Courses: past

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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.

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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.

Registration for this course is not currently available.

In this course we develop elements of the theory of Gaussian and empirical processes that have proved useful for statistical inference in high-dimensional models, i.e. statistical models in which the number of parameters is much larger than the sample size. The course consists of three parts, with the first two parts laying the foundation for the third one: an introduction to modern techniques in Gaussian processes, a recap of empirical classical process theory emphasizing weak convergence on metric spaces, and lastly, a discussion of Gaussian approximation, high-dimensional CLTs, and the conditional multiplier bootstrap.

Course Contents:

• Part 1: Elements of Gaussian processes (concentration, comparison, anti-concentration, and super-concentration inequalities, Talagrand’s Generic chaining bounds).
• Part 2: Elements of empirical processes (convergence of laws on separable metric spaces, Glivenko-Cantelli and Donsker theorems under metric and bracketing entropy, applications to bootstrap)
• Part 3: A selection of theoretical problems in high-dimensional inference (including but not limited to Gaussian approximation, high-dimensional CLTs, and multiplier bootstrap when function classes are not Donsker).

Homework and Examinations

There will be regular homework assignments and an oral examination. The oral examination will work as follows: The lecture will be divided in roughly ten topics which will be shared with the students ahead of time. At the day of the examination the students will randomly draw two topics and give two 10-15 min presentations on their topics on the blackboard (no prepared notes allowed). Each presentation will conclude with ca. 5 minutes of follow-up questions. Textbooks for the first and second part:

• Dudley, R. M. (2014). “Uniform Central Limit Theorems”. CUP.
• Giné, E. and Nickl, R. (2016). “Mathematical Foundations of Infinite-Dimensional Statistical Models”. CUP.
• van der Vaart, A. and Wellner, J. (1996). “Weak Convergence and Empirical Processes”. Springer.

Typed lecture notes of all three parts will be provided.

Please note, the WDA registration deadline for this course at UBC will be Jan 6th, 2023.

Registration for this course is not currently available.

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

Introduction

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

Hartree-Fock theory

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

Interpretation of Hartree-Fock orbitals

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

Implementation of the Hartree-Fock equations

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

Post-Hartree-Fock methods

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

The density functional theory (DFT)

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

Semi-empirical approach to studying electronic structure

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

Semi-empirical approach to studying mechanical structure

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

Please note, the WDA registration deadline for this course at UBC will be Jan 6th, 2023.

Registration for this course is not currently available.

This is a first course in analytic number theory. In this course we will focus on the theory of the Riemann zeta function and of prime numbers. The goal of this course will include proving explicit bounds for the number $\pi(x)$ of primes which are less than a given number $x$. Building analytical tools to prove the prime number theorem (PNT) will be at the core of this course. We will explore and compare explicit formulas, whether they are using smooth weights or a truncated Perron formula, to relate averages over primes and $\pi(x)$ to sums over the zeros of zeta. Another originality of this course will be to explore each topic explicitly (essentially by computing all the hidden terms implied in the asymptotic estimates). With this respect, students will get an introduction to relevant programming languages and computational software. This will be closely connected to Analytic Number Theory 2 by Greg Martin (UBC), as the sequences of topics are coordinated between us; the intention is for students at all PIMS institutions to be able to take the second analytic number theory course as a continuation of the first one with maximum benefit. In addition, these two courses will provide excellent training for students who plan to attend the “Inclusive Paths in Explicit Number Theory” CRG summer school in 2023. All these events are part of the PIMS CRG “L-functions in Analytic Number Theory” (2022-2025).

Registration for this course is not currently available.

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

https://personal.math.ubc.ca/~lior/teaching/2223/535_W23/

Other Information

Registration for this course is not currently available.

The course covers foundational mathematical tools that are useful in analyzing high-dimensional single-cell datasets, and modelling developmental stochastic processes. We cover basic probability theory, statistical inference, convex optimization, Markov stochastic processes, and advanced topics in optimal transport.

See the course website for the syllabus and other details.

https://sites.google.com/view/math612d/home

Other Information

Registration for this course is not currently available.

The course provides learning opportunities on statistical software, R, with some focus on data management and wrangling, reproducibility, and visualization. On top of that, there are basic introductions to Machine Learning such as k-NN, Naive Bayes, regression methods, etc. The focus is on hands-on skills with R and applications to real data.

Registration for this course is not currently available.

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

https://personal.math.ubc.ca/~gerg/index.shtml?613-Winter2023

Other Information

Registration for this course is not currently available.

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

Registration for this course is not currently available.

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

Registration for this course is not currently available.

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

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

Registration for this course is not currently available.

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

Registration for this course is not currently available.

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

Registration for this course is not currently available.

Cell biology provides many interesting challenges across many spatial scales. Mathematical and computational modeling are tools that can help gain a better understanding of cellular phenomena. At the small scales, there are puzzling examples of patterns formed by proteins inside cells, and dynamic rearrangement of cellular components that enable cells to actively move. At higher scales, cells sense chemical gradients, exhibit active motility, and interact with other cells to form functioning tissues and organs. Mathematical and computational models allow us to explore many of the leading questions at each of these levels. How do patterns form spontaneously? What are the limits of cell sensing? How do cells polarize and migrate in a directed manner? How does a collection of cells self-organize into a structured tissue? In this graduate course, we will explore such questions in the context of deterministic models (ordinary and partial differential equations) as well as stochastic simulations that emphasize multiscale approaches.

The course is designed to be equally suitable for mathematics graduate students looking to learn advanced modeling methods, interesting applications, and topics for further analysis, and biologists who want to understand and critically assess models and carry out advanced multiscale simulations. All participants will learn multiscale simulations (using the open source software Morpheus) to visualize behaviour that emerges from intracellular signaling systems, cell migration, and cell-cell interactions. An emphasis will be on communication across disciplines, matching students from distinct disciplines for joint presentations and projects. Learning goals, expectations, assignments, and grading will take into account the student background.