Georgia Institute of Technology College of Sciences

Macdonald polynomials are a family of symmetric functions that are known to have remarkable connections to a well-studied particle model called the asymmetric simple exclusion process (ASEP). The modified Macdonald polynomials are obtained from the classical Macdonald polynomials using an operation called plethysm. It is natural to ask whether the modified Macdonald polynomials specialize to the partition function of some other particle system.

We answer this question in the affirmative with a certain multispecies totally asymmetric zero-range process (TAZRP). This link motivated a new tableaux formula for modified Macdonald polynomials. We present a Markov process on those tableaux that projects to the TAZRP and derive formulas for stationary probabilities and certain correlations, proving a remarkable symmetry property. This talk is based on joint work with Arvind Ayyer and James Martin.

The study of representations of fundamental groups of surfaces into Lie groups is captured by the character variety. One main tool to study character varieties are Higgs bundles, a complex geometric tool. They fail to see the mapping class group symmetry. I will present an alternative approach which replaces Higgs bundles by so-called higher complex structures, given in terms of commuting nilpotent matrices. The resulting theory has many similarities to the non-abelian Hodge theory. Joint with Georgios Kydonakis and Charlie Reid.

Recent advances in data-driven modeling approaches have proven highly successful in a wide range of fields in science and engineering. In this talk, I will briefly discuss several ubiquitous challenges with the conventional model development / discretization / parameter inference / model revision loop that our methodology attempts to address. I will present our weak form methodology which has proven to have surprising performance properties. In particular, I will describe our equation learning (WSINDy) and parameter estimation (WENDy) algorithms.  Lastly, I will discuss applications to several benchmark problems illustrating how our approach addresses several of the above issues and offers advantages in terms of computational efficiency, noise robustness, and modest data needs (in an online learning context).

 In the 1950s Phil Anderson made a prediction about the effect of random impurities on the conductivity properties of a crystal. Mathematically, these questions amount to the study of solutions of differential or difference equations and the associated spectral theory of self-adjoint operators obtained from an ergodic process. With the arrival of quasicrystals, in addition to random models, nonrandom lattice models such as those generated by irrational rotations or skew-rotations on tori have been studied over the past 30 years. 

By now, an extensive mathematical theory has developed around Anderson’s predictions, with several questions remaining open. This talk will attempt to survey certain aspects of the field, with an emphasis on the theory of SL(2,R) cocycles with an irrational or  Diophantine  rotation on the circle as base dynamics. In this setting, Artur Avila discovered about a decade ago that the Lyapunov exponent is piecewise affine in the imaginary direction after complexification of the circle. In fact, the slopes of these affine functions are integer valued. This is easy to see in the uniformly hyperbolic case, which is equivalent to energies falling into the gaps of the spectrum, due to the winding number of the complexified Lyapunov exponent. Remarkably, this property persists also in the non-uniformly hyperbolic case, i.e., on the spectrum of the Schrödinger operator. This requires a delicate continuity property of the Lyapunov exponent in both energy and frequency. Avila built his global theory (Acta Math. 2015) on this quantization property. I will present some recent results with Rui HAN (Louisiana) connecting Avila’s notion of  acceleration (the slope of the complexified Lyapunov exponent in the imaginary direction) to the number of zeros of the determinants of  finite volume Hamiltonians relative to the complex toral variable. This connection allows one to answer questions arising in the supercritical case of Avila’s global theory concerning the measure of the second stratum, Anderson localization on this stratum, as well as settle a conjecture on the Hölder regularity of the integrated density of states.

The Intermediate Long Wave equation (ILW) describes long internal gravity waves in stratified fluids. As the depth parameter in the equation approaches zero or infinity, the ILW formally approaches the Kortweg-deVries equation (KdV) or the Benjamin-Ono equation (BO), respectively. Kodama, Ablowitz and Satsuma discovered the formal complete integrability of ILW and formulated inverse scattering transform solutions. If made rigorous, the inverse scattering method will provide powerful tools for asymptotic analysis of ILW. In this talk, I will present some recent results on the ILW direct scattering problem. In particular, a Lax pair formulation is clarified, and the spectral theory of the Lax operators can be studied. Existence and uniqueness of scattering states are established for small interaction potential. The scattering matrix can then be constructed from the scattering states. The solution is related to the theory of analytic functions on a strip. This is joint work with Peter Perry.

