Posted October 9, 2024

Mathematical Physics and Representation Theory Seminar

2:30 pm – 3:20 pm Lockett 233
Tobias Simon, University of Erlangen, Germany

Realizations of irreducible unitary representations of the Lorentz group in spaces of distributional sections over de Sitter space

In Algebraic Quantum Field Theory, one is interested in constructing nets of local von Neumann algebras satisfying the Haag Kastler axioms. Every such net defines a local net of standard subspaces in the corresponding Hilbert space by letting the selfadjoint elements in the local algebras act on a common cyclic and separating vector. In this talk, we discuss work by Frahm, Neeb and Olafsson which constructs nets standard subspaces on de Sitter space satisfying the corresponding axioms. Here the main tool is "realizing" irreducible unitary representations of the Lorentz group SO(1,d) in spaces of distributional sections over de Sitter space. These can be constructed from SO(1,d-1)-finite distribution vectors obtained as distributional boundary values of holomorphically extended orbit maps of SO(d)-finite vectors. Our main contribution is the proof of polynomial growth rates of these orbit maps, which guarantees the existence of the boundary values in the space of distribution vectors.

Posted October 9, 2024

11:00 am – 12:00 pm TBD"What to do in Summer"

Join us for the "Summer Opportunities Event," organized by the SIAM Student Chapter! This session will provide valuable insights on building effective CVs and resumes, as well as exploring a variety of summer opportunities such as internships, summer schools, and workshops. The event will help you enhance your academic profile, gain professional experience, and guide you in finding the right opportunities and preparing the necessary materials.

Posted October 14, 2024

Faculty Meeting Questions or comments?

3:00 pm – 3:45 pm Thursday, October 10, 2024 ZoomMeeting of the Tenured Faculty

Posted August 30, 2024

Last modified October 17, 2024

Informal Geometry and Topology Seminar Questions or comments?

1:30 pm Locket 233
Adithyan Pandikkadan, Louisiana State University

Construction of Hyperbolic Manifolds

In this talk, we will discuss two ways for constructing hyperbolic manifolds. We will begin by introducing hyperbolic surfaces, focusing on how to equip a hyperbolic structure on higher genus surfaces. Following this, we will discuss the construction of arithmetic hyperbolic manifolds which is a more general approach.

Posted October 14, 2024

Probability Seminar Questions or comments?

3:30 pm Lockett 381
Benjamin Fehrman, Louisiana State University

Lectures on Homogenization

In these lectures, we will develop a fully rigorous theory of stochastic homogenization for linear elliptic equations, beginning with the periodic case. Applications of homogenization are diverse, and include modeling the conductivity of composites with small-scale defects and the large-scale behavior of passive advected quantities like temperature in turbulent fluid flows. These systems are effectively random, to our eyes, and their study is essentially equivalent to the asymptotic behavior of a diffusion process in a random environment. Our aim is to derive an effective model that provides a good approximation of the original system with high probability.

Posted August 29, 2024

Last modified October 7, 2024

Geometry and Topology Seminar Seminar website

3:30 pm
Bin Sun, Michigan State University

$L^2$-Betti numbers of Dehn fillings

I will talk about a recent joint work with Nansen Petrosyan where we obtain conditions under which $L^2$-Betti numbers are preserved by group-theoretic Dehn fillings. As an application, we verify the Singer Conjecture for certain Einstein manifolds and provide new examples of hyperbolic groups with exotic subgroups. We also establish a virtual fibering criterion and obtain bounds on deficiency of Dehn fillings. A key step in our approach of computations of $L^2$-Betti numbers is the construction of a tailored classifying space, which is of independent interest.

Posted September 27, 2024

Last modified October 16, 2024

Mathematical Physics and Representation Theory Seminar

2:30 pm – 3:20 pm Lockett 233
Xinchun Ma, University of Chicago

Cherednik algebras, Torus knots and flag commuting varieties

In this talk, we will explore how the Khovanov-Rozansky homology of the (m,n)-torus knot can be extracted from the finite-dimensional representation of the rational Cherednik algebra at slope m/n, equipped with the Hodge filtration. Our approach involves constructing a family of coherent sheaves on the Hilbert scheme of points on the plane, arising from cuspidal character D-modules. In describing this family of coherent sheaves, the geometry of nilpotent flag commuting varieties naturally emerges, closely related to the compactified regular centralizer in type A.

