Fall 2024

Speaker: Jannatun Irana Ira

Title: Importance of extracellular vimentin in the dynamics of pathogen invasion

Abstract: Vimentin is a type III intermediate filament protein that plays many important roles in the cell's cytoskeleton, like maintaining cell structure, cell migration, cell division, wound healing, and tissue development. It has also been shown in multiple studies that, vimentin can play important roles in different disease progression. In this study, we are focusing on the extracellular vimentin. In this specific environment,  the protein can have filamentous as well as non-filamentous forms, the protein can also be found attached to the cell surface and sometimes as an exogenous soluble form. In the event of pathogen invasions, it has been found to interact and bind with pathogens. More importantly, depending on the nature of the pathogens, vimentin has been found to either facilitate or inhibit the internalization process inside healthy cells. For example, in the case of SARS-COV, vimentin helps to create a docking platform for the virus to enter the intracellular region. In contrast, in HPV invasion it has been found to trap the virus delaying the internalization process.

In this talk, we focus on understanding the dynamic changes and interactions of the pathogen and vimentin in the extracellular environment and their overall effect. We have formulated a mathematical model consisting of the pathogen, vimentin, pathogen-vimentin complex, healthy epithelial cell, and infected epithelial cell dynamics to analyze different disease situations.

Speaker: Tommy Cai
Title: Generalized torsion in amalgams and orderability

Abstract: A group is left-orderable (bi-orderable) if there is a total order $<$ invariant under multiplication from the LHS (resp. both sides)  i.e., $a<b$ implies that $ca<cb$  (resp. $cad<cbd$). For example, the group of real numbers is bi-orderable using the usual order. 

A group is torsion free all elements are of finite order. A group is generalized torsion free if no product of conjugates of any nontrivial element is trivial; i.e., if $g\neq1$, then $h_1^{-1}gh_1\dotsm h_n^{-1}gh_n\neq1$ for all $h_i$ and $n\geq1$.

It's easy to prove that if a group is left-orderable (bi-orderable) then it is torsion free (resp. generalized torsion free). It's not true vice versa, and there are known examples, although this is not as easy. An open question is whether every generalized torsion free group is  left-orderable. In fact, this is Problem 16.48 of the Kourovka Notebook (which is also Question 2.1 in Unsolved problems in ordered and orderable groups, arXiv:0906.2621).

In this talk, we explain how we solved this problem by constructing a group which is generalized torsion free and not left-orderable. We will first explain what's an amalgam of two groups. Second, we state a theorem providing a sufficient condition for an amalgam to be generalized torsion free. Then we explain how we use this theorem to construct the desired group which is generalized torsion free and not left-orderable. 

This is a joint work with Adam Clay.

Speaker: Michael Astwood
Title: The Kepler Problem on Manifolds
Abstract: The dynamics of celestial bodies have been studied by mathematical physicists for centuries. This talk introduces the key concepts of geometric mechanics, including the Lagrangian and Hamiltonian formulations of mechanics on manifolds, symmetries, and Noether's theorem. Several important properties related to orbital mechanics are discussed, such as superintegrability and Bertrand's theorem. Finally, the presenter will provide an overview of his recent MSc project, in which the Kepler problem on pseudo-Riemannian surfaces of revolution was solved explicitly.

Speaker: Brock Klippenstein
Title: Solutions of the Cosmic Ray Fokker-Planck Equation
Abstract: When energetic particles such as cosmic rays travel through magnetized plasma, they encounter turbulent magnetic fields. This in turn renders the equation of motion very difficult to apply. Hence, we instead work with the Fokker-Planck partial differential equation, which gives us the probability of finding the particle at a certain time, position, and velocity. Here, we talk about methods which allow for fast solving the Fokker-Planck equation.

Speaker: Samuel Adeshina
Title: Hilbert Functions of Projective Points
Abstract: We discuss an invariant point set in the projective space called the Hilbert function. Although a simple sequence, the Hilbert function gives both algebraic and geometric information about points in projective space. We will show a characteristic of the sequence of this function using a simple combinatorial technique.