This course is available for registration under the Western Dean's Agreement but registrations must be approved by the course instructor. Please contact the instructor (using the email link to the left) including details of how you meet the course prerequisites. Next, you must complete the Western Deans' Agreement form , with the following course details:

Completed forms should be returned to your graduate advisor who will sign it and take the required steps. For students at PIMS sites, please see this list to find your graduate advisor, for other sites, contacts can be found on the Western Deans' Agreement contact page .

The Western Deans' Agreement provides an automatic tuition fee waiver for visiting students. Graduate students paying normal required tuition fees at their home institution will not pay tuition fees to the host institution. However, students will typically be be required to pay other ancillary fees to the host institution, or explicitly request exemptions (e.g. Insurance or travel fees). Details vary by university, so please contact the graduate student advisor at your institution for help completing the form. Links to fee information and contact information for PIMS member universities is available below in the WDA section.

Students at universities not covered by the WDA but which are part of the Canadian Association for Graduate Studies (CAGS) may still be eligible to register for some courses under the terms of the Canadian University Graduate Transfer Agreement (CUGTA). Details of this program vary by university and registration is also typically subject to ancillary fees. Both your local and the hosting university must participate in the agreement (e.g. UBC does not participate in CAGS). Please contact the relevant graduate student advisors for more information.

In the past 25 years or so, the subject of “Arithmetic Statistics”, beginning with the work of Bhargava’s success in enumerating rings and fields of low degree and rank, and Bhargava and Shankar’s proof of the boundedness of algebraic rank of elliptic curves, is an enormously exciting subject. We will give an introduction to the subject centred on the work of Bhargava and his coworkers.

Remote Access

Lectures will be conducted via zoom, using electronic slides. Slides, assignments and exams will be distributed electronically. Lectures will be recorded and made available to registered students.

This course is available for registration under the Western Dean's Agreement but registrations must be approved by the course instructor. Please contact the instructor (using the email link to the left) including details of how you meet the course prerequisites. Next, you must complete the Western Deans' Agreement form , with the following course details:

Completed forms should be returned to your graduate advisor who will sign it and take the required steps. For students at PIMS sites, please see this list to find your graduate advisor, for other sites, contacts can be found on the Western Deans' Agreement contact page .

The Western Deans' Agreement provides an automatic tuition fee waiver for visiting students. Graduate students paying normal required tuition fees at their home institution will not pay tuition fees to the host institution. However, students will typically be be required to pay other ancillary fees to the host institution, or explicitly request exemptions (e.g. Insurance or travel fees). Details vary by university, so please contact the graduate student advisor at your institution for help completing the form. Links to fee information and contact information for PIMS member universities is available below in the WDA section.

Students at universities not covered by the WDA but which are part of the Canadian Association for Graduate Studies (CAGS) may still be eligible to register for some courses under the terms of the Canadian University Graduate Transfer Agreement (CUGTA). Details of this program vary by university and registration is also typically subject to ancillary fees. Both your local and the hosting university must participate in the agreement (e.g. UBC does not participate in CAGS). Please contact the relevant graduate student advisors for more information.

The purpose of this graduate course is to equip graduate students with cutting-edge techniques in data-driven mathematical and computational modelling, analysis and simulations of semi-linear parabolic partial differential equations (PDEs) of reaction-diffusion type. It will cover diverse areas in data-driven modelling using PDEs in biology. I will cover approaches on formulating models from data using first principles, mathematical analysis of reaction-diffusion systems such as linear stability analysis, basic concepts on bifurcation analysis and numerical bifurcation analysis. The second part will focus on numerical methods for PDEs including finite difference methods, and finite elements. This part will also deal with time-stepping schemes and nonlinear solvers for nonlinear PDEs. If time allows, we will look at applications of reaction diffusion theory to cell motility and pattern formation. To support theoretical modelling and numerical analysis, numerical algorithms will be developed and implemented in MATLAB as well as in open finite element source software packages such as FeNiCs, deal.ii and others. Students will be allowed to use packages of their choice as appropriate. Expertise and skills sets to be acquired through this course

