Posted January 28, 2024

Combinatorics Seminar Questions or comments?

2:00 pm – 3:00 pm Lockett Hall 233
Yiwei Ge, Louisiana State University

Oriented diameter of near planar triangulations

The oriented diameter of an undirected graph $G$ is the smallest diameter over all the strongly connected orientations of $G$. A near planar triangulation is a planar graph such that every face except possibly the outer face is a triangle. In this talk, we will show that the oriented diameter of all $n$-vertex near planar triangulations(except seven small exceptions) is bounded by $\lceil \frac{n}{2}\rceil$, and the bound is tight. Joint work with Xiaonan Liu and Zhiyu Wang.

Posted January 31, 2024

Colloquium Questions or comments?

2:30 pm – 3:30 pm Zoom
Bruno Poggi, Universitat Autònoma de Barcelona

Two problems in the mathematical physics of the magnetic Schrödinger operator and their solutions via the landscape function.

Abstract. In two papers in the 90's, Zhongwei Shen studied non-asymptotic bounds for the eigenvalue counting function of the magnetic Schrödinger operator, as well as the localization of eigenfunctions. But in dimensions 3 or above, his methods required a strong scale-invariant quantitative assumption on the gradient of the magnetic field; in particular, this excludes many singular or irregular magnetic fields, and the questions of treating these later cases had remained open, giving rise to a problem and a conjecture. This strong assumption on the gradient of the magnetic field has appeared also in the harmonic analysis related to the magnetic Schrödinger operator. In this talk, we present our solutions to these questions, and other new results on the exponential decay of solutions (eigenfunctions, integral kernels, resolvents) to Schrödinger operators. We will introduce the Filoche-Mayboroda landcape function for the (non-magnetic) Schrödinger operator, present its pointwise equivalence to the classical Fefferman-Phong-Shen maximal function (also known as the critical radius function in harmonic analysis literature), and then show how one may use directionality assumptions on the magnetic field to construct a new landscape function in the magnetic case. We resolve the problem and the conjecture of Z. Shen (and recover other results in the irregular setting) by putting all these observations together.

Posted February 2, 2024

Control and Optimization Seminar Questions or comments?

11:30 am – 12:20 pm Zoom (Click “Questions or Comments?” to request a Zoom link)
Ali Kara, University of Michigan

Stochastic Control with Partial Information: Optimality, Stability, Approximations and Learning

Partially observed stochastic control is an appropriate model for many applications involving optimal decision making and control. In this talk, we will first present a general introduction and then study optimality, approximation, and learning theoretic results. For such problems, existence of optimal policies have in general been established via reducing the original partially observed stochastic control problem to a fully observed one with probability measure valued states. However, computing a near-optimal policy for this fully observed model is challenging. We present an alternative reduction tailored to an approximation analysis via filter stability and arrive at an approximate finite model. Toward this end, we will present associated regularity and Feller continuity, and controlled filter stability conditions: Filter stability refers to the correction of an incorrectly initialized filter for a partially observed dynamical system with increasing measurements. We present explicit conditions for filter stability which are then utilized to arrive at approximately optimal solutions. Finally, we establish the convergence of a learning algorithm for control policies using a finite history of past observations and control actions (by viewing the finite window as a 'state') and establish near optimality of this approach. As a corollary, this analysis establishes near optimality of classical Q-learning for continuous state space stochastic control problems (by lifting them to partially observed models with approximating quantizers viewed as measurement kernels) under weak continuity conditions. Further implications and some open problems will also be discussed.

Posted September 22, 2023

Last modified January 25, 2024

Applied Analysis Seminar Questions or comments?

3:30 pm Lockett 232
Eduard-Wilhelm Kirr, University of Illinois Urbana-Champagne

Can one find all coherent structures supported by a wave equation?

