Courses: 2021-2022

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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

Other Information

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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.

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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

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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.

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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.

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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.

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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.

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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

Other Information

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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.

Reference texts

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

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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.

1. Introduction
   • Background and motivation
   • Review of required mathematics
2. Continuum theories
   • Oldroyd’s theory for viscoelastic fluids
   • Ericksen-Leslie theory for liquid crystals
   • Viscoplastic theories
3. Kinetic-microstructural theories
   • Dumbbell theory for polymer solutions
   • Bead-rod-chain theories
   • Doi-Edwards theory for entangled systems
   • Doi theory for liquid crystalline materials
4. 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
5. Applications
   • Polymer processing
   • Sedimentation and Fluidization
   • Bio-materials and processes: Pattern formation and self-assembly
   • Others (gels, surfactants, colloids, Marangoni flows, etc.)

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.

Intended audience

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.

Instructor

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.