Posted October 4, 2024

5:30 pm James E. Keisler Lounge (room 321 Lockett)Actuarial Student Association Meeting

We will have guest speaker Jordan Hayes from AETNA (CVS Health). Pizza will be served.

Posted August 14, 2024

Last modified October 17, 2024

Algebra and Number Theory Seminar Questions or comments?

2:30 pm Lockett 233 or click here to attend on Zoom
Brian Grove, LSU

The Explicit Hypergeometric Modularity Method

The existence of hypergeometric motives predicts that hypergeometric Galois representations are modular. More precisely, explicit identities between special values of hypergeometric character sums and coefficients of certain modular forms on appropriate arithmetic progressions of primes are expected. A few such identities have been established in the literature using various ad-hoc techniques. I will discuss a general method to prove these hypergeometric modularity results in dimensions two and three. This is joint work with Michael Allen, Ling Long, and Fang-Ting Tu.

Posted October 18, 2024

Computational Mathematics Seminar

3:30 pm DMC 1034
Xili Wang, Peking University

DL for PDEs: towards parametric, high-dimensional and PDE-constrained optimization

Despite advances in simulating multiphysics problems through numerical discretization of PDEs, mesh-based approximation remains challenging, especially for high-dimensional problems governed by parameterized PDEs. Moreover, other PDE-related problems, such as PDE-constrained shape optimization, introduce additional difficulties including mesh deformation and correction. While Physics-Informed Neural Networks (PINNs) offer an alternative, they often lack the accuracy of traditional methods like finite element methods. Relying solely on a 'black-box' approach may not be the best choice for scientific computing. Inspired by adaptive finite element methods, we propose a deep adaptive sampling approach to solve low-regularity parametric PDEs and high-dimensional committor functions in rare event simulations. Additionally, by integrating the mesh-free nature of neural networks into the direct-adjoint looping (DAL), we achieve fully mesh-independent solutions for PDE-constrained shape optimization problems.

Posted August 30, 2024

Last modified October 17, 2024

Informal Geometry and Topology Seminar Questions or comments?

1:30 pm Locket 233
Megan Fairchild, Louisiana State University

Slicing Obstructions from 4-Manifold Theory

The orientable 4 genus of a knot is defined to be the minimum genus amongst all smoothly embedded surfaces in the 4-ball with boundary the knot. A knot is called slice if it bounds a smoothly embedded disk in the 4-ball. Invariants of knots, either classical or Heegaard Floer, are commonly used as lower bounds for the orientable 4 genus of knots. We will examine a different approach to showing knots are not smoothly slice, coming from 4-manifold theory.

Posted October 22, 2024

Probability Seminar Questions or comments?

3:30 pm Lockett 237
Benjamin Fehrman, Louisiana State University

Lectures on Homogenization - Part 2

In these lectures, we will develop a fully rigorous theory of stochastic homogenization for linear elliptic equations, beginning with the periodic case. Applications of homogenization are diverse, and include modeling the conductivity of composites with small-scale defects and the large-scale behavior of passive advected quantities like temperature in turbulent fluid flows. These systems are effectively random, to our eyes, and their study is essentially equivalent to the asymptotic behavior of a diffusion process in a random environment. Our aim is to derive an effective model that provides a good approximation of the original system with high probability.

Posted October 15, 2024

Last modified October 22, 2024

Nathan Mehlhop, Louisiana State University

Ergodic averaging operators

Certain quantitative estimates such as oscillation inequalities are often used in the study of pointwise convergence problems. Here, we study these for discrete ergodic averaging operators and discrete singular integrals along polynomial orbits in multidimensional subsets of integers or primes. Because of its relevance to multiparameter averaging operators, we also consider the vector-valued setting. Several tools including the Hardy-Littlewood circle method, Weyl's inequality, the Ionescu-Wainger multiplier theorem, the Magyar-Stein-Wainger sampling principle, the Marcinciewicz-Zygmund inequality, and others, are important in this field. The talk will introduce the problem and many of these ideas, and then give some outline of how the various estimates can be put together to give the conclusion.

Posted August 19, 2024

Last modified September 27, 2024

Control and Optimization Seminar Questions or comments?

