Posted March 19, 2024
Computational Mathematics Seminar
until 3:30 pm Digital Media Center 1034
Quoc Tran-Dinh, UNC Chapel Hill
Boosting Convergence Rates for Fixed-Point and Root-Finding Algorithms
Approximating a fixed-point of a nonexpansive operator or a root of a nonlinear equation is a fundamental problem in computational mathematics, which has various applications in different fields. Most classical methods for fixed-point and root-finding problems such as fixed-point or gradient iteration, Halpern's iteration, and extragradient methods have a convergence rate of at most O(1/square root k) on the norm of the residual, where k is the iteration counter. This convergence rate is often obtained via appropriate constant stepsizes. In this talk, we aim at presenting some recent development to boost the theoretical convergence rates of many root-finding algorithms up to O(1/k). We first discuss a connection between the Halpern fixed-point iteration in fixed-point theory and Nesterov's accelerated schemes in convex optimization for solving monotone equations involving a co-coercive operator (or equivalently, fixed-point problems of a nonexpansive operator). We also study such a connection for different recent schemes, including extra anchored gradient method to obtain new algorithms. We show how a faster convergence rate result from one scheme can be transferred to another and vice versa. Next, we discuss various variants of the proposed methods, including randomized block-coordinate algorithms for root-finding problems,which are different from existing randomized coordinate methods in optimization. Finally, we consider the applications of these randomized coordinate schemes to monotone inclusions and finite-sum monotone inclusions. The algorithms for the latter problem can be applied to many applications in federated learning.
Posted January 18, 2024
Last modified April 15, 2024
Informal Geometry and Topology Seminar Questions or comments?
1:30 pm Lockett 233, Zoom
Krishnendu Kar, Louisiana State University
An Introduction to ‘Spetra’cular category
We define higher homotopy groups of a topological space $X$ by taking homotopy classes of the maps from higher dimensional spheres. These higher-homotopy groups are exponentially more difficult to compute than homology or cohomology groups due to the failure of some robust computational tools such as excision. Excision holds for these groups up to connectivity, and so does the Mayer-Vietoris sequence. The suspension map $\Sigma:X\rightarrow \Sigma X$ induces a map on higher homotopy groups $\Sigma:\pi_n(X)\rightarrow \pi_{n+1}(\Sigma X)$. A theorem by Freudenthal states that after taking enough suspensions, these homotopy groups will stabilize eventually. We call the colimit of these homotopy groups as the stable homotopy group. In the modern treatment of stable homotopy theory, spaces are replaced by spectra. In this talk, we will see some important facts, examples, and, more importantly, justification for the title.
Posted April 1, 2024
Last modified April 15, 2024
Geometry and Topology Seminar Seminar website
3:30 pm Lockett 233
Kevin Schreve, Louisiana State University
Homology growth and aspherical manifolds
Suppose we have a space X and a tower of finite covers that are increasingly better approximations to the universal cover. In this talk, we will be interested in how classical homological invariants grow as we go up the tower. In particular, I will survey various conjectures about the rational/F_p-homology growth and integral torsion growth in these towers. We'll discuss constructions of closed aspherical manifolds that have F_p-homology growth outside of the middle dimension, and give some applications to (non)-fibering of high-dimensional manifolds. This is joint work with Grigori Avramidi and Boris Okun.
Posted April 14, 2024
Combinatorics Seminar Questions or comments?
2:00 pm – 3:00 pm Lockett Hall 233
Xingxing Yu, Georgia Institute of Technology
Planar Turan Number of Cycles
The planar Turan number of a graph $H$, $ex_P(n,H)$, is the maximum number of edges in an $n$-vertex planar graph without $H$ as a subgraph. We discuss recent work on $ex_P(n,H)$, in particular when $H=C_k$ (cycle of length $k$), including our work on $ex_P(n,C_7)$. We prove an upper bound on $ex_P(n, C_k)$ for $k, n\ge 4$, establishing a conjecture of Cranston, Lidicky, Liu, and Shantanam. The discharging method and previous work on circumference of planar graphs are used.
