Seminars and Colloquia by Series

Integrability and wave turbulence for Hamiltonian partial differential equations

Series
School of Mathematics Colloquium
Time
Tuesday, February 9, 2016 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Patrick Gerard – Université Paris-Sud –
In the world of Hamiltonian partial differential equations, complete integrability is often associated to rare and peaceful dynamics, while wave turbulence rather refers to more chaotic dynamics. In this talk I will first try to give an idea of these different notions. Then I will discuss the example of the cubic Szegö equation, a nonlinear wave toy model which surprisingly displays both properties. The key is a Lax pair structure involving Hankel operators from classical analysis, leading to the inversion of large ill-conditioned matrices. .

Convexity over lattices and discrete sets: new theorems on Minkowski's Geometry of Numbers.

Series
School of Mathematics Colloquium
Time
Monday, February 8, 2016 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Jesus De Loera – University of California, Davis
Convex analysis and geometry are tools fundamental to the foundations of several applied areas (e.g., optimization, control theory, probability and statistics), but at the same time convexity intersects in lovely ways with topics considered pure (e.g., algebraic geometry, representation theory and of course number theory). For several years I have been interested interested on how convexity relates to lattices and discrete subsets of Euclidean space. This is part of mathematics H. Minkowski named in 1910 "Geometrie der Zahlen''. In this talk I will use two well-known results, Caratheodory's & Helly's theorems, to explain my most recent work on lattice points on convex sets. The talk is for everyone! It is designed for non-experts and grad students should understand the key ideas. All new theorems are joint work with subsets of the following mathematicians I. Aliev, C. O'Neill, R. La Haye, D. Rolnick, and P. Soberon.

Constancy regions for mixed test ideals

Series
Algebra Seminar
Time
Monday, February 8, 2016 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Felipe Pérez – Georgia State University
For the last four decades, mathematicians have used the Frobenius map to investigate phenomena in several fields of mathematics including Algebraic Geometry. The goal of this talk is twofold, first to give a brief introduction to the study of singularities in positive characteristic (aided by the Frobenius map). Second to define an explain the constancy regions for mixed test ideals in the case of a regular ambient; an invariant associated to a family of functions that shows a Fractal behavior.

The Kelmans-Seymour conjecture II: special separations (5-separations containing a triangle)

Series
Graph Theory Seminar
Time
Friday, February 5, 2016 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Yan Wang – Math, GT
Seymour and, independently, Kelmans conjectured in the 1970s that every 5-connected nonplanar graph contains a subdivision of K_5. This conjecture was proved by Ma and Yu for graphs containing K_4^-. In order to establish the Kelmans-Seymour conjecture for all graphs, we need to consider 5-separations and 6-separations with less restrictive structures. We will talk about special 5-separations and 6-separations whose cut contains a triangle. Results will be used in subsequently to prove the Kelmans-Seymour conjecture.

A census of Platonic manifolds

Series
Geometry Topology Seminar
Time
Friday, February 5, 2016 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Matthias Goerner – Pixar –
We call a 3-manifold Platonic if it can be decomposed into isometric Platonic solids. Many key examples in 3-manifold topology are Platonic manifolds, e.g., the Poincar\'e homology sphere, the Seifert-Weber dodecahedral space and the complements of the figure eight knot, the Whitehead link, and the minimally twisted 5-component chain link. They have a strong connection to regular tessellations and illustrate many phenomena such as hidden symmetries.I will talk about recent work on a census of hyperbolic Platonic manifolds and some new techniques we developed for its creation, e.g., verified canonical cell decompositions and the isometry signature which is a complete invariant of a cusped hyperbolic manifold.

On the Widom-Rowlinson occupancy fraction in regular graphs

Series
ACO Student Seminar
Time
Friday, February 5, 2016 - 13:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Emma Cohen – Georgia Tech
We consider the Widom-Rowlinson model of two types of interacting particles on $d$-regular graphs. We prove a tight upper bound on the occupancy fraction: the expected fraction of vertices occupied by a particle under a random configuration from the model. The upper bound is achieved uniquely by unions of complete graphs on $d+1$ vertices. As a corollary we find that $K_{d+1}$ also maximizes the normalized partition function of the Widom-Rowlinson model over the class of $d$-regular graphs, proving a conjecture of Galvin. Joint work with Will Perkins and Prasad Tetali.

Random zero sets under repeated differentiation of an analytic function

Series
Stochastics Seminar
Time
Thursday, February 4, 2016 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Sneha Subramanian – School of Mathematics, Georgia Tech
For a random (complex) entire function, what can we say about the behavior of the zero set of its N-th derivative, as N goes to infinity? In this talk, we shall discuss the result of repeatedly differentiating a certain class of random entire functions whose zeros are the points of a Poisson process of intensity 1 on the real line. We shall also discuss the asymptotic behavior of the coefficients of these entire functions. Based on joint work with Robin Pemantle.

Optimization of Network Dynamics: Attributes and Artifacts

Series
School of Mathematics Colloquium
Time
Thursday, February 4, 2016 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Prof. Dr. Adilson E. Motter – Northwestern University –
The recent interest in network modeling has been largely driven by the prospect that network optimization will help us understand the workings of evolution in natural systems and the principles of efficient design in engineered systems. In this presentation, I will reflect on unanticipated properties observed in three classes of network optimization problems. First, I will discuss implications of optimization for the metabolic activity of living cells and its role in giving rise to the recently discovered phenomenon of synthetic rescues. I will then comment on the problem of controlling network dynamics and show that theoretical results on optimizing the number of driver nodes often only offer a conservative lower bound to the number actually needed in practice. Finally, I will discuss the sensitive dependence of network dynamics on network structure that emerges in the optimization of network topology for dynamical processes governed by eigenvalue spectra, such as synchronization and consensus processes. It follows that optimization is a double-edged sword for which desired and adverse effects can be exacerbated in network systems due to the high dimensionality of their phase spaces.

Homogeneous solutions to the incompressible Euler equation

Series
PDE Seminar
Time
Wednesday, February 3, 2016 - 16:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Roman Shvydkoy – University of Illinois, Chicago
In this talk we describe recent results on classification and rigidity properties of stationary homogeneous solutions to the 3D and 2D Euler equations. The problem is motivated be recent exclusions of self-similar blowup for Euler and its relation to Onsager conjecture and intermittency. In 2D the problem also arises in several other areas such as isometric immersions of the 2-sphere, or optimal transport. A full classification of two dimensional solutions will be given. In 3D we reveal several new classes of solutions and prove their rigidity properties. In particular, irrotational solutions are characterized by vanishing of the Bernoulli function; and tangential flows are necessarily 2D axisymmetric pure rotations. In several cases solutions are excluded altogether. The arguments reveal geodesic features of the Euler equation on the sphere. We further discuss the case when homogeneity corresponds to the Onsager-critical state. We will show that anomalous energy flux at the singularity vanishes, which is suggestive of absence of extreme $0$-dimensional intermittencies in dissipative flows.

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