Tuesday

Mini-course 1. Mathematical Modeling in Ecology: Population Dynamics

Lidia Mrad

Abstract: The aim of this mini-course is to explore some models which have been central to the development of mathematical modeling, and which provide examples of how mathematics can be used in understanding ecological interactions. We focus on dynamical systems, which historically have been bound up with the calculus. However, we assume no knowledge of calculus and build up ideas using computational tools. Models we will study include the logistic growth model, competitive and symbiotic species interaction models, and predator-prey models.

Mini-course 2. What is dimension?

Sean McCurdy

Abstract: We all have the intuition that lines are one-dimensional, planes are two-dimensional, and cubes are three-dimensional. But why? What does that mean? And how do we apply that intuition to sets which are not so “nice”? It turns out, there are many different ways to make our intuitions of “dimension” rigorous. Surprisingly, these different ways don’t always agree! This mini-course invites participants into mathematics as a creative process through the question “What is dimension?” Attempts to answer this question will take us through 5 definitions of “dimension,” with plenty of opportunities for participants to come up with their own definitions along the way. We will see some strange things (fractional dimensions, infinite dimensions, even negative dimensions!) as well as some big ideas in Linear Algebra, Topology, and Analysis. The course will be structured loosely as a dialectic. Each class will develop a new definition of “dimension” and end with an exploration of its strengths and limitations. Problem Sessions for this mini-course will give participants the opportunity to A) further explore the theory in which a given definition of “dimension” is framed, or B) invent new definitions of dimension which fix the limitations. The next class will focus upon seeing where these the new definitions lead.

Talk. Spherical geometry and platonic solid

Gordon Rojas Kirby

Abstract: As the word “geometry” is derived from the Ancient Greek for earth measurement, it makes sense to understand what points, lines, and polygons are on the surface of a sphere. In this talk, we’ll explore some fun quirks of spherical geometry including why spherical triangles are fat and why you cross over Greenland on a flight from Washington to Europe. We will then apply these principles from spherical geometry to classify the five Platonic solids—convex solids in three dimensions built out of equiangular polygons with equal side lengths that look the same at each vertex. Lastly, as mathematicians love to do, we’ll consider how far we can extend these concepts into higher dimensions and other geometries.

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