Georgia Institute of Technology College of Sciences

After providing a mathematical background for some curious optical experiments in the 19th century, I will then describe how these ideas inform our understanding of the Deift conjecture for the Korteweg--de Vries equation. Specifically, in joint work with Chapouto and Visan, we showed that the evolution of almost-periodic initial data need not remain almost periodic.

The talk is postponed. The updated schedule will be announced when it is ready.

The theory of stable polynomials features a key notion called proper position, which generalizes interlacing of real-rooted polynomials to higher dimensions. In a recent paper, I introduced a Lorentzian analog of proper position and used it to give a new characterization of elementary quotients of valuated matroids. This connects the local structure of spaces of Lorentzian polynomials with the incidence geometry of tropical linear spaces. A central object in this connection is the moduli space of codimension-1 tropical linear subspaces of a given tropical linear space. In this talk, I will show some new structural results on this moduli space and their implications for Lorentzian polynomials.

In this talk, I will present a complete coarse classification of strongly exceptional Legendrian realizations of the connected sum of two Hopf links in contact 3-spheres. This is joint work with Sinem Onaran.

 We will survey a number of recent developments in the theory of completely integrable nonlinear dispersive PDE.  These include a priori bounds, orbital stability of multisolitons, well-posedness at optimal regularity, and the existence of dynamics for Gibbs distributed initial data. I will describe the basic objects that tie together these disparate results, as well as the diverse ideas required for each problem.

The talk is postponed. The updated schedule will be announced when it is ready.

Given a modular form $f$, one can construct a measure $\mu_f$ on the modular surface $SL(2,\mathbb{Z})\backslash\mathbb{H}$. The celebrated mass equidistribution theorem of Holowinsky and Soundararajan states that as $k\rightarrow\infty$, the measure $\mu_f$ approaches the uniform measure on the surface. Given a maximal order in a quaternion algebra which is non-split over $\mathbb{Q}$, a maximal order leads to a cocompact subgroup of $R^1\subseteq SL(2,\mathbb{Z})$ where the quotient $R^1\backslash\mathbb{H}$ is a Shimura curve. Given a Hecke form $f$ on this Shimura curve, one can construct the analogous measure $\mu_f$, and ask about the limit as $k\rightarrow\infty$. Recent work of Nelson relates this equidistribution problem for the cocompact case to obtaining bounds on sums of Hecke eigenvalues summed over quadratic progressions. In this talk, I will describe this problem in both the cocompact and non-cocompact case while highlighting how differences in algebras lead to differences in geometry. I will then state progress that I have made on bounds that correspond to square root cancellation on average for sums of Hecke eigenvalues summed over quadratic progressions when averaged over a basis of Hecke forms. 

Dynamical systems exhibiting some degree of hyperbolicity often admit “fractal" invariant objects.  However, extra symmetries or “randomness” in the system often preclude the existence of such fractal objects.

I will give some concrete examples of the above and then discuss problems and results related to random dynamics and group actions on surfaces.  I will especially focus on questions related to absolute continuity of stationary measures.

Applied algebraic geometry is a subfield of applied mathematics that utilizes concepts, tools, and techniques from algebraic geometry to solve problems in various applied sciences. It blends tools from algebraic geometry, optimization, and statistics to develop certifiable computational algebraic methods to address modern engineeering challenges.

In this talk, I will showcase the power of these methods in solving problems related to Gaussian mixture models (GMMs). In the first part of the talk I will discuss a statistical technique for parameter recovery called the method of moments. I will discuss how to leverage algebraic techniques to design scalable and certifiable moment-based methods for parameter recovery of GMMs. In the second part of this talk, I will discuss recent work relating to Gaussian Voronoi cells. This work introduces new geometric perspectives with implications for high-dimensional data analysis. I will also touch on how these methods complement my broader research in polynomial optimization and power systems engineering.

https://gatech.zoom.us/j/97398944571?pwd=s8S02kNZd5dyVvSY8mZzNOfbNZrqfg.1

In the 1990s, Arkadi Nemirovski asked the question: 

How hard is it to estimate a solution to an unknown homogeneous linear difference equation with constant coefficients of order S, observed in the Gaussian noise on [0,N]?

The class of all such solutions, or "signals," is parametric -- described by 2S complex parameters -- but extremely rich: it contains all exponential polynomials over C with total degree S, including harmonic oscillations with S arbitrary frequencies. Geometrically, this class corresponds to the projection onto C^n of the union of all shift-invariant subspaces of C^Z of dimension S. We show that the statistical complexity of this class, quantified by the squared minimax radius of the (1-P)-confidence Euclidean norm ball, is nearly the same as for the class of S-sparse signals, namely (S log(N) + log(1/P)) log^2(S) log(N/S) up to a constant factor. Moreover, the corresponding near-minimax estimator is tractable, and it can be used to build a test statistic with a near-minimax detection threshold in the associated detection problem. These statistical results rest upon an approximation-theoretic one: we show that finite-dimensional shift-invariant subspaces admit compactly supported reproducing kernels whose Fourier spectra have nearly the smallest possible p-norms, for all p ≥ 1 at once. 

The talk is based on the recent preprint https://arxiv.org/pdf/2411.03383.

We will discuss some surprising rigidity phenomena for Anosov flows in dimension 3. For example, in the context of generic transitive 3-dimensional Anosov flows, any continuous conjugacy is either smooth or reverses the positive and negative SRB measures.

This is joint work with Martin Leguil and Federico Rodriguez Hertz