Informal Topology Seminar
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Posted September 20, 2017

10:30 am - 12:00 pm Lockett 233

Ryan Leigon, Louisiana State University

Federico Salmoiraghi, Department of Mathematics, LSU

Equivalence of gluing maps in Heegaard Floer theory

Abstract: We show that the gluing mas in Heegaard Floer theory defined by Honda, Kazez and Matic and by Zarev are equivalent.

Applied Analysis Seminar
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Posted August 25, 2017

Last modified September 8, 2017

Changfeng Gui, University of Texas at San Antonio

The Sphere Covering Inequality and its applications

In this talk, I will introduce a new geometric inequality: the Sphere Covering Inequality. The inequality states that the total area of two {it distinct} surfaces with Gaussian curvature less than 1, which are also conformal to the Euclidean unit disk with the same conformal factor on the boundary, must be at least $4 pi$. In other words, the areas of these surfaces must cover the whole unit sphere after a proper rearrangement. We apply the Sphere Covering Inequality to show the best constant of a Moser-Trudinger type inequality conjectured by A. Chang and P. Yang. Other applications of this inequality include the classification of certain Onsager vortices on the sphere, the radially symmetry of solutions to Gaussian curvature equation on the plane, classification of solutions for mean field equations on flat tori and the standard sphere, etc. The resolution of several open problems in these areas will be presented. The talk is based on joint work with Amir Moradifam from UC Riverside.

Colloquium
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Posted August 22, 2017

Last modified August 27, 2017

Hongyu He, Department of Mathematics, LSU

Branching Laws and Interlacing Relation

Abstract: Let $H$ be a subgroup of a compact group $G$. Then any irreducible unitary representation of $G$, when restricted to $H$, decomposes into a direct sum of irreducible representations of $H$. A description of such a decomposition is often called a branching law. They are important in harmonic analysis, quantum mechanics and number theory. In this talk, I shall discuss the branching laws of the discrete series of the noncompact unitary groups and the recent progress towards the local Gan-Gross-Prasad conjectures. Discrete series representations were classified by Harish-Chandra in the sixties and played a fundamental role in Langland's program.

Computational Mathematics Seminar

Posted August 22, 2017

3:30 pm - 4:30 pm 1034 Digital Media Center
Yangyang Xu, University of Alabama

Primal-dual methods for affinely constrained problems

Abstract: Optimization has been applied in many areas including engineering, statistics, finance, and data sciences. Modern applications often have rich structure information. Traditional methods like projected subgradient and the augmented Lagrangian can be used, but they do not utilize structures of the problems and thus are not so efficient. This talk will focus on convex optimization problems with affine constraints. The first part assumes two-block structure on the problem and presents the alternating direction method of multipliers (ADMM) and its accelerated variant. With strong convexity on one block variable, the ADMM can be accelerated from O(1/k) rate to O(1/k^2). Numerical results will be given to demonstrate the improved speed. In the second part, I will present a novel primal-dual block update method for a multi-block (at least three blocks) problem. Existing works have shown that directly extending two-block ADMM to multi-block problems may diverge. To guarantee convergence, either strong assumptions are made or updating order of the blocks has to be changed. Our method uses a simple randomization technique on choosing block variables, and it enjoys O(1/k) ergodic convergence rate and also global convergence in probability. In addition, by choosing a few blocks every time and using Jacobi-type update, the method enables parallel computing with guaranteed convergence. Numerical experiments will be shown to demonstrate its efficiency compared to other methods.

Informal Topology Seminar
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Posted August 23, 2017

Last modified September 11, 2017

Yu-Chan Chang, Louisiana State University

Introduction to bounded cohomology of discrete groups

Abstract: In this introductory talk, we will focus on bounded cohomology of discrete groups with real or integer coefficient. I will talk about the bounded cohomology of Gromov hyperbolic groups and amenable groups. Also, we will discuss the comparison map, which is the map from the bounded cohomology to the usual cohomology.

Topology Seminar
Seminar website

Posted August 24, 2017

Last modified September 20, 2017

John Etnyre, Georgia Institute of Technology

Contact surgeries and symplectic fillings

Abstract: It is well known that all contact manifolds can be obtained from the standard contact structure on the 3-sphere by contact surgery on a Legendrian link. What is not so well understood is what properties of a contact structure are preserved by positive contact surgeries (the case for negative contact surgeries is fairly well understood now). In this talk we will discuss some new results about positive contact surgeries and in particular completely characterize when contact r surgery is symplectically fillable when r is in (0,1].