I will present a new mathematical technique aimed at discovering all coherent structures supported by a given nonlinear wave equation. It relies on global bifurcation analysis which shows that, inside the Fredholm domain, the coherent structures organize themselves into manifolds which either form closed surfaces or must reach the boundary of this domain. I will show how one can find all the limit points at the Fredholm boundary for the Nonlinear Schrodinger/Gross-Pitaevskii Equation. Then I will use these limit points to uncover all coherent structures and their bifurcation points.

Posted November 13, 2023

Last modified February 2, 2024

Algebra and Number Theory Seminar Questions or comments?

3:20 pm – 4:10 pm Lockett 233 or click here to attend on Zoom
Rajat Gupta, University of Texas at Tyler

On summation formulas attached to Hecke's functional equation and $p$-Herglotz functions

In this talk, we will review the work of Chandrasekharan and Narasimhan on the theory of Hecke’s functional equation (with one gamma factor) and the summation formulas of various kinds, such as the Voronoi summation formula, the Poisson summation formula, and the Abel-Plana summation formula. We will then give recent developments in this theory followed by some new results and summation formulas in the setting of Hecke’s functional equation analogous to the ones mentioned above. In particular, I will discuss these summation formulas in the case of cusp corms of weight $2k$ attached to the modular group ${\rm SL}_2(\mathbb{Z})$. Finally, I will also talk about on my recent work with Rahul Kumar on Herglotz functions and their analogues.

Posted February 1, 2024

Last modified February 2, 2024

Colloquium Questions or comments?

3:30 pm – 4:30 pm Lockett 232
Narek Hovsepyan, Rutgers University

On the lack of external response of a nonlinear medium in the second-harmonic generation process.

Abstract: Second Harmonic Generation (SHG) is a process in which the input wave (e.g. laser beam) interacts with a nonlinear medium and generates a new wave, called the second harmonic, at double the frequency of the original input wave. We investigate whether there are situations in which the generated second harmonic wave does not scatter and is localized inside the medium, i.e., the nonlinear interaction of the medium with the probing wave is invisible to an outside observer. This leads to the analysis of a semilinear elliptic system formulated inside the medium with non-standard boundary conditions. More generally, we set up a mathematical framework needed to investigate a multitude of questions related to the nonlinear scattering problem associated with SHG (or other similar multi-frequency optical phenomena). This is based on a joint work with F. Cakoni, M. Lassas and M. Vogelius.

Posted January 18, 2024

Informal Geometry and Topology Seminar Questions or comments?

1:30 pm Lockett 233
Colton Sandvick, Louisiana State University

Characteristic Classes

Posted December 5, 2023

Last modified January 31, 2024

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Steven Sivek, Imperial College London

Rational homology 3-spheres and SL(2,C) representations

Building on non-vanishing theorems of Kronheimer and Mrowka in instanton Floer homology, Zentner proved that if Y is a homology 3-sphere other than S^3, then its fundamental group admits a homomorphism to SL(2,C) with non-abelian image. In this talk, I’ll explain how to generalize this to any Y whose first homology is 2-torsion or 3-torsion, other than #^n RP^3 for any n or lens spaces of order 3. This is joint work with LSU’s own Sudipta Ghosh and with Raphael Zentner.

Posted February 3, 2024

3:30 pm – 4:30 pm Zoom
Cody Stockdale, Clemson University

On the Calderón-Zygmund theory of singular integrals

Calderón and Zygmund's seminal work on singular integral operators has greatly influenced modern harmonic analysis. We begin our discussion with some classical aspects of CZ theory, including examples and applications, and then focus on the crucial weak-type (1,1) estimate for CZ operators. We investigate techniques for obtaining weak-type inequalities that use the CZ decomposition and ideas inspired by Nazarov, Treil, and Volberg. We end with an application of these methods to the study of the Riesz transforms in high dimensions.

Posted January 28, 2024

Combinatorics Seminar Questions or comments?

