Fall 2024 schedule

Aug 26, 2024 — Ebenezer Acquah

Title: The Dynamics of Spruce Budworm and Predatory Birds: A Mathematical and ABM Approach Using Mesa

Abstract: Understanding the intricate interactions within ecological systems is critical for preserving biodiversity and ecosystem functionality. This project presents a comprehensive mathematical model elucidating the dynamics of the spruce budworm population under predation, using a dual approach that integrates traditional differential equations with modern agent-based modeling (ABM) techniques. We first establish a system of two nonlinear differential equations to simulate the interplay between budworms and their avian predators. This system incorporates both logistic growth for the budworms and a Type-III functional response to represent predation. The stability of this biological interaction is then examined through Routh–Hurwitz conditions, revealing various potential outcomes including stable coexistence based on initial population densities. Subsequently, we leverage ABM facilitated by Mesa—a cutting-edge Python framework—to recreate these dynamics from a bottom-up perspective. By modeling individual agents with distinct behaviors, we gain novel insights into emergent properties and complex phenomena not readily apparent through traditional methods. Our findings suggest that given certain initial conditions, a state of equilibrium can be achieved where both species sustainably coexist. However, future research aims to incorporate additional life stages of the budworm into our models, examining their influence on population cycles. Moreover, we plan to extend our model to include insecticide effects, offering a more nuanced understanding of human intervention in natural systems.

Sep 9, 2024 — Dalen Dockery

Title: A Tour of the Partition Function

Abstract: The partition function is one the most studied functions in number theory, and yet its incredible properties are still being unearthed. In this talk, I’ll start by discussing the history of the partition function and the discoveries that facilitated its study. Next I’ll describe the sorts of problems modern researchers investigate, and I’ll end by mentioning several tools that are useful for tackling such questions. Along the way, I’ll point out a collection of the field’s open problems and avenues for further study. 

Sep 16, 2024 — Sam Webb

Title: Fault-Free and Simple Tilings with L-Trominoes

Abstract: A fault-free tiling is a type of tiling in which the board cannot be separated into two rectangular pieces without cutting through or breaking a tile. A simple tiling is a fault-free tiling in which there are no rectangular area in the tiling composed of two or more tiles. We will determine what size boards allow us to make fault-free and simple tilings under some specific conditions.

Sep 23, 2024 — Nathan Burns

Title: Introduction to Curve Shortening Flow

Abstract: Geometric flows, sometimes known as geometric evolution equations, are (typically parabolic) PDEs that describe the evolution of certain geometric quantities on Riemannian manifolds. Geometric flows have found applications in material science, image processing, and, more recently, community detection in networks. In this talk, I will introduce the extrinsic geometry of curves, which in turn will allow us to define the curve shortening flow and its basic properties. We will then use the tools of PDE theory to prove the conservation of convexity under the flow and the isoperimetric inequality in the plane. We will finish with a survey of classical results.

Sep 30, 2024 — Andrew Gannon

Title: The Real vs. Complex-Driven Loewner Equation

Abstract: The (chordal) Loewner differential equation encodes growing families of compact sets in the upper half of the complex plane into a single real-valued function, called a driving function. But we can reverse this process, and generate growing families of compact sets, called hulls, from a given real-valued driving function. If we generalize this process by instead considering complex-valued driving functions, then we immediately encounter obstacles not present in the real-driven case. Complex-driven Loewner hulls can exhibit geometric behavior which is not possible for real-driven hulls. In this talk, we will derive the Loewner equation, explore some of its basic properties, define Loewner hulls, and explore what makes complex-driven hulls so different from real-driven hulls.

 Oct 14, 2024 — John McAlister

Title: Coordination in Continuous Spatial and Strategic Domains

Abstract: Coordination games have been of interest to game theorists, economists, and ecologists for many years to study such problems as the emergence of local conventions and the evolution of cooperative behavior. Coordination and anti-coordination games have been widely studied on discrete domains but have not previously been extended into continuous domains. Here, we present an extension of a broad class of coordination-like games into continuous spatial and strategic domains and study them dynamically using non-local partial differential equations. We prove existence and uniqueness of solutions in the free boundary case and then, for coordination games in particular, we show that, by way of a maximum principle, the existence is global in time. Moreover, a weak regularity result allows us to demonstrate some numerical findings.  Finally, we relate stationary solutions to results from the discrete case and address the model's implications on the application areas. 