1. Acquire data-driven modelling skills and techniques in PDEs and their applications to biology
2. Acquire techniques and knowledge in mathematical analysis of reaction-diffusion systems
3. Acquire expertise and skills in bifurcation analysis, numerical bifurcation, and sensitivity analysis
4. Acquire numerical analysis techniques and skills to compute approximate numerical solutions
5. Acquire expertise and knowledge in finite difference methods for semi-linear parabolic PDEs
6. Acquire expertise and knowledge in finite element methods for semi-linear parabolic PDEs
7. Gain some knowledge in bulk-surface PDEs, and their analysis (might be covered if time allows) Key

1. The art of mathematical modelling
   1. An introduction to the art of mathematical modelling
   2. The physical origins of partial differential equations and their applications
      1. Derivation of the heat equation: Heat Transfer (A taster of what to come)
      2. General classification of PDEs
   3. Mathematical Notations and Definitions
   4. Physical laws
   5. Exercises
2. Reaction-diffusion systems on stationary domains: modelling, analysis and simulations
   1. Introduction
   2. Derivation of reaction-diffusion systems on stationary domains
   3. Classical nonlinear reaction kinetics
      1. Activator-depleted reaction kinetics
      2. Gierer-Meinhard reaction kinetics
      3. Thomas reaction kinetics
   4. Non-dimensionalisation – unit free
      1. Reaction-diffusion system with activator-depleted reaction kinetics
      2. Reaction-diffusion system with Gierer–Meinhardt reaction kinetics
      3. Reaction-diffusion system with Thomas reaction kinetics
3. Stability analysis of reaction-diffusion systems on stationary domains and the generation of parameter spaces
   1. Introduction
      1. Preliminaries
   2. Linear stability analysis of reaction-diffusion systems on stationary domains
      1. Linear stability in the absence of spatial variations
      2. Linear stability in the presence of spatial variations
   3. Eigenfunctions in one dimension and on special domains in two dimensions
      1. Eigenfunctions in one dimension
      2. Eigenfunctions of a rectangle
4. Numerical Methods for Reaction-Diffusion Systems on Stationary Domains
   1. Finite Difference Methods for Reaction-Diffusion Systems on Stationary Domains
      1. Finite Difference Stencils in 2- and 3-Dimensional Domains
      2. Forward Euler Method
      3. Backward Euler Method
      4. Crank-Nicholson Method
      5. Fractional-Step 𝜃 method
      6. Implicit and explicit (IMEX) time-stepping schemes for reaction-diffusion systems on stationary domains
   2. Finite Element Methods for Reaction-Diffusion Systems on Stationary Domains
      1. Sobolev Spaces
      2. Weak Variational Form
      3. Space discretisation
      4. Mesh Generation
      5. Time discretisation
   3. Fully implicit time-stepping schemes and non-linear solvers for systems of reaction-diffusion equations
   4. Algorithm development and implementation using finite element open source software pages
      1. Introduction to PDE computing with FeNiCs
      2. Algorithm development and testing in FeNiCs
5. Introduction to reaction-diffusion systems on evolving domains and surfaces
   1. Reaction-diffusion systems on deforming domains and surfaces . . . . . .
   2. Finite element methods for reaction-diffusion systems on deforming domains and surfaces
6. Summary of the course taught.

Remote Access

We will use zoom for each lecture. Course notes will be distributed in advance and lecture notes will be distributed after each lecture.

This course is available for registration under the Western Dean's Agreement but registrations must be approved by the course instructor. Please contact the instructor (using the email link to the left) including details of how you meet the course prerequisites. Next, you must complete the Western Deans' Agreement form , with the following course details:

Completed forms should be returned to your graduate advisor who will sign it and take the required steps. For students at PIMS sites, please see this list to find your graduate advisor, for other sites, contacts can be found on the Western Deans' Agreement contact page .

