Mathematics

The heat kernel is the fundamental solution to a parabolic partial differential equation. From a probabilistic perspective, the heat kernel is the transition probability density of a stochastic process. Harnack inequalities and functional inequalities such as Poincare and Sobolev inequalities provide tools to understand the relationship between the behavior of the heat kernel and the geometry of the underlying space. An important feature of the approach using functional inequalities is its robustness under perturbations.

The study of the heat kernel and its estimates has produced fruitful interactions between the fields of Analysis, Geometry, and Probability. One of the goals of this course is to illustrate these interactions of heat kernel estimates with functional inequalities, boundary trace processes, quasisymmetric maps, circle packings, the time change of Markov processes, Doob's h-transform, and estimates of harmonic measure or exit distribution.

The setting for this course is a symmetric Markov process which is equivalentlydescribed using a Dirichlet form. This course will contain an introduction to the theory of Dirichlet forms. This theory will be used to construct and analyze Markov processes. This course will survey both classical results and recent progress in our understanding of heat kernel estimates and Harnack inequalities.

• Lecture 1
• Lecture 2
• Lecture 3
• Lecture 4
• Lecture 5
• Lecture 6
• Lecture 7
• Lecture 8
• Lecture 9
• Lecture 10
• Lecture 11
• Lecture 12
• Lecture 13
• Lecture 14
• Lecture 15
• Lecture 16

The heat kernel is the fundamental solution to a parabolic partial differential equation. From a probabilistic perspective, the heat kernel is the transition probability density of a stochastic process. Harnack inequalities and functional inequalities such as Poincare and Sobolev inequalities provide tools to understand the relationship between the behavior of the heat kernel and the geometry of the underlying space. An important feature of the approach using functional inequalities is its robustness under perturbations.

The study of the heat kernel and its estimates has produced fruitful interactions between the fields of Analysis, Geometry, and Probability. One of the goals of this course is to illustrate these interactions of heat kernel estimates with functional inequalities, boundary trace processes, quasisymmetric maps, circle packings, the time change of Markov processes, Doob's h-transform, and estimates of harmonic measure or exit distribution.

The setting for this course is a symmetric Markov process which is equivalentlydescribed using a Dirichlet form. This course will contain an introduction to the theory of Dirichlet forms. This theory will be used to construct and analyze Markov processes. This course will survey both classical results and recent progress in our understanding of heat kernel estimates and Harnack inequalities.

• Lecture 1
• Lecture 2
• Lecture 3
• Lecture 4
• Lecture 5
• Lecture 6
• Lecture 7
• Lecture 8
• Lecture 9
• Lecture 10
• Lecture 11
• Lecture 12
• Lecture 13
• Lecture 14
• Lecture 15
• Lecture 16

Concern for the impact of climate change on the spread and severity of infectious disease is widespread. For long-term management of global health, we need to consider parasite evolution under such environmentalchange. Vector-borne diseases are likely to be particularly affected bychanging climates due to the sensitivity of ectothermic vectors to temperature.Here, I present a work-in-progress of an age-structured SI model to represent the ecological dynamics of a general vector-borne disease, incorporating temperature-dependent parameters. Using sequential invasion analyses, the evolutionary trajectory of within-host parasite replication rate, and thus virulence, can then be predicted under a specified heating regime.

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.