# Courses: ongoing

The following courses were scheduled for the ongoing academic year:

Analytic Number Theory I

### Prerequisites

• Elementary Number Theory

• Real and Complex Analysis

### Registration

Registration for this course is not currently available.

## Abstract

This is a first course in analytic number theory. In this course we will focus on the theory of the Riemann zeta function and of prime numbers. The goal of this course will include proving explicit bounds for the number $\pi(x)$ of primes which are less than a given number $x$. Building analytical tools to prove the prime number theorem (PNT) will be at the core of this course. We will explore and compare explicit formulas, whether they are using smooth weights or a truncated Perron formula, to relate averages over primes and $\pi(x)$ to sums over the zeros of zeta. Another originality of this course will be to explore each topic explicitly (essentially by computing all the hidden terms implied in the asymptotic estimates). With this respect, students will get an introduction to relevant programming languages and computational software. This will be closely connected to Analytic Number Theory 2 by Greg Martin (UBC), as the sequences of topics are coordinated between us; the intention is for students at all PIMS institutions to be able to take the second analytic number theory course as a continuation of the first one with maximum benefit. In addition, these two courses will provide excellent training for students who plan to attend the “Inclusive Paths in Explicit Number Theory” CRG summer school in 2023. All these events are part of the PIMS CRG “L-functions in Analytic Number Theory” (2022-2025).

## Syllabus

Syllabus_Analytic_Number_Theory_HK.pdf

## Other Information

OT+Bio - Single Cell Analysis

### Prerequisites

• Linear algebra as in Math 307

### Registration

Registration for this course is not currently available.

## Abstract

The course covers foundational mathematical tools that are useful in analyzing high-dimensional single-cell datasets, and modelling developmental stochastic processes. We cover basic probability theory, statistical inference, convex optimization, Markov stochastic processes, and advanced topics in optimal transport.

## Syllabus

See the course website for the syllabus and other details.

## Other Information

Statistical Machine Learning for Data Science

### Prerequisites

• Students should have basic statistical theoretical knowledge

• A good understanding of linear regression

• Basic R coding skills.

### Registration

Registration for this course is not currently available.

## Abstract

The course provides learning opportunities on statistical software, R, with some focus on data management and wrangling, reproducibility, and visualization. On top of that, there are basic introductions to Machine Learning such as k-NN, Naive Bayes, regression methods, etc. The focus is on hands-on skills with R and applications to real data.

## Syllabus

Syllabus_846_PIMS.pdf

## Other Information

Mathematical Models in Cell Biology

### Prerequisites

• For math students - Some familiarity with ODEs, PDEs

• For biology students - Flexible, hopefully some math background, such as calculus

### Registration

Registration for this course is not currently available.

## Abstract

Cell biology provides many interesting challenges across many spatial scales. Mathematical and computational modeling are tools that can help gain a better understanding of cellular phenomena. At the small scales, there are puzzling examples of patterns formed by proteins inside cells, and dynamic rearrangement of cellular components that enable cells to actively move. At higher scales, cells sense chemical gradients, exhibit active motility, and interact with other cells to form functioning tissues and organs. Mathematical and computational models allow us to explore many of the leading questions at each of these levels. How do patterns form spontaneously? What are the limits of cell sensing? How do cells polarize and migrate in a directed manner? How does a collection of cells self-organize into a structured tissue? In this graduate course, we will explore such questions in the context of deterministic models (ordinary and partial differential equations) as well as stochastic simulations that emphasize multiscale approaches.

The course is designed to be equally suitable for mathematics graduate students looking to learn advanced modeling methods, interesting applications, and topics for further analysis, and biologists who want to understand and critically assess models and carry out advanced multiscale simulations. All participants will learn multiscale simulations (using the open source software Morpheus) to visualize behaviour that emerges from intracellular signaling systems, cell migration, and cell-cell interactions. An emphasis will be on communication across disciplines, matching students from distinct disciplines for joint presentations and projects. Learning goals, expectations, assignments, and grading will take into account the student background.