The Western Deans' Agreement provides an automatic tuition fee waiver for visiting students. Graduate students paying normal required tuition fees at their home institution will not pay tuition fees to the host institution. However, students will typically be be required to pay other ancillary fees to the host institution, or explicitly request exemptions (e.g. Insurance or travel fees). Details vary by university, so please contact the graduate student advisor at your institution for help completing the form. Links to fee information and contact information for PIMS member universities is available below in the WDA section.

Students at universities not covered by the WDA but which are part of the Canadian Association for Graduate Studies (CAGS) may still be eligible to register for some courses under the terms of the Canadian University Graduate Transfer Agreement (CUGTA). Details of this program vary by university and registration is also typically subject to ancillary fees. Both your local and the hosting university must participate in the agreement (e.g. UBC does not participate in CAGS). Please contact the relevant graduate student advisors for more information.

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.

Lecture Schedule

Lectures will take place Monday, Wednesday and Friday 12:30 - 1:20 PM Regina time.

Remote Access

The class will be in a hybrid format hosted in a classroom equipped with hyflex technology.

Lecture notes will be projected on the screen, shared simultaneously on Zoom, and posted afterwards on the course website.

This course is available for registration under the Western Dean's Agreement but registrations must be approved by the course instructor. Please contact the instructor (using the email link to the left) including details of how you meet the course prerequisites. Next, you must complete the Western Deans' Agreement form , with the following course details:

Completed forms should be returned to your graduate advisor who will sign it and take the required steps. For students at PIMS sites, please see this list to find your graduate advisor, for other sites, contacts can be found on the Western Deans' Agreement contact page .

The Western Deans' Agreement provides an automatic tuition fee waiver for visiting students. Graduate students paying normal required tuition fees at their home institution will not pay tuition fees to the host institution. However, students will typically be be required to pay other ancillary fees to the host institution, or explicitly request exemptions (e.g. Insurance or travel fees). Details vary by university, so please contact the graduate student advisor at your institution for help completing the form. Links to fee information and contact information for PIMS member universities is available below in the WDA section.

Students at universities not covered by the WDA but which are part of the Canadian Association for Graduate Studies (CAGS) may still be eligible to register for some courses under the terms of the Canadian University Graduate Transfer Agreement (CUGTA). Details of this program vary by university and registration is also typically subject to ancillary fees. Both your local and the hosting university must participate in the agreement (e.g. UBC does not participate in CAGS). Please contact the relevant graduate student advisors for more information.

This course provides an introduction into analytic number theoretic methods with applications to arithmetic geometry. We will study Dirichlet series with applications to distributions of prime numbers and as examples of L-series. We will also look at modular forms and their applications to the arithmetic of elliptic curves and their moduli spaces. We will also consider results in diophantine approximation, such as lower bounds on linear combinations of logs of algebraic numbers, with as application Siegel’s theorem on finiteness of integral points on elliptic curves.

Remote Access

The class will be held in a room equipped with controllable cameras. The instructor will write on whiteboards in this room and the camera controls used to provide clear views of the boards.

This course is available for registration under the Western Dean's Agreement but registrations must be approved by the course instructor. Please contact the instructor (using the email link to the left) including details of how you meet the course prerequisites. Next, you must complete the Western Deans' Agreement form , with the following course details:

Completed forms should be returned to your graduate advisor who will sign it and take the required steps. For students at PIMS sites, please see this list to find your graduate advisor, for other sites, contacts can be found on the Western Deans' Agreement contact page .

The Western Deans' Agreement provides an automatic tuition fee waiver for visiting students. Graduate students paying normal required tuition fees at their home institution will not pay tuition fees to the host institution. However, students will typically be be required to pay other ancillary fees to the host institution, or explicitly request exemptions (e.g. Insurance or travel fees). Details vary by university, so please contact the graduate student advisor at your institution for help completing the form. Links to fee information and contact information for PIMS member universities is available below in the WDA section.

Students at universities not covered by the WDA but which are part of the Canadian Association for Graduate Studies (CAGS) may still be eligible to register for some courses under the terms of the Canadian University Graduate Transfer Agreement (CUGTA). Details of this program vary by university and registration is also typically subject to ancillary fees. Both your local and the hosting university must participate in the agreement (e.g. UBC does not participate in CAGS). Please contact the relevant graduate student advisors for more information.

