Georgia Institute of Technology College of Sciences

The set of (higher) Weierstrass points on a curve of genus g > 1 is an analogue of the set of N-torsion points on an elliptic curve. As N grows, the torsion points "distribute evenly" over a complex elliptic curve. This makes it natural to ask how Weierstrass points distribute, as the degree of the corresponding divisor grows. We will explore how Weierstrass points behave on tropical curves (i.e. finite metric graphs), and explain how their distribution can be described in terms of electrical networks. Knowledge of tropical curves will not be assumed, but knowledge of how to compute resistances (e.g. in series and parallel) will be useful.

In 1665, Huygens discovered that, when two pendulum clocks hanged from a same wooden beam supported by two chairs, they synchronize in anti-phase mode. Metronomes provides a second example of oscillators that synchronize. As it can be seen in many YouTube videos, metronomes synchronize in-phase when oscillating on top of the same movable surface. In this talk, we will review these phenomena, introduce a mathematical model, and analyze the the different physical effects. We show that, in a certain parameter regime, the increase of the amplitude of the oscillations leads to a bifurcation from the anti-phase synchronization being stable to the in-phase synchronization being stable. This may explain the experimental observations.

It has been known that when an equiangular tight frame (ETF) of size |Φ|=N exists, Φ ⊂ Fd (real or complex), for p > 2 the p-frame potential ∑i ≠ j | < φj, φk > |p achieves its minimum value on an ETF over all N sized collections of vectors. We are interested in minimizing a related quantity: 1/ N2 ∑i, j=1 | < φj, φk > |p . In particular we ask when there exists a configuration of vectors for which this quantity is minimized over all sized subsets of the real or complex sphere of a fixed dimension. Also of interest is the structure of minimizers over all unit vector subsets of Fd of size N. We shall present some results for p in (2, 4) along with numerical results and conjectures. Portions of this talk are based on recent work of D. Bilyk, A. Glazyrin, R. Matzke, and O. Vlasiuk.

In this talk I will discuss the Mikhlin-H\"ormander multiplier theorem for $L^p$ boundedness of Fourier multipliers in which the multiplier belongs to a fractional Sobolev space with smoothness $s$. I will show that this theorem does not hold in the limiting case $|1/p - 1/2|=s/n$. I will also present a sharp variant of this theorem involving a space of Lorentz-Sobolev type. Some of the results presented in this talk were obtained in collaboration with Loukas Grafakos.

This talk will be an introduction to the homotopy principle (h-principle). We will discuss several examples. No prior knowledge about h-principle will be assumed.

TBA

Non-compact hyperbolic surfaces serve as a model case for quantum scattering theory with chaotic classical dynamics. In this talk I’ll explain how scattering resonances are defined in this context and discuss our current understanding of their distribution. The primary focus of the talk will be on some recent conjectures inspired by the physics of quantum chaotic systems. I will introduce these and discuss the numerical evidence as well as recent theoretical progress.

A sum of squares of real numbers is always nonnegative. This elementary observation is quite powerful, and can be used to prove graph density inequalities in extremal combinatorics, which address so-called Turan problems. This is the essence of semidefinite method of Lov\'{a}sz and Szegedy, and also Cauchy-Schwartz calculus of Razborov. Here multiplication and addition take place in the gluing algebra of partially labelled graphs. This method has been successfully used on many occasions and has also been extensively studied theoretically. There are two competing viewpoints on the power of the sums of squares method. Netzer and Thom refined a Positivstellensatz of Lovasz and Szegedy by showing that if f> 0 is a valid graph density inequality, then for any a>0 the inequality f+a > 0 can be proved via sums of squares. On the other hand, Hatami and Norine showed that testing whether a graph density inequality f > 0 is valid is an undecidable problem, and also provided explicit but complicated examples of inequalities that cannot be proved using sums of squares. I will introduce the sums of squares method, do several examples of sums of squares proofs, and then present simple explicit inequalities that show strong limitations of the sums of squares method. This is joint work in progress with Annie Raymond, Mohit Singh and Rekha Thomas.

I will discuss some elementary theory of symmetric functions and give a brief introduction to representation theory with a focus on the symmetric groups. This talk relates to the discussion of Schubert calculus in the intersection theory reading course but can be understood independent of attending the reading course.

In this talk, we introduce several models of the so-called forward-forward Mean-Field Games (MFGs). The forward-forward models arise in the study of numerical schemes to approximate stationary MFGs. We establish a link between these models and a class of hyperbolic conservation laws. Furthermore, we investigate the existence of solutions and examine long-time limit properties. Joint work with Diogo Gomes and Levon Nurbekyan.

In the continuous-time majority vote model, each vertex of a graph is initially assigned an ``opinion,'' either 0 or 1. At exponential times, vertices update their values by assuming the majority value of their neighbors. This model has been studied extensively on Z^d, where it is known as the zero-temperature limit of Ising Glauber dynamics. I will review some of the major questions and conjectures on lattices, and then explain some new work with Arnab Sen (Minnesota) on the 3-regular tree. We relate the majority vote model to a new model, which we call the median process, and use this process to answer questions about the limiting state of opinions. For example, we show that when the initial state is given by a Bernoulli(p) product measure, the probability that a vertex's limiting opinion is 1 is a continuous function of p.

At the heart of most algorithms today there is an optimization engine trying to learn online and provide the best decision, for e.g. rankings of objects, at any time with the partial information observed thus far in time. Often it becomes difficult to find near optimal solutions to many problems due to their inherent combinatorial structure that leads to certain computational bottlenecks. Submodularity is a discrete analogue of convexity and is a key property often exploited in tackling combinatorial optimization problems. In the first part of the talk, we will focus on computational bottlenecks that involve submodular functions: (a) convex function minimization over submodular base polytopes (for e.g. permutahedron) and (b) movement along a line inside submodular base polytopes. We give a conceptually simple and strongly polynomial algorithm Inc-Fix for the former, which is useful in computing Bregman projections in first-order projection-based methods like online mirror descent. For the latter, we will bound the iterations of the discrete Newton method which gives a running time improvement of at least n^6 over the state of the art. This is joint work with Michel Goemans and Patrick Jaillet. In the second part of the talk, we will consider the dual problem of (a), i.e. minimization of composite convex and submodular objectives. We will resolve Bach's conjecture from 2015 about the running time of a popular Kelley's cutting plane variant to minimize these composite objectives. This is joint work with Madeleine Udell and Song Zhou.

One of the general methods of proving h-principle is holonomic aprroximation. In this series of talks, I will give a proof of holonomic approximation theorem, and talk about some of its applications.

Spectral algorithms, such as principal component analysis and spectral clustering, typically require careful data transformations to be effective: upon observing a matrix A, one may look at the spectrum of ψ(A) for a properly chosen ψ. We propose a simple and generic construction for sparse graphs based on graph powering. It is shown that graph powering regularizes the graph and decontaminates its spectrum in the following sense: (i) If the graph is drawn from the sparse Erd˝os-R´enyi ensemble, which has no spectral gap, it is shown that graph powering produces a “maximal” spectral gap, with the latter justified by establishing an Alon-Boppana result for powered graphs; (ii) If the graph is drawn from the sparse SBM, graph powering is shown to achieve the fundamental limit for weak recovery. (Joint work with E. Abbe, E. Boix, C. Sandon.)