECH, South Korea).

##### Inverse Littlewood-Offord theory, Smooth Analysis and the Circular Law

A corner stone of the theory of random matrices is Wigner's semi-circle law,
obtained in the 1950s, which asserts that (after a proper normalization) the
limiting distribution of the spectra of a random hermitian matrix with iid
(upper diagonal) entries follows the semi-circle law. The non-hermitian case is
the famous Circular Law Conjecture, which asserts that (after a proper normalization)
the limiting distribution of the spectra of a random matrix with iid entries is
uniform in the unit circle.\\

Despite several partial results (Ginibre-Mehta, Girko, Bai, Edelman,
Gotze-Tykhomirov, Pan-Zhu etc) the conjecture remained open for more than
50 years. In 2008, T. Tao and I confirmed the conjecture in full generality. I am going to
give an overview of this proof, which relies on rather surprising connections
between various fields: combinatorics, probability and theoretical
computer science.

##### Resilience of Random Graphs

Let $G$ be a graph having property $\mathcal{P}$. We ask the
question, How strongly does $G$ possess $\mathcal{P}$?''. One way
to answer this questions is by computing the number of edges one
must delete from $G$ to obtain a graph not having property
$\mathcal{P}$. We call this quantity the \textit{resilience} of $G$
with respect to having $\mathcal{P}$. Many classical results such
as, Tur\'{a}n's Theorem, Dirac's Theorem and Bondy's Theorem can be
understood in the context of resilience of a complete graph.
Recently, Sudakov and Vu have initiated the systematic study of the
resilience of random graphs by computing the resilience of it with
respect to properties such as hamiltonicity, chromatic number,
perfect matching and symmetry. Since then, several interesting
results have been obtained by various researchers.

We will first take a survey on the developments in this area by
looking at these results and mentioning related open problems. Then
we will discuss in more depth, the resilience of random graphs with
respect to two specific properties, pancyclicity and containing an
$H$-packing. (A graph $G$ is called pancyclic if it contains a cycle
of all possible lengths. And for a fixed graph $H$, $G$ is said to
have an $H$-packing if there exists vertex disjoint copies of $H$
covering all the vertices of $G$.)

Joint work with Hao Huang, Michael Krivelevich, Wojceich Samotij,
and Benny Sudakov

##### Millennium Problem Series: The Birch and Swinnerton-Dyer conjecture

If $E$ is an elliptic curve defined over a number field $K$, then it has been known for some time now that the group of $K$-rational points on $E$ is a finitely generated abelian group. The BSD, in its weak form, offers a conjectural link between the rank of $E(K)$ and the order of vanishing of a certain complex analytic L-function associated to $E$. The strong form of the BSD goes further and links the leading term of this L-function to certain algebraic invariants associated to $E$. In this talk I will sketch enough of the definitions, etc. to make this link more precise. Despite all the fancy words, this talk should be accessible to everyone.

##### Functional Clustering For Time-Course Gene Expression Experiments Using ARMA(p,q)

Functional gene clustering is a statistical approach for identifying the
temporal patterns of gene expression measured at a series of time points. I
will discuss two approaches for functional clustering, one designed to estimate
periodic transcriptional profiles using Fourier series, and the second relying
on wavelets for dimension reduction. The covariance matrix of the serial
measurements over time for each gene is flexibly modeled through an
autoregressive moving-average process of order (p, q). An EM algorithm is used
to estimate the unknown parameters, and the model is chosen through a selection
criterion. The methods are shown to be effective in simulation studies and on
recent yeast data.

##### Real-Time Auto-Probing for Breast Cancer Detection by Magnetic Resonance Imaging

This project aims to develop a breast cancer detection system with the invention of auto-probing methodology to localize and visualize the region of cancerous lesion. This system can reduce the analysis time and enables more thorough examination for breast screening. The lesion will be coloured and exposed out on the MR images for better identification and verification of the lesion's characteristics. The region of interest selection enables the system to generate a graph which contains clearer details about the lesion. Furthermore, higher sensitivity and specificity is expected to be achieved with this system in comparison to the current practice where random probing is applied.

##### Mathematical approaches to modeling cancer treatments

What can mathematics tell us about the treatment of cancer? In this talk I will present some of work that I have done in the modeling of tumor growth and treatment over the last ten years.

Cancer is a myriad of individual diseases, with the common feature that an individual's own cells have become malignant. Thus, the treatment of cancer poses great challenges, since an attack must be mounted against cells that are nearly identical to normal cells. Mathematical models that describe tumor growth in tissue, the immune response, and the administration of different therapies can suggest treatment strategies that optimize treatment efficacy and minimize negative side-effects. However, the inherent complexity of the immune system and the spatial heterogeneity of human tissue gives rise to mathematical models that pose unique analytical and numerical challenges. In this talk I will briefly discuss two mathematical problems that we have encountered in our work: optimization of systems of delay differential equations, and the analysis of spatial models that incorporate different time scales.

No knowledge of biology will be assumed.

##### Almost spanning trees in Sparse Graph

We prove that sparse (pseudo-) random graphs contain every almost spanning bounded degree trees. We actually prove a stronger result, a local resilience type one, that the above statement holds even if from each vertex of a sparse pseudo-random graph half of the edges is removed.
The proof uses Szemeredi's Regularity Lemma for sparse graphs, and modifies a theorem of Friedman and Peppenger. It is joint work with B. Csaba, M. Pei and W. Samotij.

##### The Galois module structure of $\ell$--adic realizations of Picard $1$-motives and applications

I will report on my recent joint work with Greither
on the Galois module structure of $\ell$-adic realizations of Picard
$1$-motives in positive characteristic and their Iwasawa theoretic
analogues in characteristic 0. In the process, I will state and
sketch the proof of an equivariant Main Conjecture in Iwasawa theory.
Finally, I will discuss applications of the equivariant Main Conjecture
ranging from proofs of refinements of the Brumer-Stark and Coates-Sinnott
Conjectures to proofs of particular cases of the Equivariant Tamagawa
Number Conjecture of Bloch-Kato and the non-abelian Main Conjecture
of Coates-Fukaya-Kato-Sujatha-Venjakob. This is a two hour lecture, the first half emphasizing the more geometric aspects of our work, while
the second will focus on the number theoretic aspects and applications.

Talk time runs until 4:00 PM.

##### Solution of Kac's problem for analytic plane domains with a symmetry

Kac's hear the shape of a drum" problem is the extent to which a plane domain
is determined by its Dirichlet eigenvalues. I.e. is the map from domains to their spectra 1-1.
We show that if the spectrum map is restricted to analytic plane domains with one up down
symmetry (and an axis length fixed), then it is one-one. I.e. you can determine such a domain
from its eigenvalues among other such domains. In joint work with Hamid Hezari,
we also give a generalization to higher dimensions.

##### Support Vector Machine Classification with Indefinite Kernels

Support Vector Machines (SVMs) are binary classifiers that have
been used in a wide variety of applications. One of the
qualities that make SVMs popular is their ability to utilize
a diverse class of similarity measures (kernel functions),
which makes them sufficiently flexible to handle many
different classification problems. This talk will offer an
introduction to both linear and nonlinear SVMs, and discuss
techniques that allow for the use of kernels that do not
satisfy Mercer's condition, primarily the method proposed by
Luss and d'Aspermont in their paper of the same name. No
knowledge of SVMs is assumed.