Posted January 6, 2024
Last modified March 4, 2024
Control and Optimization Seminar Questions or comments?
11:30 am – 12:20 pm Zoom (click here to join)
Madalena Chaves, Centre Inria d'Université Côte d'Azur
Coupling, Synchronization Dynamics, and Emergent Behavior in a Network of Biological Oscillators
Biological oscillators often involve a complex network of interactions, such as in the case of circadian rhythms or cell cycle. Mathematical modeling and especially model reduction help to understand the main mechanisms behind oscillatory behavior. In this context, we first study a two-gene oscillator using piecewise linear approximations to improve the performance and robustness of the oscillatory dynamics. Next, motivated by the synchronization of biological rhythms in a group of cells in an organ such as the liver, we then study a network of identical oscillators under diffusive coupling, interconnected according to different topologies. The piecewise linear formalism enables us to characterize the emergent dynamics of the network and show that a number of new steady states is generated in the network of oscillators. Finally, given two distinct oscillators mimicking the circadian clock and cell cycle, we analyze their interconnection to study the capacity for mutual period regulation and control between the two reduced oscillators. We are interested in characterizing the coupling parameter range for which the two systems play the roles "controller-follower".
Posted January 28, 2024
Last modified April 1, 2024
Mathematical Physics and Representation Theory Seminar
2:30 pm – 3:20 pm Lockett 233
Greg Parker, Stanford University
$\mathbb Z_2$-harmonic spinors as limiting objects in geometry and topology
$\mathbb Z_2$-harmonic spinors are singular solutions of Dirac-type equations that allow topological twisting around a submanifold of codimension 2. These objects arise as limits at the boundary of various moduli spaces in several distinct areas of low-dimensional topology, gauge/Floer theory, and enumerative geometry. The first part of this talk will introduce these objects, and discuss the various contexts in which they arise and the relationship between them. The second part of the talk will focus on the deformations of $\mathbb Z_2$-harmonic spinors when varying background parameters as a model for the novel analytic problems presented by these objects. In particular, the deformations of the singular submanifold play a role, giving the problem some characteristics similar to a free-boundary-value problem and leading to a hidden elliptic pseudo-differential operator that governs the geometry of the moduli spaces.
Posted April 21, 2024
Probability Seminar Questions or comments?
3:30 pm – 4:30 pm Lockett 232
Ben Seeger, The University of Texas at Austin
Equations on Wasserstein space and applications
The purpose of this talk is to give an overview of recent work involving differential equations posed on spaces of probability measures and their use in analyzing mean field limits of controlled multi-agent systems, which arise in applications coming from macroeconomics, social behavior, and telecommunications. Justifying this continuum description is often nontrivial and is sensitive to the type of stochastic noise influencing the population. We will describe settings for which the convergence to mean field stochastic control problems can be resolved through the analysis of the well-posedness for a certain Hamilton-Jacobi-Bellman equation posed on Wasserstein spaces, and how this well-posedness allows for new convergence results for more general problems, for example, zero-sum stochastic differential games of mean-field type.
Posted February 21, 2024
Last modified April 12, 2024
Applied Analysis Seminar Questions or comments?
3:30 pm Lockett 232
Ben Seeger, The University of Texas at Austin
Equations on Wasserstein space and applications
The purpose of this talk is to give an overview of recent work involving differential equations posed on spaces of probability measures and their use in analyzing mean field limits of controlled multi-agent systems, which arise in applications coming from macroeconomics, social behavior, and telecommunications. Justifying this continuum description is often nontrivial and is sensitive to the type of stochastic noise influencing the population. We will describe settings for which the convergence to mean field stochastic control problems can be resolved through the analysis of the well-posedness for a certain Hamilton-Jacobi-Bellman equation posed on Wasserstein spaces, and how this well-posedness allows for new convergence results for more general problems, for example, zero-sum stochastic differential games of mean-field type.
Posted January 18, 2024
Last modified April 22, 2024
Informal Geometry and Topology Seminar Questions or comments?
