Seminars and Colloquia by Series

Gevrey smoothing of weak solutions of the homogeneous Boltzmann equation for Maxwellian molecules

Series
Math Physics Seminar
Time
Tuesday, March 14, 2017 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Tobias Ried – Karlsruhe Institute of Technology –
We study regularity properties of weak solutions of the homogeneous Boltzmann equation. While under the so called Grad cutoff assumption the homogeneous Boltzmann equation is known to propagate smoothness and singularities, it has long been suspected that the non-cutoff Boltzmann operator has similar coercivity properties as a fractional Laplace operator. This has led to the hope that the homogenous Boltzmann equation enjoys similar smoothing properties as the heat equation with a fractional Laplacian. We prove that any weak solution of the fully nonlinear non-cutoff homogenous Boltzmann equation (for Maxwellian molecules) with initial datum $f_0$ with finite mass, energy and entropy, that is, $f_0 \in L^1_2(\R^d) \cap L \log L(\R^d)$, immediately becomes Gevrey regular for strictly positive times, i.e. it gains infinitely many derivatives and even (partial) analyticity.This is achieved by an inductive procedure based on very precise estimates of nonlinear, nonlocal commutators of the Boltzmann operator with suitable test functions involving exponentially growing Fourier multipliers.(Joint work with Jean-Marie Barbaroux, Dirk Hundertmark, and Semjon Vugalter)

Joint GT-UGA Seminar at GT: Gluck twists and trisections

Series
Geometry Topology Seminar
Time
Monday, March 13, 2017 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
David Gay – UGA
This is joint work with Jeff Meier. The Gluck twist operation removes an S^2XB^2 neighborhood of a knotted S^2 in S^4 and glues it back with a twist, producing a homotopy S^4 (i.e. potential counterexamples to the smooth Poincare conjecture, although for many classes of 2-knots theresults are in fact known to be smooth S^4's). By representing knotted S^2's in S^4 as doubly pointed Heegaard triples and understanding relative trisection diagrams of S^2XB^2 carefully, I'll show how to produce trisection diagrams (a.k.a. Heegaard triples) for these homotopy S^4's.(And for those not up on trisections I'll review the foundations.) The resulting recipe is surprisingly simple, but the fun, as always, is in the process.

Low-rank approximations of binary forms

Series
Algebra Seminar
Time
Monday, March 13, 2017 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Hwangrae Lee – Auburn University
For a given generic form, the problem of finding the nearest rank-one form with respect to the Bombieri norm is well-studied and completely solved for binary forms. Nonetheless, higher-rank approximation is quite mysterious except in the quadratic case. In this talk we will discuss such problems in the binary case.

Involutive Heegaard Floer homology

Series
Geometry Topology Seminar
Time
Monday, March 13, 2017 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Kristen Hendricks – Michigan State University
We use the conjugation symmetry on the Heegaard Floer complexes to define a three-manifold invariant called involutive Heegaard Floer homology, which is meant to correspond to Z_4-equivariant Seiberg-Witten Floer homology. From this we obtain two new invariants of homology cobordism, explicitly computable for surgeries on L-space knots and quasi-alternating knots, and two new concordance invariants of knots, one of which (unlike other invariants arising from Heegaard Floer homology) detects non-sliceness of the figure-eight knot. We also give a formula for how this theory behaves under connected sum, and use it to give examples not homology cobordant to anything computable via our surgery formula. This is joint work with C. Manolescu; the last part of is also joint with I. Zemke.

Polynomial convergence rate to nonequilibrium steady-state

Series
Applied and Computational Mathematics Seminar
Time
Monday, March 13, 2017 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Prof. Yao Li – University of Massachusetts Amherst –
In this talk I will present my recent result about the ergodic properties of nonequilibrium steady-state (NESS) for a stochastic energy exchange model. The energy exchange model is numerically reduced from a billiards-like deterministic particle system that models the microscopic heat conduction in a 1D chain. By using a technique called the induced chain method, I proved the existence, uniqueness, polynomial speed of convergence to the NESS, and polynomial speed of mixing for the stochastic energy exchange model. All of these are consistent with the numerical simulation results of the original deterministic billiards-like system.

An Application of Combinatorics on Posets to Topological Graph Theory

Series
Combinatorics Seminar
Time
Friday, March 10, 2017 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Tom Trotter – Georgia Tech
Researchers here at Georgia Tech initiated a "Ramsey Theory" on binary trees and used the resulting tools to show that the local dimension of a poset is not bounded in terms of the tree-width of its cover graph. Subsequently, in collaboration with colleagues in Germany and Poland, we extended these Ramsey theoretic tools to solve a problem posed by Seymour. In particular, we showed that there is an infinite sequence of graphs with bounded tree-chromatic number and unbounded path-chromatic number. An interesting detail is that our research showed that a family conjectured by Seymour to have this property did not. However, the insights gained in this work pointed out how an appropriate modification worked as intended. The Atlanta team consists of Fidel Barrera-Cruz, Heather Smith, Libby Taylor and Tom Trotter The European colleagues are Stefan Felsner, Tamas Meszaros, and Piotr Micek.

Conjugacy of circle maps to rotations

Series
Dynamical Systems Working Seminar
Time
Friday, March 10, 2017 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 254
Speaker
Rafael de la Llave – GT Math
A classical theorem of Arnold, Moser shows that in analytic families of maps close to a rotation we can find maps which are smoothly conjugate to rotations. This is one of the first examples of the KAM theory. We aim to present a self-contained version of Moser's proof and also to present some efficient numerical algorithms.

Lagrangian Floer Theory I

Series
Geometry Topology Working Seminar
Time
Friday, March 10, 2017 - 14:00 for 1.5 hours (actually 80 minutes)
Location
Skiles 006
Speaker
John Etnyre – Georgia Tech

Please Note: This will be a 1.5 hour seminar.

Following up on the previous series of talks we will show how to construct Lagrangian Floer homology and discuss it properties.

Hardness Results for Solving Graph-Structured Linear Systems

Series
ACO Student Seminar
Time
Friday, March 10, 2017 - 13:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Peng Zhang – College of Computing, Georgia Tech
Spielman and Teng (2004) showed that linear systems in Laplacian matrices can be solved in nearly linear time. Since then, a major research goal has been to develop fast solvers for linear systems in other classes of matrices. Recently, this has led to fast solvers for directed Laplacians (CKPPRSV'17) and connection Laplacians (KLPSS'16).Connection Laplacians are a special case of PSD-Graph-Structured Block Matrices (PGSBMs), block matrices whose non-zero structure correspond to a graph, and which additionally can be expressed as a sum of positive semi-definite matrices each corresponding to an edge in the graph. A major open question is whether fast solvers can be obtained for all PGSBMs (Spielman, 2016). Fast solvers for Connection Laplacians provided some hope for this. Other important families of matrices in the PGSBM class include truss stiffness matrices, which have many applications in engineering, and multi-commodity Laplacians, which arise when solving multi-commodity flow problems. In this talk, we show that multi-commodity and truss linear systems are unlikely to be solvable in nearly linear time, unless general linear systems (with integral coefficients) can be solved in nearly linear time. Joint work with Rasmus Kyng.

Similarities and differences: faculty positions at research universities versus highly selective liberal arts colleges

Series
Professional Development Seminar
Time
Thursday, March 9, 2017 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 114
Speaker
Julianna Tymoczko – Smith College
A conversation with Julianna Tymoczko, associate professor and chair of the Department of Mathematics & Statistics at Smith, who received her BS from Harvard and PhD from Princeton and was a postdoc at the University of Michigan and assistant professor at the University of Iowa.

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