Title: Applications of Linear Algebra to Combinatorics

Abstract: In Babai and Frankl’s “Linear Algebra Techniques in Combinatorics”, they detail several ways in which turning collections of finite sets into collections of finite dimensional vectors helps to reveal otherwise difficult to detect combinatorial information about those collections. In this talk, I will go through three surprising results from that book that use linear algebraic techniques in their proofs and discuss some more generally useful linear algebraic tricks for solving related problems.

Title: Flattened Stirling Permutations and Type B Set Partitions

Abstract: A Stirling permutation is a permutation on the multiset {1, 1, 2, 2, . . . , n, n} such that any numbers appearing between repeated values of i must be greater than i. We call a Stirling permutation “flattened” if the leading terms of maximal chains of ascents (called runs) are in weakly increasing order. Our main result establishes a bijection between flattened Stirling permutations and type B set partitions of {0, ±1, ±2, . . . , ±(n−1)}, which are known to be enumerated by the Dowling numbers, and we give an independent proof of this fact. We conclude with some conjectures and generalizations worthy of future investigation.

Title: Fractal Geometry and Non-Integer Dimensions

Abstract: Popularized in the 1980s, fractals have become something of a household name. These fractal sets often demonstrate peculiar topological properties. One such property is the notion of a fractal dimension. Sets such as the Cantor set, Sierpinski Gasket (SG), and the von Koch curve are traditionally visualized in 2D images. However, these sets actually exist in-between dimensions 1 and 2! 

Certain fractals can be built using what is known as an Iterated Function System (IFS), and there is a powerful theorem stating that having an IFS representation of a fractal provides a simple means of determining the fractal dimension. I will begin by stating the IFS that generates the Sierpinski Gasket. There are two transformations on the Gasket to which creates the Level-n Stretched Sierpinski Gasket (SSGn). I will demonstrate how one constructs the IFS for SSGn, as well as provide the highlights to a theorem in my published paper in which I prove the fractal dimension of SSGn.

Title: Introduction to Persistant Homology 

Abstract: Persistent diagrams are powerful mathematical tools used in the field of topological data analysis (TDA). These diagrams provide a concise and intuitive representation of topological features in data sets, particularly those that exhibit complex and multi-scale structures. Unlike traditional approaches, persistent diagrams capture the evolution of topological features across a range of spatial resolutions, allowing for the identification of robust and stable patterns amidst noise and variation.

In this talk, we explore the fundamental concepts behind persistent diagrams, such as persistence homology and barcode representations (No prior background will be assumed). In next week's talk by Jillian Cervantes, we will see an application of persistent diagrams with time series data to explore gravitational wave data.

Title: Optimizing Gravitational Wave Detection Using Topological Data Analysis

Abstract: The interactions between black holes create decaying periodic "chirp'' signals known as gravitational waves (GW). The detection of GW is an important problem in physics, but proves difficult as the signals are embedded in background noise. In 2019, Bresten and Jung determined that using persistence vectors as topological features for input to a convolutional neural network (CNN) results in a better classifier than the traditional CNN method. Our aim is to determine whether a CNN is necessary, or if better topological features with a simpler classifier can achieve similar results. We evaluate persistence vectors in comparison with other topological features, including persistence landscapes, persistence images, and persistence entropy.  We determine that the persistence vectors feature extraction method with a support vector classifier achieves the best results without using a CNN, at 0.943 AUC. This classifier is much easier to train and interpret than a CNN, and is competitive even with a significantly smaller sample size.

Title: Pinnacles on Graphs

Abstract: We consider graphs that are simple, finite, connected, and undirected, and a bijective labeling on the vertices of the graph with the integers 1, 2, ..., n where n is the number of vertices. Whenever a label on a vertex is larger than the label of all of its neighbors, then we say it is a pinnacle. In this talk, we explore pinnacle sets and enumerate labelings that have specific pinnacle sets. We present results on various graph families including complete graphs and star graphs. We also explore the method used to enumerate these labelings algorithmically. This talk is meant to be an accessible introduction to the research area of generalizing pinnacles on graphs.

Title: Changepoint Detection Through Segmented Autoregressive Moving Average (ARMA) Models for the Study of Epidemics - A Bayesian Approach

Abstract: Changepoint detection often involves the discovery of sudden abrupt and sustained fluctuations in population dynamics over time. This may reflect a change in population parameters, or it may be that we are unknowingly drawing data from separate populations or groups. If we assume data following a changepoint to be independent of previous data, we may analyze data on segments delimited by changepoints. If data within segments are considered correlated, it may be appropriate to use an Autoregressive Moving Average (ARMA) model on each segment. Publications on segmented ARMA modeling are relatively limited, and the application of such a model for the study of epidemics is surprisingly absent from the literature. The challenges that arise when implementing such models will be demonstrated, and potential for applications in epidemiology will be discussed.

Title: An Examination of an ODE Model of Braitenberg Vehicles

Abstract: Braitenberg Vehicles are simple agents that respond to stimuli according to a small set of rules. Despite this simplicity, they can exhibit fairly complex behavior. I. Rano (2012) proposed a system of Ordinary Differential Equations that can be used to simulate the behavior of type 2 and 3 Braitenberg Vehicles. We wish to use this system to potentially model the feeding behavior of parasitic marine isopods and will examine how altering the parameters of the system effects its output when modeling a Vehicle of type 3a. We will also look at possible alterations or alternative systems that may better simulate the behavior we are trying to emulate. 

Title: Mixed Volumes of Networks with Binomial Steady-States

Abstract: Mass-action kinetics on a chemical reaction network gives rise to a parameterized polynomial dynamical system. The number of complex steady-states of this system for generic parameters is called the steady-state degree; it is a measure of the algebraic complexity of solving for the steady-states. While the steady-state degree is difficult to compute in general, the mixed volume of the system can provide a good upper bound. We exploit the geometry of partitionable binomial networks to give a method for computing the mixed volume via a matrix determinant.

 Title: Structure-Randomness and The Green-Tao Theorem

Abstract: In this talk we will give a brief introduction of the dichotomy between structure and randomness in the context of Szemeredi's Theorem and the celebrated Green-Tao theorem. We will look at the definition of a positive upper density set and look at a few examples exploring the dichotomy between structure and randomness. We will then look at a sketch proof of one case of Szemeredi's Theorem and touch upon the techniques used to expand this result to the Green-Tao Theorem.