Georgia Institute of Technology College of Sciences

I will highlight recent interplay between problems in extremal combinatorics and real algebraic geometry. This sheds a new light on undecidability of graph homomorphism density inequalities in extremal combinatorics, trace inequalities in linear algebra, and symmetric polynomial inequalities in real algebraic geometry. All of the necessary notions will be introduced in the talk. Joint work with Jose Acevedo, Sebastian Debus and Cordian Riener.

I'll talk about some 2D billiards, the most visual class of dynamical systems, where orbits (rays) move along straight lines within a billiard table with elastic reflections off the boundary.  Elliptic flowers are built “around" convex polygons, and the boundary of corresponding billiard tables consists of the arcs of ellipses. It will be explained why some classes of such elliptic flowers demonstrate a never expected before dynamics, and why it raises a variety of (seemingly new) questions in geometry (particularly in 3D), in bifurcation theory (particularly about singularities of wave fronts and creation of wave trains), in statistical mechanics,  quantum chaos, and perhaps some more. The talk will be concluded by showing a free movie. Everything (including various definitions of ellipses) will be explained/reminded.

We study the local and global dynamics of mean curvature flow with cylindrical singularities. We find the most generic dynamic behavior of such singularities, and show that the singularities with the most generic dynamic behavior are robust. We also show that the most generic singularities are isolated and type-I. Among applications, we prove that the singular set structure of the generic mean convex mean curvature flow has certain patterns, and the level set flow starting from a generic mean convex hypersurface has low regularity. This is joint work with Jinxin Xue (Tsinghua University)

Through the pioneering numerical computations of Fermi-Pasta-Ulam (mid 50s) and Kruskal-Zabusky (mid 60s) it was observed that nonlinear equations modeling wave propagation asymptotically decompose as a superposition of “traveling waves” and “radiation”. Since then, it has been a widely believed (and supported by extensive numerics) that “coherent structures” together with radiations describe the long-time asymptotic behavior of generic solutions to nonlinear dispersive equations. This belief has come to be known as the “soliton resolution conjecture”.  Roughly speaking it tells that, asymptotically in time, the evolution of generic solutions decouples as a sum of modulated solitary waves and a radiation term that disperses. This remarkable claim establishes a drastic “simplification” to the complex, long-time dynamics of general solutions. It remains an open problem to rigorously show such a description for most dispersive equations.  After an informal introduction to dispersive equations, I will illustrate how to understand the long-time behavior solutions to dispersive waves via various results I obtained over the years.

We consider a random walk on the $d\ge 3$ dimensional discrete torus starting from vertices chosen independently and uniformly at random. In this talk, we discuss the fluctuation behavior of the size of the range of the random walk trajectories at a time proportional to the size of the torus. The proof relies on a refined analysis of tail estimates for hitting time. We also discuss related results and open problems. This is based on joint work with Partha Dey.

The mathematically rigorous derivation of nonlinear Boltzmann equations from first principles in interacting physical systems is an extremely active research area in Analysis, Mathematical Physics, and Applied Mathematics. In classical physical systems, rigorous results of this type have been obtained for some models. In the quantum case on the other hand, the problem has essentially remained open. In this talk, I will explain how a cubic quantum Boltzmann equation arises within the fluctuation dynamics around a Bose-Einstein condensate, within the quantum field theoretic description of an interacting Boson gas. This is based on joint work with Michael Hott.

Join Zoom Meeting at https://gatech.zoom.us/j/92873362365

We will introduce some basic notions needed to talk about the question of decidability for roots of polynomials with coefficients in a specified ring R in the sense of Hilbert's tenth problem with an emphasis on rings of number theoretic interest. We will also attempt to give an overview of the literature on the topic and recent lines of work.

https://gatech.zoom.us/j/95197085752?pwd=WmtJUVdvM1l6aUJBbHNJWTVKcVdmdz09

Oscillations are ubiquitous in the brain, but their role is not completely understood. In this talk we will focus on the study of oscillations in neuronal networks. I will introduce some neuronal models and I will show how tools from dynamical systems theory, such as the parameterization method for invariant manifolds or the separatrix map, can be used to provide a thorough analysis of the oscillatory dynamics. I will show how the conclusions obtained may contribute to unveiling the role of oscillations in certain cognitive tasks.
 

In recent years, there has been increased interest in using quantum computing for the purposes of solving problems in combinatorial optimization. No prior knowledge of quantum computing is necessary for this talk; in particular, the talk will be divided into three parts: (1) a gentle high-level introduction to the basics of quantum computing, (2) a general framework for solving combinatorial optimization problems with quantum computing (the Quantum Approximate Optimization Algorithm introduced by Farhi et al.), (3) and some recent results that my colleagues and I have found. Our group has looked at the Max-Cut problem and have developed a new quantum algorithm that utilizes classically-obtained warm-starts in order to improve upon already-existing quantum algorithms; this talk will discuss both theoretical and experimental results associated with our approach with our main results being that we obtain a 0.658-approximation for Max-Cut, our approach provably converges to the Max-Cut as a parameter (called the quantum circuit depth) increases, and (on small graphs) are approach is able to empirically beat the (classical) Goemans-Williamson algorithm at a relatively low quantum circuit-depth (i.e. using a small amount of quantum resources). This work is joint with Jai Moondra, Bryan Gard, Greg Mohler, and Swati Gupta.

Let $A$ be drawn uniformly at random from the set of all $n \times n$ symmetric matrices with entries in $\{-1,1\}$. What is the probability that $A$ is singular? This is a classical problem at the intersection of probability and combinatorics. I will give an introduction to this type of question and sketch a proof that the singularity probability of $A$ is exponentially small in $n$. This is joint work with Marcelo Campos, Marcus Michelen and Julian Sahasrabudhe.

We show that for any natural number n, the set of domains containing absolutely periodic orbits of order n are dense in the set of bounded strictly convex domains with smooth boundary. The proof that such an orbit exists is an extension to billiard maps of the results of a paper by Gonchenko, Shilnikov, and Turaev, where it is proved that such maps are dense in Newhouse domains in regions of real-analytic area-preserving two-dimensional maps. Our result is a step toward disproving a conjecture that no absolutely periodic billiard orbits of infinite order exist in Euclidean billiards and is also an indication that Ivrii's Conjecture about the measure set of periodic orbits may not be true.

One of the most beautiful aspects of math is the interplay between its different fields. We will discuss one such interaction by studying topology using tools from combinatorics and group theory. In particular, given a surface (two-dimensional manifold) S, we construct the curve complex of S, which is a graph that encodes topological data about the surface. We will then state a seminal result of Ivanov: the symmetries of a surface S are in a natural bijection with the symmetries of its curve complex. In the direction of the proof of Ivanov's result, we will touch on some tools we have when working with infinite graphs.