Georgia Institute of Technology College of Sciences

In this talk, we explore the fractional log-concavity property of generating polynomials of discrete distributions. This property is an analog to the Lorentzian [Branden-Huh’19]/log-concavity [Anari-Liu-OveisGharan-Vinzant’19] property of the generating polynomials of matroids. We show that multivariate generating polynomials without roots in a sector of the complex plane are fractionally log-concave. Furthermore, we prove that the generating polynomials of linear delta matroids and of the intersection between a linear matroid and a partition matroid have no roots in a sector, and thus are fractionally log-concave. Beyond root-freeness, we conjecture that for any subset F of {0,1}^n such that conv(F) has constantly bounded edge length, the generating polynomial for the uniform distribution over F is fractionally log-concave.

Based on joint works with Yeganeh Alimohammadi , Nima Anari and Kirankumar Shiragur.

In high dimensional contact and symplectic topology, finding interesting constructions for Legendrian submanifolds is an active area of research. Further, it is desirable that the constructions lend themselves nicely to computation of invariants. The doubling construction was defined by Ekholm, which uses Lagrangian fillings of a Legendrian knot in standard contact R^{2n-1} to produce a closed Legendrian submanifold in standard contact R^{2n+1}. Later Courte-Ekholm showed that symmetric doubles of embedded fillings are "uninteresting". In recent work the symmetric doubling construction was generalised to any contact manifold, giving two isotopic constructions related to open book decompositions of the ambient manifold. In a separate joint work with James Hughes, we explore the asymmetric doubling construction through Legendrian weaves.

In this talk, we present new gradient sliding results for constrained convex optimization with applications in image reconstruction and decentralized distributed optimization. Specifically, we will study classes of large-scale problems that minimizes a convex objective function over feasible set with linear constraints. We will show that by exploring the gradient sliding technique, the number of gradient evaluations of the objective function can be reduced by exploring the smoothness structure. Our results could lead to new decentralized algorithms for multi-agent optimization with graph topology invariant gradient/sampling complexity and new ADMM algorithms for solving total variation image reconstruction problems with accelerated gradient complexity.

Transport equations arise in the modelling of several complex systems, including mean field games. Such equations often involve nonlinear dependence of the solution in the drift. These nonlinear transport equations can be understood by developing a theory for transport equations with irregular drifts. In this talk, I will outline the well-posedness theory for certain transport equations in which the drift has a one-sided bound on the divergence, yielding contractive or expansive behavior, depending on the direction in which the equation is posed. The analysis requires studying the relationship between the transport and continuity equations and the associated ODE flow. The theory is then used to discuss certain nonlinear transport equations arising in the study of finite state-space mean field games. This is joint work with P.-L. Lions.

We show that if each edge of the complete bipartite graph $K_{n,n}$ is given a random list of $C(\log n)$ colors from $[n]$, then with high probability, there is a proper edge coloring where the color of each edge comes from the corresponding list. We also prove analogous results for Latin squares and Steiner triple systems. This resolves several related conjectures of Johansson, Luria-Simkin, Casselgren-Häggkvist, Simkin, and Kang-Kelly-Kühn-Methuku-Osthus. I will discuss some of the main ingredients which go into the proof: the Kahn-Kalai conjecture, absorption, and the Lovasz Local Lemma distribution. Based on joint work with Huy Tuan Pham. 

We will look at a number of interesting examples — some proven, others merely conjectured — of Hamburger moment sequences in combinatorics. We will consider ways in which this positivity may be expected, for instance in different types of combinatorial statistics on perfect matchings that turn out to encode moments in noncommutative analogues of the classical Central Limit Theorem. We will also consider situations in which this positivity may be surprising, and where proving it would open up new approaches to a class of very hard open problems in combinatorics.

I'll talk about the structure of the collection of all n-ples of eigenvalues of elements of Zariski-dense subgroups D of SL(n,R). Subgroups like this appear, for instance, as the images of holonomy representations of geometric structures. Our focus is a deep and useful result of Benoist, which states that the natural cone one is led to consider here has strong convexity and non-degeneracy properties, and a succinct, qualitative characterization of the cones that so arise from Zariski-dense subgroups. The theorem comes out of a study of the dynamics of the actions of D on spaces of flags such as RP^n and the collection of open subsemigroups of SL(n,R). Everything in this talk is from Benoist’s paper Propriétés Asymptotiques des Groupes Linéaires (GAFA, 2002), and holds in far more generality than I'll state.

We will consider the inverse problem of determining the sound
speed or index of refraction of a medium by measuring the travel times of
waves going through the medium. This problem arises in global seismology
in an attempt to determine the inner structure of the Earth by measuring
travel times of earthquakes. It also has several applications in optics
and medical imaging among others.

The problem can be recast as a geometric problem: Can one determine
the Riemannian metric of a Riemannian manifold with boundary by
measuring the distance function between boundary points? This is the
boundary rigidity problem.

We will also describe some recent results, joint with Plamen Stefanov
and Andras Vasy, on the partial data case, where you are making
measurements on a subset of the boundary.