##### Tutte polynomial for toric arrangements

A toric arrangement is a finite family of hypersurfaces in a
torus, every hypersurface being the kernel of a character. We describe
some properties of such arrangements, by comparing them with hyperplane
arrangements. The Tutte polinomial is an invariant which encodes a rich
description of the topology and the combinatorics of a hyperplane
arrangement, and satisfies a simple recurrence. We introduce the
analogue of this polynomial for a toric arrangement. Furthermore, we
show that our polynomial computes the volume of the related zonotope,
counts its integral points, and provides the graded dimension of a space
of quasipolynomials introduced by Dahmen and Micchelli to study
partition functions.

##### On the asymptotic size of testing procedures I

The talk is about tests and confidence intervals based on a test
statistic that has a limit distribution that is discontinuous in a
nuisance parameter or the parameter of interest. It is shown that standard
fixed critical value (FCV) tests and subsample tests often have asymptotic
size - defined as the limit of the finite sample size - that is greater
than the nominal level of the test. A precise formula for the asymptotic
size of such tests is provided under a general set of high-level
conditions that are relatively easy to verify. The asymptotic size is
determined by a sequence of parameter values that approach the point of
discontinuity of the asymptotic distribution. The problem is not a small
sample problem. For every sample size, there can be parameter values for
which the test over-rejects the null hypothesis. Analogous results hold
for confidence intervals.
The talk also covers a hybrid subsample/FCV test that alleviates the
problem of over-rejection asymptotically and in some cases eliminates
it. In addition, size-corrections to the FCV, subsample, and hybrid
tests are discussed that eliminate over-rejection asymptotically.
Many examples will be given.\\

Talk time runs until 3:00 PM.

##### Towards an l-modular Jacquet-Langlands correspondence

Let F be a non-Archimedean locally compact field of residue characteristic p,
and let G be an inner form of GL(n,F), that is a group of the form GL(m,D) where D is a
division algebra of centre F. Given R an algebraically closed field of characteristic different from p, I will explain how to classify the irreducible smooth representations of G with coefficients in R, in terms of parameters involving the supercuspidal representations of the Levi subgroups of G. We will discuss about the possibility of having a Jacquet-Langlands correspondence modulo $l$. This is a joint work with Vincent S$\mathrm{\acute{e}}$cherre.

##### On the asymptotic size of testing procedures II

The talk is about tests and confidence intervals based on a test
statistic that has a limit distribution that is discontinuous in a
nuisance parameter or the parameter of interest. It is shown that standard
fixed critical value (FCV) tests and subsample tests often have asymptotic
size - defined as the limit of the finite sample size - that is greater
than the nominal level of the test. A precise formula for the asymptotic
size of such tests is provided under a general set of high-level
conditions that are relatively easy to verify. The asymptotic size is
determined by a sequence of parameter values that approach the point of
discontinuity of the asymptotic distribution. The problem is not a small
sample problem. For every sample size, there can be parameter values for
which the test over-rejects the null hypothesis. Analogous results hold
for confidence intervals.
The talk also covers a hybrid subsample/FCV test that alleviates the
problem of over-rejection asymptotically and in some cases eliminates
it. In addition, size-corrections to the FCV, subsample, and hybrid
tests are discussed that eliminate over-rejection asymptotically.
Many examples will be given. \\

Talk time runs until 3:30 PM.

##### From Inadmissibility to Effective Regularization

Charles Stein (1956) discovered that, under quadratic loss, the usual
unbiased estimator for the mean vector of a multivariate normal
distribution is inadmissible if the dimension $n$ of the mean vector
exceeds two. Contemporaries claimed that Stein's results and the subsequent
James-Stein estimator are counter-intuitive, even paradoxical, and not very
useful. This talk reexamines such assertions in the light of arguments
presented, sketched, or foreshadowed in Stein's beautifully written 1956
paper. Among often overlooked aspects of the paper are the asymptotic
geometry of quadratic loss in high dimensions that makes Stein estimation
transparent; asymptotic optimality results associated with Stein
estimators; the explicit mention of practical multiple shrinkage
estimators; and the foreshadowing of confidence balls centered at Stein
estimators. Implications of these ideas underlie effective modern
regularization estimators, among them, penalized least squares estimators
with multiple quadratic penalties, running weighted means, nested submodel
fits, and more.

##### Some statistical procedures for boundary estimation and image analysis

The problem of estimating a set S from a random sample of points arises in connection with some applications in statistical quality control, clustering, image analysis and statistical learning. This problem can be established in a more formal way as the problem of estimating the support of an absolutely continuous probability measure P from n independent observations drawn from P. So, the goal here is to estimate a set, not a parameter or a function.

Assuming that the set of interest belongs to a certain family of sets can be useful in order to find an efficient estimation method. The case where S is assumed to be convex has received a special attention. If we assume that S is the support of the distribution which generates the sample points, there is a quite obvious estimator: the convex hull of the sample, that is, the smallest convex set which contains the sample. However, if S is not convex, the convex hull of the sample can be a bad choice. In this talk support estimation under the assumption that the set satisfies a much more flexible shape restriction, which is named alpha-convexity, will be presented. It will be showed that the new estimator can achieve, in a much more general setting, the same convergence rate as the convex hull.

Support estimation is also connected to another interesting problem: the estimation of certain geometric characteristics of the set such as the volume or the surface area. It seems natural to think that the volume or the surface area of a good support estimator should provide good approximations of these geometrical quantities. Here we analyze the problem of boundary length estimation when it is assumed that the set is alpha-convex.

Joint work with Beatriz Pateiro (Universidad de Santiago de Compostela), Antonio Cuevas (Universidad Autónoma de Madrid) and Ricardo Fraiman (Universidad de San Andres).

##### Extremal Graph Theory and its applications

In typical extremal problem one wants to determine maximum cardinality of
discrete structure with certain prescribed properties. Probably the
earliest such result was obtain 100 years ago by Mantel who computed the
maximum number of edges in a triangle free graph on n vertices. This was
generalized by Turan for all complete graphs and became a starting point
of Extremal Graph Theory. In this talk we survey several classical
problems and results in this area and present some interesting
applications of Extremal Graph Theory to other areas of mathematics. We
also describe a recent surprising generalization of Turan's theorem which
was motivated by question in Computational Complexity.

##### Noncommutative Partial Differential Equations

This talk classifies all harmonic noncommutative polynomials, as well as all polynomial solutions to other selected noncommutative partial differential equations. The directional derivative of a noncommutative polynomial in the direction $h$ is defined as $D[p(x_1,\ldots,x_g),x_i,h]:= \frac{d}{dt}[p(x_1, \ldots, (x_i+th), \ldots, x_g)]_{|_{t=0}}$. From this noncommutative derivative, one may define differential equations which take as solutions polynomials in free variables. A noncommutative harmonic polynomial is a polynomial such that its noncommutative Laplacian is zero.

##### Introduction to LaTeX Typesetting

I will be giving a hands-on class on LaTeX typesetting today in AP&M 6402, from 1:30 to 3:00 PM. The material covered is appropriate for beginners and those with intermediate knowledge of the language. Individual questions are
welcome and will be answered as time permits.

A second class covering more advanced topics is in the works and will be announced shortly.

##### Mathematical Finance

Ever since Fischer Black and Myron Scholes first determined a successful
way to price options in 1973, Wall Street has increasingly been interested
in using mathematics to price derivatives and hedge risk. Mathematical
finance has since emerged as an interesting blend of many different fields
of mathematics, ranging from probability and statistics to partial
differential equations and numerical analysis.

In this talk, will cover some of the basic definitions and concepts of
mathematical finance, including puts, calls, and arbitrage. We will then
look at a simple model for option pricing, known as the
Cox-Ross-Rubinstein Binomial model. Finally, time permitting, we will
discuss some more advanced topics such as the Black-Scholes formula. This
talk should be accessible to everyone.

##### Fully nonlinear equations, elementary symmetric functions and convexity property of solutions

We discuss convexity properties of solutions of fully nonlinear partial differential equations. The classical examples indicate that the level-sets of equilibrium potential in a convex domain is convex and the first eigenfunction of the Laplace equation in a convex domain is log-concave. Recently, there emerge two differente types of methods in the study of convexity of solutions of nonlinear equations. The macroscopic convexity principle is based on the convex hull of the solution. The microscopic convexity principle is based on constant rank type theorem for the Hessian matrix of the solution. The microscopic convexity principle is effective to treat nonlinear geometric differential equations on general manifolds. In the talk, we will explain how elementary symmetric functions can be used in a crucial way in this direction and what kind of "convexity" structural conditions are involved. The microscopic convexity principle shares close relationship with the Hamilton's maximum principle for general evolution equations. We will also discuss some related open problems.

##### A counterexample to the Alon-Saks-Seymour conjecture and related problems

Consider a graph obtained by taking an edge disjoint union of $k$ complete bipartite graphs, Alon, Saks, and Seymour conjectured that such graphs have chromatic number at most k+1. This well known conjecture remained open for almost twenty years. In this talk, we will show a counterexample to this conjecture. This construction will also lead to some related results in combinatorial geometry and communication complexity. In particular, it implies a nontrivial lower bound of the non-deterministic communication complexity of the clique versus independent set'' problem.\\

Joint work with Benny Sudakov.

##### Contracting Exceptional Curves by the Kahler-Ricci Flow

We give a criterion under which a solution g(t) of the
Kahler-Ricci flow contracts exceptional divisors on a compact manifold and can be uniquely continued on a new manifold. This is a joint work with Jian Song.

##### The twisted symmetric square L-function on GL(r)

In this talk, we will discuss the twisted symmetric square
L-function of a cuspidal automorphic representation on GL(r). In
particular, we will show that the L-function is holomorphic except at
s=0 and 1. This work generalizes the previous work by Bump and
Ginzburg, who considered the non-twisted case.

##### Model Selection for Generalized Linear Mixed Models

This talk is about model selection for clustered data. A conditional Akaike Information, cAI, has been defined for linear mixed models when the inference is on cluster, rather than population, parameters. We extend this definition to generalized linear mixed models and derive an asymptotically unbiased estimator of cAI. This estimator is applied to a cancer data set. Alternatively, cAI can be estimated using a nonparametric bootstrap. We compare the two estimators in simulation

##### Counting graphs without a fixed complete bipartite subgraph

A graph is called $H$-free if it contains no copy of $H$. Denote by $f_n(H)$ the number of (labeled) $H$-free graphs on $n$ vertices. Since every subgraph of an $H$-free graph is also $H$-free, it immediately follows that $f_n(H) \geq 2^{\mathrm{ex}(n,H)}$. Erd{\H o}s conjectured that, provided that $H$ contains a cycle, this trivial lower bound is asymptotically tight, i.e.,
$f_n(H) = 2^{(1+o(1))\mathrm{ex}(n,H)}.$

The conjecture was resolved in the affirmative for graphs with chromatic number at least $3$ by Erd{\H o}s, Frankl, and R{\"o}dl (1986)

##### An Augmented Lagrangian Method for Image Restoration Problems

This talk concerns the classical total variation (TV)
image deblurring problems, which involves an unconstrained
minimization problem consisting of a least-squares term and
a total variation regularization term. We transform the
original unconstrained problem into an equivalent constrained
problem, and use an augmented Lagrangian method to handle the
constraints. The transformation allows the differentiable and
non-differentiable parts of the objective function to be
treated using separate subproblems. Each subproblem may be
solved efficiently and an alternating strategy is used to
combine the solutions. The new algorithm is faster than
several state-of-the-art TV algorithms.

##### Ricci flow on quasiprojective varieties

Singularities occur in Ricci flow because of curvature blowup. For dimensional reasons, when approaching a singularity, one expects the curvature to blow up like the inverse of the time to the singularity. If this does not happen, the singularity is said to be type II. The first example of a type II singularity, studied by Daskalopoulos-Del Pino-Hamilton-Sesum, occurs on a noncompact surface which is the result of capping off a hyperbolic cusp. The analysis in the surface case uses isothermal coordinates. It is not immediately clear whether it extends to higher dimensions. We look at the Ricci flow on finite-volume metrics that live on the complement of a divisor in a compact Kahler manifold. We compute the blowup time in terms of cohomological data and give sufficient conditions for a type II singularity to emerge. This is joint work with Zhou Zhang.

##### Convexity estimates for level sets of solutions to nonlinear PDEs

This is a joint work with Lu Xu. We establish a geometric lower bound for the principal curvature of the level surfaces of solutions to $F(D^2u, Du, u, x)=0$ in convex ring domains, under a refined structural condition introduced by Bianchini-Longinetti-Salani.\\

Talk time starts at 4:15 PM.

##### A polynomial identity for the Hilbert series of diagonal harmonics

A special case of Haiman's identity for the character of the quotient ring of diagonal coinvariants under the diagonal action of the symmetric group yields a formula for the bigraded Hilbert series as a sum of rational functions in $q,t$. In this talk I will show how a summation identity of Garsia and Zabrocki for Macdonald polynomial Pieri coefficients
can be used to transform Haiman's formula for the Hilbert series into an explicit polynomial in $q,t$ with integer coefficients. An equivalent formulation expresses the Hilbert series as the
constant term in a certain multivariate Laurent series.

##### Subsampling p-values and the Linear Process Bootstrap

In the first portion of the talk, I will discuss the use of subsampling to construct p-values for hypothesis
tests. The p-values are based on a modification of the usual subsampling hypothesis tests that involves
centering the subsampled test statistics as in the construction of confidence intervals. This modification makes
the hypothesis tests more powerful, and as a consequence provides meaningful p-values. The new p-values are
shown to be asymptotically uniform under the null hypothesis and to converge zero under the alternative. \\

The second half of the talk addresses the problem of estimating the autocovariance matrix of a stationary
process. The proposed estimator is a gradually tapered version of the sample autocovariance matrix in which the
main diagonals are fully weighted, and the off-diagonal entries are tapered towards zero. Under short range
introduce a new resampling scheme for stationary processes, which we call the Linear Process Bootstrap (LPB). The
LPB is asymptotically valid for the sample mean and related statistics, and conjectured to be valid for all
statistics which depend only on the first two moments of the data.

##### The Refined Gross-Prasad Conjecture for Unitary Groups

Let $V_n\subset V_{n+1}$ be orthogonal spaces of dimensions $n$ and $n+1$ over a number field $F$, and let $G_n\subset G_{n+1}$ be the associated special orthogonal groups. Let $\pi_n$ and $\pi_{n+1}$ be irreducible, cuspidal, tempered, automorphic representations of $G_n(\mathbb{A}_F)$ and $G_{n+1}(\mathbb{A}_F)$. In the early 1990s, Gross and D. Prasad conjectured that a certain period integral attached to $\pi_n$ and $\pi_{n+1}$ is non-zero if and only if a certain automorphic $L$-function is non-zero at $s=1/2$. Recently, A. Ichino and T. Ikeda have proposed a refinement of this conjecture; they give an explicit formula relating the period integral to the $L$-value. In this talk, we state a similar conjecture for unitary groups, as well as sketch the proof of the first case.

##### The Kahler-Ricci flow and scalar curvature

I will talk about some results of Tian-Zhang and Zhang on the
Kahler-Ricci flow on a compact Kahler manifold with arbitrary initial Kahler class.

##### Experimental Investigations into the Zeros of Approximates to the Riemann Xi-function

Reimann introduced the study of the zeta function of a
complex variable, and showed how multiplying zeta by a gamma factor and
other simple factors resulted in an even, entire function called the
Riemann Xi-function. He conjectured that all the zeros of the
Xi-function are real, now known as the Riemann Hypothesis. In this talk
we introduce the study of the zeros of the partial sums in Riemann's
uniformly convergent series expansion for the Xi function in terms of
incomplete gamma functions, and discuss how various known or conjectured
properties of the Xi-function seem to be reflected by these
approximates.

##### Issues of the Transition to College Mathematics

Over the past quarter century, 2- and 4-year college enrollment in first semester
calculus has remained constant while high school enrollment in
calculus has grown tenfold, from 60,000 to 600,000, and continues to grow at
6\% per year. We have passed the cross-over point where each year more
students study first semester calculus in US high schools than in all 2- and
4-year colleges and universities in the United States. In theory, this
should be an engine for directing more students toward careers in science,
engineering, and mathematics. In fact, it is having the opposite effect.
This talk will present what is known about the effects of this growth and
what needs to happen in response within our high schools and universities. \\

\noindent David Bressoud is the current President of the Mathematical Association of America.

##### A Large deviation bound for the cover time

For random walk on a graph, the cover time $T$ is the number of
steps required to visit every vertex at least once. We prove the following
large deviation bound for the cover time: For every $A>0$ and $d>0$ there
is a constant $c>0$ such that for all $n$ and graphs on $n$ vertices of
maximum degree $d$, we have $P(T < An) < \exp(-cn)$. \\

\noindent Joint work with Itai Benjamini and Ori Gurel-Gurevich.

##### Parametrization of mappings between certain nonminimal hypersurfaces

We construct an analytic jet parametrization for hypersurfaces of the form im $w =$ re $w (|z|^2 + \dots)$. In particular, we obtain a Lie group structure on the automorphism group of such hypersurfaces.\\

(Joint work with Robert Juhlin).

##### Sparse solutions in standard quadratic optimization

Sparse solutions in optimization has been a major concern for optimization problems from various applications such as image processing and portfolio selection. It is well-known that several classes of linear optimization problems such as the linear assignment problem, transportation problem and the $L_1$ minimization problem with linear equality constraints, have sparse solutions.\\

It has also been observed in experiments for long that several classes of quadratic optimization problems (QP) such as Markowitz's mean-variance model for portfolio selection always have sparse optimal solutions. However, so far little is known from a theoretical perspective.\\

In this talk, we present a new theoretical framework to interpret why certain classes of QPs do have sparse solutions. For this, we first use the optimality conditions for the so-called standard quadratic optimization problem to establish an intrinsic relation between the sparsity of the optimal solution of the QP and some prpbability events. Then we show that with a very high probability, the underlying QP has a very sparse solution if the input data of the associated QP follows certain distributions such as uniform and normal distributions. If time allows, we shall also discuss some extensions and open questions.

##### Exceptional holonomy and calibrated submanifolds.

We give an introduction to recent developments in the geometry
of compact manifolds with exceptional holonomy, focusing on recent work
with Corti, Nordstrom and Pacini; we prove the existence of many compact
7-manifolds with holonomy G2 that contain rigid associative submanifolds.
The main ingredients in the proof are: an appropriate noncompact version
of the Calabi conjecture, gluing methods and a certain class of complex
projective 3-folds (weak Fano 3-folds).

##### Formal and finite order equivalences

We compare formal equivalences with equivalences of any finite order and show that they coincide for finite collections of real-analytic sets as well as for local dynamical systems (maps and vector fields) but may not coincide for infinite collections.

##### Size doesn't matter: heights in number theory

How complicated is a
rational number? Its size is not a very good indicator for this. For
instance, 1987985792837/1987985792836 is approximately 1, but so much
more complicated than 1. We'll explain how to measure the complexity of
a rational number using various notions of height. We'll then see how
heights are used to prove some basic finiteness theorems in number
theory. One example will be the Mordell-Weil theorem: that on any
rational elliptic curve, the group of rational points is finitely
generated.

##### A Spectrum of Colorful Diameters

For any finite simple graph $G$, we can form its adjacency matrix $A$ to be defined as $A_{ij} = 1$ if $i$ is adjacent to $j$ and $=0$ otherwise. This matrix has several interpretations, and because of these interpretations, the eigenvalues (i.e., the spectrum) determine a lot of information about the graph. Let $\lambda_{min}$ and $\lambda_{max}$ denote the minimal and maximal eigenvalues of $A$ respectively, and let $\chi(G)$ denote the chromatic number of $G$. Hoffman proved (1969) $\chi(G) \ge 1- \frac{\lambda_{max}}{\lambda{min}}$. In this talk, we present a new simple probabilistic proof of Hoffman's Theorem. We will then follow up on how to apply Hoffman's theorem, and it's connection to the Lovasz Theta function, to yield a spectral bound on the diameter of the graph.

The intention of the talk (and its seemingly ridiculous title) is to showcase how several different basic mathematical tools- including those from probability, linear algebra, and combinatorics- can be used to in concert to yield beautiful results.

##### Sparse SDP Relaxation and Moment Methods for Approximating Solutions of Nonlinear Differential Equations

\indent To solve a system of nonlinear differential equations numerically, we formulate it as a polynomial optimization problem (POP) by discretizing it via a finite difference approximation. The resulting POP satisfies a structured sparsity, which we can exploit to apply the sparse semidefinite programming (SDP) relaxation of Waki et al. to the POP to obtain a discrete approximation for a solution of the differential equation. The main features of this approach are:

(a) we can choose an appropriate objective function, and (b) we can add inequality constraints on the unknown variables and their derivatives, in order to compute specific solutions of the system of differential equations.

High resolution grid discretizations of differential equations result in high dimensional POPs and large-scale SDP relaxations. We discuss techniques to reduce the size of sparse SDP relaxations by exploiting sparsity and structure of the POPs.

Finally, we propose a further method, which is based on sparse SDP relaxations, moments and maximum entropy estimation to detect smooth approximations for solutions of nonlinear differential equations. This method utilizes the SDP relaxation method for finding discrete approximations and can be understood as an attempt to overcome the curse of dimensionality in grid based approaches for solving differential equations.

##### The Probabilistic Method

\indent In a seminal paper on Ramsey numbers in 1947, Erd\H{o}s introduced a technique which is in a broad sense referred to as
{\em the probabilistic method}. This method is now used in many branches of mathematics, especially for existence proofs.

In this talk, I will outline the basic method and give some remarkable applications to problems from combinatorics, geometry, number theory and analysis.

##### Ascent Sequences, 2+2-Free Posets, Upper Triangular Matrices, and Genocchi Numbers

The combinatorics of the Genocchi numbers was developed by
Dumont and various co-authors in the 70's and 80's. More recently,
Bousquet-Melou, Claesson, Dukes, Kitaev and Parviainen showed that the
2+2-free posets are in bijection with so-called ascent sequences and
with non-negative integer valued upper triangular matrices which have no
zero rows or columns. We will show how the Genocchi numbers can be
interpreted as the number of up-down ascent sequences thus connecting
these various classes of combinatorial objects.