10:30 am – 11:20 am Zoom (click here to join)
Andrii Mironchenko, University of Klagenfurt
IEEE CSS George S. Axelby Outstanding Paper Awardee

Superposition Theorems for Input-to-State Stability of Time-Delay Systems

We characterize input-to-state stability (ISS) for nonlinear time-delay systems (TDS) in terms of stability and attractivity properties for systems with inputs. Using the specific structure of TDS, we obtain much tighter results than those possible for general infinite-dimensional systems. The subtle difference between forward completeness and boundedness of reachability sets (BRS) is essential for the understanding of the ISS characterizations. As BRS is important in numerous other contexts, we discuss this topic in detail as well. We shed light on the differences between the ISS theories for TDS, generic infinite-dimensional systems, and ODEs.

Posted October 1, 2024

Combinatorics Seminar Questions or comments?

2:30 pm – 3:30 pm Friday, October 4, 2024 Zoom Link
Andrew Fulcher, University College Dublin

The cyclic flats of L-polymatroids

In recent years, $q$-polymatroids have drawn interest because of their connection with rank-metric codes. For a special class of $q$-polymatroids called $q$-matroids, the fundamental notion of a cyclic flat has been developed as a way to identify the key structural features of a $q$-matroid. In this talk, we will see a generalization of the definition of a cyclic flat that can apply to $q$-polymatroids, as well as a further generalization, $L$-polymatroids. The cyclic flats of an $L$-polymatroid is essentially a reduction of the data of an $L$-polymatroid such that the $L$-polymatroid can be retrieved from its cyclic flats. As such, in matroid theory, cyclic flats have been used to characterize numerous invariants.

Posted September 25, 2024

Last modified October 7, 2024

(Originally scheduled for Tuesday, October 8, 2024)

Summer Opportunities

TBA DATE is still to be determined!

Posted September 27, 2024

Last modified October 24, 2024

Mathematical Physics and Representation Theory Seminar

2:30 pm – 3:20 pm Lockett 233
Nikolay Grantcharov, University of Chicago

Infinitesimal structure of BunG

Given a semisimple group G and a smooth projective curve X over an algebraically closed field of arbitrary characteristic, let Bun_G(X) denote the moduli space of principal G-bundles over X. For a bundle P without infinitesimal symmetries, we describe the n^th order divided-power infinitesimal jet spaces of Bun_G(X) at P for each n. The description is in terms of differential forms on the Fulton-Macpherson compactification of the configuration space, with logarithmic singularities along the diagonal divisor. We also briefly discuss applications into constructing Hitchin's flat connection on the vector bundle of conformal blocks.

Posted September 26, 2024

Last modified October 25, 2024

Applied Analysis Seminar Questions or comments?

3:30 pm Lockett 233
Matias Delgadino, University of Texas at Austin

Generative Adversarial Networks: Dynamics

Generative Adversarial Networks (GANs) was one of the first Machine Learning algorithms to be able to generate remarkably realistic synthetic images. In this presentation, we delve into the mechanics of the GAN algorithm and its profound relationship with optimal transport theory. Through a detailed exploration, we illuminate how GAN approximates a system of PDE, particularly evident in shallow network architectures. Furthermore, we investigate known pathological behaviors such as mode collapse and failure to converge, and elucidate their connections to the underlying PDE framework through an illustrative example.

Posted August 21, 2024

Last modified October 25, 2024

Algebra and Number Theory Seminar Questions or comments?

2:30 pm – 3:30 pm Lockett 233 or click here to attend on Zoom
Brett Tangedal, University of North Carolina, Greensboro

Real Quadratic Fields and Partial Zeta-Functions

We focus on real quadratic number fields and explain an approach to the partial zeta-functions associated with the various ideal class groups of such fields dating back to the original work of Zagier, Stark, Shintani, David Hayes, and others. Along the way, we will give a brief introduction to Stark's famous first order zero conjecture.

Posted August 30, 2024

Last modified October 27, 2024

Informal Geometry and Topology Seminar Questions or comments?

1:30 pm Locket 233
Matthew Lemoine, Louisiana State University

A Brief Introduction to Khovanov Homology through an example

In this talk, we will discuss Khovanov Homology and how to compute this homology using an example with the trefoil knot. We will also discuss the relations between Khovanov Homology and the Jones Polynomial.

