Georgia Institute of Technology College of Sciences

The flag variety G/B plays an outsized role in representation theory, combinatorics, geometry and algebra. Hessenberg varieties form a special class of subvarietes of the flag variety, arising in diverse contexts. The cohomology ring of a semisimple Hessenberg variety is recognized to be a representation of an associated finite group, and is related to the expansion of some special polynomials in terms of other well-known polynomial bases. These varieties may have pathological behavior, and their basic properties have been characterized only in restricted cases. Matrix Hessenberg schemes in type A consist of a lift of these varieties to G = Gl(n, C), where we can use the coordinate ring of matrices to study them.

In this talk, we present a full characterization of matrix Hessenberg schemes over the minimal sheet of Lie(G) in type A. We show that each semisimple matrix Hessenberg scheme lies in a flat family with a nilpotent matrix Hessenberg scheme, which in turn allows us to study their geometric properties. We describe the schemes fully in terms of Schubert varieties and opposite Schubert varieties, both well-known subvarieties of G/B. More subtly we characterize combinatorially which matrix Hessenberg schemes are reduced. These results are joint with Martha Precup at Washington University, St. Louis. 

The fundamental group is one of the most powerful invariants to distinguish closed three-manifolds, and the existence of non-trivial homomorphisms $\pi_1(M)\to SU(2)$ is a great way of measuring the non-triviality of a three-manifold $M$. It is known that if an integer homology 3-sphere is either Seifert fibered or toroidal, then irreducible representations do exist. In contrast, the existence of SU(2)-representations for hyperbolic homology spheres has not been completely established. With this as motivation, I will talk about partial progress made in the case of hyperbolic homology spheres realized as branched covers. This is joint work with Sudipta Ghosh and Zhenkun Li.

Reservoir computing is a branch of neuromorphic computing, which is usually realized in the form of ESNs (Echo State Networks). In this talk, I will present some fundamentals of reservoir computing from both the mathematical and the computational points of view. While reservoir computing was designed for sequential/time-series data, we recently observed its great performances in dealing with static image data once the reservoir is set to process certain image features, not the images themselves. Hence, I will discuss possible applications and open questions in reservoir computing.

We prove that negative energy solutions to the modified Benjamin-Ono (mBO) equation, which is L^2 critical, with mass slightly above the ground state mass, blow-up in finite or infinite time.   These blow-up solutions lie adjacent to those constructed by Martel & Pilod (2017) that have mass exactly equal to the ground state mass.  The solutions that we construct, with mass slightly above the ground state mass, are numerically observable and expected to be stable.  This is joint work with Svetlana Roudenko and Kai Yang.

For \( c\in(1,2) \) we consider the following operators
\[
\mathcal{C}_{c}f(x) \colon = \sup_{\lambda \in [-1/2,1/2)}
\bigg| \sum_{n \neq 0} f(x-n) \frac{e^{2\pi i\lambda \lfloor |n|^{c} \rfloor}}{n} \bigg|\text{,}
\]
\[
\mathcal{C}^{\mathsf{sgn}}_{c}f(x) \colon = \sup_{\lambda \in [-1/2,1/2)}
\bigg| \sum_{n \neq 0} f(x-n) \frac{e^{2\pi i\lambda \mathsf{sign}(n) \lfloor |n|^{c} \rfloor}}{n} \bigg| \text{,}
\]
and prove that both extend boundedly on \( \ell^p(\mathbb{Z}) \), \( p\in(1,\infty) \). 

The second main result is establishing almost everywhere pointwise convergence for the following ergodic averages
\[
A_Nf(x)\colon =\frac{1}{N}\sum_{n=1}^N f(T^n S^{\lfloor n^c\rfloor} x) \text{,}
\]
where $T,S\colon X\to X$ are commuting measure-preserving transformations on a  $\sigma$-finite measure space $(X,\mu)$, and $f\in L_{\mu}^p(X), p\in(1,\infty)$. 

