August 30, 2024
Renling Jin, College of Charleston
The simplest proof of Ramsey's Theorem
A new proof of Ramsey’s Theorem with the least amount of combinatorial argument modulo some nonstandard techniques will be presented. The nonstandard idea with multiple levels of infinities will be introduced informally.
September 6, 2024
Michael Levet, College of Charleston Computer Science Department
Canonizing Graphs of Bounded Rank-Width in Parallel via Weisfeiler–Leman
The Graph Isomorphism problem takes as input two finite graphs G and H, and asks if G ≅ H. It is
a longstanding open problem whether Graph Isomorphism is solvable in polynomial-time. In this talk, we will discuss recent advances in the computational complexity of identifying graphs of bounded rank-width, a class of dense graphs for which isomorphism testing has only recently been shown to belong to P. Precisely, we will show that O(log n) rounds of the constant-dimensional Weisfeiler–Leman algorithm serves as a complete isomorphism test for this family. This, in particular, improves the complexity-theoretic upper bound from P to TC1. We then leverage the Weisfeiler–Leman algorithm as a subroutine, to obtain a TC2 algorithm to compute canonical labelings for this family. This is joint work with Puck Rombach and Nicholas Sieger.
September 13, 2024
Garrett Mitchener, College of Charleston
Symbolic Regression in JuliaSymbolic regression is a family of machine learning techniques in which the goal is to discover a relatively simple algebraic expression that captures a relationship between several variables. Evolutionary algorithms are one way of searching the set of symbolic expressions. The Julia programming language was originally designed for numerical methods, but it can also handle symbolic expressions. I'll give an introduction to Julia and explain the algorithm and its implementation. This project comes out of work done over the past summer with undergraduate Aidan Riordan, who will speak about other parts of the project at a future colloquium.
September 20, 2024
Behrang Forghani, College of Charleston
Random walks and boundaries
One of the central questions in the theory of random walks on groups asks how the long-term behavior of random walks on a group couples with its algebraic or geometric structure. In the 60s, Furstenberg introduced the Poisson boundary to describe the stochastically significant behavior of a random walk at infinity. In this talk, we will begin with an introduction to random walks on groups, construction of the Poisson boundary, and characterizations of groups according to the Poisson boundary. We will discuss methods to identify the Poisson boundary of random walks on groups, particularly, groups endowed with additional geometrical, algebraic, or combinatorial structures (e.g., free groups or, more generally, groups acting on a hyperbolic space).
*Most of this talk is accessible to undergraduate and graduate students*
September 27, 2024
Stéphane Lafortune, College of Charleston
On the Stability of Smooth Solutions to Peakon Equations
The Camassa-Holm equation with linear dispersion was originally derived as an asymptotic equation in shallow water wave theory. Among its many interesting mathematical properties, perhaps the most striking is the fact that it admits weak multi-soliton solutions - ‘peakons’- with a peaked shape corresponding to a discontinuous first derivative. Since the discovery of the Camassa-Holm equation, several peakon equations with similar properties, both in the integrable and non-integrable cases have been studied. Among them, there is the integrable Novikov equation, which can be regarded as a generalization to a cubic nonlinearity of the Camassa-Holm equation. Furthermore, both the Camassa-Holm and the Novikov equations each admit a generalization taking the form of a one-parameter family of peakon equations, most of which are not integrable.
In this talk, we study the spectral and orbital stability of various smooth solutions to peakon equations. One of the main difficulties when dealing with the linear operators arising from peakon equation is that they often include a non-local term. An additional challenge is the fact that the localized smooth solutions admit a nonzero background.
October 4, 2024
TBA
TBA
October 11, 2024
Hans Riess, Duke University
Graph diffusion with enriched category theory
This talk explores diffusion through different levels of abstraction, starting from random walks on graphs. After introducing quantale-enriched categories (Q-categories) and weighted meets and joins, we discuss the notion of a Laplacian in this context. We arrive at a theory of diffusion on network presheaves with Q-category stalks, with a fixed point theorem describing global sections.