The anomaly cancellation is a basic property of the Standard Model, crucial for its consistence. We consider a lattice chiral gauge theory of massless Wilson fermions interacting with a non-compact massiveU(1) field coupled with left- and right-handed fermions in four dimensions. We prove in the infinite volume limit, for weak coupling and inverse lattice step of the order of boson mass, that the anomaly vanishes up to subleading corrections and under the same condition as in the continuum. The proof is based on a combination of exact Renormalization Group, non-perturbative decay bounds of correlations and lattice symmetries.

The talk can be accessed via zoom: Meeting ID: 989 6686 9205

The general subject of the talk is spectral theory of discrete (tight-binding) Schrodinger operators on d-dimensional lattices. For operators with periodic potentials, it is known that the spectra of such operators are purely absolutely continuous. For random i.i.d. potentials, such as the Anderson model, it is expected and can be proved in many cases that the spectra are almost surely purely point with exponentially decaying eigenfunctions (Anderson local- ization). Quasiperiodic operators can be placed somewhere in between: depending on the potential sampling function and the Diophantine properties of the frequency and the phase, one can have a large variety of spectral types. We will consider quasiperiodic operators

(H(x)ψ)n =ε(∆ψ)n +f(x+n·ω)ψn, n∈Zd,

where ∆ is the discrete Laplacian, ω is a vector with rationally independent components, and f is a 1-periodic function on R, monotone on (0,1) with a positive lower bound on the derivative and some additional regularity properties. We will focus on two methods of proving Anderson localization for such operators: a perturbative method based on direct analysis of cancellations in the Rayleigh-Schr ̈odinger perturbation series for arbitrary d, and a non?perturbative method based on the analysis of Green?s functions for d = 1, originally developed by S. Jitomirskaya for the almost Mathieu operator.

The talk is based on joint works with S. Krymskii, L. Parnovski, and R. Shterenberg (per- turbative methods) and S. Jitomirskaya (non-perturbative methods).

Zoom link to the talk: https://gatech.zoom.us/j/94387417679

We will present the notion of Stein kernel, which provides generalizations of the integration by parts, a.k.a. Stein's formula, for the normal distribution (which has a constant Stein kernel, equal to its covariance). We will first focus on dimension one, where under good conditions the Stein kernel has an explicit formula. We will see that the Stein kernel appears naturally as a weighting of a Poincaré type inequality and that it enables precise concentration inequalities, of the Mills' ratio type. In a second part, we will work in higher dimensions, using in particular Max Fathi's construction of a Stein kernel through the so-called "moment maps" transportation. This will allow us to describe the performance of some shrinkage and thresholding estimators, beyond the classical assumption of Gaussian (or spherical) data. This presentation is mostly based on joint works with Max Fathi, Larry Goldstein, Gesine Reinert and Jon Wellner.

We will actually finish our proof of the key technical lemma for the quantitative unique continuation principle of Ding-Smart, reviewing briefly the volumetric bound from the theory of \varepsilon-coverings/nets/packings. From there, we will outline at a high level the strategy for the rest of the proof of the unique continuation principle using this key lemma, before starting the proof in earnest.

We consider the problem of estimating the factors of a rank-1 matrix with i.i.d. Gaussian, rank-1 measurements that are nonlinearly transformed and corrupted by noise. Considering two prototypical choices for the nonlinearity, we study the convergence properties of a natural alternating update rule for this nonconvex optimization problem starting from a random initialization. We show sharp convergence guarantees for a sample-split version of the algorithm by deriving a deterministic recursion that is accurate even in high-dimensional problems. Our sharp, non-asymptotic analysis also exposes several other fine-grained properties of this problem, including how the nonlinearity and noise level affect convergence behavior.

On a technical level, our results are enabled by showing that the empirical error recursion can be predicted by our deterministic sequence within fluctuations of the order n−1/2 when each iteration is run with n observations. Our technique leverages leave-one-out tools originating in the literature on high-dimensional M–estimation and provides an avenue for sharply analyzing higher-order iterative algorithms from a random initialization in other high-dimensional optimization problems with random data.

We study quantizations of SL_n-character varieties, which appears as moduli spaces for many geometric structures. Our main goal is to establish the existence of several quantum trace maps. In the classical limit, they reduce to the Fock-Goncharov trace maps, which are coordinate charts on moduli spaces of SL_n-local systems used in higher Teichmuller theory. In the quantized theory, the algebras are replaced with non-commutative deformations. The domains of the quantum trace maps are the SL_n-skein algebra and the reduced skein algebra, and the codomains are quantum tori, which are non-commutative analogs of Laurent polynomial algebras. In this talk, I will review the classical theory and sketch the definition of the quantum trace maps.

Link: https://gatech.zoom.us/j/91390791493?pwd=QnpaWHNEOHZTVXlZSXFkYTJ0b0Q0UT09

Abstract: In this talk I will explain a new numerical framework, employing physics-informed neural networks, to find a smooth self-similar solution for different equations in fluid dynamics. The new numerical framework is shown to be both robust and readily adaptable to several situations.

Joint work with Yongji Wang, Ching-Yao Lai and Tristan Buckmaster.

I will discuss some problems in geometric topology, and relate them to graph-theoretic properties. I will give some open problems, and answer questions of Kalai, Belolipetski, Gromov and others.