2020-21

Modeling Ecological Populations

Ecologists and wildlife managers want to know how they can keep biological populations healthy; that is, they don't want a population to die out or to explode. However, they can't run experiments on wild populations because that usually costs too much money, or they risk irreparably damaging a population. Thus, biologists use mathematical models and computers to study populations. In this talk, I will introduce a number of "structured" population models. These models make use of physiological differences, like the size of a fish or the developmental stage of a grasshopper. Biologists use matrices, differential equations, and more recently ``integral operators", to simulate structured populations in the wild. I will give examples of each of these models, with lots of pictures. The mathematical details should be accessible to anyone who has taken calculus.

Algebraic Sets and the Relative Point of View

Algebraic sets, like parabolas and lines in the plane, are familiar objects of study from college algebra. Rather than study the internal properties of these sets, the "relative point of view" suggests that we consider functions defined upon these sets. This approach leads to Hilbert's Famous "Nullstellensatz" (german for "theorem of zeros"), which provides a dictionary for converting an algebraic set into a new algebraic object called a coordinate ring. Properties of the algebraic sets, like being differentiable or having a particular dimension, are converted to properties of the coordinate rings. This conversion allows notions like differentiability of a curve to be generalized, even to settings in which we have no notion of calculus.

Games, Graphs, and Managing Expectations

You've probably heard of such famous games as tic-tac-toe, dots and boxes, and connect four.  Maybe you've even played some of them yourself.  If so, you're in good company!  Many mathematicians have a remarkable love for games and puzzles, and even dare to analyze games mathematically.  In the case of tic-tac-toe, we know that if neither player makes a mistake in their strategy, the game will always end in a draw.  But for other games, we can prove that a certain player always has an advantage.  In this talk, we will explore one particular type of "positional" game in which (for any instance of the game) exactly one player has a winning strategy.  In particular, we will learn about random graphs and try to use them to give our favorite plahyer an advantage in the game.

Cryptography: The Art of Making and Breaking Codes

People have been keeping secrets since the beginning of civilization, and others have been trying to find out those secrets for just as long. Cryptography lies in the intersection of mathematics, computer science, and other disciplines and is devoted to making information secure while breaking into others' secrets. We will look at the development of cryptography from simple substitution ciphers to the Enigma in WWII to today's quantum-resistant cryptosystems, which are designed to keep our information safe with the rise of quantum computers.

A topological journey through spaces and knots

Topology is the study of abstract spaces which can be stretched or moved around. I will talk about some examples of these kinds of spaces in 1, 2, 3, and even 4 dimensions. In particular, I will introduce some foundational concepts in knot theory. We will color knots, stick things together and pull them apart, and all kinds of topological activities. There will be lots of pictures and examples.

Accurately Modeling Fluid Flow: Data Assimilation, Parameter Recovery, & Ocean Modeling

Scientists and mathematicians apply the continuum hypothesis to model fluid flow, i.e. the flow of substances like air or water. One of the challenges of the accurate simulation of turbulent flows is that initial data is often incomplete. Data assimilation circumvents this issue by continually incorporating the observed data into the model.  In this talk, I will discuss my work using a new approach to data assimilation in order to both accurately model fluid flow and to identify certain physical properties of the fluid being modeled.  I will also discuss implementation of this algorithm in large-scale climate models.

Properties for Solutions to Integral and Differential Equations

Partial differential equations (PDEs) and integral equations (IEs) model a variety of physical phenomena, for which they offer predictive behavior through theoretical and numerical studies. Due to a high degree of complexity, it is usually not feasible to obtain exact solutions to PDEs or IEs, thus we aim to investigate the behavior through their properties, or through estimates on the magnitude of different quantities (generally called energies). We will discuss the uniqueness, existence, and continuous dependence of solutions in the context of PDEs and IEs, by looking at the ideas behind classical tools, such as energy minimization, which is useful for showing uniqueness and existence. Finally, we present some ideas behind continuous dependence or stability of solutions.