October 18, 2024
Whitney Kitchen, College of Charleston
Aidan Riordan, College of Charleston
Stafford Yerger, College of Charleston
Data-Driven Discovery of Governing Equations of the Belousov-Zhabotinsky Reaction
Whitney Kitchen, Dr. Alex Brummer
College of Charleston, Department of Physics and Astronomy
Partial differential equation (PDE) models are necessary for many physical systems, from the Schrödinger equation to the Navier-Stokes system. In this project, we utilize PDEFind, a branch of a mathematical modeling tool called the Sparse Identification of Nonlinear Dynamics (SINDy). SINDy uses a data-driven approach, in which data is fed to a library of potential terms and multiple best-fit models are generated. PDEFind is a new technique, and little research has been done on its efficacy on experimentally collected data. Our project refined and used this tool to model the Belousov-Zhabotinsky (BZ) chemical reaction, which creates concentric rings that vary spatially and temporally. We collected our own experimental data and cleaned it using Independent Component Analysis (ICA) before running it through PDEFind. By comparing our discovered models to known reaction kinetic models, we are testing and refining the novel model discovery technique and contributing to research on the BZ reaction.
Explainable Machine Learning through Symbolic Regression and Genetic Algorithms
Aidan Riordan and Garrett Mitchener
College of Charleston, Department of Mathematics
Machine learning algorithms uncover meaningful patterns in data, enabling computers to make predictions and support decision-making. However, many state-of-the-art algorithms like neural networks are black boxes; they make accurate predictions, but offer no explanation of how they arrive at them. In contrast, symbolic regression techniques search for mathematical formulas that both fit given data points and are simple enough for humans to understand. This project will develop new symbolic regression algorithms inspired by biology. Just as networks of genes regulate processes in living cells, networks of math operations will form an iterative calculation to transform input data into output predictions. A selection-mutation process will discover mathematical expressions that balance accuracy against simplicity. These algorithms will be tested on publicly available datasets and compared to existing methods. Success will advance new tools for interpretable machine learning. The resulting open-source software library will enable others to build understandable prediction models.
Embedding Equitable (s, p)-Edge-Colorings of Kn
Stafford R. Yerger, Dr. Stacie Baumann, Mika Olufemi
College of Charleston, Department of Mathematics
An (s, p)-edge-coloring of a graph G is a coloring of the edges of G with s colors such that p colors appear at each vertex. In order to generalize the notion of proper edge-coloring, such colorings are defined to be equitable: the number of edges of each color appearing at a vertex are fairly distributed.
We find the necessary and sufficient conditions to embed an equitable (s1, p1)-edge-coloring of Kn1 into an equitable (s2, p2)-edge-coloring of Kn2. We focus on a set of values that cannot be solved by the traditional technique of switching colors along an alternating path. We employ the proof technique of amalgamations, amalgamating vertices that are missing the same color, coloring the amalgamation graph, and then detaching the graph to obtain the desired coloring
October 25, 2024
Leila Setayeshgar, College of Charleston
Large Deviations for a Class of Stochastic Semilinear Partial Differential Equations
Standard approaches to large deviations analysis for stochastic partial differential equa- tions (SPDEs) are often based on approximations. These approximations are mostly technical and often onerous to carry out. In 2008, Budhiraja, Dupuis and Maroulas, employed the weak convergence approach and showed that these approximations can be avoided for many infinite dimensional models. Large deviations analysis for such systems instead relied upon demonstrating existence, uniqueness and tightness properties of certain perturbations of the original process. In this talk, we use the weak convergence approach, and establish the large deviation principle for the law of the solutions to a class of semilinear SPDEs. Our family of semilinear SPDEs contains, as special cases, both the stochastic Burgers’ equation, and the stochastic reaction-diffusion equation (Joint work with M. Foondun).
November 1, 2024
TBA
TBA
November 8, 2024
Nic Jones, College of Charleston
On covering systems in analytic number theoryA covering system is a finite set of congruences x Ξ ri (mod ni) such that the union of their equivalence classes covers the set of integers. In 1950, Erdős introduced the concept of covering systems. At the time he was particularly interested in distinct covering systems, that is covering systems with all distinct moduli. This talk will begin with an introduction to covering systems, constructing distinct covering systems, and the current literature on covering systems with a focus on the non-existence of a distinct covering system in the interval [n, kn], where k is some positive integer greater than 1. The talk will conclude with a survey of open questions pertaining to covering systems such as the non-existence of a covering system with distinct odd moduli.
*This talk will be accessible to undergraduate and graduate students*
November 15, 2024
TBA
TBA
November 22, 2024
TBA
TBA