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The course will provide an introduction to algebraic and probabilistic techniques in combinatorics and graph theory. The main topics included will be: Eigenvalues of graphs and their applications, probabilistic methods (first order, second order, Lovasz local lemma), Szemeredi regularity lemma. Recent discoveries like the proof of the Sensitivity conjecture, the use of eigenvalues for equiangular lines, etc., will be part of the course. '

Lecture Times

• Time: Tuesday 10:30-12:20 and Thursday 10:30-12:20
  • First day of classes: January 9
  • Reading break: February 20-25
  • Last day of classes: April 11

Delivery details

Note: This course is also offered through PIMS and WDA (Western Dean’s Agreement) as an online course. A Zoom link will be shared with registered students.

Course outline

Part I

• Introduction (warmup application of graph eigenvalues)
• Eigenvalue basics (including Perron-Frobenius Theorem)
• Eigenvalue interlacing (bounds on the maximum clique and chromatic number)
• Wilf’s Theorem, proof of sensitivity conjecture
• Graph Laplacians (Matrix-tree Theorem, Cheeger inequality)
• Random walks, effective resistance
• Spectral sparsifiers

Part II

• Random graphs, probabilistic method (including Lovasz local lemma)
• Quasirandom graphs
• Eigenvalues of random graphs (Wigner, Tao-Vu)
• Regularity Lemma
• Finding regular partitions
• Random covers and Ramanujan graphs

Grading scheme:

• Homework assignments 30%
• Midterm 30%
• Final exam 40%

Registration for this course is not currently available.

This will be a standard graduate number theory course. Topics will include:

• Number fields, rings of integers, ideals and unique factorization. Finiteness of the class group.
• Valuations and completions; local fields.
• Ramification theory, the different and discriminant.
• Geometry of numbers: Dirichlet’s Unit Theorem. and discriminant bounds.
• Other topics if time permits

The main pre-requisites are basic algebra (rings and fields, rings of polynomials, unique factorization in Euclidean\ndomains), basic number theory (modular arithemtic, factorization into primes) and Galois Theory, but no specific courses are required.

https://personal.math.ubc.ca/~lior/teaching/2324/538_W24/

Other Information

Registration for this course is not currently available.

Ergodic theory is the study of measure-preserving transformations. These occur naturally in an array of areas of mathematics (e.g. probability, number theory, geometry, information theory). The course will introduce measure-preserving transformations, give a range of basic examples, prove a number of general theorems (including the Poincare recurrence theorem, the Birkhoff ergodic theorem and sub-additive ergodic theorem). Entropy, one of the principal invariants of ergodic theory will be introduced. From there, the course will focus on applications to other areas.

Lecture Times

Lectures will take place every Monday and Thursday from from 8:30-9:50 (Pacific time).

Remote Access

Lectures will be shared via zoom. Students will need to have installed the zoom app on their desktop/laptop.

Registration for this course is not currently available.

Students taking this course will be exposed to the latest developments in the field of regulators algebraic cycles. This course was taught to advanced graduate students and experts alike at the University of Alberta in 2013. It was later taught at the University of Science and Technology in China, in 2014. A detailed syllabus can be extracted from the table of contents of the uploaded pdf file.

Lecture Times

Lectures will take place on Mondays, Wednesdays and Fridays from 13:00-13:50 (Mountain Time)

Remote Access

These lectures will take place via zoom. Students should have zoom installed on their laptop or other device.

Registration for this course is not currently available.

In this course we will study the theory of hyperbolic systems of conservation laws.

Hyperbolic systems arise in many areas of applied mathematics, including gas dynamics, thermodynamics, population dynamics, or traffic flow. In contrast to dissipative systems (like reaction-diffusion equations), solutions of hyperbolic systems with smooth initial data can generate “shocks” in finite time. The solution is no longer differentiable and weak solutions have to be studied.

