Department of Mathematics,
University of California San Diego

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Algebra Seminar

Nolan Wallach
UCSD

$\bf{GK}$ Dimensions of $\bf{gK}$-modules

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AP&M 7218

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Department of Mathematics,
University of California San Diego

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Math 268 - Logic and Computation

Sam Buss

Toda's Theorem V

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AP&M 5402

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Department of Mathematics,
University of California San Diego

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Center for Computational Mathematics Seminar

Stefan Sauter
University of Zurich

Convergence Analysis for Finite Element Discretizations of Highly Indefinite Helmholtz Problems

Abstract:

\indent A rigorous convergence theory for Galerkin methods for a model Helmholtz problem in $R^{d}, d=1,2,3,$ is presented.
General conditions on the approximation properties of the approximation space are stated that ensure quasi-optimality of the method. As an application of the general theory, a full error analysis of the classical hp-version of the finite element method (hp-FEM) is presented where the dependence on the mesh width $h$, the approximation order $p$, and the wave number $k$ is given explicitly. In particular, it is shown that quasi-optimality is obtained under the conditions that $kh/p$ is sufficiently small and the polynomial degree $p$ is at least $O(log k)$. This result improves existing stability conditions substantially.

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AP&M 2402

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Department of Mathematics,
University of California San Diego

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Math 269 - Combinatorics

Mark Tiefenbruck
UCSD

Using an Extension of the Garsia-Milne Involution Principle to Find Bijections

Abstract:

\indent We will consider two recent open problems stating that
certain statistics on various sets of combinatorial objects are
equidistributed. The first, posed by Anders Claesson and Svante
Linusson, relates nestings in matchings of $2n$ points on a line to
occurrences of a certain pattern in permutations in $S_n$. The second, posed by Miles Jones and Jeffrey Remmel, relates occurrences of a large class of consecutive permutation patterns to occurrences of the same pattern in the cycles of permutations. We will prove an extension of the Garsia-Milne involution principle and use it to solve both problems.

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AP&M 7321

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Department of Mathematics,
University of California San Diego

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Math 268 - Logic and Computation

Sam Buss

Toda's Theorem VI

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AP&M 5402

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Department of Mathematics,
University of California San Diego

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Algebraic Geometry Seminar

Anton Geraschenko

When is a variety a quotient of a smooth variety by a finite group?

Abstract:

\indent If a variety $X$ is a quotient of a smooth variety by a finite group, it has quotient singularities---that is, it is \emph{locally} a quotient by a finite group. In this talk, we will see that the converse is true if $X$ is quasi-projective and is known to be a quotient by a torus. In particular, all quasi-projective simplicial toric varieties are global quotients by finite groups! Though the proof is stack-theoretic, the construction of a smooth variety $U$ and finite group $G$ so that $X=U/G$ can usually be made explicit purely scheme-theoretically.

\indent To illustrate the construction, I'll produce a smooth variety $U$ with an action of $G=\mathbb{Z}/2$ so that $U/G$ is the blow-up of $\mathbb{P}(1,1,2)$ at a smooth point. This example is interesting because even though $U/G$ is toric, $U$ cannot be taken to be toric. This is joint work with Matthew Satriano.

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AP&M 7218

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Department of Mathematics,
University of California San Diego

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Math 288 - Probability and Statistics Seminar

Tom Laetsch
UCSD

An $\bf{L^2}$ metric limit theorem for Wiener measure on manifolds with non-positive sectional curvature

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AP&M 6402

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Department of Mathematics,
University of California San Diego

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Mathematical and Numerical Relativity Seminar

Jim Isenberg
University of Oregon

Solutions of the Initial Value Constraint Equations of General Relativity

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AP&M 2402

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Department of Mathematics,
University of California San Diego

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Food For Thought Seminar

Alex Eustis
UCSD

Independence Number of Hypergraphs

Abstract:

\indent An $r$-graph is like a graph except that every edge contains $r$ vertices instead of two. We'll talk about how to find a large independent set in an $r$-graph, which means a set of vertices not containing any edge. Conversely, we'll also discuss how to generate a hypergraph where large independent sets do not exist. Some of research is joint with Jacques Verstraete.

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AP&M 7321

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Department of Mathematics,
University of California San Diego

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Informal Seminar on Mathematics and Biochemistry-Biophysics

Zhenli Xu
Department of Mathematics and Institute of Natural Sciences \newline Shanghai Jiaotong University, China \newline Department of Mathematics, UCSD

Image Effects in Colloidal Suspensions

Abstract:

\indent It is widely known the classical Poisson-Boltzmann theory fails to describe electrostatic interactions in the cases of multivalent counterions, highly charged surfaces, and low temperatures, as the theory ignores many-body ion-ion correlations. Surprising phenomena due to correlation effects include the charge inversion and the attraction between two likely charged colloids, both of which are recent focuses of theoretical and experimental study. In this talk, we will present some theoretical and simulation results on these problems by studying the role of image charges on spherical colloids, and discuss the method to include the correlation effects in the mean-field PB theory.

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AP&M 5829

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Department of Mathematics,
University of California San Diego

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Number Theory Seminar

Michael Skirvin
Northwestern University

Geometric Langlands and global Springer theory

Abstract:

\indent I will begin by giving a broad overview of the geometric Langlands progam, with emphasis on the local to global results of Beilinson, Drinfeld, Frenkel, and Gaitsgory. Their methods are particularly compelling because they do not have analogues in the original Langlands program. Using geometric Langlands as motivation, I will introduce the Hitchin fibration and describe recent results regarding the geometry of the global nilpotent cone (i.e., Hitchin fiber over zero). These results may be viewed in the
context of a global analogue of Springer theory, which suggests many future directions. If there is time, I will also explain relations to classical and higher rank Brill-Noether theory.

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AP&M 7321

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