Title: modeling diseases 
Abstract: We will talk about my journey into mathematics and learn how to use math to model the spread of diseases.

Title: Maximum Likelihood Degree of Brownian motion tree models

Abstract: A Brownian motion tree model is a Gaussian model whose associated set of covariance matrices is linearly constrained according to common ancestry in a phylogenetic tree. In joint work with Jane Coons, Aida Maraj, and Ikenna Nometa, we study the complexity of inferring the maximum likelihood (ML) estimator for a Brownian motion tree model by computing its ML-degree. Our main result is that the ML-degree of the BMT model on a star tree with n + 1 leaves is 2^(n+1) – 2n – 3, which was previously conjectured by Amendola and Zwiernik. We also prove that the ML-degree of a Brownian motion tree model is independent of the choice of root and we find a combinatorial formula for the likelihood function. In this talk, I will introduce Brownian motion tree models and the tools from computational algebraic geometry that we use to compute the ML-degree.

Title: Fluid dynamics within symbiotic systems

Abstract: To understand the fluid dynamics of marine symbiotic systems, systems composed of multiple organisms, fluid-structure interaction problems must be solved. Challenges exist in developing numerical methods to solve these flow problems with boundary conditions at fluid-structure interfaces. We are studying the impact of fluid flow on symbiotic systems within two different biological systems. (1) Pulsing soft corals, Xeniidae corals, that have internal algal endosymbionts, zooxanthellae, which provide them with much of their energy and (2) the bobtail squid, Euprymna scolopes, that depend on the luminous bacteria, Vibrio fischeri, to protect them from predators through counterillumination. Both of these problems are motivated by field and experimental work in the marine sciences. I will discuss these related data, mathematical models and numerical methods developed to study these problems and provide comparisons with the modeling.

Lynette Agcaoili’s MA presentation is scheduled for April 30, 2024.  Everyone is welcome and graduate students are especially encouraged to attend.

Tuesday, April 30, 2024, 3:00 – 5:00 pm, Keller 404

Title:  An Introduction to Inverse Limits

Abstract:  The goal of this presentation is to give an introduction to inverse limits in a way that is (hopefully) accessible to advanced undergraduates/incoming graduate students. We will, of course, define what inverse limits are, and then construct injective resolutions for both abelian groups and inverse systems. We will then talk about flasque resolutions and some properties of flasque to construct a short exact sequence of inverse systems. Finally we will give explicit constructions of the inverse limit of a system and its first derived functor (i.e. varprojlim_{leftarrow}^(1) A_i), and show that if our indexing set is the natural numbers, then the second derived functor and higher are all 0 (i.e. varprojlim_{leftarrow}^(n) A_i = 0 for any n>1).