Courses: past

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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

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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.

Textbooks

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).

Course grade

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.

Registration for this course is not currently available.

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.

Learning Outcomes:

1. Recognize security and privacy threats, and enumerate possible defense mechanisms and their effectiveness in a distributed computer system
2. Identify mechanisms for controlling access to a computer system, and compare and contrast their effectiveness in practice.
3. Basic understanding of cryptographic tools and techniques and their applications in securing computer systems.
4. 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.
5. Identify network and software related attacks, and distinguish the role of different mechanisms in protecting the system.

Tentative Outline:

1. Introduction
2. Authentication
3. Access control
4. Malware
5. Introduction to cryptography
6. Modern cryptography - symmetric key
7. Modern cryptography - public-key
8. Web security
9. Introduction to blockchain
10. Network security

Registration for this course is not currently available.

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.

Registration for this course is not currently available.

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.

Registration for this course is not currently available.

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.

Registration for this course is not currently available.

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

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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.

Registration for this course is not currently available.

This course will cover measure theoretic probability with applications to statistics, including measure theory, measurable functions and random variables, expectation and integration, product spaces, independence, derivatives, conditional probability, characteristic functions, and limit theorems. The material is based on a course that has been taught to students in statistics, mathematics, engineering and science for many years. While rigorous proofs are discussed, the emphasis is on developing an understanding of how measure theory is used as a model for probability theory and how probability theory is used as a physical model. The statistics applications are used to motivate the development. Because it emphasizes foundations, it is paced differently than a common graduate probability theory course, e.g., covers more measure theory, and because it covers probability, it is different than a standard measure theory course.

The plan is to have recorded videos for the longer proofs. Instead of covering proofs in class, I will trim the class time by the length of the videos of proofs of theorems covered in a class and then answer questions about the proofs in the next class. This lets the students go through the proofs at a speed and level of detail that they like but maintains the total time allotted to lectures and still getting to ask questions.

Course work will be based on homework assignments given out every 1.5-2 weeks (so about 7-9 total). I have a resubmission policy in which I let students resubmit selected problems based on feedback received from the first submission.

Shedule

• Location: This class will meet remotely
• Course times:
  • Wed 4:30PM - 6:20 PM (Pacific)
  • Fri 4:30PM - 6:20 PM (Pacific)

See also the course outline at sfu, for more details.

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.

Reference texts:

• 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

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.

Reference texts

• [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.

Course Delivery

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.

Grading Scheme

• 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).

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.

Textbook

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

• https://bookstore.ams.org/ulect-70

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

Grading Scheme

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.

Schedule

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.

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

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

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.

https://research-groups.usask.ca/hydrology/training-education/intensive-courses/geog-825.php#CourseObjectives

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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.

Reference Texts

• 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

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.

Reference texts

• 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.

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.

Registration

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.

Reference texts

• Ergodic Theory by Karl Petersen

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.

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.

1. Formal Calculus
2. Axioms of vertex algebras and modules.
3. Representations of vertex algebras.
4. Local systems and the construction theorem.
5. Examples: vertex algebras constructed from
   1. Virosoro algebra;
   2. Affine Lie algebras;
   3. Lattices
6. Intertwining operators and tensor products of modules.
7. Cofiniteness conditions and convergence problems.
8. Vertex tensor categories of modules for rational vertex operator algebras.

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

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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

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