The Pacific Institute for the Mathematical Sciences is pleased to announce the following network-wide graduate courses in mathematical sciences. These courses are available online and provide access to experts from throughout the PIMS network.

Students at Canadian PIMS member universities may apply for graduate credit via the Western Deans’ Agreement (WDA). Please be advised, in some cases students must enroll 6 weeks in advance of the term start date and will typically be required to pay ancillary fees to the host institution (as much as $270) or explicitly request exemptions. Please see the WDA section for details of fees at specific sites, and check the individual courses below for registration details. Courses hosted at UW must nominate a co-instructor at a Canadian PIMS site. That site will be used to process WDA student applications.

Students at universities not covered by the WDA but which are part of the Canadian Association for Graduate Studies (CAGS) may still be eligible to register for this course under the Canadian University Graduate Transfer Agreement (CUGTA). Details of this program vary by university and are also typically subject to ancillary fees. Please contact your local Graduate Student Advisor for more information.

The courses in this section are currently open for registration. Expand each item to see the course details.

Habiba Kadiri : habiba.kadiri@uleth.ca

University of Lethbridge

Elementary Number Theory

Real and Complex Analysis

This course is available for registration under the Western Dean's Agreement but registrations must be approved by the course instructor. Please contact the instructor (using the email link to the left) including details of how you meet the course prerequisites. Next, you must complete the Western Deans' Agreement form , with the following course details:

- Course Name
- Analytic Number Theory I
- Course Number
- MATH 5460
- Section Number
- Section A
- Section Code
- 30910

Completed forms should be returned to your graduate advisor who will sign it and take the required steps. For students at PIMS sites, please see this list to find your graduate advisor, for other sites, contacts can be found on the Western Deans' Agreement contact page .

The Western Deans' Agreement provides an automatic tuition fee
waiver for visiting students. Graduate students paying normal
required tuition fees at their home institution will not pay
tuition fees to the host institution. **However, students will typically be be
required to pay other ancillary fees to the host institution, or
explicitly request exemptions (e.g. Insurance or
travel fees)**. Details vary by university, so please contact the
graduate student advisor at your institution for help completing the
form. Links to fee information and contact information for PIMS
member universities is available below in the WDA
section.

Students at universities not covered by the WDA but which are part of the Canadian Association for Graduate Studies (CAGS) may still be eligible to register for some courses under the terms of the Canadian University Graduate Transfer Agreement (CUGTA). Details of this program vary by university and registration is also typically subject to ancillary fees. Both your local and the hosting university must participate in the agreement (e.g. UBC does not participate in CAGS). Please contact the relevant graduate student advisors for more information.

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).

Geoff Schiebinger : geoff@math.ubc.ca

University of British Columbia

Linear algebra as in Math 307

This course is available for registration under the Western Dean's Agreement but registrations must be approved by the course instructor. Please contact the instructor (using the email link to the left) including details of how you meet the course prerequisites. Next, you must complete the Western Deans' Agreement form , with the following course details:

- Course Name
- OT+Bio - Single Cell Analysis
- Course Number
- Math 612D
- Section Number
- Section Code

Completed forms should be returned to your graduate advisor who will sign it and take the required steps. For students at PIMS sites, please see this list to find your graduate advisor, for other sites, contacts can be found on the Western Deans' Agreement contact page .

The Western Deans' Agreement provides an automatic tuition fee
waiver for visiting students. Graduate students paying normal
required tuition fees at their home institution will not pay
tuition fees to the host institution. **However, students will typically be be
required to pay other ancillary fees to the host institution, or
explicitly request exemptions (e.g. Insurance or
travel fees)**. Details vary by university, so please contact the
graduate student advisor at your institution for help completing the
form. Links to fee information and contact information for PIMS
member universities is available below in the WDA
section.

Students at universities not covered by the WDA but which are part of the Canadian Association for Graduate Studies (CAGS) may still be eligible to register for some courses under the terms of the Canadian University Graduate Transfer Agreement (CUGTA). Details of this program vary by university and registration is also typically subject to ancillary fees. Both your local and the hosting university must participate in the agreement (e.g. UBC does not participate in CAGS). Please contact the relevant graduate student advisors for more information.

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://personal.math.ubc.ca/~geoff/courses/W2019T1/Math612.html

Li Xing : lix491@usask.ca

University of Saskatchewan

Students should have basic statistical theoretical knowledge

A good understanding of linear regression

Basic R coding skills.

