Abstracts

Abstract: The theory of diffeology emerged arounds the 80s due to Jean-Marie Souriau, which is still in development today (mostly led by Souriau's former student, Patrick Iglesias-Zemmour). A diffeology on a set is a certain, abstract smooth structure which is comprised of functions which one wants to consider "smooth." We present the basic ideas of diffeology and discuss how differential geometry naturally emerges from this more general structure. No background information in geometry or topology is assumed!

Abstract: Have you ever thought that limits are the worst part of calculus? Ever wanted to remake calculus in a way that avoids limits entirely? Well now you can! In this talk, we will develop an analogue of ordinary calculus for sequences instead of functions. Our main tool: finite differences. Using this tool, we will prove discrete analogues of many theorems including the Fundamental Theorem of Calculus. Finally, we will apply our newly developed techniques to find “nice” formulas for common partial sum expressions.

Abstract: Broadly, Ramsey theory is the study of the emergence of “order” in seemingly random structures of sufficiently large size. Frank Ramsey’s now-famous theorem is a prime example: for any positive integer n, any graph with enough vertices must contain either a clique of size n or an independent set of size n. Classical results—such as Van der Waerden’s theorem and Szemerédi’s theorem—together with more recent breakthroughs like the Green–Tao theorem, showcase this phenomenon in an arithmetic setting.

In this talk, we give a brief history of the subject, sketch Ramsey’s original argument, and present both combinatorial and topological-dynamics viewpoints on Van der Waerden’s theorem. We’ll end with a snapshot of the current state of the area and highlight a few open problems.

Abstract: A long history of applying mathematical principles to the fields of ecology and evolution has generated robust interdisciplinary scholarship. I build on this history to address the evolution of sociality, adapting Voronoi neighborhood mathematics to understand territorial dynamics in endangered birds. Based on my analysis of two decades of aerial survey data, I will share what we have learned from looking at a population recovering from low density exploring how density mediates social dynamics. Broadly, this talk will inspire new perspectives on the interactions between mathematics and biology.

Abstract: Simplicial complexes are a useful tool from topology that turn geometric problems into purely combinatorial ones, hence making them much easier to compute. The study of these objects started simple with 3-dimensional polyhedra. However, making our way into higher dimensions, we quickly find that these objects are quite complex. In particular, since these objects exist to make computations more viable, one might ask: just how much can we compute and how easily? We'll take a look at simplicial complexes through the lens of computability theory, which focuses on studying questions about when problems are actually solvable algorithmically, and more importantly why most problems are not. In particular, we will study how hard it is to determine properties of a simplicial complex.

Abstract: Due to the spherical geometry of the globe, if four travelers took off from the same location in four different directions, all at the same speed, following “geodesic” paths, they would all meet up again exactly at the antipodal point of wherever they started. If we started here in Storrs, Connecticut, we would end up several hundred miles off the western coast of Australia, in the middle of the ocean. This is a rudimentary example of the computation of the “cut locus” in geometry, or the set of points where geodesic paths cease to be strictly optimal.

The cut locus is a critical object of study in differential geometry with a variety of applications, which we will explore in this talk through concrete examples using familiar surfaces.

Mathematics Continued Conference

Session A1

Diffeology: A Smooth Analogue of Topology

Session A2:

The Geometry of Black Holes

Removing Limits from Calculus: A Discrete Approach

Session B1:

Arithmetic Ramsey Theory

Session B2:

Session A3:

Territorial Geometries of Hornbills in Southern Africa

Session A4:

How Complex Is a Simplicial Complex?

Geodesics and applications of the Cut Locus

Session B3:

Graph Coloring

Session B4: