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.

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 accepting registrations. Expand each item to see the course details and registration information.

Yuriy Zinchenko : yzinchen@ucalgary.ca

University of Calgary

This course can be though of as interdisciplinary and is suitable to well-prepared engineering, CS and physics students besides grad students in mathematiacs

No prior knowledge of convex optimization is required

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
- Convex Optimization with Applications
- Course Number
- MATH 661
- 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 this course under the terms of 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 main goal of the course is to equip students with practical modeling and computational skills needed to solve optimization problems arising in various applications, with major emphasis on the so-called convex optimization. Some theoretical aspects of convex optimization will be covered. Roughly half of the course will be dedicated to modeling techniques and optimization software. A particular emphasis will be made on CVX package: Matlab Software for Disciplined Convex Programming.

At the end of the course (MATH661: Convex Optimization) the students will be able to

- Define constrained optimization problem,
- Define convex optimization problem,
- Test Euclidean subsets for convexity,
- Test univariate (not necessarily differentiable) functions for convexity,
- Define and recognize basic sub-types of convex optimization problems, such as least-squares, linear programming, QP, QCQP, SOCP, and possibly SDP,
- Give concrete examples of convex optimization applications in areas like statistical estimation (MLE), math finance (portfolio optimization), approximation of hard combinatorial problems (boolean LP relaxation), etc.
- Convert and solve exemplary (real-world) word problems into abstract convex programs, and use MATLAB/Python CVX environment to solve the problems above.

Michael Jacobson, Jr. : jacobs@ucalgary.ca

University of Calgary

This course is

*not*intended for students specializing in information security and privacy technologies.

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
- Explorations in Information Security and Privacy
- Course Number
- CPSC 602
- Section Number
- L01
- 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 this course under the terms of 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.

Surveys topics in information security and privacy, with the purposes of cultivating an appropriate mindset for approaching security and privacy issues and developing basic familiarity with related technical controls.

This course may not be repeated for credit.

- Recognize security and privacy threats, and enumerate possible defense mechanisms and their effectiveness in a distributed computer system
- Identify mechanisms for controlling access to a computer system, and compare and contrast their effectiveness in practice.
- Basic understanding of cryptographic tools and techniques and their applications in securing computer systems.
- Outlining opinions and views about ethical and legal issues related to information security, their effect on digital and privacy rights, and research and development in this domain.
- Identify network and software related attacks, and distinguish the role of different mechanisms in protecting the system.

- Introduction
- Authentication
- Access control
- Malware
- Introduction to cryptography
- Modern cryptography - symmetric key
- Modern cryptography - public-key
- Web security
- Introduction to blockchain
- Network security

Karen Gunderson : karen.gunderson@umanitoba.ca

University of Manitoba

Karen Meagher

University of Regina

Venkata Raghu Tej Pantangi

University of Lethbridge

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
- Extremal Combinatorics
- Course Number
- MATH 8210
- Section Number
- T01
- 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 this course under the terms of 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.

We will be exploring topics in extremal combinatorics from problems for set systems to graph theory and hypergraphs. These include extremal results for chains and antichains, intersecting set systems, isoperimetric problems, extremal numbers for graphs, extremal properties of matchings, extremal numbers for small hypergraphs, graph eigenvalues, extremal problems for graph diameter, distance transitive graphs, and some extremal results from combinatorial matrix theory.

A more detailed list of topics is available in the preliminary syllabus.

Alexander Litvak : alitvak@ualberta.ca

University of Alberta

Vladyslav Yaskin : yaskin@ualberta.ca

University of Alberta

- Course Name
- High-Dimensional Geometric Analysis
- Course Number
- Math 617 - Topics in Functional Analysis
- Section Number
- Section Code

Asymptotic Geometric Analysis (AGA) lies at the border between geometry and analysis stemming from the study of geometric properties of finite dimensional normed spaces, especially the characteristic behavior that emerges when the dimension is suitably large or tends to infinity. Time permitting we plan to cover Banach-Mazur distance between convex bodies; John’s theorem; Dvoretsky’s theorem; properties of sections and projections of convex bodies; $MM^*$-estimate; M-ellipsoids, volumetric, entropic, and probabilistic methods for finite-dimensional convex bodies. We will also discuss methods of Fourier analysis in convex geometry. The idea of this approach is to express certain geometric quantities (such as sections or projections of convex bodies) in terms of the Fourier transform and then apply methods of harmonic analysis to solve geometric problems. In particular, we will talk about the following topics: the Fourier transform and sections of convex bodies; the Busemann-Petty problem; the Fourier transform and projections of convex bodies; Shephard’s problem; extremal sections of $l_p$-balls.

Banach-Mazur distance; John’s theorem and applications; Dvoretsky’s theorem; M-ellipsoids; the Fourier transform of distributions; the Busemann-Petty problem; Shephard’s problem; Additional topics at the discretion of the instructors.