The course will be an introduction to the behaviour of fluids (liquids and gases) from an applied math perspective, starting with an introduction to the Navier-Stokes equations and other PDEs used to model fluids. The emphasis will be on physically relevant properties of solutions that can be deduced mathematically. The course will have more mathematics than a typical physics or engineering fluids course, including basic functional analysis and variational methods, and it will have more physics than a pure PDE analysis course. Through detailed study of several fundamental model systems, we will see PDE examples of topics that may be more familiar in the context of ODE dynamical systems, such as linear stability, nonlinear stability, bifurcations and chaos. Undergraduate knowledge of PDEs and real analysis are assumed.

There will be no exams, only assignments access and submitted online via Crowdmark.

Class Schedule

• M/Th 11:30 AM - 12:50 PM PST

Remote Access

The lecturer will use zoom for each lecture. Typed lecture notes will be distributed electronically.

This course is available for registration under the Western Dean's Agreement but registrations must be approved by the course instructor. Please contact the instructor (using the email link to the left) including details of how you meet the course prerequisites. Next, you must complete the Western Deans' Agreement form , with the following course details:

Completed forms should be returned to your graduate advisor who will sign it and take the required steps. For students at PIMS sites, please see this list to find your graduate advisor, for other sites, contacts can be found on the Western Deans' Agreement contact page .

The Western Deans' Agreement provides an automatic tuition fee waiver for visiting students. Graduate students paying normal required tuition fees at their home institution will not pay tuition fees to the host institution. However, students will typically be be required to pay other ancillary fees to the host institution, or explicitly request exemptions (e.g. Insurance or travel fees). Details vary by university, so please contact the graduate student advisor at your institution for help completing the form. Links to fee information and contact information for PIMS member universities is available below in the WDA section.

Students at universities not covered by the WDA but which are part of the Canadian Association for Graduate Studies (CAGS) may still be eligible to register for some courses under the terms of the Canadian University Graduate Transfer Agreement (CUGTA). Details of this program vary by university and registration is also typically subject to ancillary fees. Both your local and the hosting university must participate in the agreement (e.g. UBC does not participate in CAGS). Please contact the relevant graduate student advisors for more information.

This course presents classical mechanics to a mixed audience of mathematics and physics undergraduate and graduate students. It is complementary to regular phsyics courses in that while the physics background will be developed the emphasis will be on the resulting mathematical analysis. Physics topics may include Newtonian mechanics and Galilean symmetry, Lagrangian mechanics, conservation laws and Noether’s Theorem, rigid body motion, Hamiltonian mechanics. Mathematical topics may include existence and uniqueness of solutions to ODE, calculus of variations, convexity and Legendre transformations, manifolds, tangent and cotangent vectors, rotations and the orthogonal group.

Topics to be discussed may include:

1. Kinematics; coordinates 2. Newtonian mechanics; existence and uniqueness of solutions to ODE. Examples: simple harmonic motion; central potentials; 3. Symmetry and the Galilean group; change of coordinates 4. Lagrangian mechanics; calculus of variations, Lagrange multipliers. 5. Constraints and manifolds; the tangent bundle. 6. Symmetry, rigid body motion and angular momentum; Noether’s Theorem, the orthogonal group, and its Lie algebra
2. Hamiltonian mechanics; convexity and Legendre transformation, the cotangent bundle.
3. Conserved quantities and the Poisson bracket; symplectic structure 9. Liouville’s Theorem and Poincaré recurrence

Remote Access

Lectures will be held in-person on the UBC campus and on Zoom. Lectures will be recorded and the videos posted to an unlisted but openly accessible YouTube playlist. There will be Zoom office hours and a Piazza discussion board.