We will develop the existence and uniqueness theory for solutions of conservation laws in spaces of functions of “bounded variation" (BV-spaces). At the beginning we will recall distributions and weak limits of measures. Then we study “broad” solutions (solutions which do not form shocks). After that we investigate discontinuous solutions in detail, we will derive the Rankine-Hugoniot conditions, the entropy conditions, the Lax-condition and we will discuss the vanishing viscosity method. We will classify strictly hyperbolic systems into genuinely nonlinear or linear degenerate systems. Then we use solutions to the Riemann problem to define a front tracking algorithm. This method is merely an\ analytical tool to obtain results on local and global existence and on uniqueness.

Lecture Times

Lectures will take place Monday, Wednesday and Friday from 13:00-13:50 (Mountain Time).

Remote Access

Lectures are online on zoom.

Registration for this course is not currently available.

The objective of this course is to provide an introduction to modern algebraic geometry in the language of schemes, with an emphasis on arithmetic schemes, integral models and applications to L-functions, and resolutions of singularities. The course also introduces the etale site on varieties, and sheaves on this site.

Registration for this course is not currently available.

We will cover the advances in decoupling theory beginning with Bourgain and Demeter’s 2014 proof of the $l^2$ decoupling conjecture. We will also cover Fourier restriction theory, and in particular the recent use of tools such as the polynomial method.

Lecture Times

Lectures will take place every Monday, Wednesday and Friday 11:00am-12:00pm (Pacific Time)

Remote Access

Lectures will be shared via zoom. Students will need to have installed the zoom app on their desktop/laptop.

Registration for this course is not currently available.

A course on algorithms for algebraic computation and tools for computing with multivariate polynomials, polynomial ideals, exact linear algebra, and algebraic numbers. Tools include the Fast Fourier Transform, Groebner bases, and the Schwartz-Zippel Lemma. We will use Maple as a calculator and as a programming language to implement algorithms. Instruction in Maple usage and programming will be provided.

Lecture Times

Lectures are on Tuesdays and Thursdays 9:30am to 11:20am (Pacific Time) .

Remote Access

Lectures will be shared via zoom. Students will need to have installed the zoom app on their desktop/laptop.

Registration for this course is not currently available.

The last few years have seen amazing advances in interactive proof assistants and their use in mathematics. For example, Lean’s mathematics library mathlib now has over one million lines of code and is still growing in a significant rate. Furthermore, recent highly celebrated successes in the subject, such as the completion of the sphere eversion project and the liquid tensor experiment, suggest that we are approaching a paradigm shift in mathematics, where cutting edge research can be formally verified in a relatively short amount of time. This course will serve as an introduction to the formalization of mathematics, using the Lean4 interactive proof assistant and its mathematics library Mathlib4. See the attached syllabus for an outline of the topics we expect to cover.

Lecture Times

Lectures are Tuesdays and Thursdays, 11am to 12:20pm, Mountain time. All lectures will take place electronically using zoom (or similar software).

Registration for this course is not currently available.

Matrices are ubiquitous in many aspects of mathematics. They show up, for instance, when considering the local asymptotic stability of equilibria of systems of ordinary differential equations, the long term behaviour of Markov chains, the study of graphs and the discretization of reaction-diffusion equations.

Objectives of the course:

1. explore the role of matrices in several fields of mathematics;
2. study properties of these matrices;
3. develop a toolbox to study some matrix properties computationally.

https://julien-arino.github.io/math-4370-7370/

Other Information

Registration for this course is not currently available.

This course is an advanced graduate course in number theory, designed to follow Analytic Number Theory I taught by Prof. Habiba Kadiri (University of Lethbridge) in Fall 2022 and Analytic Number Theory II taught by Prof. Greg Martin (UBC) in Winter 2023. All three of these courses are part of the current PIMS CRG “L-functions in Analytic Number Theory”. In this course, we will establish estimates for moments of L-functions and explore the tools needed to study them including approximate functional equations, zero density estimates, zero free regions, mean value estimates for Dirichlet polynomials, large sieve inequalities, Poisson and Voronoi summation formulae, shifted convolution sums, holomorphic modular forms and associated L-functions, trace formulae, and the spectral theory of automorphic forms.