Kristine Bauer : bauerk@ucalgary.ca

University of Calgary

An undergraduate course in Abstract Algebra including introduction to groups, rings and fields (University of Calgary Math 431 or equivalent).

- Course Name
- Introduction to Algebraic Topology
- Course Number
- Math 625
- Section Number
- Lecture 1 (tentative)
- Section Code
- MATH 625 L01

Introduction to the algebraic invariants that distinguish topological spaces. Focuses on the fundamental group and its applications, and homology. Introduction to the basics of homological algebra.

This course introduces the algebraic invariants that distinguish topological spaces. The course will focus on the fundamental group and its applications and homology. Students will also learn the basics of homological algebra. Over the last few decades, algebraic topology has developed many applications to data science, materials science, and robotics. Whenever possible, connections to these emerging research fields will be discussed.

Eric Cytrynbaum : cytryn@math.ubc.ca

University of British Columbia

Some experience with differential equations

Some familiarity with the ideas of probability and/or statistics

- Course Name
- Introduction to Mathematical Biology
- Course Number
- Math 560
- Section Number
- Section Code

MATH 560 provides a broad overview of Mathematical Biology at an introductory level. The scope is obviously subject to the limitations of time and instructor knowledge and interests - this is a HUGE area of research.

It is intended for early stage math bio grad students, general applied math grad students interested in finding out more about biology applications, and grad students in other related departments interested in getting some mathematical and computational modelling experience.

The course is organized around a sample of topics in biology that have seen a significant amount of mathematical modelling over the years. Currently, I’m including content from ecology, evolution and evolutionary game theory, epidemiology, biochemistry and gene regulation, cell biology, electrophysiology, developmental biology. However, this list changes gradually from year to year, to reflect students’ and my own interests. The mathematical modelling methods and techniques covered are those that typically arise in the biological applications listed above. For example, I will cover models using ordinary and partial differential equations, stochastic processes, agent-based models and introduce techniques from bifurcation theory, asymptotics, dimensional analysis, numerical solution methods, and parameter estimation. An emphasis will be placed on reading and discussing classic and current papers.

A complete syllabus is available on the course website

Michael Ward : ward@math.ubc.ca

University of British Columbia

Students should have a working knowledge of Applied PDE (such as MATH 400 at UBC), a basic Complex Variables Course (such as M300 at UBC), and preferably the course M550 at UBC on an introduction to perturbation theory. Some experience with mathematical modeling in either continuum mechanics or biology is also desirable.

- Course Name
- Perturbation Methods for Partial Differential Equations and Applications
- Course Number
- Math 551
- Section Number
- 201
- Section Code

This is a course in modern techniques in applied mathematics, focusing on perturbation methods for partial differential equations. The material provides valuable skills and resources complementary to scientific computations, mathematical modeling in applications, analysis of PDE’s and dynamical systems. The general concepts and methods are illustrated and developed for a wide variety of specific problems arising in math biology, fluid mechanics, materials science, and wave propagation.

Donald Estep : destep@sfu.ca

Simon Fraser University

- Course Name
- Theory of Probability and Applications
- Course Number
- STAT 831
- Section Number
- Section Code

This course will cover measure theoretic probability, random variables, expectation, product spaces, independence, derivatives, conditional probability, characteristic functions, and limit theorems. While rigorous proof is emphasized as the way to understand the material, the material is based on a course that has been taught to students in statistics, mathematics, engineering and science for many years. The proofs are elementary and build a foundation to learn how to conduct analysis. The close connection of measure and probability theory to computation and approximation are emphasized. The book covers the core material. It does not cover some standard advance topics, e.g., martingales, but (time permitting) does cover topics like disintegration and exchangeability that are important in applications.

Course work will be based on homework assignments.

The courses in this section are currently running and may not be accepting registrations. Expand each item to see the course details.

Zaid Harchaoui (teaching)

University of Washington

Soumik Pal (teaching) : soumikpal@gmail.com

University of Washington

Young-Heon Kim (WDA administrator) : yhkim@math.ubc.ca

University of British Columbia

Prior knowledge of real analysis

Prior knowledge of probability

Prior knowledge of statistics

Prior knowledge of and machine learning

Familiarity with coding in Python or R is a plus

- Course Name
- Optimal Transport + Machine Learning (OT + ML)
- Course Number
- Math 566 - Topics in Optimal Transport
- Section Number
- Section Code

In the second installment of OT+X series we take X=ML or machine learning. A number of problems equivalent or related to the Monge-Kantorovich Optimal Transport (OT) problem have appeared in recent years in machine learning, and data science at large. The fruitful connections between the two fields have led to several important advances impacting both. The Wasserstein metric defines a metric between probability measures, used to describe distributions over data or distributions over models, that improves upon existing metrics based on Hilbertian metrics and f-divergences, and that is now more easily amenable to efficient numerical computation.

