Seminars and Colloquia by Series

A solution to the Burr-Erdos problems on Ramsey completeness

Series
School of Mathematics Colloquium
Time
Thursday, November 21, 2019 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jacob FoxStanford University

A sequence A of positive integers is r-Ramsey complete if for every r-coloring of A, every sufficiently large integer can be written as a sum of the elements of a monochromatic subsequence. Burr and Erdos proposed several open problems in 1985 on how sparse can an r-Ramsey complete sequence be and which polynomial sequences are r-Ramsey complete. Erdos later offered cash prizes for two of these problems. We prove a result which solves the problems of Burr and Erdos on Ramsey complete sequences. The proof uses tools from probability, combinatorics, and number theory. 

Joint work with David Conlon.

The condition number of square random matrices

Series
High Dimensional Seminar
Time
Wednesday, November 20, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Michail SarantisGeorgiaTech

The condition number of a matrix A is the quantity κ(A) = smax(A)/smin(A), where smax(A), smin(A) are the largest and smallest singular values of A, respectively. Let A be a random n × n matrix with i.i.d, mean zero, unit variance, subgaussian entries. We will discuss a result by Litvak, Tikhomirov and Tomczak-Jaegermann which states that, in this setting, the condition number satisfies the small ball probability estimate

P{κ(A) ≤ n/t} ≤ 2 exp(−ct^2), t ≥ 1, where c > 0 is a constant depending only on the subgaussian moment.

Prime Decomposition of 3-Manifolds

Series
Geometry Topology Student Seminar
Time
Wednesday, November 20, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Thomas RodewaldGeorgia Tech

I will discuss the prime decomposition of three-manifolds. First I will define the connect sum operation, irreducible and prime 3-manifolds. Then using the connect sum operation as "multiplication," I will show any closed oriented three-manifold decomposes uniquely into prime factors using spheres. If time permits, I will show another way of decomposing using discs.

The Shape of Things: Organizing space using algebra

Series
Research Horizons Seminar
Time
Wednesday, November 20, 2019 - 12:20 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Miriam Kuzbary

Determining when two objects have “the same shape” is difficult; this difficulty depends on the dimension we are working in. While many of the same techniques work to study things in dimensions 5 and higher, we can better understand dimensions 1, 2, and 3 using other methods. We can think of 4-dimensional space as the “bridge” between low-dimensional behavior and high-dimensional behavior.

 

One way to understand the possibilities in each dimension is to examine objects called cobordisms: if an (n+1)-dimensional space has an ``edge,” which is called a boundary, then that boundary is itself an n-dimensional space. We say that two n-dimensional spaces are cobordant if together they form the boundary of an (n+1)-dimensional space. Using the idea of spaces related by cobordism, we can form an algebraic structure called a group. In this way, we can attempt to understand higher dimensions using clues from lower dimensions.

 

In this talk, I will discuss different types of cobordism groups and how to study them using tools from a broad range of mathematical areas.

Comparing high-dimensional neural distributions with computational geometry and optimal transport 

Series
Mathematical Biology Seminar
Time
Wednesday, November 20, 2019 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Eva DyerGeorgia Tech (BME & ECE)

In both biological brains and artificial neural networks, the representational geometry - the shape and distribution of activity - at different layers in an artificial network or across different populations of neurons in the brain, can reveal important signatures of the underlying computations taking place. In this talk, I will describe how we are developing strategies for comparing and aligning neural representations, using a combination of tools from computational geometry and optimal transport.

Invariant Gibbs measures and global strong solutions for 2D nonlinear Schrödinger equations

Series
PDE Seminar
Time
Tuesday, November 19, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Andrea R. NahmodUniversity of Massachusetts Amherst

In this talk I'll first give an background overview of Bourgain's approach to prove the invariance of the Gibbs measure for the periodic cubic nonlinear Schrodinger equation in 2D and of the para-controlled calculus of Gubinelli-Imkeller and Perkowski in the context of parabolic stochastic equations. I will then present our resolution of the long-standing problem of proving almost sure global well-posedness (i.e. existence /with uniqueness/) for the periodic nonlinear Schrödinger equation (NLS) in 2D on the support of the Gibbs measure, for any (defocusing and renormalized) odd power nonlinearity. Consequently we get the invariance of the Gibbs measure. This is achieved by a new method we call /random averaging operators /which precisely captures the intrinsic randomness structure of the problematic high-low frequency interactions at the heart of this problem. This is work with Yu Deng (USC) and Haitian Yue (USC).

Physical Periodic Ehrenfests' Wind-Tree Model

Series
Dynamical Systems Working Seminar
Time
Tuesday, November 19, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Hassan AttarchiGT, School of Math

We consider a physical periodic Ehrenfests' Wind-Tree model where a moving particle is a hard ball rather than (mathematical) point particle. Some dynamics and statistical properties of this model are studied. Moreover, it is shown that it has a new superdiffusive regime where the diffusion coefficient $D(t)\sim(\ln t)^2$ of dynamics seems to be never observed before in any model.

Free resolutions of function classes via order complexes

Series
Algebra Seminar
Time
Tuesday, November 19, 2019 - 13:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Justin ChenGeorgia Institute of Technology

Function classes are collections of Boolean functions on a finite set. Recently, a method of studying function classes via commutative algebra, by associating a squarefree monomial ideal to a function class, was introduced by Yang. I will describe this connection, as well as some free resolutions and Betti numbers for these ideals for an interesting collection of function classes, corresponding to intersection-closed posets. This is joint work with Chris Eur, Greg Yang, and Mengyuan Zhang.

Multiscale analysis of sets and measures

Series
Job Candidate Talk
Time
Tuesday, November 19, 2019 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Ben JayeClemson University

In this talk I will give an introduction to certain aspects of geometric Littlewood-Paley theory, which is an area of harmonic analysis concerned with deriving regularity properties of sets and measures from the analytic behavior of associated operators. The work we shall describe has been carried out in collaboration with Fedor Nazarov, Maria Carmen Reguera, Xavier Tolsa, and Michele Villa.

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