2:00 pm – 3:00 pm Zoom (Please email zhiyuw at lsu.edu for Zoom link)
Sam Spiro, Rutgers University

The Random Tur\'an Problem

Let $G_{n,p}$ denote the random $n$-vertex graph obtained by including each edge independently and with probability $p$. Given a graph $F$, let $\mathrm{ex}(G_{n,p},F)$ denote the size of a largest $F$-free subgraph of $G_{n,p}$. When $F$ is non-bipartite, the asymptotic behavior of $\mathrm{ex}(G_{n,p},F)$ was determined in breakthrough work done independently by Conlon-Gowers and by Schacht. In this talk we discuss some recent results for bipartite $F$ (where much less is known), as well as for the analogous problem for $r$-partite $r$-graphs.

Posted January 25, 2024

Last modified January 31, 2024

Colloquium Questions or comments?

3:30 pm Lockett 232
Ali Kara, University of Michigan

Reinforcement Learning in Non-Markovian Environments under General Information Structures

Abstract: For decision-making under uncertainty, typically only an ideal model is assumed, and the control design is based on this given model. However, in reality, the assumed model may not perfectly reflect the underlying dynamics, or there might not be an available mathematical model. To overcome this issue, one approach is to use the past data of perceived state, cost and control trajectories to learn the model or the optimal control functions directly, a method also known as reinforcement learning. The majority of the existing literature has focused on methods structured for systems where the underlying state process is Markovian and the state is fully observed. However, there are many practical settings where one works with data and does not know the possibly very complex structure under which the data is generated and tries to respond to the environment. In this talk, I will present a convergence theorem for stochastic iterations, particularly focusing on Q-learning iterates, under a general, possibly non-Markovian, stochastic environment. I will then discuss applications of this result to the decision making problems where the agent's perceived state is a noisy version of some hidden Markov state process, i.e. partially observed MDPs, and when the agent keeps track of a finite memory of the perceived data. I will also discuss applications for a class of continuous-time controlled diffusion problems.

Posted February 11, 2024

Combinatorics Seminar Questions or comments?

2:00 pm – 3:00 pm Zoom (Please email zhiyuw at lsu.edu for Zoom link)
Youngho Yoo, Texas A&M University

Minimum degree conditions for apex-outerplanar minors

Motivated by Hadwiger's conjecture, we study graphs H for which every graph with minimum degree at least |V(H)|-1 contains H as a minor. We prove that a large class of apex-outerplanar graphs satisfies this property. Our result gives the first examples of such graphs whose vertex cover numbers are significantly larger than a half of its vertices, and recovers all known such graphs that have arbitrarily large maximum degree. If time permits, we discuss how our proof can be adapted to directed graphs to show that every directed graph with minimum out-degree at least t contains a certain orientation of the wheel and of every apex-tree on t+1 vertices as a subdivision and a butterfly minor respectively. Joint work with Chun-Hung Liu.

Posted February 15, 2024

Faculty Meeting Questions or comments?

3:30 pm – 4:30 pm ZoomMeeting of the Professorial Faculty

Posted February 12, 2024

Colloquium Questions or comments?

3:30 pm – 4:30 pm 232 Lockett Hall
Ke Chen, University of Maryland

Towards efficient deep operator learning for forward and inverse PDEs: theory and algorithms

Abstract: Deep neural networks (DNNs) have been a successful model across diverse machine learning tasks, increasingly capturing the interest for their potential in scientific computing. This talk delves into efficient training for PDE operator learning in both the forward and inverse PDE settings. Firstly, we address the curse of dimensionality in PDE operator learning, demonstrating that certain PDE structures require fewer training samples through an analysis of learning error estimates. Secondly, we introduce an innovative DNN, the pseudo-differential auto-encoder integral network (pd-IAE net), and compare its numerical performance with baseline models on several inverse problems, including optical tomography and inverse scattering. We will briefly mention some future works at the end, focusing on the regularization of inverse problems in the context of operator learning.

