Abstracts

Exploring the Power and Boundaries of Machine Learning  (Lara Kassab) Machine learning plays a significant role in numerous aspects of our society from personalized music recommendations to influencing court decisions. In this introductory talk, we first explore some key concepts and techniques in machine learning. Then, we will discuss serious and critical issues such as bias, fairness, and transparency in machine learning models by examining real-world examples and case studies.

Algebraic geometry in spectral theory (Jordy Lopez Garcia) Bloch varieties arise in the spectral theory of operators on periodic media. Although these varieties are analytic in nature, by discretizing to operators on periodic graphs, they can be regarded as algebraic varieties. Thus, a number of physically important properties can be studied using strong methods from combinatorial algebraic geometry. In this talk, I will describe some of the tools we use to study these varieties. This is joint work with Matt Faust.

How nonlocal applied math helps model damage and learning processes (Cynthia Flores) Do you remember those operators from vector calculus: gradient, divergence, and curl? This talk will discuss some applied math and overcoming obstacles in modeling when discontinuous phenomena. Moreover, we introduce nonlocal theories in the mechanics of solids where the propagation of cracks and fractures hinders the use of classical differential operators.  By replacing these with integral operators, nonlocal frameworks allow the consideration of solutions to the peridynamics equation of motion with little to no differentiability of solutions. Additionally, a collection of nonlocal tools can be identified that is useful for analyzing the Helmholtz-Hodge Decompositions (HHD) of a vector field into its nonlocal divergence-free and curl-free components. Finally, we will discuss industrial applications, including a recent materials model for capturing the damage and fractures of metals due to corrosion done by undergraduate researchers and new avenues, including machine learning models, to apply nonlocal ideas.

Phylogenetic diversity and the geometry of point configurations (Luca Schaffler) In biology, phylogenetic trees illustrate the evolutionary relationships among different species. The evolutive distances between pairs of species can be recorded in a vector called dissimilarity vector. During the talk, we will explore the geometry of the set of dissimilarity vectors, leading to a new weighted variant of these which is better behaved when estimating phylogenetic trees from data on arbitrary sets of species instead of just pairs. At the same time, we will discover unexpected connections between phylogenetics, tropical geometry, and point configurations on specific types of curves. The original results presented are in collaboration with Alessio Caminata, Noah Giansiracusa, and Han-Bom Moon.