This course is available for registration under the Western Dean's Agreement but registrations must be approved by the course instructor. Please contact the instructor (using the email link to the left) including details of how you meet the course prerequisites. Next, you must complete the Western Deans' Agreement form , with the following course details:

- Course Name
- Statistical Machine Learning for Data Science
- Course Number
- Stat 846
- Section Number
- Section Code
- 82213

Completed forms should be returned to your graduate advisor who will sign it and take the required steps. For students at PIMS sites, please see this list to find your graduate advisor, for other sites, contacts can be found on the Western Deans' Agreement contact page .

The Western Deans' Agreement provides an automatic tuition fee
waiver for visiting students. Graduate students paying normal
required tuition fees at their home institution will not pay
tuition fees to the host institution. **However, students will typically be be
required to pay other ancillary fees to the host institution, or
explicitly request exemptions (e.g. Insurance or
travel fees)**. Details vary by university, so please contact the
graduate student advisor at your institution for help completing the
form. Links to fee information and contact information for PIMS
member universities is available below in the WDA
section.

Students at universities not covered by the WDA but which are part of the Canadian Association for Graduate Studies (CAGS) may still be eligible to register for some courses under the terms of the Canadian University Graduate Transfer Agreement (CUGTA). Details of this program vary by university and registration is also typically subject to ancillary fees. Both your local and the hosting university must participate in the agreement (e.g. UBC does not participate in CAGS). Please contact the relevant graduate student advisors for more information.

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.

Mahsa Faizrahnemoon : mfaizrah@sfu.ca

Simon Fraser University

Applied Linear Algebra Math 232 or Linear Algebra Math 240

Linear Optimization Math 308

Some basic programming knowledge in Python or MATLAB is required.

- Course Name
- Optimal Public Transport
- Course Number
- Section Number
- Section Code

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.

Leah Edelstein-Keshet : keshet@math.ubc.ca

University of British Columbia

For math students - Some familiarity with ODEs, PDEs

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

- Course Name
- Mathematical Models in Cell Biology
- Course Number
- UBC Math 563
- Section Number
- 101
- Section Code
- 2022W:563:101

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.

The courses in this section are not yet accepting registrations.

Alireza Nojeh : alireza.nojeh@ubc.ca

University of British Columbia

The course does not have formal prerequisites, but assumes background knowledge of linear algebra and calculus, including some degree of vector calculus, at the undergraduate engineering/physics level. Familiarity with the calculus of variations, as well as basic quantum mechanics, would also be helpful, but those are not formal requirements since the key necessary concepts will be presented.

Registration for this course is not currently available.

This course provides an extensive theoretical foundation for as well as hands-on introduction to several widely used methods for studying the properties of materials and structures, in particular at the nanoscale and mesoscale. The majority of the time is spent on quantum-mechanical methods: the first-principles approaches (starting from the Hartree-Fock theory and building up to Configuration Interaction and the Møller–Plesset Perturbation Theory) and, in particular, the Density Functional Theory, which are derived and discussed in detail. Semi-empirical methods such as Tight Binding and Molecular Dynamics are also covered, as well as strategies for modelling material properties (electronic, mechanical, optical, etc.). Practical activities include implementing some of the above theories in computer code, in addition to using established software (Gaussian, SIESTA, VASP, LAMMPS, etc.). Each student also works on a project of their choice using the methods discussed.

- Modelling quantum systems and phenomena
- The many-body wave function and the Schrödinger equation
- The Born-Oppenheimer approximation
- Spin and the Pauli exclusion principle
- Representation of functions

- Hartree products and Slater determinants
- The variation principle
- The expectation value of the Hamiltonian with a single Slater-determinant
- Lagrange’s method of undetermined multipliers
- Exchange interaction, the Fock operator, and the Hartree-Fock equations

- Unitary transformations and the diagonalization of the Hartree-Fock equations
- The Koopmans theorem and the significance of canonical Hartree-Fock orbitals

- Basis functions and basis sets
- The Roothaan equations
- Mulliken population analysis

- Many-electron excitations
- Basis set for many-electron wave functions
- Configuration interaction
- The Møller-Plesset perturbation theory

- Functional derivatives
- The theorems of Hohenberg and Kohn
- The Kohn-Sham method
- Total energy in DFT, and the significance of Kohn-Sham orbitals
- Correlation energy and exchange-correlation functionals
- The connection between DFT and the Thomas-Fermi-Dirac and Hartree-Fock theories
- Periodicity, the Bloch theorem, and band structure in DFT
- Finite-temperature DFT
- Time-dependent DFT

- Linear combination of atomic orbitals
- The Hückel method
- The Pariser-Parr-Pople method
- The tight-binding method

- Molecular mechanics and molecular dynamics
- Force fields
- Time propagation
- Temperature, pressure, thermostats, and barostats

Lior Silberman : lior@math.ubc.ca

University of British Columbia

There will be no formal pre-requisites. Ideally students would have a general graduate background including real analysis and integration, point set topology, and functional analysis. Familiarity with the classification of complex semisimple Lie algebras (e.g. by taking UBC MATH 534) would be an advantage.