##### Modeling Cell Migration with Phase Field Method

Cell migration is important in many biological processes. Many eukaryotic cells can move using a crawling motion powered by the actin-myosinII machinery. We developed a phase field model and investigated the relation between actin-myosinII distribution, cell shape and cell speed.

##### Primal and Dual Active-Set Methods for Quadratic Programming

We present an active-set quadratic programming (QP) method based on inertia control. The method is appropriate for problems with many degrees of freedom and problems that are not necessarily convex, making it particularly useful in sequential quadratic programming (SQP) methods that use exact second derivatives. In the convex case, the method is applied to the dual QP, which may be suitable for QPs arising in mixed integer nonlinear programming, where points may be dual feasible but primal infeasible. The inertia-controlling property prevents singularity in the associated linear systems, which allows the straightforward application of modern "off-the-shelf" linear algebra software.

##### Is 100000000000000000000000000\\000000000000000000000027 prime?

We will talk about how to tell if very large numbers are prime or composite, and what this has to do with online banking.

##### Locally Restricted Compositions

Ordinary compositions and Carlitz compositions (adjacent parts differ) have been studied extensively. Locally restricted compositions are a broad generalization studied by Rod Canfield and myself in an ongoing series of papers. I'll discuss some results and open problems. These include total number, distribution of various counts, largest part, runs and gaps. Methods will be discussed, but technical details will be avoided.

##### Indications of de Sitter Spacetime from a Discrete Causal Dynamics

One of the greatest challenges facing theoretical physics is the
reconciliation of Quantum Theory with General Relativity. I will
propose a discrete mathematical structure to underlie the continuum,
a causal set, and show that a classical precursor to its dynamics
yields causal sets which share many features in common with de Sitter
spacetime. This leads to a sketch of how quantum gravity may resolve
some of the current puzzles of cosmology.

##### Separating Invariants and Degree Bounds

If $V$ is a representation of a linear algebraic group $G$, a set $S$ of $G$-invariant regular functions on $V$ is called separating if the following holds: If two elements $v,v'$ from $V$ can be separated by an invariant function, then there is an f from S such that $f(v)$ is different from $f(v')$.

It is known that there always exist finite separating sets, even though the invariant ring might not be finitely generated. Moreover, if the group $G$ is finite, then the invariant functions of degree $\le |G|$ always form a separating set. So the degree bounds are definitely smaller than for the generators of the invariants.

Jointly with Martin Kohls we have shown that for a non-finite linear algebraic group G such an upper bound for the degrees of a separating set does not exist. Moreover, for a finite group G we define b(G) to be the minimal number d such that for every G-module V there is a separating set of degree less or equal to d. We then show that for a subgroup H of G we have $b(H) \le b(G) \le [G:H] b(H)$, and that $b(G) \le b(G/H) b(H)$ in case H is normal. In addition, we calculate $b(G)$ for some specific finite groups.

##### Macdonald polynomials, parking functions, trees, and zonotopal algebras

These objects are related to each other through their connection to the
Shi hyperplane arrangement. We will give definitions of each of these
objects, and through six bijections, show how the basis for a Gordon
module consisting of Macdonald polynomials is connected to the basis for
the
central space of a zonotopal algebra. We will also see Catalan numbers
appear in this story of six bijections.

##### Short-time existence of the Ricci flow on noncompact Riemannian manifolds

In this talk, using the local Ricci flow, we prove the
short-time existence of the Ricci flow on noncompact manifolds, whose
Ricci curvature has global lower bound and sectional curvature has
only local average integral bound. The short-time existence of the
Ricci flow on noncompact manifolds was studied by Wan-Xiong Shi in
1990s, who required a point-wise bound of curvature tensors. As a
corollary of our main theorem, we get the short-time existence part of
Shi's theorem in this more general context.

##### Zero-Divisor Graphs

We shall investigate $\gamma(R)$, the zero-divisor graph of a commutative ring $R$, where $0 \ne 1$.

##### Oscillatory component recovery and separation in images by Sobolev norms

It has been suggested by Y. Meyer and numerically confirmed by many others that dual spaces are good for texture recovery. Among the dual spaces, our work focuses on Sobolev spaces of negative differentiability to recover texture from noisy blurred images. Such Sobolev spaces are good to model oscillatory component, on the other hand, the spaces themselves hardly distinguishes texture component from noise component because noise is also considered to be a highly oscillatory component. In this talk, in addition to oscillatory component recovery, we will further investigate a one-parameter family of Sobolev norms to achieve such a separation task.

##### The Overconvergent de Rham-Witt complex

The aim of the talk is to describe the overconvergent de Rham-Witt complex. It is a subcomplex of the de Rham-Witt complex and it can be used to compute Monsky-Washnitzer cohomology for affine varieties, and rigid cohomology in general. (All our varieties are over a perfect field of
characteristic p.)

We will begin by reviewing Monsky-Washnitzer cohomology and the de Rham-Witt complex. Next we will define overconvergent Witt vectors and then the overconvergent de Rham-Witt complex. As time permits, we will say something about the proof of the comparison theorem between Monsky-Washnitzer cohomology and overconvergent de Rham-Witt ohomology.

This is joint work with Andreas Langer and Thomas Zink.

##### Exploiting Structured Sparsity in Linear and Nonlinear Semidefiite Programs

This talk summarizes conversion of large scale linear and nonlinear SDPs, which satisfies the sparsity characterized by a chordal graph structure , into smaller scale SDPs. The sparsity is classified in two types, the domain-space sparsity (d-space sparsity) for the symmetric matrix variable in the objective and/or constraint functions of the SDP, which is required to be positive semidefinite, and the range-space sparsity (r-space sparsity) for a linear or nonlinear matrix-inequality constraint of the SDP. Some numerical results on the conversion methods indicate their potential for improving the efficiency of solving various problems.

##### The algebra, analysis and combinatorics of box splines

Box splines are multivariate compactly supported piecewise
polynomial functions, initially introduced in the 1980s to tackle
problems of interpolation and approximation in several variables.
It quickly transpired, through the work of Dahmen, Micchelli,
de Boor, Ron, and others, and that box splines naturally give rise
to a fascinating algebraic structure, with connections to such
diverse areas as polynomial ideals, D-modules, hyperplane
arrangements, tiling and enumeration of integer points in zonotopes,
parking functions on graphs, and even partition functions of
statistical mechanics. Some of these connections are established
only recently in joint work with Ron and Xu as well as in the
work of de Concini, Procesi, Vergne, Ardila, Postnikov,
Sturmfels and others. The speaker will try to give an overview
of the area, with the focus on some exciting new developments.

##### Polynomials constant on a line (or a plane).

Suppose we have a polynomial $p(x,y)$ such that $p(x,y) = 1$ whenever
$x+y=1$, and such that all coefficients of $p$ are nonnegative. If $N$ is
the number of nonzero coefficients and $d$ is the degree, then $d \leq 2N-3$.
For example, if the degree is 3, then you have to have at least 3 terms in $p.$
I will talk about proving this bound and also about proving similar bounds in
higher dimensions (polynomials in more than two variables). While the
statement of the problem above is elementary, the class of polynomials
considered appears as a special case of a hard problem in complex analysis.

##### Differential Geometry Seminar

In this talk, we present a proof of a weaker version of the
Candelas de la Ossa's conjecture, i.e. conifold transitions and flops
for Ricci-flat Calabi-Yau manifolds are continuous in the
Gromov-Hausdorff sense.

