Seminars and Colloquia by Series

Contagion in random graphs and systemic risk

Series
Combinatorics Seminar
Time
Friday, February 15, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Hamed Amini – Georgia State University
We provide a framework for testing the possibility of large cascades in random networks. Our results extend previous studies on contagion in random graphs to inhomogeneous directed graphs with a given degree sequence and arbitrary distribution of weights. This allows us to study systemic risk in financial networks, where we introduce a criterion for the resilience of a large network to the failure (insolvency) of a small group of institutions and quantify how contagion amplifies small shocks to the network.

The Proof of an Abstract Nash-Moser Implicit Function Theorem

Series
Dynamical Systems Working Seminar
Time
Friday, February 15, 2019 - 03:05 for 1 hour (actually 50 minutes)
Location
Skiles 246
Speaker
Yian Yao – GT Math
I will present a proof of an abstract Nash-Moser Implicit Function Theorem. This theorem can cope with derivatives which are not boundly invertible from one space to itself. The main technique is to combine Newton steps - which loses derivatives with some smoothing that restores them.

A tight net with respect to a random matrix norm and applications to estimating singular values

Series
Stochastics Seminar
Time
Thursday, February 14, 2019 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
G. Livshyts – SOM, GaTech
In this talk we construct a net around the unit sphere with strong properties. We show that with exponentially high probability, the value of |Ax| on the sphere can be approximated well using this net, where A is a random matrix with independent columns. We apply it to study the smallest singular value of random matrices under very mild assumptions, and obtain sharp small ball behavior. As a partial case, we estimate (essentially optimally) the smallest singular value for matrices of arbitrary aspect ratio with i.i.d. mean zero variance one entries. Further, in the square case we show an estimate that holds only under simply the assumptions of independent entries with bounded concentration functions, and with appropriately bounded expected Hilbert-Schmidt norm. A key aspect of our results is the absence of structural requirements such as mean zero and equal variance of the entries.

Polynomial to exponential transition in Ramsey theory

Series
Joint School of Mathematics and ACO Colloquium
Time
Thursday, February 14, 2019 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Dhruv Mubayi – University of Illinois at Chicago
After a brief introduction to classical hypergraph Ramsey numbers, I will focus on the following problem. What is the minimum t such that there exist arbitrarily large k-uniform hypergraphs whose independence number is at most polylogarithmic in the number of vertices and every s vertices span at most t edges? Erdos and Hajnal conjectured (1972) that this minimum can be calculated precisely using a recursive formula and Erdos offered $500 for a proof. For k=3, this has been settled for many values of s, but it was not known for larger k. Here we settle the conjecture for all k at least 4. Our method also answers a question of Bhatt and Rodl about the maximum upper density of quasirandom hypergraphs. This is joint work with Alexander Razborov.

Convex Geometry of the Truncated Moment Problem

Series
High Dimensional Seminar
Time
Wednesday, February 13, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Greg Blekherman – Georgia Tech

Moment problem is a classical question in real analysis, which asks whether a set of moments can be realized as integration of corresponding monomials with respect to a Borel measure. Truncated moment problem asks the same question given a finite set of moments. I will explain how some of the fundamental results in the truncated moment problem can be proved (in a very general setting) using elementary convex geometry. No familiarity with moment problems will be assumed. This is joint work with Larry Fialkow.

Some results for functionals of Aharanov-Bohm type

Series
Analysis Seminar
Time
Wednesday, February 13, 2019 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Michael Loss – Georgia Tech
In this talk I present some variational problems of Aharanov-Bohm type, i.e., they include a magnetic flux that is entirely concentrated at a point. This is maybe the simplest example of a variational problems for systems, the wave function being necessarily complex. The functional is rotationally invariant and the issue to be discussed is whether the optimizer have this symmetry or whether it is broken.

Tropical h-vectors of polytopes

Series
Research Horizons Seminar
Time
Wednesday, February 13, 2019 - 12:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Josephine Yu – Georgia Tech
For a polytope P, the h-vector is a vector of integers which can be calculated easily from the number of faces of P of each dimension. For simplicial polytopes, it is well known that the h-vector is symmetric (palindromic) and unimodal. However in general the h-numbers may even be negative. In this talk I will introduce the tropical h-vector of a polytope, which coincides with the usual h-vector of the dual polytope, if the polytope is simple. We will discuss how they are related to toric varieties, tropical geometry, and polytope algebra. I will also discuss some open problems.

Global solutions of incompressible viscoelastic fluids with large velocity on low frequency part

Series
PDE Seminar
Time
Tuesday, February 12, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
skiles 006
Speaker
Ting Zhang – Zhejiang University –

Abstract: In this talk, we consider the Cauchy problem of the N-dimensional incompressible viscoelastic fluids with N ≥ 2. It is shown that, in the low frequency part, this system possesses some dispersive properties derived from the one parameter group e∓itΛ. Based on this dispersive effect, we construct global solutions with large initial velocity concentrating on the low frequency part. Such kind of solution has never been seen before in the literature even for the classical incompressible Navier-Stokes equations. The proof relies heavily on the dispersive estimates for the system of acoustics, and a careful study of the nonlinear terms. And we also obtain the similar result for the isentropic compressible Navier-Stokes equations. Here, the initial velocity with arbitrary B⋅N 2 −1 2,1 norm of potential part P⊥u0 and large highly oscillating are allowed in our results. (Joint works with Daoyuan Fang and Ruizhao Zi)

Simplification of singularities of Lagrangian and Legendrian fronts

Series
Geometry Topology Seminar
Time
Monday, February 11, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Daniel Álvarez-Gavela – IAS
We will present an h-principle for the simplification of singularities of Lagrangian and Legendrian fronts. The h-principle says that if there is no homotopy theoretic obstruction to simplifying the singularities of tangency of a Lagrangian or Legendrian submanifold with respect to an ambient foliation by Lagrangian or Legendrian leaves, then the simplification can be achieved by means of a Hamiltonian isotopy. We will also discuss applications of the h-principle to symplectic and contact topology.

Convex-Nonconvex approach in segmentation and decomposition of scalar fields defined over triangulated surfaces

Series
Applied and Computational Mathematics Seminar
Time
Monday, February 11, 2019 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Martin Huska – University of bologna, Italy
In this talk, we will discuss some advantages of using non-convex penalty functions in variational regularization problems and how to handle them using the so-called Convex-Nonconvex approach. In particular, TV-like non-convex penalty terms will be presented for the problems in segmentation and additive decomposition of scalar functions defined over a 2-manifold embedded in \R^3. The parametrized regularization terms are equipped by a free scalar parameter that allows to tune their degree of non-convexity. Appropriate numerical schemes based on the Alternating Directions Methods of Multipliers procedure are proposed to solve the optimization problems.

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