Registration for this course is not currently available.

The space of probability distributions with finite second moments can be made into a natural metric space, called the Wasserstein space, whose metric is defined by using the optimal transportation between probability distributions. On this metric space one can draw curves that represent motion along the steepest descent (AKA gradient flow) of functionals of probability measures. This is a very fruitful way to view many important families of probability measures that arise from PDEs and stochastic processes. For example, using this geometric framework, one may derive functional inequalities and infer rates of convergence of Markov processes. A striking example is that of the heat equation, whose solution can be interpreted as the family of marginal distributions of Brownian motion. In the Wasserstein space, this curve of probability laws is the gradient flow of the Shannon entropy.

We will discuss the theory of Wasserstein gradient flows, including the formal Riemannian calculus due to Otto, and the modern techniques of metric measures spaces. Apart from the classical examples, we will also discuss many modern variations such as Wasserstein mirror gradient flows that come up in statistical applications. A fruitful interaction between probability, geometry, and PDE theory will be developed simultaneously. This is a continuation of the sequence of OT+X courses under the Kantorovich Initiative.

Delivery Details

The course is being offered simultaneously at Korea Advanced Institute of Science and Technology (KAIST) and the PIMS network, including the University of Washington, Seattle. Due to different time schedules for individual campuses and the time zones, the course has an unusual structure. Please read the details below carefully.

Lectures

• Lecture hours 6:30pm - 8pm Pacific on Tuesdays and Thursdays. Thus we will have two classes per week, each for 90 mins.

• Lectures will be taught over Zoom and videos and notes will be made available to everyone afterwards.

• A Slack channel will be used to communicate with students and distribute teaching material.

  There will be no exams in this course. Occasional homework problems will be provided.

Registration

Students at Canadian PIMS Member Universities may register through the Western Deans Agreement. Students at UW may register directly for the UW course. Course codes and other registration details for students in either of these cases are listed in the registration section above. Students at other institutions should contact one of the instructors to attend the course as a non-registered student.

Course Structure

Part I

Part I is a recap of the basics of Monge-Kantorovich optimal transport theory. You do NOT need to take this part if you are already familiar with the basics. This will be covered between AUG 28 and SEP 26. Topics covered during this period are:

• linear programming
• Monge-Kantorovich problem
• Kantorovich duality
• Monge-Ampère PDE
• Brenier’s Theorem
• Wasserstein-2 metric

Part II

This will start on SEP 27 and continue through DEC 7. A rough syllabus of topics covered are presented below in the order they will be covered. There might be some changes depending on our progress.

core topics

• Wasserstein space
  • metric property
  • geodesics, displacement interpolation, generalized geodesic
  • Geodesic convexity
• AC curves in the Waserstein space and the continuity equation
• Benamou-Brenier and dynamic OT
• Otto calculus
  • tangent spaces to the Wasserstein space
  • Riemannian gradient
• Diffusions as gradient flows via Otto calculus
  • Brownian motion
  • Langevin diffusions

Modern research topics that will be surveyed

• log-Sobolev and other functional inequalities
• Convergence of finite dimensional gradient flow of particles to the McKean-Vlasov diffusions and gradient flow in the Wasserstein space.
• The implicit Euler or JKO scheme
• Entropy regularization and gradient flows
  • Schrödinger bridges
  • Large deviation and gradient flows
• Mirror gradient flows, parabolic Monge-Ampere and the Sinkhorn algorithm

Registration for this course is not currently available.

Spectral methods are numerical methods for solving PDEs. When the solution is analytic, the convergence rate is exponential. The first part of this course gives an introduction to spectral methods. The emphasis is on the analysis of these methods including truncation and interpolation error estimates, and condition number estimates. The second part of the course focuses on fast algorithms for orthogonal polynomials. These algorithms leverage data-sparsities that are present in many of the problems when solved by orthogonal polynomial expansions.