The point of departure for both proofs is the study of exponential sums with phases  $\xi_2 \lfloor |n^c|\rfloor+ \xi_1n$ through the use of a simple variant of the circle method.

This talk is based on joint work with Leonidas Daskalakis.
 

The Jones polynomial was first defined by Vaughan Jones as a "trace function" on an algebra discovered via operator algebras. It was discovered that the polynomial satisfies certain skein relations. The HOMFLY polynomial was discovered through both skein relations and a "lift" of the trace function on the Jones algebra to the "Hecke algebra". Another 2-variable polynomial called the Kauffman polynomial was discovered purely via skein relations. In this talk, we discuss how the process started by Jones was reversed for this polynomial. More precisely, we will show how Birman, Wenzl, and Murakami constructed the BMW algebra and a trace function that yields the Kauffman polynomial. We will discuss the significance of the Kauffman polynomial as well as some relationships between the BMW, Hecke, and Jones algebras.

N. Zabusky coined the word "soliton" in 1965 to describe a curious feature he and M. Kruskal observed in their numerical simulations of the initial-value problem for a simple nonlinear PDE. The first part of the talk will be a broad introduction to the theory of solitons/solitary waves and integrable PDEs (the Korteweg-de Vries equation in particular), describing classical results in the field. The second (and main) part of the talk will focus on some new developments and growing interest into a special case of solutions defined as "soliton gas".

We study random configurations of N soliton solutions q_N(x,t) of the KdV equation. The randomness appears in the scattering (linear) problem, which is used to solve the PDE: the complex eigenvalues are chosen to be (1) i.i.d. random variables sampled from a probability distribution with compact support on the complex plane, or (2) sampled from a random matrix law. 

Next, we consider the scattering problem for the expectation of the random measure associated to the spectral data, in the limit as N -> + infinity. The corresponding solution q(x,t) of the KdV equation is a soliton gas. 

We are then able to prove a Law of Large Numbers and a Central Limit Theorem for the differences q_N(x,t)-q(x,t).

This is a collection of works (and ongoing collaborations) done with K. McLaughlin (Tulane U.), T. Grava (SISSA/Bristol), R. Jenkins (UCF), A. Minakov (U. Karlova), J. Najnudel (Bristol).

The generative modeling of data on manifolds is an important task, for which diffusion models in flat spaces typically need nontrivial adaptations. This article demonstrates how a technique called `trivialization' can transfer the effectiveness of diffusion models in Euclidean spaces to Lie groups. In particular, an auxiliary momentum variable was algorithmically introduced to help transport the position variable between data distribution and a fixed, easy-to-sample distribution. Normally, this would incur further difficulty for manifold data because momentum lives in a space that changes with the position. However, our trivialization technique creates a new momentum variable that stays in a simple fixed vector space. This design, together with a manifold preserving integrator, simplifies implementation and avoids inaccuracies created by approximations such as projections to tangent space and manifold, which were typically used in prior work, hence facilitating generation with high-fidelity and efficiency. The resulting method achieves state-of-the-art performance on protein and RNA torsion angle generation and sophisticated torus datasets. We also, arguably for the first time, tackle the generation of data on high-dimensional Special Orthogonal and Unitary groups, the latter essential for quantum problems.

Let $d\geq 2$, and $n\geq 2d+2$. Frankl and Pach initiated the study of the maximum size of a $(d+1)$-uniform set system in $[n]$ with VC-dimension at most $d$. The best-known upper bound is essentially $\binom{n}{d}$, and the best-known lower bound is $\binom{n-1}{d} + \binom{n-4}{d-2}$. In this talk, I will discuss some recent improvements on the upper bound and some interesting connections between this problem and the celebrated Erdős--Ko--Rado theorem. In particular, I will discuss our conjecture, which can be viewed as a generalization of the EKR as well as an "uniform version" of the disproved Erdős--Frankl--Pach conjecture, and highlight some of our partial progress. Joint work with Ting-Wei Chao, Zixiang Xu, and Shengtong Zhang.