Registration for this course is not currently available.

This is a graduate course on the structure and representation theory of real Lie groups. The course will have four parts: an introduction to topological and compact groups, the basics of Lie groups and differential geometry, the structure and representation theory of compact Lie groups, and (as time allows) the structure and representation theory of semisimple Lie groups.

Greg Martin : gerg@math.ubc.ca

University of British Columbia

A course in analytic number theory that includes Dirichlet series and a complex-analytic proof of the prime number theorem (preferably Analytic Number Theory I taught by Kadiri in Fall 2022), or other exposure to those topics

Registration for this course is not currently available.

This course is a second graduate course in number theory, designed to follow
Analytic Number Theory I taught by Prof. Habiba Kadiri
(University of Lethbridge) in Fall 2022. We will learn about Dirichlet
characters and sums involving them, Dirichlet *L*-functions and their zeros,
and the prime number theorem in arithmetic progressions. With the explicit
formula for the error term in this theorem, we will continue into limiting
distributions of error terms and comparative prime number theory (“prime
number races”). This course also precedes the summer school “Inclusive Paths
in Explicit Number Theory” in Summer 2023 and is designed to give students the
ideal preparation for that summer school program. All three of these events
are part of the current PIMS CRG “*L*-functions in Analytic Number
Theory” (2022-2025).

Kevin J. Keen : Kevin.Keen@unbc.ca

University of Northern British Columbia

For STAT 475-3, undergraduates must have completed 3 credit hours of linear algebra (equivalent to UNBC MATH 150-3 or UNBC MATH 220-3) and 3 credit hours of linear modelling (UNBC STAT 471-3). There are no formal prerequisites for STAT 673-3 as a graduate student enrolling without the undergraduate pre-requisites must be prepared to make up any deficiencies in linear algebra, probability, statistics, and statistical linear modelling.

Registration for this course is not currently available.

This course discusses practical techniques for the analysis of multivariate data. Topics covered include estimation and hypothesis testing for multivariate means and variances; partial, multiple and canonical correlations; principal components analysis and factor analysis for data reduction; multivariate analysis of variance; discriminant analysis; and cluster analysis.

Yaozhong Hu : yaozhong@ualberta.ca

University of Alberta

Some preparation on mathematical analysis and probability theory

Prerequisites may be waived at the discretion of the instructor

Registration for this course is not currently available.

This is a one semester three credit hour course. It is about the theory and applications of stochastic differential equations driven by Brownian motion. A stochastic differential equation (SDE) is a differential equation in which the rate of change is determined by the state of the system itself, some external known forces and some unknown external forces as well. The noise in the system is given by random white noise calculated as the derivative of Brownian motion or the Wiener process. However, other types of random behaviour are possible, such as jump processes. Random differential equations are conjugate to stochastic differential equations. This course will concentrate on stochastic differential equations driven by Brownian motions. The stochastic differential equations are used to model various phenomena such as unstable stock prices or physical systems subject to thermal fluctuations. They have found applications in finance, signal processing, population dynamics and many other fields. It is the basis of some other applied probability areas such as filtering theory, stochastic control and stochastic differential games. To balance the theoretical and applied aspects and to include as much audience as possible, we shall focus on the stochastic differential equations driven only by Brownian motion (white noise). We will focus on the theory and not get into specic applied area (finance, signal processing, filtering, control and so on). We shall first briefly introduce some basic concepts and results on stochastic processes, in particular, the Brownian motions. Then we will discuss stochastic integrals, Ito formula, the existence and uniqueness of stochastic differential equations, some fundamental properties of the solution. We will concern with the Markov property, Kolmogorov backward and forward equations, Feynman-Kac formula, Girsanov formula. We will also concern with the ergodic theory and other stability problems. We may also mention some results on numerical simulations, Malliavin calculus and so on.

Rouslan Krechetnikov : krechet@ualberta.ca

University of Alberta

Equivalent of MA PH 343 (basic introduction to Classical Mechanics)

pre-requisites may be waived with the consent of instructor

Registration for this course is not currently available.

This course offers a concise, but self-contained, introduction to the subject of mechanics, which combines geometrical view and physical insights. We will start with a formulation of classical mechanics in the framework of variational principles, translate from point to continuous systems, and analyze the effects of holonomic and nonholonomic constraints. The discussion of effects of friction and collision will naturally lead us to ergodic theory. A significant part of the course will be devoted to the geometric language of mechanics including analysis on manifolds, Lie groups, and differential topology. Among its applications, we will focus on symmetries, reduction, and geometric phase both in finite and infinite dimensions including fluid mechanics. Two key references which define the spirit of the course are “Lectures on Mechanics” by Jerrold Marsden and “Mathematical Methods of Classical Mechanics” by Vladimir Arnold.