##### Einstein, the Moon and the Long Lost Soviet Reflector

One of the greatest successes of the former Soviet space program was a lunar rover called Lunokhod 1—Russian for “moonwalker.”Landing on the moon on November 17, 1970 with a laser reflector, it wandered around the moon’s surface for 11 months then mysteriously disappeared -- until last spring. On April 22, nearly 40 years after Lunokhod 1 disappeared, a team headed by Tom Murphy found the reflector and pinpointed its distance from earth to within one centimeter. \\

The discovery came as part of a long-term project Murphy heads to send pulses of laser light to the moon from a telescope in New Mexico. The purpose, which he will describe in his talk, is to look for deviations of Einstein’s theory of general relativity by measuring the shape of the lunar orbit to within the accuracy of one millimeter, or about the thickness of a paperclip. \\

The talk is free and the public welcome. Light refreshments will be served afterwards. If you have questions, please contact physcievents@ucsd.edu.

##### The Metropolis-Hastings Algorithm

The Metropolis-Hastings algorithm is a widely applicable procedure for
drawing samples from a specified distribution on a large finite set. The
talk will describe the Metropolis-Hastings algorithm and some of its
properties. There will be a discussion of some of the applications and an
example.

##### Explicit Coleman integration for hyperelliptic curves

The Coleman integral is an analytic tool that serves as the
p-adic analogue of the usual real-valued line integral. The theory of
Coleman integration plays a prominent role in the arithmetic of curves
and abelian varieties, allowing us to find torsion and rational points
and giving us a means to compute p-adic heights and regulators. We
describe algorithms for computing Coleman integrals on hyperelliptic
curves as well as some applications of these algorithms. Throughout,
we illustrate our methods with numerical examples, computed using our
Sage implementation.

This is joint work with Robert Bradshaw and Kiran Kedlaya.

##### Energetic Variational Approaches in the Modeling of Ionic Solutions and Ion Channels

Ion channels are key components in a wide variety of biological processes. The selectivity of ion channels is the key to many biological process. Selectivities in both calcium and sodium channels can be described by the reduced models, taking into consideration of dielectric coefficient and ion particle sizes, as well as their very different primary structure and properties. These self-organized systems will be modeled and analyzed with energetic variational approaches (EnVarA) that were motivated by classical works of Rayleigh and Onsager. The resulting/derived multiphysics-multiscale systems automatically satisfy the Second Laws of Thermodynamics and the basic physics that are involved in the system, such as the microscopic diffusion, the electrostatics and the macroscopic conservation of momentum, as well as the physical boundary conditions. In this talk, I will discuss the some of the related biological, physics, chemistry and mathematical issues arising in this area.

##### Invitation to MathStorm

The Mathematics Department's Graduate Consulting Group, Mathstorm,
has been providing free consulting service for the campus research community
since 1999. The goal is to offer campus researchers a resource for resolving
mathematical issues that arise in their research while at the same time
providing an opportunity for mathematics graduate students to gain experience
consulting on solutions to real-world problems. In this meeting we'll discuss
how MathStorm works, what kinds of problems are posed, and encourage a new
generation of graduate students to join.

##### The Symplectic Geometry of Fish

Fish, or in general any mechanical object moving in an inviscid fluid, can be described by means of a number of interesting differential-geometric structures, amongst other bundles and connections, groups of diffeomorphisms, and symplectic reduction. I will describe some of these structures and outline their role in fluid dynamics. Along the way, a number of parallels will appear with other dynamical systems: time permitting, we will describe a Kaluza-Klein description of fluid-structure interactions (making the link with magnetic particles), and we will see how the flux homomorphism from symplectic geometry makes an appearance through an old construction of Kelvin.

##### What would Hilbert do? Undecidability and decidabilty in mathematics.

Hilbert's vision of mathematics was of a vast game governed by simple rules, where all facts and proofs could be deduced systematically by finitely many applications of these rules. Throughout the twentieth century, we have seen dramatic counterexamples to this vision. However, it is still meaningful to study that part of mathematics that can be described in this way. In this talk, we will discuss the notion of a decision procedure and the related idea of computability. We will see examples of interesting mathematical problems that are decidable and, on the flip side, think about what it would mean for a problem to be undecidable. Coming full circle, we return to Hilbert and to his famous list of problems. In particular, his 10th problem proved to be a milestone in undecidability theory. We will trace through the history of its solution and notice the various consequences of undecidability that crop up.

##### Orienting the Edges of the Hypercube With Only Two In-degrees

\indent We show that it is possible to orient the edges of the
$n$-dimensional cube so that only the in-degrees $a$ and $b$ occur if and only if the two obvious necessary conditions hold, namely there are nonnegative integers $s$ and $t$ so that $s+t=2^n$ and $as+bt=n2^{n-1}$. This is connected to a question arising from constructing strategies for a type of hat game with $n$ players so that regardless of placements the number of correct guesses is $a$ or $b$.

Joint work with Joe Buhler, Ron Graham and Eric Tressler

##### Warped Product Einstein Structures

We will discuss old and new questions about when a fixed
Riemannian manifold is the base of a warped product Einstein manifold.
This problem has been completely solved when the base is 1 or
2-dimensional and much progress has been made in higher dimensions as
well. There are also many interesting extensions to the case where the
base might have boundary and when we allow for warping functions that
change sign.

##### Markov Processes On Time-like Graphs

I will discuss Markov processes where the "time" parameter is replaced by paths in a directed graph from an initial vertex to a terminal one. Along each directed path the process is
Markov and has the same distribution as the one along any other directed path. If two directed paths do not interact, in a suitable sense, then the distributions of the processes on the two paths are conditionally independent, given their values at the common endpoint of the two paths. Conditions on graphs that support such processes (e.g., hexagonal lattice) are established. Next we analyze a particularly suitable family of Markov processes, called harnesses, which includes Brownian motion and other Levy processes, on such time-like graphs. Finally we investigate continuum limits of harnesses on a sequence of time-like graphs that admits a limit in a suitable sense. This talk is based on joint work with Soumik Pal.

##### Canonical Metrics

I will talk about canonical metrics in geometry and how to

##### On the Minimal Ramification Problem for Semiabelian Groups

Let G be a finite group and d(G) the minimal number of conjugacy
classes that generate G. In any tame realization of G as a Galois group over Q there are at least d(G) ramified primes. The (tame) minimal ramification problem asks whether any group G can be realized (tamely) over Q with exactly d(G) ramified primes.
It has been recently proved that this problem has an affirmative answer for a substantial class of finite nilpotent groups (all finite semiabelian nilpotent groups). (Joint work with Hershy Kisilevsky and Jack Sonn)

##### How Many Points Can A Curve Have?

\indent Diophantine equations, one of the oldest topics of mathematical research, remains the object of intense and fruitful study. A rational solution to a system of algebraic equations is tantamount to a point with rational coordinates (briefly, a "rational point") on the corresponding algebraic variety $V$. Already for $V$ of dimension 1 (an "algebraic curve"), many natural theoretical and computational questions remain open, especially when the genus $g$ of $V$ exceeds 1. (The genus is a natural measure of the complexity of $V$; for example, if $P$ is a nonconstant polynomial without repeated roots then the equation $y^2 = P(x)$ gives a curve of genus $g$ iff $P$ has degree $2g+1$ or $2g+2$.) Faltings famously proved that if $g>1$ then the set of rational points is finite (Mordell's conjecture), but left open the question of how its size can vary with $V$, even for fixed $g$. Even for $g=2$ there are curves with literally hundreds of points; is the number unbounded?