The first part of the course will cover the mathematical basics of OT and introduce the geometry of Wasserstein spaces. The second part of the course will cover computational aspects of OT and describe the central role played by OT in convergence analysis of stochastic algorithms for deep learning, in distributionally robust statistical learning, and in combinatorial or geometrical problems arising in data science applications. The course is meant for a wide audience including graduate students and industry professionals. Prior knowledge of real analysis, probability, statistics, and machine learning will be particularly helpful. The course will be interspersed with numerical illustrations. Familiarity with coding in Python or R is a plus.

https://sites.math.washington.edu/~soumik/OTML.html

Charles Doran : Charles.Doran@ualberta.ca

University of Alberta

- Course Name
- Differential Equations in Algebraic Geometry
- Course Number
- Math 676 - Topics in Geometry I
- Section Number
- LEC-A1-57019
- Section Code

What can differential equations tell us about the solutions to systems of algebraic equations? Conversely, what are the special properties of differential equations, and their solutions, that “come from geometry”?

In this course, we will combine tools from both algebra and analysis in our concrete introduction to transcendental algebraic geometry. This includes the theory of differential forms and integration on families of algebraic curves, complex surfaces, and even Calabi-Yau threefolds. Along the way we will present the general theory of Fuchsian differential equations, their isomonodromic deformations, and associated completely integrable Pfaffian systems. Techniques of computation will be emphasized along with the theory.

What can differential equations tell us about the solutions to systems of algebraic equations? Conversely, what are the special properties of differential equations, and their solutions, that “come from geometry”?

In this course, we will combine tools from both algebra and analysis in our concrete introduction to transcendental algebraic geometry. This includes the theory of differential forms and integration on families of algebraic curves, complex surfaces, and even Calabi-Yau threefolds. Along the way we will present the general theory of Fuchsian differential equations, their isomonodromic deformations, and associated completely integrable Pfaffian systems. Techniques of computation will be emphasized along with the general theory.

Course notes and excerpts from classic papers; For general differential equations content, the excellent new textbook “Linear Differential Equations in the Complex Domain: From Classical Theory to Forefront” by Yoshishige Haraoka (Springer Lecture Notes in Mathematics, Volume 2271).

The course grade will be based on a research project/paper, tuned to each student’s background and interests, that will be completed during the term in consultation with the professor.

The courses in this section are no longer accepting registrations. Expand each item to see a course abstract

Martin Frankland : Martin.Frankland@uregina.ca

University of Regina

A course in general topology, or metric space topology.

A course in group theory.

Registration for this course is not currently available.

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

**Main reference:**Algebraic Topology, by Allen Hatcher. Available for free on the author’s website: http://pi.math.cornell.edu/~hatcher/AT/ATpage.html**Secondary reference:**A Concise Course in Algebraic Topology, by J. Peter May. Available for free on the author’s website: https://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf

Bojan Mohar : mohar@sfu.ca

Simon Fraser University

Topological spaces

Continuous maps

Metric space topology

Quotient topology

Compactness

Basic notions about simplicial complexes, fundamental groups and covering spaces will be helpful, but students will also be given opportunity to self-study about these notions during the first month of the course and help will be offered during tutorials.

Registration for this course is not currently available.

This is a basic level graduate course with introduction to algebraic topology and its applications in combinatorics, graph theory and geometry. The course will cover introductory chapters from [1] and parts of [2]. With a guest lecture by Nati Linial from Israel, we will also touch some recent topics like the topology of random simplicial complexes. The instructor expects that students with interests in topology and those with interests in discrete mathematics and geometry would find the course suitable.

This is a basic level graduate course with introduction to algebraic topology and its applications in combinatorics, graph theory and geometry. The course will start with a brief review of the basic notions of topology, including the notions mentioned as prerequisites. It will continue with introductory chapters from Hatcher’s textbook [1]. Simplicial complex. Cell complex. Homotopy and fundamental group (Sections 1.1-1.3 and 1.A). Homology (Sections 2.1-2.2 and parts of 2.A-2.C). The second part of the course will concentrate on various applications of algebraic topology in combinatorics, graph theory, and geometry. We will follow relevant chapters from Matousek’s book [2]. Some of those applications use Borsuk-Ulam Theorem, which will be covered first. Time permitting, we may touch a recent flourishing topics on the topology of random simplicial complexes.

- [1] A. Hatcher, Algebraic Topology, Cambridge University Press, 2002. (Available for free download from http://pi.math.cornell.edu/~hatcher/AT/ATpage.html).
- [2] J. Matousek, Using the Borsuk–Ulam Theorem - Lectures on Topological Methods in Combinatorics and Geometry, Springer, 2003.