Posted October 7, 2024

Last modified October 28, 2024

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Monika Kudlinska, University of Cambridge

Solving equations in free-by-cyclic groups

A group G is said to be free-by-cyclic if it maps onto the infinite cyclic group with free kernel of finite rank. Free-by-cyclic groups form a large and widely-studied class with close links to 3-manifold topology. A group G is said to be equationally Noetherian if any system of equations over G is equivalent to a finite subsystem. In joint work with Motiejus Valiunas we show that all free-by-cyclic groups are equationally Noetherian. As an application, we deduce that the set of exponential growth rates of a free-by-cyclic group is well ordered.

Posted October 29, 2024

Probability Seminar Questions or comments?

3:30 pm Lockett 237
Benjamin Fehrman, Louisiana State University

Lectures on Homogenization - Part 3

In these lectures, we will develop a fully rigorous theory of stochastic homogenization for linear elliptic equations, beginning with the periodic case. Applications of homogenization are diverse, and include modeling the conductivity of composites with small-scale defects and the large-scale behavior of passive advected quantities like temperature in turbulent fluid flows. These systems are effectively random, to our eyes, and their study is essentially equivalent to the asymptotic behavior of a diffusion process in a random environment. Our aim is to derive an effective model that provides a good approximation of the original system with high probability.

Posted August 26, 2024

Last modified October 24, 2024

Control and Optimization Seminar Questions or comments?

10:30 am – 11:20 am Zoom (click here to join)
Angelia Nedich, Arizona State University

Resilient Distributed Optimization for Cyber-physical Systems

This talk considers the problem of resilient distributed multi-agent optimization for cyber-physical systems in the presence of malicious or non-cooperative agents. It is assumed that stochastic values of trust between agents are available which allows agents to learn their trustworthy neighbors simultaneously with performing updates to minimize their own local objective functions. The development of this trustworthy computational model combines the tools from statistical learning and distributed consensus-based optimization. Specifically, we derive a unified mathematical framework to characterize convergence, deviation of the consensus from the true consensus value, and expected convergence rate, when there exists additional information of trust between agents. We show that under certain conditions on the stochastic trust values and consensus protocol: 1) almost sure convergence to a common limit value is possible even when malicious agents constitute more than half of the network, 2) the deviation of the converged limit, from the nominal no attack case, i.e., the true consensus value, can be bounded with probability that approaches 1 exponentially, and 3) correct classification of malicious and legitimate agents can be attained in finite time almost surely. Further, the expected convergence rate decays exponentially with the quality of the trust observations between agents. We then combine this trust-learning model within a distributed gradient-based method for solving a multi-agent optimization problem and characterize its performance.

Posted October 29, 2024

LSU AWM Student Chapter LSU AWM Student Chapter Website

12:30 pm – 1:20 pmAWM Student Chapter Q & A Session with Prof. Angelia Nedich

Join us for a QA session hosted by the Association for Women in Mathematics (AWM) Student Chapter. We are honored to have Prof. Angelia Nedich from Arizona State University, a leading convex analysis and optimization researcher. Prof. Nedich will share insights from her research and discuss her academic experiences.

Posted November 1, 2024

5:30 pm James E. Keisler Lounge (room 321 Lockett)Actuarial Student Association Meeting

Guest speaker Jack Berry from Cigna health will speak. Pizza will be served.

Posted October 8, 2024

Last modified October 30, 2024

Algebra and Number Theory Seminar Questions or comments?

2:30 pm – 3:30 pm Virtual talk: click here to attend on Zoom
Linli Shi, University of Connecticut

On higher regulators of Picard modular surfaces

The Birch and Swinnerton-Dyer conjecture relates the leading coefficient of the L-function of an elliptic curve at its central critical point to global arithmetic invariants of the elliptic curve. Beilinson’s conjectures generalize the BSD conjecture to formulas for values of motivic L-functions at non-critical points. In this talk, I will relate motivic cohomology classes, with non-trivial coefficients, of Picard modular surfaces to a non-critical value of the motivic L-function of certain automorphic representations of the group GU(2,1).

Posted August 30, 2024

Last modified November 4, 2024

Informal Geometry and Topology Seminar Questions or comments?

