## Spring 2023

Time & Location: All talks are on Tuesdays in TBA at TBA PM unless otherwise noted.

Organizer: Nathaniel Vaduthala

**September 5**

**Title**: *Flag Matroids as Greedoids***Nathaniel Vaduthala - Tulane University**

**Abstract**: Matroids were developed in the 1930s in order to provide a combinatorial abstraction of linear subspaces and have become ubiquitous in modern combinatorics. In a similar manner, flag matroids can be seen as combinatorial abstractions of flags of linear subspaces. Despite their name however, flag matroids are not matroids, but rather can be viewed as greedoids, which themselves are generalizations of matroids. In this talk, we will briefly discuss the basics of matroid and greedoid theory and look at some properties and results related to flag matroids.

**Time**: 3:30pm**Location: **126

**September 12**

**Title**: *A Journey From Linear Algebra to Statistics***John Argentino - Tulane University**

**Abstract**: Principle Component Analysis is a common technique of dimension reduction employed in a host of fields, one that is rooted in linear algebra and optimization and utilized commonly in statistics. In this exploration we’ll examine PCA’s derivation, it’s alternate interpretations, it’s more sophisticated applications, and hopefully perform some rudimentary implementations…ations…ations…ations.

**Time**: 3:30pm**Location: **126

**September 19**

**Title**: *Strong Compactness in L^p Spaces***Ketan Kalgi | Tulane University**

**Abstract**:The idea of compactness is central to Analysis and to many other areas of mathematics. Right from the times when people thought of closed and bounded subsets of *R* as compact to the generalizations of this idea to function spaces, we shall see what additional conditions apart from being closed and bounded give us compactness in L^p. This compactness has many applications in PDEs, Fourier transform methods, etc. This is supposed to be an expository talk and would only rely on knowledge from undergraduate mathematics.ations.

**Time**: 3:30pm**Location: **126

**September 26**

**Title**: *Euler’s identity: “The most beautiful equation”***Vinh Pham | Tulane University**

**Abstract**:In 1990, a poll of readers conducted by The Mathematical Intelligencer named Euler’s identity as the “most beautiful theorem in mathematics”. Euler’s identity shows a deep relation between the most important constants in mathematics. It is represented simply and beautifully. In this talk, I would like to present briefly the history and one of the proofs of Euler’s identity.

**Time**: 3:30pm**Location: **126

**October 3**

**Title**: *A Brief Introduction to the Theory of Water Waves***Daniela Florez Pineda - Tulane University**

**Abstract**: The theory of water waves plays a significant role in applied mathematics, and it is the main subject in coastal hydrodynamics. Euler equations are the governing equations of water waves, but due to the great difficulties studying the theoretical and numerical aspects of these equations, approximate models have been derived instead. More precisely, these models have been simplified depending upon specific features of a wave: amplitude and wavelength with respect to the water depth. We will present the basic concepts behind this theory, describe different types of waves and discuss open problems.

**Time**: 3:30 pm**Location: **126

**October 10**

**Title**: *Integrable Systems and the Lax Pair***Conrad Alleman - Tulane University**

**Abstract**: In 1968, Peter Lax published groundbreaking work introducing Lax pairs – two matrices with special properties that can be used to generate integrable systems. But what does it mean for a system to be integrable? This presentation will introduce the notion of integrability, describe how Lax derived a methodology to produce these special systems, and provide a physical example to which his work can be applied.

**Time**: 3:30 pm**Location: **126

**October 17**

**Title**: *Pattern Formation in 2D Gray-Scott Model Using Finite Difference Schemes***Rubaiyat B. Islam - Tulane University**

**Abstract**: The Gray-Scott Model is a reaction-diffusion system that models chemical reactions between two substances that diffuse over time. Chemical reactions using this model give rise to the formation of interesting patterns. We will discuss two finite-difference schemes, ADI and Hopscotch method, to solve the reaction-diffusion PDEs numerically. Additionally, we will use the same schemes for another reaction-diffusion system that describes a predator-prey interaction model. Simulations will be shown where patterns develop from this system as well.

**Time**: 3:30 pm**Location: **126

**October 24**

**Title**: *Intro to Computability***Haoxi Hu - Tulane University**

**Abstract**: I will continue with my previous talk, which only discussed the definition of computable functions and computability. This talk will focus on the Rice theorem and Turing Degree; they are essential background knowledge for people who work on theoretical computer science and math logic.

**Time**: 3:30 pm**Location: **126

**October 31**

**Title**:* Stochastic Representation of the Solution to a Specific ODE System and Sensitivity Analysis of Quantities of Interest***Borui Zhao - Tulane University**

**Abstract**: While first-order ODE theorems are well-established, explicit solutions for arbitrary systems, especially with unique boundary conditions, remain challenging. We introduce a method to simulate models described by such ODEs. When these models involve random processes, our approach yields a stochastic representation. For particles confined within a finite tube, certain quantities are dictated by an ODE-related boundary problem. We construct a Piecewise Deterministic Markov process for this model, providing a stochastic solution. Additionally, we analyze the solution's sensitivity to parameter perturbations.

**Time**: 3:30 pm**Location: **126

**November 7**

**Title**: *Introduction to Tensor Decomposition and its Application***Zheng Wang - Tulane University**

**Abstract**: Introduction to Tensor Decomposition and its Application

Abstract: In this talk, we discuss the higher-order tensor decomposition and its applications. A tensor refers to a multi-dimensional or N-way array, and its decomposition method sheds light on bountiful fields of area such as computer vision, numerical analysis, data mining, graph analysis, and so on. We consider two types of the tensor decomposition: the CP-decomposition and the Tucker Decomposition.

**Time**: 3:30 pm**Location: **126

**November 14**

**Title**: *Roots, Groups, and Decompositions: Exploring Fundamental Concepts in Algebraic Combinatorics***Corey Wolfe - Tulane University**

**Abstract**: Root systems, Weyl groups, and the Bruhat decomposition are powerful mathematical tools that arise in various contexts, from Lie theory and algebraic geometry to combinatorics and physics. In this presentation, we will define root systems and discuss their properties, such as their classification and geometric interpretation. We will then introduce Weyl groups and show how they relate to root systems. Finally, we will discuss the Bruhat decomposition, a powerful tool for decomposing algebraic groups into double cosets of a Borel subgroup. We will show how the Bruhat decomposition relates to Weyl groups and their combinatorial structures and discuss some of its applications in representation theory, algebraic geometry, and combinatorics.

**Time**: 3:30 pm**Location: **126

**November 28**

**Title**: *PDE Models for Chemotaxis and Their Application in Self-organizing Phenomena***Sinchita Lahiri - Tulane University**

**Abstract**: Chemotaxis is an attract-repellent mechanism of substances and become a fundamental tool for understanding the relationship between cells and organisms in biology. For instance, the chemotactic mechanism is responsible for attracting microbes to its food, degrading embryonic cells into developing tissues, and leading immune cells to infection sites. From a mathematical perspective, chemotactic phenomena can be written as one or a system of partial differential equations. In this talk, we will introduce some PDE models governing chemotaxis and study results related to the stability of these models.

**Time**: 3:30 pm**Location: **126