Variational principles; celestial mechanics; holonomic and nonholonomic constraints; effects of friction and collision, ergodic theory and chaos; analysis on manifolds, Lie groups, differential topology; symmetries, reduction, and geometric phase; infinite-dimensional systems.

Martin Frankland : Martin.Frankland@uregina.ca

University of Regina

A course in general topology or metric space topology is required

A course in group theory is strongly recommended

Registration for this course is not currently available.

The course is a first semester of algebraic topology. Broadly speaking, algebraic topology studies spaces and shapes by assigning algebraic invariants to them. Topics will include the fundamental group, covering spaces, CW complexes, homology (simplicial, singular, cellular), cohomology, and some applications.

Ailene MacPherson : ailenem@sfu.ca

Simon Fraser University

A Dynamical Systems course (e.g. Math 467) is required

A course in Probability Theory or Stochastic Processes (e.g. Stat 380) is recommended

Registration for this course is not currently available.

Much of our understanding of evolution, the process shaping the beautiful biological diversity in our world, is grounded in equally elegant mathematics. In this course we will cover the mathematical description of evolution. Involving a wide range of topics, from the analysis of non-linear dynamics to stochastic processes and partial differential equations, this course will challenge you to take mathematical principals and apply them to the natural world. Throughout this course we will focus particularly on addressing important contemporary existential questions with mathematical models, for example applications of evolution to conservation and public health.

Nikita Karpenko : karpenko@ualberta.ca

University of Alberta

Any basic course of algebraic geometry

Registration for this course is not currently available.

Classifying spaces of algebraic groups do not exist in classical algebraic geometry. However, they can be approximated by algebraic varieties. Using this approach, Burt Totaro defined the Chow ring CH(BG) of algebraic cycles on the classifying space BG of an affine algebraic group G. The representation ring R(G) can be viewed as the Grothendieck ring of BG. It turns out that the rings CH(BG) and R(G) are related the same way as the Chow ring and the Grothendieck ring of a smooth algebraic variety are. Namely, for elements of R(G) one can define their Chern classes with values in CH(BG). And there is a surjective ring homomorphism of CH(BG) onto the graded ring associated with certain filtration on R(G). We discuss these and related objects, consider examples of their computation, review some applications. We also take a look at the rationality problem of BG. We start with a quick introduction into algebraic groups and into the theory of Chow rings of algebraic varieties.

Classifying spaces of algebraic groups do not exist in classical algebraic geometry. However, they can be approximated by algebraic varieties. Using this approach, Burt Totaro defined the Chow ring CH(BG) of algebraic cycles on the classifying space BG of an affine algebraic group G. The representation ring R(G) can be viewed as the Grothendieck ring of BG. It turns out that the rings CH(BG) and R(G) are related the same way as the Chow ring and the Grothendieck ring of a smooth algebraic variety are. Namely, for elements of R(G) one can define their Chern classes with values in CH(BG). And there is a surjective ring homomorphism of CH(BG) onto the graded ring associated with certain filtration on R(G). We discuss these and related objects, consider examples of their computation, review some applications. We also take a look at the rationality problem of BG. We start with a quick introduction into algebraic groups and into the theory of Chow rings of algebraic varieties. Textbook: Burt Totaro. Group cohomology and algebraic cycles. Cambridge Tracts in Mathematics, 204. Cambridge University Press, Cambridge, 2014.

In order to register in a PIMS digital course for the Western Deans’ agreement you must obtain the approval of the course instructor. Once you have obtained their approval please complete the Western Deans’ agreement form . The exact process and deadlines vary by site, but the general steps for students at PIMS member universities are

- Obtain the approval of the course instructor.
- Contact your home department and obtain the necessary signatures.
- Follow the procedures at your host institution to complete and submit the application form, CC the PIMS Site Admin at your university.
- The PIMS Site Admins will be available to assist you with document tracking, fee payments and waivers, ordering transcripts, etc.

Select your university and the university hosting the course you are interested
in below. Read both sets of instructions *carefully* before proceeding.

Please note: The Western Deans’ Agreement provides an automatic tuition fee waiver for visiting students. Graduate students paying normal required tuition fees at their home institution will not pay tuition fees to the host institution. However, students will typically be required to pay other ancillary fees to the host institution (as much as $270) or explicitly request exemptions (e.g. Insurance or travel fees).

For help completing the Western Deans’ agreement form, please contact the graduate advisor at your institution. For more information about the agreement, please see the Western Deans’ Agreement website .