This course is available for registration under the Western Dean's Agreement but registrations must be approved by the course instructor. Please contact the instructor (using the email link to the left) including details of how you meet the course prerequisites. Next, you must complete the Western Deans' Agreement form , with the following course details:

Completed forms should be returned to your graduate advisor who will sign it and take the required steps. For students at PIMS sites, please see this list to find your graduate advisor, for other sites, contacts can be found on the Western Deans' Agreement contact page .

The Western Deans' Agreement provides an automatic tuition fee waiver for visiting students. Graduate students paying normal required tuition fees at their home institution will not pay tuition fees to the host institution. However, students will typically be be required to pay other ancillary fees to the host institution, or explicitly request exemptions (e.g. Insurance or travel fees). Details vary by university, so please contact the graduate student advisor at your institution for help completing the form. Links to fee information and contact information for PIMS member universities is available below in the WDA section.

Students at universities not covered by the WDA but which are part of the Canadian Association for Graduate Studies (CAGS) may still be eligible to register for some courses under the terms of the Canadian University Graduate Transfer Agreement (CUGTA). Details of this program vary by university and registration is also typically subject to ancillary fees. Both your local and the hosting university must participate in the agreement (e.g. UBC does not participate in CAGS). Please contact the relevant graduate student advisors for more information.

In this course we are learning to build and analyse nonlinear differential equation models. The focus of the course will be models of ecological systems, but the techniques learned apply broadly across application areas. We learn a wide variety of analytic, graphic, and simplification techniques which elucidate the behaviour of these mathematical models, whether or not a closed-form solution is avalable. By the end of the class, the students will be able to competently read and follow a research paper presenting and analysing a differential equation model from a wide variety of application areas.

Class Schedule

The class will meet on Monday, Wednesday and Friday during term from 12pm-1pm (Pacific Time).

Textbook

• Mathematical Biology, 3rd Edition, volumes I and II, by James Murray

Remote Access

Lectures will be livestreamed via zoom. The lecturer will be writing on a whiteboard.

This course is available for registration under the Western Dean's Agreement but registrations must be approved by the course instructor. Please contact the instructor (using the email link to the left) including details of how you meet the course prerequisites. Next, you must complete the Western Deans' Agreement form , with the following course details:

Completed forms should be returned to your graduate advisor who will sign it and take the required steps. For students at PIMS sites, please see this list to find your graduate advisor, for other sites, contacts can be found on the Western Deans' Agreement contact page .

The Western Deans' Agreement provides an automatic tuition fee waiver for visiting students. Graduate students paying normal required tuition fees at their home institution will not pay tuition fees to the host institution. However, students will typically be be required to pay other ancillary fees to the host institution, or explicitly request exemptions (e.g. Insurance or travel fees). Details vary by university, so please contact the graduate student advisor at your institution for help completing the form. Links to fee information and contact information for PIMS member universities is available below in the WDA section.

Students at universities not covered by the WDA but which are part of the Canadian Association for Graduate Studies (CAGS) may still be eligible to register for some courses under the terms of the Canadian University Graduate Transfer Agreement (CUGTA). Details of this program vary by university and registration is also typically subject to ancillary fees. Both your local and the hosting university must participate in the agreement (e.g. UBC does not participate in CAGS). Please contact the relevant graduate student advisors for more information.

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 convergence 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.

Class Schedule

This class will meet Mondays and Wednesdays from 11am-12:15pm (CDT)

Remote Access

Lectures will be delivered via Zoom using iPad with GoodNotes.

This course is available for registration under the Western Dean's Agreement but registrations must be approved by the course instructor. Please contact the instructor (using the email link to the left) including details of how you meet the course prerequisites. Next, you must complete the Western Deans' Agreement form , with the following course details:

Completed forms should be returned to your graduate advisor who will sign it and take the required steps. For students at PIMS sites, please see this list to find your graduate advisor, for other sites, contacts can be found on the Western Deans' Agreement contact page .

The Western Deans' Agreement provides an automatic tuition fee waiver for visiting students. Graduate students paying normal required tuition fees at their home institution will not pay tuition fees to the host institution. However, students will typically be be required to pay other ancillary fees to the host institution, or explicitly request exemptions (e.g. Insurance or travel fees). Details vary by university, so please contact the graduate student advisor at your institution for help completing the form. Links to fee information and contact information for PIMS member universities is available below in the WDA section.