1:30 pm Lockett 233, Zoom
Megan Fairchild, Louisiana State University
The Double Branched Cover of the Three-Sphere Over a Knot.
In this talk, we will examine how the double branched cover of the three-sphere over a knot is constructed and the linking form defined on its first homology. We will discuss how to calculate first homology, defining the linking form, and calculating the linking form given a knot diagram. The main goal of the talk is to better understand this object and examine its connection to non-orientable 4-genus of knots.
Posted January 31, 2024
Last modified April 23, 2024
Informal Geometry and Topology Seminar Questions or comments?
3:30 pm Lockett 233
Morgan Weiler, Cornell University
TBA
Posted April 19, 2024
Combinatorics Seminar Questions or comments?
2:00 pm – 3:00 pm Lockett 233
Ryan Martin, Iowa State University
Counting cycles in planar graphs
Basic Tur\'an theory asks how many edges a graph can have, given certain restrictions such as not having a large clique. A more generalized Tur\'an question asks how many copies of a fixed subgraph $H$ the graph can have, given certain restrictions. There has been a great deal of recent interest in the case where the restriction is planarity. In this talk, we will discuss some of the general results in the field, primarily the asymptotic value of ${\bf N}_{\mathcal P}(n,H)$, which denotes the maximum number of copies of $H$ in an $n$-vertex planar graph. In particular, we will focus on the case where $H$ is a cycle. It was determined that ${\bf N}_{\mathcal P}(n,C_{2m})=(n/m)^m+o(n^m)$ for small values of $m$ by Cox and Martin and resolved for all $m$ by Lv, Gy\H{o}ri, He, Salia, Tompkins, and Zhu. The case of $H=C_{2m+1}$ is more difficult and it is conjectured that ${\bf N}_{\mathcal P}(n,C_{2m+1})=2m(n/m)^m+o(n^m)$. We will discuss recent progress on this problem, including verification of the conjecture in the case where $m=3$ and $m=4$ and a lemma which reduces the solution of this problem for any $m$ to a so-called ``maximum likelihood'' problem. The maximum likelihood problem is, in and of itself, an interesting question in random graph theory. This is joint work with Emily Heath and Chris (Cox) Wells.
Posted January 17, 2024
Last modified March 4, 2024
Control and Optimization Seminar Questions or comments?
11:30 am – 12:20 pm Zoom (click here to join)
Tobias Breiten, Technical University of Berlin
On the Approximability of Koopman-Based Operator Lyapunov Equations
Computing the Lyapunov function of a system plays a crucial role in optimal feedback control, for example when the policy iteration is used. This talk will focus on the Lyapunov function of a nonlinear autonomous finite-dimensional dynamical system which will be rewritten as an infinite-dimensional linear system using the Koopman operator. Since this infinite-dimensional system has the structure of a weak-* continuous semigroup in a specially weighted Lp-space one can establish a connection between the solution of an operator Lyapunov equation and the desired Lyapunov function. It will be shown that the solution to this operator equation attains a rapid eigenvalue decay, which justifies finite rank approximations with numerical methods. The usefulness for numerical computations will also be demonstrated with two short examples. This is joint work with Bernhard Höveler (TU Berlin).
Posted February 1, 2024
Last modified April 30, 2024
Geometry and Topology Seminar Seminar website
3:30 pm Lockett 233
Jean-François Lafont, The Ohio State University
Strict hyperbolizations produce linear groups
Strict hyperbolization is a process developed by Charney--Davis, which inputs a simplicial complex, and outputs a negatively curved piecewise hyperbolic space. By applying this process to interesting triangulations of manifolds, one can create negatively curved manifolds with various types of pathological large scale behavior. I will give a gentle introduction to strict hyperbolization, and will explain why the fundamental groups of the resulting spaces are always linear over Z. This is joint work with Lorenzo Ruffoni (Tufts University).
Posted April 16, 2024
Last modified April 29, 2024
Faculty Meeting Questions or comments?