In this talk we discuss our proof of a recent conjecture of Guo and Poznanovi\'{c} concerning chains in certain 01-fillings of moon polyominoes. A key ingredient of our proof is a correspondence between words $w$ and pairs $(\mathcal{W}(w), \mathcal{M}(w))$ of increasing tableaux such that $\mathcal{M}(w)$ determines the lengths of the longest strictly increasing and strictly decreasing sequences in every subinterval of $w$.  (It will be noted that similar and well-studied correspondences like RSK insertion and Hecke insertion fail in this regard.) To define our correspondence we make use of Thomas and Yong's K-infusion operator and then use it to obtain the bijections that prove the conjecture of Guo and Poznanovi\'{c}.    (Joint work with D. Saracino.)

Given a discrete set $\Lambda\subseteq\mathbb{R}$ and an interval $I$, define the sequence of complex exponentials in $L^2(I)$, $\mathcal{E}(\Lambda)$, by $\{e^{2\pi i\lambda t}\colon \lambda\in\Lambda\}$.  A fundamental result in harmonic analysis says that if $\mathcal{E}(\frac{1}{b}\mathbb{Z})$ is an orthogonal basis for $L^2(I)$ for any interval $I$ of length $b$.  It is also well-known that there exist sets $\Lambda$, which may be irregular, such that sets $\mathcal{E}(\Lambda)$ form nonorthogonal bases (known as Riesz bases) for $L^2(S)$, for $S\subseteq\mathbb{R}$ not necessarily an interval.

Given $\mathcal{E}(\Lambda)$ that forms a Riesz basis for $L^2[0,1]$ and some 0 < a < 1, Avdonin showed that there exists $\Lambda'\subseteq \Lambda$ such that $\mathcal{E}(\Lambda')$ is a Riesz basis for $L^2[0,a]$ (called basis extraction).  Lyubarskii and Seip showed that this can be done in such a way that $\mathcal{E}(\Lambda \setminus \Lambda')$ is also a Riesz basis for $L^2[a,1]$ (called basis splitting).  The celebrated result of Kozma and Nitzan shows that one can extract a Riesz basis for $L^2(S)$ from $\mathcal{E}(\mathbb{Z})$ where $S$ is a union of disjoint subintervals of $[0,1]$.

In this talk we construct sets $\Lambda_I\subseteq\mathbb{Z}$ such that the $\mathcal{E}(\Lambda_I)$ form Riesz bases for $L^2(I)$ for corresponding intervals $I$, with the added compatibility property that unions of the sets $\Lambda_I$ generate Riesz bases for unions of the corresponding intervals.  The proof of our result uses an interesting assortment of tools from analysis, probability, and number theory.  We will give details of the proof in the talk, together with examples and a discussion of recent developments.  The work discussed is joint with Shauna Revay (GMU and Accenture Federal Services (AFS)), and Goetz Pfander (Catholic University of Eichstaett-Ingolstadt).

The braid group (which encodes the braiding of n strands) has a canonical projection to the symmetric group (recording where the ends of the strands go). We ask the question: what are the extensions of the symmetric group by abelian groups that arise as quotients of the braid group, by a refinement of this canonical projection? To answer this question, we study a particular twisted coefficient system for the symmetric group, called the integral pair module. In this module, we find the maximal submodule in each commensurability class. We find the cohomology classes characterizing each such extension, and for context, we describe the second cohomology group of the symmetric group with coefficients in the most interesting of these modules. This is joint work with Trevor Nakamura.

 We report on recent results regarding the limiting absorption principle for multi-dimensional, massless Dirac-type operators (implying absence of singularly continuous spectrum) and continuity properties of the associated spectral shift function.

We will motivate our interest in this circle of ideas by briefly describing the connection to the notion of the Witten index for a certain class of non-Fredholm operators.

This is based on various joint work with A. Carey, J. Kaad, G. Levitina, R. Nichols, D. Potapov, F. Sukochev, and D. Zanin.

