Friday

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.

Informal conversation

Jake Levison

Talk. Curvature and polyhedra

Jake Levinson

Abstract: What does it mean for a surface to be curved? One way to answer this question is in terms of triangles drawn on the surface, and there's a neat way to approach it for polyhedra -- surfaces with planar faces, like the cube and the octahedron. We'll also encounter an invariant called the Euler characteristic: a glimpse of the area of mathematics called topology.

Worktime