Students at universities not covered by the WDA but which are part of the Canadian Association for Graduate Studies (CAGS) may still be eligible to register for some courses under the terms of the Canadian University Graduate Transfer Agreement (CUGTA). Details of this program vary by university and registration is also typically subject to ancillary fees. Both your local and the hosting university must participate in the agreement (e.g. UBC does not participate in CAGS). Please contact the relevant graduate student advisors for more information.

This is a one semester three credit hour course. 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.

Remote Access

We will use zoom for each lecture. The eclass website will be used to post lecture slides, homework collections, monitor midterm and final examinations

This course is available for registration under the Western Dean's Agreement but registrations must be approved by the course instructor. Please contact the instructor (using the email link to the left) including details of how you meet the course prerequisites. Next, you must complete the Western Deans' Agreement form , with the following course details:

Completed forms should be returned to your graduate advisor who will sign it and take the required steps. For students at PIMS sites, please see this list to find your graduate advisor, for other sites, contacts can be found on the Western Deans' Agreement contact page .

The Western Deans' Agreement provides an automatic tuition fee waiver for visiting students. Graduate students paying normal required tuition fees at their home institution will not pay tuition fees to the host institution. However, students will typically be be required to pay other ancillary fees to the host institution, or explicitly request exemptions (e.g. Insurance or travel fees). Details vary by university, so please contact the graduate student advisor at your institution for help completing the form. Links to fee information and contact information for PIMS member universities is available below in the WDA section.

Students at universities not covered by the WDA but which are part of the Canadian Association for Graduate Studies (CAGS) may still be eligible to register for some courses under the terms of the Canadian University Graduate Transfer Agreement (CUGTA). Details of this program vary by university and registration is also typically subject to ancillary fees. Both your local and the hosting university must participate in the agreement (e.g. UBC does not participate in CAGS). Please contact the relevant graduate student advisors for more information.

Translation surfaces and their moduli spaces have been the objects of extensive recent study and interest, with connections to widely varied fields including (but not limited to) geometry and topology; Teichmüller theory; low-dimensional dynamical systems; homogeneous dynamics and Diophantine approximation; and algebraic and complex geometry. This course will serve as an introduction to some of the big ideas in the field, centered on the ergodic properties of translation flows and counting problems for saddle connections, and associated renormalization techniques, without attempting to reach the full state of the art (an aim that is in any case impossible given the speed at which the field is evolving).

We will start by introducing the important motivating example of the flat torus, exploring its geometry, and its associated dynamical and counting problems. The linear flow on the torus and its associated first return map, a rotation of a circle, are amongst the first dynamical systems ever studied. The counting of closed orbits is intricately tied to number theory. We discuss, as motivation, the moduli space of translation surfaces on a torus, a bundle over the well-known modular curve and the action of $GL^+(2,\mathbb R)$ on this space of translation surfaces. Translation surfaces are higher-genus generalizations of flat tori. We will define translation surfaces from three perspectives (Euclidean geometry, complex analysis, and geometric structures), and show how some translation surfaces arise from unfolding billiards in rational polygons. We will give a short introduction to Teichmüller theory and its relation to the study of translation surfaces, and discuss the natural dynamical systems associated to translation surfaces, namely, linear flows and their first return maps, interval exchange transformations. We will explore their ergodicity and mixing properties, and will study an important example of a translation surface flow for which every orbit is dense but not every orbit is equidistributed with respect to Lebesgue measure, a phenomenon that does not occur in the case of linear flows on the torus. We will show how information about the recurrence properties of an orbit of a translation surface under the positive diagonal subgroup of $SL(2, \mathbb R)$ (the Teichmüller geodesic flow) can be used to get information about the ergodic properties of the associated linear flow on an individual translation surface. As another example of the strength of renormalization ideas, we will show how the ergodic properties of the $SL(2, \mathbb R)$-action can be used to obtain counting results for saddle connections and, subsequently. Finally we will discuss examples, characterizations, and properties of surfaces with large affine symmetry groups, known as lattice or Veech surfaces.

