Courses: ongoing

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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).

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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.

See the course website for the syllabus and other details.

https://sites.google.com/view/math612d/home

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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.

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The goal of this course is to teach the students to use mathematical models to improve and optimize public transport networks. The first part of the course is about using Markov Chains and dealing with big amount of public transport data. The students will learn how to use Markov Chains to model public transport networks, and how to validate the model by using the data. Important quantitates that can be extracted from the transition matrix of the Markov Chains will be studied with their related theorems. These quantities will be used to improve and optimize the network. The second part of the course is about using Linear Optimization in Public Transport Delay Management. Different Delay Management problems such as delay management problem with fixed connections, total delay management problem, bicriteria delay management problem, and general delay management problem will be studied. Especially perturbed timetables will be discussed and two integer programming descriptions of the set of all feasible perturbed timetables will be given. The first one is based on the “intuitive” description of the problem, while the second one uses the concept of event-activity networks. Assignment: The students should submit and present a project with some computer programming tasks for this course. They should use SUMO, Simulator of Urban Mobility, to simulate the public transport network and extract the data. They need to import the data extracted from the simulation to MATLAB or Python and implement the Markov Chain and Linear Optimization models. Guest lecturers: There will be two or three guest lecturers for this course Professor Robert Shorten from Imperial College London, Dr Emanuele Crisostomi from University of Pizza, and/or Professor Tarek Sayed from University of British Columbia. Grading: Assignment (Project presentation): 40% Midterm 1: 15% Midterm 2: 15% Final: 30% Required Reading: • Optimization in Public Transportation, Springer Optimization and Its Applications, 2006th edition, by Anita Schobel. • A big-data model for multi-modal public transportation with application to macroscopic control and optimization, International Journal of Control, vol. 88, Issue. 11, pp. 1-28, 2015 The course can be presented remotely. The recorded lectures are going to be posted.

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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.