1:30 pm Locket 233
Nilangshu Bhattacharyya, Louisiana State University

Gromov norm of a compact manifold and straightening

We will define the Gromov norm of a compact manifold and straightening (every singular chain is naturally homotopic to a straight one).

Posted October 12, 2024

Last modified October 30, 2024

Vishwa Dewage, Clemson University

The Laplacian of an operator and applications to Toeplitz operators

Werner's quantum harmonic analysis (QHA) provides a set of tools that are applicable in many areas of analysis, including operator theory. As noted by Fulsche, QHA is particularly suitable to study Toeplitz operators on the Fock space. We explore the Laplacian of an operator and a heat equation for operators on the Fock space using QHA. Then we discuss some applications to Toeplitz operators. This talk is based on joint work with Mishko Mitkovski.

Posted September 6, 2024

Last modified October 11, 2024

Control and Optimization Seminar Questions or comments?

10:30 am – 11:20 am Zoom (click here to join)
Laura Menini, Università degli Studi di Roma Tor Vergata

Distance to Instability for LTI Systems under Structured Perturbations

The talk will present a procedure to compute the distance to instability for linear systems subject to structured perturbations, in particular perturbations that affect polynomially the dynamics of the system. The procedure is based on classical notions from stability of linear systems, optimization and algebraic geometry, some of which will be reviewed briefly. The application to the design of fixed-structure controllers to deal with robust control problems will also be outlined, with the goal of choosing the controller which obtains the best conservative estimate of the region of stability. The results will be illustrated on some academic examples.

Posted November 5, 2024

Combinatorics Seminar Questions or comments?

2:30 pm – 3:30 pm Lockett Hall 233 (simulcasted via Zoom)
Matthew Mizell, LSU

Structures That Arise From Nested Sequences of Flats in Projective and Affine Geometries

In a vector space $V$, take a sequence of subspaces $V_1,V_2,\dots,V_n$ such that $V_1 \subseteq V_2 \subseteq \ldots \subseteq V_n = V$. Color the non-zero elements of $V_1$ green, the elements of $V_2 \backslash V_1$ red, the elements of $V_3 \backslash V_2$ green and so on. We call the resulting set of green elements a target. The study of targets was initiated by Nelson and Nomoto in 2018 for vector spaces over the $2$-element field. In this talk, we will discuss targets over arbitrary finite fields. We will also consider the graph analogue of targets as well as targets over affine geometries. Our main results will characterize each type of target in terms of its forbidden substructures. This is joint work with James Oxley.

Posted October 8, 2024

Last modified November 4, 2024

Algebra and Number Theory Seminar Questions or comments?

2:30 pm – 3:30 pm Lockett 233 or click here to attend on Zoom
Michael Allen, Louisiana State University

An infinite family of hypergeometric supercongruences

In a recent series of papers with Brian Grove, Ling Long, and Fang-Ting Tu, we explore the relationship between modular forms and hypergeometric functions in the particular settings of complex, finite, and $p$-adic fields, and unify these perspectives through Galois representations. In this talk, we focus primarily on the $p$-adic aspects, where this relationship arises in the form of congruences between truncated hypergeometric sums and Fourier coefficients of modular forms. Such congruences are predicted to hold modulo $p$ by formal commutative group law, we refer to a congruence modulo a higher power of $p$ as a supercongruence. In this talk, we briefly survey results and methods in the area of supercongruences before establishing an infinite family of supercongruences which hold modulo $p^2$ for all primes in certain arithmetic progressions depending on the parameters of the corresponding hypergeometric functions.

Posted October 22, 2024

Computational Mathematics Seminar

3:30 pm DMC 1034
Jeremy Shahan, Louisiana State University

Shape Optimization with Unfitted Finite Element Methods

We present a formulation of a PDE-constrained shape optimization problem that uses an unfitted finite element method (FEM). The geometry is represented (and optimized) using a level set approach and we consider objective functionals that are defined over bulk domains. For a discrete objective functional (i.e. one defined in the unfitted FEM framework), we derive the exact Frechet, shape derivative in terms of perturbing the level set function directly. In other words, no domain velocity is needed. We also show that the derivative is (essentially) the same as the shape derivative at the continuous level, so is rather easy to compute. In other words, one gains the benefits of both the optimize-then-discretize and discretize-then-optimize approaches.