Remote Access

The instructor will use a tablet and Zoom. The tablet will be displayed locally in the classroom and via zoom. Lecture notes will be distributed in PDF format.

Registration for this course is not currently available.

This course covers classical problems and modern techniques in extremal combinatorics. The first part of the course is on extremal properties of families of sets: e.g.

• What is the largest size of a collection of k-element subsets of a set of size n in which any two sets in the collection intersect?
• What is the largest size of a collection of subsets of a set of size n in which no set is properly contained within another?

Other topics may include VC dimension, Kneser’s Conjecture, the Kruskal-Katona Theorem and the Littlewood Offord Problem. The rest of the course is on extremal graph theory: e.g.

• What is the maximum number of edges in a triangle-free graph on n vertices?
• What is the minimum number of 6-cycles in a graph with n vertices and m edges?
• What is the minimum size of an independent set in a triangle-free graph?

Other topics may include the Szemerédi Regularity Lemma, Shannon Capacity, the Entropy Method, the Container Method and Stability. The course webpage, which includes a link to a preliminary version of the course notes, can be found here.

Lecture Schedule

This course will run Sept. 4th-Dec. 4th, 2024. Lectures will take place every Tuesday, Wednesday and Friday from 10:30am-11:20am (Pacific Time). See the UVic course catalog entry for more details.

Remote Access

Lectures will be livestreamed via Zoom. The lecturer will write on chalkboards which will be shared via Zoom. Recordings of the lectures will be available for asynchronous viewing. Preliminary lecture notes are available on the course website and assignments will be distributed electronically.

Registration for this course is not currently available.

The most basic example of an arithmetic group is $\Gamma=SL_n(Z),$ and understanding the cohomology of this group (and its close relatives) will be the basic theme of this course. The cohomology we are interested in can also be identified with that of the locally symmetric space $\Gamma \setminus X$ where, in this case, $X= SL_n(R)/ SO(n)$ is a generalization of the (complex) upper half plane. As such, a diverse set of techniques, stemming from geometry, topology, harmonic analysis, and number theory can be used to analyze the situation. After carefully developing the basics of the subject, we will present some of the major developments in this area (mostly from the 1960s-1970s), and then end with an overview of modern directions.

Lecture Times

• Dates: Sep. 3 - Dec. 9
• Class Time: Tuesday/Thursday, 16-17:20 (Mountain Time)

Remote Access

The lecturer will use a tablet connected to zoom/camera to live stream lectures and notes. Hand written (from table) and typed lecture notes will be distributed.

Registration for this course is not currently available.

Advances in imaging techniques have enabled the access to 3D shapes present in a variety of biological structures: organs, cells, organelles, and proteins. Since biological shapes are related to physiological functions, biological studies are poised to leverage such data, asking a common statistical question: how can we build mathematical and statistical descriptions of biological morphologies and their variations? In this course, we will review recent attempts to use advanced mathematical concepts to formalize and study shape heterogeneity, covering a wide range of imaging methods and applications. The main mathematical focus will be on basics of image processing (segmentation, skeletonization, meshing), Diffeomorphisms and metrics over shape space, optimal transport theory with application for image analysis, manifold learning, with some other concepts covered in specific applications (e.g. quasiconformal mapping theory for shape representation, 3D reconstruction in Fourier space…). Students will be encourage to work in groups to present research papers and do a small project to pass the course. This course will also build on the recent BIRS workshop, Joint Mathematics Meetings, and the upcoming SIAM workshops (LSI 2024, SIMODS 2024) on this topic, with some participants to these events invited to contribute to this course and present their research.

Lecture Schedule

Remote Access

Remote access will be via zoom. A combination of prepared slides and hand written notes will be used. The hand written notes will be on a blackboard or tablet depending on room availability. The lecturer will distribute lecture notes online.

Registration for this course is not currently available.

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.

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.

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.

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.

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.

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.

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.