The Fermi variety plays a crucial role in the study of    periodic operators.  In this talk, I will  first discuss recent works on the irreducibility of  the Fermi variety  for discrete periodic Schr\"odinger  operators.   I will then  discuss the applications to  solve  problems of embedded eigenvalues, isospectrality and quantum ergodicity. 

The celebrated Hajnal-Szemerédi theorem gives best possible minimum degree conditions for clique-factors in graphs. There have been some recent variants of this result into several settings, each of which has some sort of randomness come into play. We will give a survey on these problems and the recent developments.

The estimation of distributions of complex objects from high-dimensional data with low-dimensional structures is an important topic in statistics and machine learning. Deep generative models achieve this by encoding and decoding data to generate synthetic realistic images and texts. A key aspect of these models is the extraction of low-dimensional latent features, assuming data lies on a low-dimensional manifold. We study this by developing a minimax framework for distribution estimation on unknown submanifolds with smoothness assumptions on the target distribution and the manifold. The framework highlights how problem characteristics, such as intrinsic dimensionality and smoothness, impact the limits of high-dimensional distribution estimation. Our estimator, which is a mixture of locally fitted generative models, is motivated by differential geometry techniques and covers cases where the data manifold lacks a global parametrization. 

We will prove the key lemma underlying the probabilistic unique continuation result of Ding-Smart, namely that for "thin" tilted rectangles, boundedness on all of one of the long edges and on a 1-\varepsilon proportion of the opposite long edge implies a bound (in terms of the dimensions of the rectangle) on the whole rectangle (with high probability). 

A link of an isolated complex surface singularity is the intersection of the surface with a small sphere centered at the singular point. The link is a smooth 3-manifold that carries a natural contact structure (given by complex tangencies); one might then want to study its symplectic or Stein fillings. A special family of Stein fillings, called Milnor fillings, can be obtained by smoothing the singular point of the original complex surface.  We will discuss some properties and constructions of Milnor fillings and general Stein fillings, and ways to detect whether the link of singularity has Stein fillings that do not arise from smoothings.

We study the minimum number of paths needed to express the edge set of a given graph as the symmetric difference of the edge sets of the paths. This can be seen as a weakening of Gallai’s path decomposition problem, and a variant of the “odd cover” problem of Babai and Frankl which asks for the minimum number of complete bipartite graphs whose symmetric difference gives the complete graph. We relate this “path odd-cover” number of a graph to other known graph parameters and prove some bounds. Joint work with Steffen Borgwardt, Calum Buchanan, Eric Culver, Bryce Frederickson, and Puck Rombach.

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

G. W. Hill made major contributions to Celestial Mechanics. One of them is to develop his lunar theory as an alternative approach for the study of the motion of the Moon around the Earth, which is the classical Lunar Hill problem. The mathematical model we study is one of the extensions of the classical Hill approximation of the restricted three-body problem. Considering a restricted four body problem, with a hierarchy between the bodies: two larger bodies, a smaller one and a fourth infinitesimal body, we encounter the shapes of the three heavy bodies via oblateness. We first find that the triangular central configurations of the three heavy bodies is a scalene triangle. Through the application of the Hill approximation, we obtain the limiting Hamiltonian that describes the dynamics of the infinitesimal body in a neighborhood of the smaller body. As a motivating example, we identify the three heavy bodies with the Sun, Jupiter and the Jupiter’s Trojan asteroid Hektor. 

The Deligne category of symmetric groups is the additive Karoubi closure of the partition category. The partition category may be interpreted, following Comes, via a particular linearization of the category of two-dimensional oriented cobordisms. In this talk we will use a generalization of this approach to the Deligne category coupled with the universal construction of two-dimensional topological theories to construct their multi-parameter monoidal generalizations, one for each rational function in one variable. This talk is based on joint work with M. Khovanov.

Mathapalooza! is back at this year's Atlanta Science Festival! Come join us on Saturday, March 18, for an afternoon of mathematical fun beginning at 1:00pm at the Paideia School.  There will be interactive puzzles and games, artwork, music, stage acts, and mathematics in motion.