Oct 21, 2024 — Sagnik Jana

Title: Understanding Large-Scale Geometry

Abstract: This talk introduces key concepts in Geometric Group Theory, focusing on δ-hyperbolicity, boundaries, and the notion of hyperbolic directions in spaces. We will explore how quasi-isometries allow us to compare the large-scale structures of groups and spaces, and how δ-hyperbolic spaces, characterized by "thin" geodesic triangles, capture the essence of negative curvature. Finally, we will discuss the Gromov boundary, which describes the "ends" or asymptotic geometry of hyperbolic spaces, offering insights into how these spaces behave at infinity.

Oct 28, 2024 — Caden Farley

Title: Ricci solitons in 2 dimensions

Abstract: Ricci flow, the canonical heat equation for Riemannian metrics, is a geometric flow often studied with the goal of smoothing out arbitrary metrics to obtain metrics with more uniform geometry. Ricci solitons are geometric structures on Riemannian manifolds which model the singularities that occur in Ricci flow, and the classification of Ricci solitons is instrumental in the Ricci flow proof of the uniformization theorem and in the Hamilton-Perelman proof of the geometrization conjecture. This talk will motivate Ricci flow and the Ricci soliton equation in the simpler 2-dimensional setting. I will discuss a few examples of Ricci solitons in dimension 2 as well as the most important classification results, including recent results of Topping and Yin on the Ricci flow on Riemann surfaces with rough, measure initial data.

 Nov 4, 2024 — Charlotte Beckford

Title: A Poisson-Nernst Planck Model with Cross-Diffusion

Abstract: Batteries are a key element of a sustainable energy future. We are interested in predicting what is the maximum current which can be applied without causing early battery death and in determining what material properties influence this limiting current. In 2005 Bazant et al. extended the widely used Poisson-Nernst Planck (PNP) Model of ionic transport under an applied electric field by incorporating realistic nonlinear boundary conditions to represent the redox reactions occurring at electrodes. In 2024 Kumar and Zhu presented a dynamic density functional theory to construct a thermodynamically consistent model of this system by starting with the free energy functional. By making a specific choice of the free energy functional, we derived Bazant et al.’s model from the dynamic density functional theory developed by Kumar and Zhu, establishing that the former is a specific case of the latter. By relaxing the assumptions, we obtained a generalized PNP model which includes cross-diffusion. The effect of cross-diffusion has not been previously investigated. Model derivation, validation, and preliminary results will be presented, including the sign of the cross-diffusion term’s asymmetrical effect on the limiting current.

Nov 11, 2024 — Anusrika Datta

Title: Introduction to the Drury-Arveson space and its multipliers

Abstract: This presentation provides an introduction to the Drury-Arveson space, a significant function space within the context of several complex variables and operator theory, and examines its multipliers. The Drury-Arveson space, a Hilbert space of analytic functions on the unit ball in complex n-dimensional space, is particularly notable for its connections to multivariable operator theory and the theory function spaces. We begin by exploring its foundational structure, including the reproducing kernel, inner product, and norm properties that distinguish it from the well known classical Hardy space. Special attention is given to the multiplier algebra of the Drury-Arveson space. These multipliers play a crucial role in applications to control theory, dilation theory, and the study of invariant subspaces. We conclude by discussing current research directions, highlighting the interplay between the Drury-Arveson space and related function spaces in multivariate operator theory. 

Nov 18, 2024 — Sam Wilson

Title: You're telling me a Function Generated this sequence?

Abstract: From as early as any of us can remember, we have always counted things. Nothing does this task better than the generating function, who, among other properties, counts various combinatorial objects. We will review a few different methods for counting and define generating functions. Then, we will see how generating functions are useful and learn how to find them for simple examples.

Nov 25, 2024 — Ivy Dey

Title: Morse theory on cell complexes and posets

Abstract: Morse theory provides a powerful framework to study the topology of spaces by analyzing smooth functions on them. In this talk, we introduce the basics of discrete Morse theory, an adaptation of Morse theory to combinatorial settings such as cell complexes and partially ordered sets (posets). We will explore how discrete Morse functions can simplify complex spaces by reducing them to essential critical cells, providing insights into their topology. No prior familiarity with Morse theory is required, making this an accessible introduction for all.

Dec 2, 2024 — Isuru Abeyratne

Title: The Dirac Equation

Abstract: The most fundamental equation in quantum mechanics, Schrödinger's equation is non-relativistic. Although relativistic, the Klein-Gordon equation is restricted to spin-0 particles. Formulated by Paul Dirac in 1928, the Dirac equation leads to a quantum field theory of spin-1/2 particles such as electrons. In this talk we will explore the mathematical background of the Dirac equation while emphasizing the historical intuition which inspired Paul Dirac. No familiarity with quantum mechanics is expected.