The weekly schedule will consist of four 50-minute lectures. Two to three of them will be giving new material, with some details left for the students to cover by themselves from the provided textbooks. The remaining weekly time will be used for tutorials, covering problems and examples, explaining details of proofs, and having students work in small groups and report on their solutions. The online platform used will be Zoom, with synchronous teaching that will be recorded for asynchronous viewing.

- Homework 20%
- Midterm 30%
- Final 50%

The instructor reserves the right to limit the number of students from outside of SFU. He will allow for additional students who will not take the course for credit (their homework and exams will not be graded).

Ian F. Putnam : ifputnam@uvic.ca

University of Victoria

A good course in abstract algebra, up to the first isomorphism theorem and a good course in general topology. The course is accessible to advanced undergraduates with a good background.

Registration for this course is not currently available.

The official title ‘Topology’ of this course is misleading. A better one would be ‘Topics in Dynamical Systems’. Dynamical systems is the mathematical study of models based on the idea of a topological space, representing the possible configurations of a system and a continuous map (or maps) which represent its time evolution. The systems considered in this course have two additional features: the space is compact and totally disconnected while the map is minimal in the sense that every trajectory formed by iteration on a single point is dense. Such spaces have a strongly combinatorial feel to them and one of our main goals is o provide a complete model for such systems based purely on combinatorial data called a Bratteli diagram. This model has been used extensively in topological dynamics over the last thirty years. The second main topic is to introduce a purely algebraic invariant for such systems. So the course becomes an interesting mix, moving between combinatorics, algebra and topology or dynamical systems. The overall goal is a theorem which classifies such systems up to a notion of orbit equivalence. Primarily, we will aim to understand all of the ingredients for the theorem and have some idea of how to prove it.

The text is the book Cantor MInimal Systems, written by the lecturer and published by the AMS:

It is my intention to cover all 14 Chapters, at least partially.

The grading scheme for the course will be six assignments, due roughly every two weeks. They will be weighted equally and the lowest score will be dropped before computing a final grade. There will be no tests. Students will be expected to submit their own work only, but may feel free to discuss the problems with others.

The course will be online: lectures Monday and Thursday from 11:30 am to 12:50 pm. I intend to use the first part of each lecture as a discussion for the entire class. Depending on how long these take, it may be necessary to supplement the material with recorded (i.e. asynchronous) lectures.

Greg Martin : gerg@math.ubc.ca

University of British Columbia

Solid course (preferably graduate-level) in elementary number theory

Graduate level course in analytic number theory, one that includes a proof of the prime number theorem and the corresponding “explicit formula”

Undergraduate course in probability would also be helpful

Registration for this course is not currently available.

We will begin with a quick review of the prime number theorem and the “explicit formula”, then develop the theory of Dirichlet characters, and combine these two sets of tools to prove the prime number theorem in arithmetic progressions. We will then move into comparing two counting functions of primes in arithmetic progressions, going through the history of such comparisons and learning how the normalized difference can be modeled by random variables, thus giving us a way to understand its limiting distribution. Student assessment will consist of some modest combination of presentations and reviews of research articles.

Recommended prerequisites are a solid course (preferably graduate-level) in elementary number theory, and a graduate-level course in analytic number theory, one that included a proof of the prime number theorem and the corresponding explicit formula. An undergraduate course in probability would also be helpful. Reference texts would be standard analytic number theory books by Iwaniec & Kowalski, by Montgomery & Vaughan, and by Titchmarsh. Students who are willing to learn some of this background as they go are welcome.

Classes will be held live (synchronously) on Zoom and regular attendance will be important. The current tentative schedule is to meet at 10am Pacific time on Mondays and Wednesdays and possibly Fridays. Students can join from any physical location.

Reference texts would be standard analytic number theory books by Iwaniec & Kowalski, by Montgomery & Vaughan, and by Titchmarsh.

Martyn P. Clark : martyn.clark@usask.ca

University of Saskatchewan

A firm foundation in calculus and physics at the first year university level

Some experience in computing (e.g. Familiarity with python, R, matlab)

A strong background in hydrology e.g. As obtained by taking Geography 827 “Principles of Hydrology” at the University of Saskatchewan or a similar graduate-level course in hydrology.

Registration for this course is not currently available.

The University of Saskatchewan Centre for Hydrology is offering an intensive course on the fundamentals of process-based hydrological modelling, including model development, model application, and model evaluation. The course will explain the model constructs that are necessary to simulate dominant hydrological processes, the assumptions that are embedded in models of different type and complexity, and best practices for model development and model applications. The course will cover the full model ecosystem, including the spatial discretization of the model domain, input forcing data generation, model evaluation, parameter estimation, post-processing, uncertainty characterization, data assimilation, and ensemble streamflow forecasting methods. The overall intent of the course is to provide participants with the understanding and tools that are necessary to develop and apply models across a broad range of landscapes. Specifically, the course will convey an understanding of how to represent existing process understanding in numerical models, how to devise meaningful model experiments, and how to evaluate these experiments in a systematic way. Along the way, participants will have the opportunity to build models, run models, make changes, and analyze model output.