3:00 pm – 4:00 pm Lockett 232Meeting with Dean Cynthia Peterson
Posted April 19, 2024
Combinatorics Seminar Questions or comments?
2:00 pm – 3:00 pm Zoom (Please email zhiyuw at lsu.edu for Zoom link)
Peter Nelson, University of Waterloo
Infinite matroids on lattices
There are at least well-studied ways to extend matroids to more general objects - one can allow the ground set to be infinite, or instead define the concept of independence on a lattice other than a set lattice. I will discuss some nice ideas that arise when combining these two generalizations. This is joint work with Andrew Fulcher.
Posted January 16, 2024
Last modified March 4, 2024
Control and Optimization Seminar Questions or comments?
11:30 am – 12:20 pm Zoom (click here to join)
Jorge Poveda, University of California, San Diego
Donald P. Eckman, NSF CAREER, and AFOSR Young Investigator Program Awardee
Multi-Time Scale Hybrid Dynamical Systems for Model-Free Control and Optimization
Hybrid dynamical systems, which combine continuous-time and discrete-time dynamics, are prevalent in various engineering applications such as robotics, manufacturing systems, power grids, and transportation networks. Effectively analyzing and controlling these systems is crucial for developing autonomous and efficient engineering systems capable of real-time adaptation and self-optimization. This talk will delve into recent advancements in controlling and optimizing hybrid dynamical systems using multi-time scale techniques. These methods facilitate the systematic incorporation and analysis of both "exploration and exploitation" behaviors within complex control systems through singular perturbation and averaging theory, resulting in a range of provably stable and robust algorithms suitable for model-free control and optimization. Practical engineering system examples will be used to illustrate these theoretical tools.
Posted May 3, 2024
Last modified May 8, 2024
Probability Seminar Questions or comments?
11:00 am – 12:00 pm Zoom
Olga Iziumtseva, University of Nottingham
Asymptotic and geometric properties of Volterra Gaussian processes
In this talk we discuss asymptotic and geometric properties of Gaussian processes defined as $U(t) = \int_0^t K(t, s)dW(s),\ t \geq 0$, where $W$ is a Wiener process and $K$ is a continuous kernel. Such processes are called Volterra Gaussian processes. It forms an important class of stochastic processes with a wide range of applications in turbulence, cancer tumours, energy markets and epidemic models. Le Gall’s asymptotic expansion for the volume of Wiener Sausage shows that local times and self-intersection local times can be considered as the geometric characteristics of stochastic processes that look like a Wiener process. In this talk we discuss the law of the iterated logarithm, existence of local times and construct Rosen renormalized self-intersection local times for Volterra Gaussian processes.
Posted March 21, 2024
Last modified May 3, 2024
Order, Algebra, Logic, and Real Algebraic Geometry (Day 1 of 3)
https://www.math.lsu.edu/OAL-RAG2024
Posted March 21, 2024
Last modified May 3, 2024
Order, Algebra, Logic, and Real Algebraic Geometry (Day 2 of 3)
https://www.math.lsu.edu/OAL-RAG2024
Posted March 21, 2024
Last modified May 3, 2024
Order, Algebra, Logic, and Real Algebraic Geometry (Day 3 of 3)
https://www.math.lsu.edu/OAL-RAG2024
Posted April 29, 2024
Control and Optimization Seminar Questions or comments?
11:30 am – 12:20 pm Zoom (click here to join)
Giovanni Fusco, Università degli Studi di Padova
A Lie-Bracket-Based Notion of Stabilizing Feedback in Optimal Control
With reference to an optimal control problem where the state has to asymptotically approach a closed target while paying a non-negative integral cost, we propose a generalization of the classical dissipative relation that defines a control Lyapunov function by a weaker differential inequality. The latter involves both the cost and the iterated Lie brackets of the vector fields in the dynamics up to a certain degree $k\ge 1$, and we call any of its (suitably defined) solutions a degree-k minimum restraint function. We prove that the existence of a degree-k minimum restraint function allows us to build a Lie-bracket-based feedback which sample stabilizes the system to the target while regulating (i.e., uniformly bounding) the cost.