Part I: Introduction to Spectral Methods (Shaun Lui)

1. Trigonometric and orthogonal polynomials (truncation and interpolation error estimates, aliasing, Lebesgue constants)
2. Fourier spectral (FFT), spectral Galerkin and spectral tau methods
3. Spectral collocation for Poisson equation with Dirichlet BCs (convergence and condition number estimates)
4. Neumann problems and fourth-order PDEs
5. Other topics (Ultraspherical spectral methods, time-dependent PDEs)

Part II: Fast Algorithms for Orthogonal Polynomials (Mikael Slevinsky)

1. Synthesis and analysis
2. Chebyshev polynomials and the fast discrete sine and cosine transforms
3. Modification algorithms for orthogonal polynomials (d) Fast approximation of the connection coefficients
4. Multivariate orthogonal polynomials via Koornwinder’s construction (f) Time evolution with exponential integrators

Tentative Time

• Tues, Thurs 3 - 4:15 (CDT)

Location

• MH416 and Zoom

Textbook

• Course notes will be provided.

References:

1. J. Shen T. Tao and L.-L. Wang, Spectral methods. Algorithms, analysis and applications, Springer, 2011.
2. L.N. Trefethen, Spectral Methods in Matlab, SIAM, 2000.
3. L.N. Trefethen, Approimation Theory and Approximation Practice (Extended Ed.), SIAM, 2020.
4. S. Olver, R. M. Slevinsky, and A. Townsend, Fast algorithms using orthogonal polynomials, Acta Numerica, 29: 573–699, 2020.

Grading Scheme

There are 4 Homeworks (each contributing 17% toward the grade) and a project (32%).

Academic Integrity

The Department of Mathematics, the Faculty of Science and the University of Manitoba regard acts of academic dishonesty in quizzes, tests, examinations or assignments as serious offenses and may assess a variety of penalties depending on the nature of the offense. Acts of academic dishonesty include bringing unauthorized materials into a test or exam, copying from another student, plagiarism and examination personation. Students are advised to read section 7 (Academic Integrity) and section 4.2.8 (Examinations: Personations) in the “General Academic Regulations and Requirement” of the current Undergraduate Calendar. Note, in particular that cell phones and pagers are explicitly listed as unauthorized materials, and hence may not be present during tests or examinations. Penalties for violation include being assigned a grade of zero on a test or assignment, being assigned a grade of “F” in a course, compulsory withdrawal from a course or program, suspension from a course/program/faculty or even expulsion from the University. For specific details about the nature of penalties that may be assessed upon conviction of an act of academic dishonesty, students are referred to University Policy 1202 (Student Discipline Bylaw) and to the Department of Mathematics policy concerning minimum penalties for acts of academic dishonesty. The Student Discipline Bylaw is printed in its entirety in the Student Guide, and is also available on-line or through the Office of the University Secretary. Minimum penalties assessed by the Department of Mathematics for acts of academic dishonesty are available on the Department of Mathematics web-page. All Faculty members (and their teaching assistants) have been instructed to be vigilant and report incidents of academic dishonesty to the Head of the Department.

Registration for this course is not currently available.

Based on a mathematical and statistical theory foundation, the course introduces statistical methods for supervised and unsupervised learning, focusing on hands-on skills with statistical software, R, and applications to real data. The course covers resampling methods, regression and classification, tree-based methods, dimension reduction and clustering. It embeds R training throughout the entire class.

Course Schedule

• Lecture Section: Wednesday 6:00pm-9:00pm CST (online via zoom)
• Lab Section: Thursday 3:30pm-4:50pm CST (online via zoom for outside USask students, or onsite in a lab room to be announced).
• Office Hours: Friday 5:00pm-6:00pm CST, and by appointment (online via zoom, or onsite by appointment)

Other Information

Please see the syllabus document for more details, including required reading, learning objective and evaluation components.