We briefly review the structure of rational points on curves of genus 0 and 1, and then report on relevant work since Faltings on points on curves of genus $g > 1$.

##### Jet Engine Controller Certification - A Day At The (Math) Zoo

\indent A hard practical problem will be described associated with the experimental certification of a jet engine together with its feedback controller. This has recently become much more difficult as engine controllers have become fully MIMO (multiple-input/multiple-output), which requires evaluation of vector input and output signals and their relative gains to establish a quantitative acceptance criterion. Linear algebra and matrix operator norms need to introduced to provide a rigorous foundation to the approach. The requirement to scale the data strongly affects the numerical answers, so scaling arises as a central issue. Finding the correct scaling then becomes a constrained optimization problem associated with minimizing the operator norms. Thus, interior point methods for linear matrix inequalities are introduced into the mix. Finally, to accommodate the sampled nature of the data, these scaling matrices must be extended to a smooth matrix function of a single complex variable, which requires the inclusion of interpolation theory and complex analysis. This brings us back to linear algebra and the so-called Nevanlinna-Pcik problem.\\

This brief excursion through the math zoological garden is intended to demonstrate the importance of rigorous theory providing underpinnings to hard practical approaches to engineering problems. The presence of this foundation permits advancing with a significantly greater degree of surety and confidence than would the case relying solely on experience and guesswork, particularly for problems of a complexity exceeding simple human capabilities. The end result is an experimental protocol for the acceptance of engine-controller pairs.

• ##### 11/09/10

One of the most important choices a graduate student will make will be
choosing a thesis advisor. It is never too early for students to begin
thinking about choosing an area of specialty and choosing among the
faculty who might supervise them. How did other students find a
thesis advisor? What are the key factors to consider when choosing
an advisor? What do professors look for before they accept a student
as their thesis student? How does finding a thesis advisor lead to
finding a thesis problem? We will discuss these questions.
We will have four graduate students describe their experiences finding a
thesis advisor and what happens after a thesis advisor is found. We
will also have one faculty member, Mark Gross, describe what he looks for
in a graduate student before he accepts him or her as a thesis student.
All students, especially first, second and third year PhD students, and
masters students who are considering transferring to the PhD program,
are cordially invited to attend.

##### The Shape of Water, Metamorphosis and Infinite-Dimensional Geometric Mechanics

\indent Whenever we say the words "fluid flows" or "shape changes" we enter the realm of infinite-dimensional geometric mechanics. Water, for example, flows. In fact, Euler's equations tell us that water flows a particular way. Namely, it flows to get out of its own way as adroitly as possible. The shape of water changes by smooth invertible maps called diffeos (short for diffeomorphisms). The flow responsible for this optimal change of shape follows the path of shortest length, the geodesic, defined by the metric of kinetic energy. Not just the flow of water, but the optimal morphing of any shape into another follows one of these optimal paths.

\indent The lecture will be about the commonalities between fluid dynamics and shape changes and will be discussed in the language most suited to fundamental understanding -- the language of geometric mechanics. A common theme will be the use of momentum maps and geometric control for steering along the optimal paths using emergent singular solutions of the initial value problem for a nonlinear partial differential equation called EPDiff, that governs metamorphosis along the geodesic flow of the diffeos. The main application will be in the registration and comparison of Magnetic Resonance Images for clinical diagnosis and medical procedures.

##### Graph Theory in the Information Age

\indent Nowadays we are surrounded by numerous large information networks, such as the WWW graph, the telephone graph and various social networks. Many new questions arise. How are these graphs formed? What are basic structures of such large networks? How do they evolve? What are the underlying principles
that dictate their behavior? How are subgraphs related to the large host graph? What are the main graph invariants that capture the myriad properties of such large sparse graphs and subgraphs.
In this talk, we discuss some recent developments in the study of large sparse graphs and speculate about future directions in graph theory.

##### The Advanced Mathematics Behind Various Children's Games

We will talk about the more advanced mathematics behind various children's games including the algebraic properties of Liar (a.k.a. B.S."), the probabilistic applications toward Chutes and Ladders, and the algorithm dynamics behind Guess Who?. Hopefully, after this talk, you will be able to beat your little brother or sister at their favorite game(s). If you have any games that you would like to be discussed, you also may provide me with your suggestions.

##### Radial Random Walks on Matrix Cones

\indent The present lecture is devoted to recent developments on random walks with spherical symmetry, a topic which was opened to research by J.F.C. Kingman in 1963, and which has developed wide-ranging applications through the work of W. Hazod, M. Rösler, and M. Voit. The analytic method to be described in the talk concerns generalized convolutions of measures on hypergroups, in particular on the self-dual commutative hypergroup of positive semidefinite (Hermitian) matrices. These hypergroups are defined via Bessel functions of higher rank. A typical application of the hypergroup setting is the study of Bessel random walks on matrix cones and their convergence to Wishart distributions.

##### Downsizing the US Nuclear Weapons Stockpile and Modeling an Intelligent Adversary

\indent We discuss two topics that I have worked on at Lawrence Livermore National Laboratory, which give a sense of the breadth of the mathematical work that is done at a national lab. The first of these involves using optimization techniques to assess policy options for downsizing the US nuclear weapons stockpile. We discuss consolidation of the weapons complex in general, and our implementation of a mixed-integer linear programming model that is currently being used to evaluate policy alternatives. The second topic addresses mathematical methods for modeling an intelligent adversary, and how techniques from the field of artificial intelligence can be used to solve such problems. Using the crime of money laundering as a motivating example, we specifically seek to address the gap between the theory and practical application of such methods.

##### Working for a National Laboratory in Operations Research, The Science of Better'

\indent Are you curious as to the kind of work that is done at a national laboratory? Have you heard of the field of operations research, or are you interested in learning about it and how it is applied to real problems? In this talk I will describe the kinds of math I use in my job at Lawrence Livermore National Laboratory, as well as giving an introduction to the discipline of operations research. The talk will focus on three projects I have worked on: disaster preparedness, modeling an intelligent adversary, and downsizing the US nuclear weapons stockpile. These projects span a range of time frames, sponsors, and team sizes, and hopefully will give a flavor of the diverse work that is done at a national laboratory. This talk is intended for undergraduates or anyone interested in applications of math in the real world.

##### Certification via Symbolic-Numeric Computations

\indent We present a hybrid symbolic-numeric algorithm for certifying a polynomial or a rational function with rational coefficients to be non-negative for all real values of the variables by computing a representation for it as a fraction of two polynomial sum-of squares (SOS) with rational coefficients. We can either perform high precision Newton iterations on the numerical SOS computed by SDP solvers in Matlab or use the high precision SDP solver in Maple to get the SOS with necessary precision, then we can convert the numerical SOS into an exact rational SOS by orthogonal projection or rational coefficient vector recovery. Sums-of-squares rational lower bound certificates for the radius of positive semidefiniteness of a multivariate polynomial also offer an alternative SOS proof for those positive definite polynomials that are not SOS but have a positive distance to the nearest polynomial with a real root. Moreover, we show that a random linear transformation of the variables allows with probability one for certifying the positive semidefinteness of a multivariate polynomial by representing it as an SOS over the variety defined by partial derivatives of the polynomial with respect to each variable except one.

Joint work with Feng Guo, Sharon E. Hutton, Erich L. Kaltofen, Bin Li, Mohab Safey El Din and Zhengfeng Yang.