Reading/Textbooks

- Clark, M. P., Y. Fan, D. M. Lawrence, J. C. Adam, D. Bolster, D. J. Gochis, . . . X. Zeng, 2015a: Improving the representation of hydrologic processes in Earth System Models. Water Resources Research, 51, 5929-5956, doi: 10.1002/2015WR017096.
- Clark, M. P., B. Nijssen, J. D. Lundquist, D. Kavetski, D. E. Rupp, R. A. Woods, . . . R. M. Rasmussen, 2015b: A unified approach for process-based hydrologic modeling: 1. Modeling concept. Water Resources Research, 51, 2498-2514, doi: 10.1002/2015WR017198.
- Clark, M. P., B. Nijssen, J. D. Lundquist, D. Kavetski, D. E. Rupp, R. A. Woods, . . . D. G. Marks, 2015c: A unified approach for process-based hydrologic modeling: 2. Model implementation and case studies. Water Resources Research, 51, 2515-2542, doi: 10.1002/2015WR017200.
- Clark, M. P., B. Schaefli, S. J. Schymanski, L. Samaniego, C. H. Luce, B. M. Jackson, . . . S. Ceola, 2016: Improving the theoretical underpinnings of process-based hydrologic models. Water Resources Research, 52, 2350-2365, doi: 10.1002/2015WR017910
- Clark, M. P., M. F. P. Bierkens, L. Samaniego, R. A. Woods, R. Uijlenhoet, K. E. Bennett, . . . C. D. Peters-Lidard, 2017: The evolution of process-based hydrologic models: historical challenges and the collective quest for physical realism. Hydrology and Earth System Sciences, 21, 3427-3440, doi: 10.5194/hess-21-3427-2017

Yaozhong Hu : yaozhong@ualberta.ca

University of Alberta

Some preparation on mathematical analysis and probability theory

Registration for this course is not currently available.

This is a one semester three credit hour course and meet twice a week, tentatively Tuesdays and Thursdays from 11:00-12:20. It is about the theory and applications of stochastic differential equations driven by Brownian motion. The stochastic differential equations 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 specific applied area (such as 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.

- The main reference book for this course is
- Øksendal, B. Stochastic differential equations. An introduction with applications. Sixth edition. Universitext. Springer-Verlag, Berlin, 2003. xxiv+360 pp. ISBN: 3-540-04758-1
- Karatzas, Ioannis; Shreve, Steven E. Brownian motion and stochastic calculus. Second edition. Graduate Texts in Mathematics, 113. Springer-Verlag, New York, 1991. xxiv+470 pp. ISBN: 0-387-97655-8
- Klebaner, Fima C. Introduction to stochastic calculus with applications. Third edition. Imperial College Press, London, 2012. xiv+438 pp. ISBN: 978-1-84816-832-9; 1-84816-832-2

- Other references
- Ikeda, N.; Watanabe, S. Stochastic differential equations and diffusion processes. Second edition. North-Holland Mathematical Library, 24. North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo, 1989. xvi+555 pp. ISBN: 0-444-87378-3 * Protter, P. E. Stochastic integration and differential equations. Second edition. Version 2.1. Corrected third printing. Stochastic Modelling and Applied Probability, 21. Springer-Verlag, Berlin, 2005. xiv+419 pp. ISBN: 3-540-00313-4
- Revuz, D.; Yor, M. Continuous martingales and Brownian motion. Third edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 293. Springer-Verlag, Berlin, 1999. xiv+602 pp.
- Durrett, R. Stochastic calculus. A practical introduction. Probability and Stochastics Series. CRC Press, Boca Raton, FL, 1996. x+341 pp. ISBN: 0-8493-8071-5
- Jeanblanc, M.; Yor, M.; Chesney, M. Mathematical methods for financial markets. Springer Finance. Springer-Verlag London, Ltd., London, 2009. xxvi+732 pp. ISBN: 978-1-85233-376-8
- Hasminskii, R. Z. Stochastic stability of differential equations. Translated from the Russian by D. Louvish. Monographs and Textbooks on Mechanics of Solids and Fluids: Mechanics and Analysis, 7. Sijthoff & Noordhoff, Alphen aan den RijnGermantown, Md., 1980. xvi+344 pp. ISBN: 90-286-0100-7
- Hu, Y. Analysis on Gaussian spaces. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2017. xi+470 pp. ISBN: 978-981-3142-17-6
- Kloeden, P. E.; Platen, E. Numerical solution of stochastic differential equations. Applications of Mathematics (New York), 23. Springer-Verlag, Berlin, 1992. xxxvi+632 pp. ISBN: 3-540-54062-8

Christopher Hoffman : hoffman@math.washington.edu

University of Washington

Anthony Quas : aquas@uvic.ca

University of Victoria

Graduate Real Analysis

Measure Theory

Registration for this course is not currently available.

