Math and Stats Student Colloquium (Oct 2)
Join us on Thursday, October 2nd, from 4-5 pm in IES 123 (with refreshments to be served at 3:30 pm in the 5th floor lobby). We will have three talks on exciting math topics given by our math students. The speakers will be Will Bales, Julia Williamson, and Ramsay Barlow. See the abstracts below.
Will Bales
Title: Dynamic Bayesian Networks
Dynamic Bayesian Networks have long been a ubiquitous tool for understanding probability spaces from a graph-theoretic perspective. However, inference algorithms for Dynamic Bayesian Networks have historically fallen short in capturing information about the topological structures of these graphical models. With the more recent discovery of the surprising effectiveness of persistent homology in analyzing graphical structures, there is a strong argument for leveraging this method in the context of Dynamic Bayesian Networks. It is therefore of central importance to create a formal framework for integrating topological data analysis with Dynamic Bayesian Networks. In this vein, we use a method that assigns edge strengths to a Dynamic Bayesian Network using the total variation distance of a stochastic matrix, and then apply an algorithm from topological data analysis, called a dynamic Dowker filtration, to extract a meaningful persistence diagram. This persistence diagram is the first step toward computing persistent homology, a key algebraic tool in TDA that has the potential to reveal underlying topological structures in sufficiently complex Dynamic Bayesian Networks that traditional methods often miss.
Ramsay Barlow
Title: Dynamic Latent Space Models for Global Migration
Abstract: Global migration flows have far-reaching implications, and effective models are necessary for understanding population movements over time. While country-reported data suffers from inconsistency and incompleteness, we utilize Facebook location data from over three billion active Facebook users to model migration flows between 181 countries on a monthly basis from January 2019 to December 2022. Our data defines a migration event as an individual residing in a new country for more than one year. This dataset is treated as network data, with countries serving as nodes in a network, and population flows as weighted edges between nodes. Latent space models are a class of network models that position each node in an unobserved space, thereby attempting to reveal hidden connections between nodes that explain the observed data. Dynamic versions of such models track the position of these nodes over time to understand changes in network dynamics. In this paper, we fit various dynamic latent space models to the data, attempting to capture connections between countries. We compare the strength and simplicity of different models, such as Bernoulli and Poisson likelihood methods, to efficiently capture trends in global migration without unnecessary complexity. Then, we modify traditional models to account for covariates and time-dependent parameters and evaluate their effectiveness in capturing major global events, such as the Russia-Ukraine War and the COVID-19 pandemic.
Julia Williamson
Title: Sliding a convex set under a convex function
Abstract: An existing form of approximating a non-differentiable function, f, is to slide a convex set, A, along the graph of f and take the lower boundary to form a differentiable function g. Our work reveals that sliding a convex set, A, under the graph of f gives the same approximation function, g, as sliding the set A-A along f. Additionally, sliding a convex set, A, under f is the same as sliding -A under f, or A-A along f. Due to this, if A is smooth from above or below, the approximation function g is essentially differentiable.
Join us on Thursday, October 2nd, from 4-5 pm in IES 123 (with refreshments to be served at 3:30 pm in the 5th floor lobby). We will have three talks on exciting math topics given by our math students. The speakers will be Will Bales, Julia Williamson, and Ramsay Barlow. See the abstracts below.
Will Bales
Title: Dynamic Bayesian Networks
Dynamic Bayesian Networks have long been a ubiquitous tool for understanding probability spaces from a graph-theoretic perspective. However, inference algorithms for Dynamic Bayesian Networks have historically fallen short in capturing information about the topological structures of these graphical models. With the more recent discovery of the surprising effectiveness of persistent homology in analyzing graphical structures, there is a strong argument for leveraging this method in the context of Dynamic Bayesian Networks. It is therefore of central importance to create a formal framework for integrating topological data analysis with Dynamic Bayesian Networks. In this vein, we use a method that assigns edge strengths to a Dynamic Bayesian Network using the total variation distance of a stochastic matrix, and then apply an algorithm from topological data analysis, called a dynamic Dowker filtration, to extract a meaningful persistence diagram. This persistence diagram is the first step toward computing persistent homology, a key algebraic tool in TDA that has the potential to reveal underlying topological structures in sufficiently complex Dynamic Bayesian Networks that traditional methods often miss.
Ramsay Barlow
Title: Dynamic Latent Space Models for Global Migration
Abstract: Global migration flows have far-reaching implications, and effective models are necessary for understanding population movements over time. While country-reported data suffers from inconsistency and incompleteness, we utilize Facebook location data from over three billion active Facebook users to model migration flows between 181 countries on a monthly basis from January 2019 to December 2022. Our data defines a migration event as an individual residing in a new country for more than one year. This dataset is treated as network data, with countries serving as nodes in a network, and population flows as weighted edges between nodes. Latent space models are a class of network models that position each node in an unobserved space, thereby attempting to reveal hidden connections between nodes that explain the observed data. Dynamic versions of such models track the position of these nodes over time to understand changes in network dynamics. In this paper, we fit various dynamic latent space models to the data, attempting to capture connections between countries. We compare the strength and simplicity of different models, such as Bernoulli and Poisson likelihood methods, to efficiently capture trends in global migration without unnecessary complexity. Then, we modify traditional models to account for covariates and time-dependent parameters and evaluate their effectiveness in capturing major global events, such as the Russia-Ukraine War and the COVID-19 pandemic.
Julia Williamson
Title: Sliding a convex set under a convex function
Abstract: An existing form of approximating a non-differentiable function, f, is to slide a convex set, A, along the graph of f and take the lower boundary to form a differentiable function g. Our work reveals that sliding a convex set, A, under the graph of f gives the same approximation function, g, as sliding the set A-A along f. Additionally, sliding a convex set, A, under f is the same as sliding -A under f, or A-A along f. Due to this, if A is smooth from above or below, the approximation function g is essentially differentiable.