Mathematics

Following a vaccine inoculation or disease exposure an immune response develops in time, where the description of its time evolution poses an interesting problem in dynamical systems. The principal goal of theoretical immunology is to construct models capable of describing long term immunological trends from the properties and interactions of its elementary components. In this talk I will give a brief description of the human immune system and introduce a simplified version of its elementary components. I will then discuss our contributions to the field achieved through my postdoctoral work with Dr. Jane Heffernan at York University. I will focus on our mechanistic modelling work describing vaccine-generated SARS-CoV-2 immunity and applications of our work towards understanding vaccination responses in people living with HIV. Finally, I will discuss our on-going work towards developing a machine learning public health platform capable of predicting immune response outcomes from repeated-dose immunological data.

Denoising Diffusion models have revolutionized generative modeling. Conceptually, these methods define a transport mechanism from a noise distribution to a data distribution. Recent advancements have extended this framework to define transport maps between arbitrary distributions, significantly expanding the potential for unpaired data translation. However, existing methods often fail to approximate optimal transport maps, which are theoretically known to possess advantageous properties. In this talk, we will show how one can modify current methodologies to compute Schrödinger bridges—an entropy-regularized variant of dynamic optimal transport. We will demonstrate this methodology on a variety of unpaired data translation tasks.

A set {𝑎1,𝑎2,…,𝑎𝑚}of distinct positive integers is a Diophantine 𝑚-tuple if the product of any two distinct elements in the set is one less than a square. In this talk, I will discuss some recent results related to Diophantine tuples and their generalizations. Joint work with Ernie Croot, Seoyoung Kim, and Semin Yoo.

Mathematical modeling, when combined with diverse epidemiological datasets, provides valuable insights for understanding and controlling infectious diseases. In this talk, I will present a series of case studies demonstrating how the synergy between modeling and serological, case-based, and wastewater surveillance data can enhance disease monitoring and inform public health strategies.

Serological data measures biomarkers of infection or vaccination, offering direct estimates of population immunity. This approach complements case data by providing a broader understanding of disease epidemiology. Wastewater surveillance, which detects pathogen genomes in sewage, captures infections across the entire population, including symptomatic, asymptomatic, and pre-/post-symptomatic individuals. This approach complements traditional case reporting by providing a broader, community-wide perspective on disease transmission.

In the first case study, I will discuss how we utilized serological data and a static cohort model to quantify the relative contributions of natural infection, routine vaccination, and supplementary immunization activities (SIAs) to measles seroconversion in Kenyan children. In the second case study we developed a simple static model combined with serological data to evaluate the effectiveness of SIAs in reducing the risk of a measles outbreak in the post-pandemic period. Finally, I will introduce my current work on wastewater-based epidemic modeling for mpox surveillance in British Columbia, demonstrating how wastewater and case data can be integrated within a dynamic transmission model to predict future scenarios of mpox outbreak.

These case studies illustrate the power of mathematical modeling in integrating multiple data sources to inform public health strategies and improve infectious disease control efforts.

We will talk about Drinfeld modules, and how they compare to elliptic curves for algorithms and computations.

Drinfeld modules can be seen as function field analogues of elliptic curves. They were introduced in the 1970's by Vladimir Drinfeld, to create an explicit class field theory of function fields. They were instrumental to prove the Langlands program for GL2 of a function field, or the function field analogue of the Riemann hypothesis.

Elliptic curves, to the surprise of many theoretical number theorists, became a fundamental computational tool, especially in the context of cryptography (elliptic curve Diffie-Hellman, isogeny-based post-quantum cryptography) and computer algebra (ECM method).

Despite a rather abstract definition, Drinfeld modules offer a lot of computational advantages over elliptic curves: one can benefit from function field arithmetics, and from objects called Ore polynomials and Anderson motives.

We will use two examples to highlight the practicality of Drinfeld modules computations, and mention some applications.