Ergodic theory is the study of dynamical systems from a measurable or statistical point of view. Starting with Poincaré’s recurrence theorem and the ergodic theorems of Birkoff and von Neumann ergodic theory in the early twentieth century. The field has applications to many other areas of mathematics including probability, number theory and harmonic analysis. Among the topics covered will be

- examples of ergodic systems
- the mean and pointwise ergodic theorems
- mixing conditions
- recurrence
- entropy and
- the Ornstein’s Isomorphism Theorem.

This course will run between **March 29th and June 6th of 2021**, and is *now
open for registration*. Please note that this course is shared between the
University of Washington and the University of Victoria. The course will taught
primarily by by Prof. Hoffman (UWashington) **Canadian students wishing to
register for credit under the WDA should use the details above for the course at
the University of Victoria and should direct any registration enquiries to Prof.
Quas (UVic).** Please note that for some sites students must register at least 6
weeks before the course start date, for this course that deadline is **February
15th, 2021**.

- Ergodic Theory by Karl Petersen

Adam Kashlak : Kashlak@ualberta.ca

University of Alberta

Linear Algebra: vectors, matrices, quadratic forms, orthogonality, projections, eigenvalues.

Calculus: basic multivariate differential calculus such as computing gradients and finding critical points.

Statistics: an understanding of estimation and hypothesis testing, knowledge of linear regression is helpful.

Discrete Math: familiarity with topics like basic group theory and combinatorics can help, but are not required

Registration for this course is not currently available.

We will cover classical and modern methods of experimental design starting with one-way ANOVA and Cochran’s Theorem. From there, we will consider multi-factor ANOVA using a variety of combinatorial tools such as Graeco-Latin squares and incomplete block designs. There will be a brief interlude on multiple testing followed by 2 and 3 level factorial designs, fractional factorial designs, and blocking within such designs. Then, response surface designs—i.e. quadratic polynomial surfaces used for optimization of industrial processes–will be discussed. Lastly, more advanced topics will be touched on such as prime-level factorial designs and the Plackett-Burman design, which involves Hadamard matrices. Interesting datasets, connections to optimal coding theory, and at-home experiments will also be discussed. For study purposes, discussion questions will be included with the lectures and solutions will be discussed in class.

- My online lecture notes at https://sites.ualberta.ca/~kashlak/data/stat568.pdf.
- Supplementary texts:
- Wu, CF Jeff, and Michael S. Hamada. Experiments: planning, analysis, and optimization. Vol. 552. John Wiley & Sons, 2011.

Karen Meagher

University of Regina

Joy Morris

University of Lethbridge

Karen Gunderson

University of Manitoba

Registration for this course is not currently available.

The Fall 2020 offering of Math 827, Graph Theory will consist of three units on advanced graph theory topics.

The first unit will be 6 weeks will be on algebraic techniques in graph theory taught by Dr. Karen Meagher of the University of Regina. The focus will be on spectral graph theory, adjacency matrices and eigenvalues of graphs. We will consider important families of transitive graphs where algebraic methods are particularly effective.

The second unit will be 3 weeks on Cayley graphs, taught by Dr. Joy Morris from the University of Lethbridge. This unit will focus on automorphisms, isomorphisms and the isomorphism problem, and Hamilton cycles, all in the context of Cayley graphs.

The third unit will be 3 weeks on the topic of random graphs taught by Dr Karen Gunderson from the University of Manitoba. This unit will cover various models of random graphs and some types of pseudorandomness.

Fei Qi : Fei.Qi@umanitoba.ca

University of Manitoba

Graduate level abstract algebra and complex analysis. Knowledge to Lie algebras would be helpful but not essential.

Registration for this course is not currently available.

Vertex algebras are algebraic structures formed by the vertex operators that appear both in mathematics and in physics. In mathematics, vertex algebras are used to study the Monster group, the largest finite simple group. The representation theory of vertex algebras gives a mathematical construction to two-dimensional conformal field theories. In this course, we will take an axiomatic approach and focus on the definition, axioms, properties and examples. If time permits, we will also introduce the theory of vertex tensor categories associated to the modules for the vertex operator algebras.

- 1 - 5 are core materials of the course and will be evaluated in the problem sets and final exam.
- 6 - 8 are advanced topics that can possibly lead to research papers.