Posted February 5, 2024

Last modified February 14, 2024

Algebra and Number Theory Seminar Questions or comments?

3:20 pm – 4:10 pm Lockett 233 or click here to attend on Zoom
Hasan Saad, University of Virginia

Distributions of points on hypergeometric varieties

In the 1960's, Birch proved that the traces of Frobenius for elliptic curves taken at random over a large finite field is modeled by the semicircular distribution (i.e., $SU(2),$ the usual Sato-Tate for non-CM elliptic curves). In this talk, we show how the theory of harmonic Maass forms and modular forms allow us to determine the limiting distribution of normalized traces of Frobenius over families of varieties. For Legendre elliptic curves, the limiting distribution is $SU(2),$ whereas for a certain family of $K3$ surfaces, the limiting distribution is $O(3).$ Since the $O(3)$ distribution has vertical asymptotes, we show how to obtain an explicit result by bounding the error. Additionally, we show how to count "matrix" points on these varieties and therefore determine the limiting distributions for these "matrix points."

Posted January 18, 2024

Informal Geometry and Topology Seminar Questions or comments?

1:30 pm Lockett 233
Shea Vela-Vick, Louisiana State University

Characteristic Classes

Posted February 20, 2024

Faculty Meeting Questions or comments?

1:30 pm – 2:00 pm ZoomMeeting of the Professorial Faculty

Posted January 15, 2024

Last modified February 18, 2024

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Jake Murphy, LSU

Subgroups of Coxeter groups and Stallings folds

Stallings introduced a technique called a fold to study subgroups of free groups. These folds allow us to associate labeled graphs to subgroups of free groups, which in turn provide solutions to algorithmic questions about these subgroups, and Dani and Levcovitz generalized these techniques to the setting of right-angled Coxeter groups. In this talk, we will generalize these techniques to subgroups of general Coxeter groups by creating a labeled cell complex for a given subgroup. We will show that these complexes characterize the index of a subgroup and whether a subgroup is normal. Finally, we will construct a complex corresponding to the intersection of two subgroups and use this to determine whether subgroups of right-angled Coxeter groups are malnormal.

Posted February 17, 2024

Combinatorics Seminar Questions or comments?

2:00 pm – 3:00 pm Zoom (Please email zhiyuw at lsu.edu for Zoom link)
Yixuan Huang, Vanderbilt University

Even and odd cycles through specified vertices

Cycles through specified vertices generalize Hamilton cycles. Given a vertex subset of a graph , we define the local connectivity on $\kappa_G(X)$ by $\min_{x,y \in X} \kappa_G(x,y)$, where $\kappa_G(x,y)$ is the minimum number of vertices or edges separating $x$ and $y$, and by Menger’s theorem, equal to the maximum number of internally disjoint $xy$-paths. We prove that if a vertex subset $X$ satisfies $\kappa_G(X) \ge k \ge3$ and $|X| > k$, then there is an even cycle through any $k$ vertices of $X$. In addition, if the block containing $X$ is non-bipartite, there is an odd cycle through any $k$ vertices of $X$. Our results extend the results based on ordinary connectivity due to Bondy and Lovász. As a corollary, we prove the existence of cycles through a particular subset in the prism graph.

Posted December 28, 2023

Last modified February 20, 2024

Control and Optimization Seminar Questions or comments?

11:30 am – 12:20 pm Zoom (Click “Questions or Comments?” to request a Zoom link)
Huyên Pham
Editor-in-Chief for SIAM Journal on Control and Optimization, 2024-

A Schrödinger Bridge Approach to Generative Modeling for Time Series

We propose a novel generative model for time series based on Schrödinger bridge (SB) approach. This consists in the entropic interpolation via optimal transport between a reference probability measure on path space and a target measure consistent with the joint data distribution of the time series. The solution is characterized by a stochastic differential equation on finite horizon with a path-dependent drift function, hence respecting the temporal dynamics of the time series distribution. We estimate the drift function from data samples by nonparametric, e.g. kernel regression methods, and the simulation of the SB diffusion yields new synthetic data samples of the time series. The performance of our generative model is evaluated through a series of numerical experiments. First, we test with autoregressive models, a GARCH Model, and the example of fractional Brownian motion, and measure the accuracy of our algorithm with marginal, temporal dependencies metrics, and predictive scores. Next, we use our SB generated synthetic samples for the application to deep hedging on real-data sets.