##### The Compensating Polar Planimeter

How does one measure area? As an example, how can one
determine the area of a region on a map for the purpose of real
estate appraisal? Wouldn't it be great if there were an instrument
that would measure the area of a region by simply tracing its
boundary? It turns out that there is such an instrument: it is
called a planimeter. In this talk we will discuss a particular type
of planimeter called the compensating polar planimeter. There will
be a little bit of history and some analysis involving line integrals
and Green's theorem. Finally, there will be a chance to see and
touch actual examples of these fascinating instruments from the
speaker's collection.

##### Parking Function Bijection Suggested by the Haglund-Morse-Zabrocki Conjecture

In recent work Jim Haglund, Jennifer Morse and Mike Zabrocki
introduce a new statistic on Parking Functions, the diagonal
composition,'' which gives the lengths of the intervals between
successive diagonal hits of the Dyck path. They conjectured that the
nabla operator, when applied to certain modified Hall-Littlewood
functions indexed by compositions, yields the weighted sum of the
corresponding Parking Functions by area, dinv, and Gessel
quasisymmetric function. This conjecture then gives a sharpening of
the shuffle conjecture'' and suggests several combinatorial
conjectures about the parking functions. In particular, we discuss a
bijective map on the parking functions implied by the commutativity
properties of the modified Hall-Littlewood polynomials that appear in
their conjecture.

##### Regularization for Cox's Proportional Hazards Model With NP-Dimensionality

High throughput genetic sequencing arrays with thousands of
measurements per sample and a great amount of related censored
clinical data have increased demanding need for better measurement
specific model selection. In this paper we establish strong oracle
properties of non-concave penalized methods for non-polynomial (NP)
dimensional data with censoring in the framework of Cox's proportional
hazards model. A class of folded-concave penalties are employed and
both LASSO and SCAD are discussed specifically. We unveil the question
under which dimensionality and correlation
restrictions can an oracle estimator be constructed and grasped. It is
demonstrated that non-concave penalties lead to significant reduction
of the irrepresentable condition" needed for LASSO model selection
consistency. The large deviation result for martingales, bearing
interests of its own, is developed for characterizing the strong
oracle property. Moreover, the non-concave regularized estimator, is
shown to achieve asymptotically the information bound of the oracle
estimator. A coordinate-wise algorithm is developed for finding the
grid of solution paths for penalized hazard regression problems, and
its performance is evaluated on simulated and gene association study
examples. \\

Joint work with Jianqing Fan (Princeton University) and Jiancheng
Jiang (University of North Carolina at Charlotte)

##### Numerical Methods for Geometric Elliptic Partial Differential Equations

\indent Geometric Partial Differential Equations (PDEs) are at the forefront of current research in mathematics, as evidenced by Perelman's use of these equations in his proof of the Poincare Conjecture and Cedric Villani's Fields Medal in 2010 for his work on Optimal Transportation. They can be used to describe, manipulate and construct shapes based on intrinsic geometric properties such as curvatures, volumes, and geodesic lengths.

These equations are useful in modern applications (Image Registration, Computer Animation) which require geometric manipulation surfaces and volumes. Convergent numerical schemes are important in these applications in order to capture geometric features such as folds and corners, and avoid artificial singularities which arise from bad representations of the operators.

In general these equations are considered too difficult to solve, which is why linearized models or other approximations are commonly used. Progress has recently been made in building solvers for a class of Geometric PDEs. I'll discuss a few important geometric PDEs which can be solved using a numerical method called Wide Stencil finite difference schemes: Monge-Ampere, Convex Envelope, Infinity Laplace, Mean Curvature, and others.

Focusing in on the Monge-Ampere equation, which is the seminal geometric PDE, I'll show how naive schemes can work well for smooth solutions, but break down in the singular case. Several groups of researchers have proposed numerical schemes which fail to converge, or converge only in the case of smooth solutions. I'll present a convergent solver which which is fast: comparable to solving the Laplace equation a few times.

##### Asynchronous Distributed Systems: Modeling and Analysis

Over the past few decades, the rapid evolution of computing, communication, and sensor technologies has brought about the proliferation of asynchronous distributed dynamic systems, mostly artificially constructed and often highly complex, such as intelligent transportation systems, computer and communication networks, automated manufacturing systems, air traffic control systems, and distributed software systems. In such man-made systems, most activities are governed by operational rules, and their dynamics are therefore characterized by occurrences of discrete events. Modeling and analyzing such systems are far beyond the scope of traditional system theory that is built on differential/difference equations. In this talk, I will first present modeling tools that are appropriate for asynchronous distributed systems, and then focus on specific estimation and diagnosis problems to demonstrate techniques that can be used for system analysis.

##### A new proof of Haglund's Diagonal Harmonics Hilbert Series Result

By a tour de force argument Jim Haglund opened up a new
approach to the computation of several Hilbert series
formulas for the Diagonal Harmonics and other closely
related Sn modules. Recent joint work with Haglund
revealed that the same formulas can be obtained
by less painful but more sophisticated uses of
pre-existing Macdonald Polynomial identities.
In this talk we present a variety of new manipulatorial
and combinatorial problems that naturally emerge
from these new discoveries.

##### Conifold transitions via tropical geometry

The process of degenerating a complex variety $X$ to a
singular variety $X_0$
and then resolving to obtain $X'$ is called a geometric transition. The case where
the singularities are just double points is called a conifold transition. There
are known obstructions to either resolving a set of nodes or smoothing
them, depending
on whether we want to preserve respectively the symplectic or complex structure.
Moreover mirror symmetry is thought to reverse this process, i.e. the
mirror of a
smoothing is expected to be a resolution and vice versa. I will
explain an interpretation
of these facts in terms of tropical geometry", which encodes
information of both symplectic
and complex geometry in terms of discrete data.

##### Charge-based Approach to the Poisson-Boltzmann Continuum Solvation Treatment

Continuum modeling of electrostatic interactions based upon numerical solutions of the Poisson-Boltzmann equation has been widely adopted in biomolecular applications. To extend their applications to molecular dynamics and energy minimization, accurate and robust methodologies to compute solvation energies and forces must be developed. Indeed, energy conservation is still not possible in molecular dynamics simulations with any numerical Poisson-Boltzmann methods at realistic spatial discretization resolutions of 1/4 to 1/2 Angstrom. It was observed a while ago that the use of surface polarization charges in the computation of solvation energies may greatly enhance the energy convergence in the finite-difference solution of the Poisson equation. In this talk, I will discuss our recent efforts to generalize the strategy to the finite-difference solutions of the full Poisson-Boltzmann equation and to the computation of solvation forces. Our preliminary studies show that a combination of the charge-based strategy, the harmonic average treatment of the dielectric interface, and the charge singularity removal offers the convergence quality in energies and forces on a par with higher-order finite-difference numerical solutions.

##### On the arithmetic of $1$--motives

$1$--motives have been introduced by Deligne in the 1970s as the
simplest kind of mixed motives; basically, they are the only ones having
a simple and concrete geometric description. In recent years there has been a
resurgence of interest in them, and they have been successfully applied to
arithmetic questions as well, such as generalizations of the classical
duality theorems of Poitou and Tate, or finding rational points on
algebraic varieties. I shall explain some of these results.

##### Schr\"odinger Maps"

I will introduce and motivate the Schr\"odinger Map problem.

##### Galois Theory: Past and Present

In 2011 we shall celebrate the 200th anniversary of the birth of
Evariste Galois. On this occasion I shall explain what Galois himself
discovered in the theory of algebraic equations and how his ideas are
still alive in present-day research.