- Formal Calculus
- Axioms of vertex algebras and modules.
- Representations of vertex algebras.
- Local systems and the construction theorem.
- Examples: vertex algebras constructed from
- Virosoro algebra;
- Affine Lie algebras;
- Lattices

- Intertwining operators and tensor products of modules.
- Cofiniteness conditions and convergence problems.
- Vertex tensor categories of modules for rational vertex operator algebras.

https://server.math.umanitoba.ca/~qif

- Lepowsky-Li, Introduction to vertex algebras and its representation theory
- Vertex Operator Algebras and the Monster by Igor Frenkel, James Lepowsky, and Arne Meurman
- A series of papers by Yi-Zhi Huang, Jim Lepowsky and Lin Zhang on intertwining operators and vertex tensor categories.

Lele Wang : lelewang@ece.ubc.ca

University of British Columbia

Working knowledge of probability and linear algebra

No prior knowledge on graph theory is assumed

Registration for this course is not currently available.

A large variety of data science and machine learning problems use graphs to characterize the structural properties of the data. In social networks, graphs represent friendship among users. In biological networks, graphs indicate protein interactions. In the World Wide Web, graphs describe hyperlinks between web pages. In recommendation systems, graphs reveal the economic behaviors of users. Unlike the one-dimensional linear data sequence, data appearing in the form of a graph can be viewed as a two-dimensional matrix with special structures. How to compress, store, process, estimate, predict, and learn such large-scale structural information are important new challenges in data science. This course will provide an introduction to mathematical and algorithmic tools for studying such problems. Both information-theoretic methods for determining the fundamental limits as well as methodologies for attaining these limits will be discussed. The course aims to expose students to the state- of-the-art research in mathematical data science, statistical inference on graphs, combinatorial statistics, among others, and prepare them with related research skills.

- Random graphs (basic notions in graph theory, Erdös–Rényi graph, threshold phenomenon)
- Tools from the probabilistic method (first and second moment method, the method of moments)
- Vertex degrees (degree distribution, graph isomorphism algorithm
- Connectivity
- Small subgraphs (thresholds, asymptotic distributions)
- Spectral method (graph Laplacian, graph cut interpretation, perturbation of eigenstructures)
- Basic random matrix theory, pertubation theory
- Semidefinite programming
- Applications (Planted clique problem, community detection, graph matching, sorting and ranking)

https://canvas.ubc.ca/courses/59429

All ebooks are available at https://www.library.ubc.ca/.

- Alan Frieze and Michał Karon ́ ski, Introduction to Random Graphs, Cambridge University Press, 2015
- Béla Bollobás, Random Graphs, 2nd Edition, Cambridge University Press, 2001.
- Svante Janson, Tomasz Łuczak, and Andrzej Rucinski, Random Graphs, John Wiley & Sons, Inc., 2000.
- Noga Alon and Joel H. Spencer, The Probabilistic Method, 4th Edition, Wiley, 2016.

**Grading**: Homework 50% and paper reading 50% (presentations 20%, critical reviews 15%, in-class participation in discussing the paper 15%).**Homework assignment**: In the first half of the course, homework will be assigned every other week on Tuesday and due the Tuesday in two weeks. You are allowed, even encouraged, to work on the homework in small groups, but you must write up your own homework to hand in.**If you use materials other than the textbooks and lecture notes — this applies to having discussions with classmates or searching the Internet — please acknowledge the source clearly.****Paper reading seminar**: The second half of the course will be paper reading seminars. One research paper will be discussed per lecture. Students work in groups. One group is responsible in thoroughly understanding the paper and giving a 40 min summary in class. Remaining groups write critical reviews of the paper before the lecture. Each lecture, there will be a presentation around an hour (40 min technical summary with questions during the presentation), followed by a 20 min discussion about limitations, comparisons, potential improvements, future directions of the paper.- Paper list and assignment will be provided.
- Depending on registration numbers, each group presents 1 paper and writes critical reviews for the remaining papers (one review per group per paper). Guidance on how to structure a presentation and how to review a paper will be provided.
- The presenting group is required to meet the instructor during office hour (or by appointment) to discuss the planned presentation at least two weeks before the lecture.
- Both the presenting group and the reviewing groups should attend the paper reading seminars.

Raymond Spiteri : spiteri@cs.usask.ca

University of Saskatchewan

Basic background in programming and numerical analysis desirable

Registration for this course is not currently available.

Despite the extraordinary advances in computing technology, we continue to need ever greater computing power to address important fundamental scientific questions. Because individual compute processors have essentially reached their performance limits, the need for greater computing power can only be met through the use of parallel computers. This course is intended for students who are interested in learning how to take advantage of high-performance computing with the focus of writing parallel code for processor-intensive applications to be run on local clusters, the cloud, or shared infrastructure such as that provided by Compute Canada. Extensive use of pertinent and practical examples from scientific computing will be made throughout. Allowable programming languages include Julia, Matlab, Maple, sage, python, Fortran, or C/C++. Various paradigms of parallel computing will be covered via the OpenMP, MPI, and OpenCL libraries. By the end of the course, students will be expected to be able to correctly solve non-trivial problems involving parallel programming as well as appreciate the issues involved in solving such problems.

