Mini-Courses – 2022 CMS Summer Meeting

CMS is organizing three-hour mini-courses to add more value to meetings and make them attractive for students and researchers to attend.

The mini-courses will be held on Friday, June 3 and include topics suitable for any interested parties. You don’t have to be registered for the meeting in order to register for mini courses.

Registration fees for the mini-courses are as follows.

Regular rate (fees are for one mini course)
Student/ Postdocs (members)	$50
Student/ Postdocs (non-members)	$75
CMS Members	$100
CMS Non-Members	$150
RBC Sponsored Closing the Gap (Black, Indigenous, female-identifying, LGBTQ+, or person with disabilities; must be 15-29 years of age) – CMS student member ***Canadian citizens and permanent residents only	$25

Geometry of Black Hole Mergers

Friday, June 3, 2022 | 9:00 - 12:00 NDT

Presenter: Ivan Booth, Memorial University

Black holes are astrophysical objects. They play a central role in many of the most dramatic astrophysical processes in our universe including supernova, active galactic nuclei and black hole mergers. In dramatic developments in the last few years, new observational techniques have not only “listened” to the gravitational waves emitted during a black hole merger (LIGO, VIRGO) but also actually taken a horizon-scale X-ray picture of the 2.4 billion solar mass black hole at the centre of M87 (the Event Horizon Telescope).

Black holes are also geometrical objects. As solutions of Einstein’s equations, black hole spacetimes are four-dimensional manifolds with Lorentzian-signature metrics. The horizon of a black hole boundary at an “instant in time” is then a Riemannian signature, two-dimensional, closed manifold. Geometrically, these two-dimensional boundaries are very closely related to minimal surfaces from classical differential geometry (even being identical in some special cases) and so they can be studied and classified with many of the same mathematical tools.

This series of lectures will begin with a review of geodesics and minimal surfaces in Riemannian geometry, including the information contained in their Jacobi/stability operators. We will then consider the closely related marginally outer trapped surfaces (MOTS) of general relativity, which feature prominently in mathematical and numerical relativity studies of black holes and see how they relate to the better known apparent and event horizons. Finally, using these tools along with both numerical and exact examples, we will see what a study of MOTS can tell us about the intricacies of black hole mergers. In particular we will address the problem as to how, during merger of a pair of black holes, two apparent horizons can become one.

Introduction to arithmetics in function fields

Friday, June 3, 2022 | 9:00 - 12:00 NDT

Presenter: Matilde Lalin, Université de Montréal

Number theory was initially concerned with the arithmetic properties of the integers and the rationals. It turns out that many of these properties translate to a different context, namely those of polynomials with coefficients over a finite field, $\mathbb{F}_q[T]$, and the corresponding rational functions $\mathbb{F}_q(T)$. For example, both the integers and $\mathbb{F}_q[T]$ satisfy unique factorization, both contain infinitely many primes, etc. In fact, for some questions, including the Riemann Hypothesis, we know more about the function field setting than the rational setting.

In this mini-course we will start by exploring the basics of arithmetics of $\mathbb{F}_q(T)$ (including arithmetic functions, the Riemann zeta function, reciprocity, and arithmetic progressions). We will then progress towards more general settings (global function fields and their zeta functions), by stressing similarities with the rationals, but also differences. We will finish by giving an overview of some of the questions of current research in the topic.

Modes of Motion by a Coronavirus

Friday, June 3, 2022 | 13:00 - 16:00 NDT

Presenter: Goong Chen, Texas A&M University;

Organizer: Jie Xiao, Memorial University

In this mini-course, we study coronavirus from a structural molecular biology point of view, with certain emphasis on how the mechanical motions of the coronavirus may play in the invasion process of healthy cells.

We first give a brief introduction and survey of the biological properties of a coronavirus concerning how it moves and how it invades healthy cells. We then proceed to build our model by continuum mechanics in lieu of an atom-by-atom model. A spherical shell with many spikes mimicking the shape of coronavirus has been chosen as the elasto-plastic continuum. We then analyze its eigenmodes of vibration by Modal Analysis. We have found the six degree of freedom rigid body modes and several thousands more nontrivial modes, some peculiar ones of which can play important roles in the invasion process. Related problems are also posed.

Many animation videos from supercomputer computations and simulations will be shown to illustrate the motions of a coronavirus.

Introduction to non-Archimedean Analysis

Friday, June 3, 2022 | 13:00 - 16:00 NDT

Presenter: Khodr Shamseddine, University of Manitoba

In this mini-course, I will first review basic properties of ultrametric spaces, valued fields and ordered fields as well as the connection between these different mathematical objects. In particular, I will present elements of the algebraic, topological and metric structures of non-Archimedean valued fields which are different from what we know for the Archimedean fields of real and complex numbers. As examples of non-Archimedean valued fields, I will introduce the p-adic fields as well as the so-called general Hahn fields and Levi-Civita fields, and I will present a summary of their key properties.

Then, I will focus on two special Levi-Civita fields: R and its complex counterpart C. Among all the non-Archimedean fields surveyed in the first part of the course, R and C are unique from a pure mathematical point of view as well as from a computational point of view. I will give a brief summary of my research group’s work on R and C, and show one computational application.

Basic knowledge of Analysis, Algebra and Topology at the undergraduate level will be assumed.

An invitation to high-dimensional approximation: from sparse polynomials to deep learning

Friday, June 3, 2022 | 13:00 - 16:00 NDT

Presenter: Simone Brugiapaglia, Concordia University

Approximating functions of many variables from limited data is a key task in modern scientific computing and machine learning. This task is made intrinsically difficult by the curse of dimensionality, a concept introduced by Richard Bellman in the 1950s that refers to phenomena and computational challenges that arise in high dimensions. The objective of this mini-course is to introduce recent approximation techniques able to mitigate the curse of dimensionality based on sparse polynomials and Monte Carlo sampling.

We will start by illustrating the essentials of sparse polynomial approximation theory, motivating our study with examples of parametric differential equations employed in uncertainty quantification. Then, we will present numerical methods for computing sparse polynomial approximations from limited Monte Carlo samples. We will focus on techniques based on least squares and compressed sensing and discuss under what circumstances they are provably able to mitigate the curse of dimensionality. We will conclude by demonstrating how these ideas can be applied to deep learning theory and illustrate so-called practical existence theorems for deep neural networks. We will also present open problems and current research directions in the field.

Although this mini-course will be mostly focused on theoretical aspects, we will also illustrate numerical examples. We will assume at least some familiarity with the basics of real and complex analysis, approximation theory, numerical analysis, and probability.