Posted November 14, 2023

Last modified February 27, 2024

Algebra and Number Theory Seminar Questions or comments?

3:20 pm – 4:10 pm Lockett 233 or click here to attend on Zoom
Eleanor McSpirit, University of Virginia

Infinite Families of Quantum Modular 3-Manifold Invariants

In 1999, Lawrence and Zagier established a connection between modular forms and the Witten-Reshetikhin-Turaev invariants of 3-manifolds by constructing q-series whose radial limits at roots of unity recover these invariants for particular manifolds. These q-series gave rise to some of the first examples of quantum modular forms. Using a 3-manifold invariant recently developed Akhmechet, Johnson, and Krushkal, one can obtain infinite families of quantum modular invariants which realize the series of Lawrence and Zagier as a special case. This talk is based on joint work with Louisa Liles.

Posted January 12, 2024

Last modified February 21, 2024

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Maddalena Pismataro, University of Bologna

Cohomology rings of abelian arrangements

Abelian arrangements are generalizations of hyperplane and toric arrangements, whose complements cohomology rings have been studied since the 70’s. We introduce the complex hyperplane case, proved by Orlik and Solomon (1980), and the real case, Gelfand-Varchenko (1987). Then, we describe toric arrangements, showing results due to De Concini and Procesi (2005) and to Callegaro, D ’Adderio, Delucchi, Migliorini, and Pagaria (2020). Finally, we discuss a new technique to prove the Orlik-Solomon and De Concini-Procesi relations from the Gelfand-Varchenko ring and to provide a presentation of the cohomology ring of the complements of all abelian arrangements. This is a join work with Evienia Bazzocchi and Roberto Pagaria.

Posted February 26, 2024

3:30 pm – 4:30 pm Online Zoom
Yaghoub Rahimi, Georgia Institute of Technology

AVERAGES OVER THE GAUSSIAN PRIMES: GOLDBACH’S CONJECTURE AND IMPROVING ESTIMATES

In this discussion we will establish a density version of the strong Goldbach conjecture for Gaussian integers, restricted to sectors in the complex plane.

Posted February 25, 2024

Combinatorics Seminar Questions or comments?

2:00 pm – 3:00 pm Lockett Hall 233
Scott Baldridge, Louisiana State University

Using quantum states to understand the four-color theorem

The four-color theorem states that a bridgeless plane graph is four-colorable, that is, its faces can be colored with four colors so that no two adjacent faces share the same color. This was a notoriously difficult theorem that took over a century to prove. In this talk, we generate vector spaces from certain diagrams of a graph with a map between them and show that the dimension of the kernel of this map is equal to the number of ways to four-color the graph. We then generalize this calculation to a homology theory and in doing so make an interesting discovery: the four-color theorem is really about all of the smooth closed surfaces a graph embeds into and the relationships between those surfaces. The homology theory is based upon a topological quantum field theory. The diagrams generated from the graph represent the possible quantum states of the graph and the homology is, in some sense, the vacuum expectation value of this system. It gets wonderfully complicated from this point on, but we will suppress this aspect from the talk and instead show a fun application of how to link the Euler characteristic of the homology to the famous Penrose polynomial of a plane graph. This talk will be hands-on and the ideas will be explained through the calculation of easy examples! My goal is to attract students and mathematicians to this area by making the ideas as intuitive as possible. Topologists and representation theorists are encouraged to attend also—these homologies sit at the intersection of topology, representation theory, and graph theory. This is joint work with Ben McCarty.