- D.L. Chopp, Introduction to High Performance Scientific Computing, Society for Industrial and Applied Mathematics, 2019.

James Feng : james.feng@ubc.ca

University of British Columbia

Registration for this course is not currently available.

This course will give students an overview of Non-Newtonian Fluid Dynamics, and discuss two approaches to building constitutive models for complex fluids: continuum modeling and kinetic- microstructural modeling. In addition, it will provide an introduction to multiphase complex fluids and to numerical models and algorithms for computing complex fluid flows.

- Introduction
- Background and motivation
- Review of required mathematics

- Continuum theories
- Oldroyd’s theory for viscoelastic fluids
- Ericksen-Leslie theory for liquid crystals
- Viscoplastic theories

- Kinetic-microstructural theories
- Dumbbell theory for polymer solutions
- Bead-rod-chain theories
- Doi-Edwards theory for entangled systems
- Doi theory for liquid crystalline materials

- Heterogeneous/multiphase systems
- Suspension theories (Einstein, Taylor, Batchelor, etc.)
- Kinetic theory for emulsions and drop dynamics
- Energetic formalism for interfacial dynamics
- Numerical methods for moving boundary problems

- Applications
- Polymer processing
- Sedimentation and Fluidization
- Bio-materials and processes: Pattern formation and self-assembly
- Others (gels, surfactants, colloids, Marangoni flows, etc.)

Brendan Pass : pass@ualberta.ca

University of Alberta

Registration for this course is not currently available.

This course is part of a long-term initiative to develop integrated teaching and
learning optimal transport infrastructure connecting the various PIMS sites. The
plan is to offer this course several times over the next few years; in each
iteration, ‘X’ will be chosen from the many disciplines in which optimal
transport places an important role, including data science/statistics,
computation, biology,finance, etc. **In Fall, 2020 we will take
‘X’=“economics”.**

This course is part of a long-term initiative to develop integrated teaching and
learning optimal transport infrastructure connecting the various PIMS sites. The
plan is to offer this course several times over the next few years; in each
iteration, ‘X’ will be chosen from the many disciplines in which optimal
transport places an important role, including data science/statistics,
computation, biology,finance, etc. **In Fall, 2020 we will take
‘X’=“economics”.**

This course has two main objectives: first, to introduce a wide range of students to the exciting and broadly applicable research area of optimal transport, and second, to explore more closely its applications in a particular field, which will vary from year to year (represented by ‘X’ in the title). Optimal transport is the general problem of moving one distribution of mass to another as efficiently as possible (for example, think of using a pile of dirt to fill a hole of the same volume, so as to minimize the average distance moved). This basic problem has a wealth of applications within mathematics (in PDE, geometry, functional analysis, probability…) as well as in other fields (comparing images in image processing, comparing and interpolating between data sets in statistics, matching partners in economics, aligning electrons in chemical physics…) and is currently an extremely active research area.

The first part of the course surveys the basic theory of optimal transport. Topics covered include: formulation of the problem, Kantorovich duality theory, existence and uniqueness theory, c-monotonicity and structure of solutions, discrete optimal transport. This is the core part of the course, which is important for all areas of application, and will be largely the same each year, although the presentation of some topics may vary slightly from year to year, to ensure compatibility with ‘X’.

The second part of the course develops applications in a particular area (corresponding to ‘X’ in the title), which rotates from year to year. In Fall, 2020, we will take ‘X’ = ”economics.” A surprisingly wide variety of problems in economic theory, econometrics and operations research are naturally formulated in terms of optimal transport. As a simple, illustrative example, consider an employer assigning a large number of heterogeneous employees to a diverse set of tasks. The employees have different skill sets which affect their proficiency at different jobs in different ways; matching a particular worker with a particular job results in a surplus which depends on the characteristics of both the worker and job. Assigning the workers to tasks to maximize the overall surplus is an optimal transport problem.

Many other examples arise in econometrics (where optimal transport can be used to optimize the estimation of incomplete information, or where multi-variate generalizations of quantiles, constructed using optimal transport, can be used to study dependence structures between distributions), matching problems (matching spouses on the marriage market, or employees and employers on the labour market, for instance) industrial organization (screening problems), contract theory (hedonic or discrete choice models), and financial engineering (estimating model free bounds on derivative prices and optimizing portfolios).

In both parts, we aim to keep the presentation accessible to non-experts, so that students with no prior background in either optimal transport or economics can follow the course.

Senior undergraduates, master’s and PhD students in quantitative disciplines, such as pure and applied mathematics, statistics, computer science, economics and engineering. The course potentially may also be attractive to those working in industry with a strong background in one of these areas.

This iteration of the course will be taught by Brendan Pass, and enhanced by guest lectures from experts in applications of optimal transport in economics and finance.

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 .