Paul Hurtado, University of Nevada, Reno
Title: A generalized linear chain trick (GLCT) for improving ODE model derivation & analysis
Abstract: ODE models are ubiquitous among mathematical modelling applications in the sciences, with many of those models having been derived using “rule of thumb” rather than more rigorous approaches. While such shortcuts greatly simplify the process of model derivation, they can lead to unintentionally over-simplified (or erroneous) assumptions and can lead to systematic biases. The so-called “linear chain trick” is one technique used to address some of those oversimplifications, and in this talk I will introduce this technique and our extension, the generalized linear chain trick (GLCT). I will provide some examples from population ecology and infectious disease modelling to illustrate how this technique can not only help modelers to efficiently derive systems of ODEs from first principles when they can be framed using continuous time Markov chains, but also how viewing ODE models through the lens of the GLCT opens up new opportunities for how we interpret mathematical analyses of such models using concepts from stochastic processes.
Luis David Garcia Puente, Colorado College
Title: Estimating Gaussian Mixtures: An introduction to algebraic statistics
Abstract: A fundamental problem in statistics is to estimate the parameters of a density from samples. This problem is called density estimation. To have any hope of solving this problem we need to assume our density lives in a family of distributions. One family of densities known as Gaussian mixture models are a popular choice due to their broad expressive power. In particular, these models have applications ranging from speech recognition, image segmentation and biometrics to the spread of COVID-19.
In this talk, we will give an introduction to this problem from a historical perspective. We will discuss the very first paper on this subject: Pearson’s 1894 “Contributions to the Mathematical Theory of Evolution’’ where he introduces the Methods of Moments to solve the density estimation problem. Along the way, we will reinterpret Pearson’s original work in terms of modern computational algebra techniques; and we will discuss how this translation provides a powerful framework to obtain new results on this subject. No mathematical background beyond calculus and introductory statistics will be assumed.
Cigole Thomas – Colorado State University
Title: Dynamics of Group Action on Varieties
pdf of abstract
Abstract: In this talk, we will be exploring the “symmetry” or “shape” of a “set” as the size of the set gets increasingly bigger by applying a collection of transformations to the set. Varieties are objects that arise as simultaneous zero sets of a collection of polynomials. The “sets” considered here are character varieties – equivalence classes of homomorphisms – while the set of transformations is an outer automorphism group. For a specific finitely generated group, F, and a complex reductive algebraic group, G – group with a variety structure – we explore the dynamics of the action of Out(F) on the G-character variety of F (read as equivalence classes of homomorphisms from F to G). In particular, we will be working with the finite field points of this variety. The talk will be accessible to a general audience and will not assume any knowledge of algebraic geometry.