Courses: 2024-2025

The following courses were scheduled for the 2024-2025 academic year:

A Primer to Arithmetic Statistics

Instructor(s)

Prerequisites

  • Group and ring theory

  • linear algebra

  • real analysis

Registration

Registration for this course is not currently available.

Abstract

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.

Other Information

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.

Advanced studies in Theoretical and Computational Biology

Instructor(s)

Prerequisites

  • Ordinary differential equations

  • Numerical methods (Numerical Analysis I and II)

  • Partial differential equations

  • Matrix theory

  • Linear systems

Registration

Registration for this course is not currently available.

Abstract

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

Syllabus

  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.

Other Information

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.

Algebraic Topology

Instructor(s)

Prerequisites

  • A course in general topology or metric space topology (required)

  • A course in group theory (strongly recommended)

Registration

Registration for this course is not currently available.

Abstract

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.

Syllabus

syllabus.pdf

Other Information

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.

Analytic and diophantine number theory with applications to arithmetic geometry

Instructor(s)

Prerequisites

  • Required

    • Undergraduate algebra (groups, rings, fields)
    • Undergraduate complex analysis

    • Galois theory
    • Undergraduate introduction to algebraic geometry

Registration

Registration for this course is not currently available.

Abstract

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.

Other Information

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.

Extremal Combinatorics

Instructor(s)

Prerequisites

  • An undergraduate course on discrete mathematics, combinatorics or graph theory. It is recommended that students have taken at least two such courses.

Registration

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:

Course Name
Extremal Combinatorics
Course Number
Math426 (undergraduate) or MATH529 (graduate)
Section Number
A01
Section Code

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.

Abstract

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.

Other Information

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.

Fundamental models in fluid dynamics

Instructor(s)

Prerequisites

  • Introductory PDEs

  • Introductory analysis

Registration

Registration for this course is not currently available.

Abstract

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.

Other Information

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

Remote Access

The lecturer will use zoom for each lecture, and lectures will be recorded. Typed lecture notes will be distributed electronically.

Introduction to Cohomology of Arithmetic Groups

Instructor(s)

Prerequisites

  • We will aim to make this course accessible to students with a basic background in algebra and analysis (at the level of introductory graduate courses) and basic topology (having seen cohomology before would be useful, but is not absolutely essential). Although no specific knowledge from differential geometry, Lie theory, or number theory are required, additional familiarity or interest in these fields will be useful, especially in the latter parts of the course.

Registration

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:

Course Name
Topics in Algebra
Course Number
MATH681
Section Number
LECTURE B1
Section Code

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.

Abstract

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.

Syllabus

syllabus.pdf

Other Information

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.

Mathematical Classical Mechanics

Instructor(s)

Prerequisites

Registration

Registration for this course is not currently available.

Abstract

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.

Syllabus

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

Other Information

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.

Mathematical Ecology - Nonlinear PDE Models

Instructor(s)

Prerequisites

Registration

Registration for this course is not currently available.

Abstract

Mathematical modelling in ecology, including population dynamics, epidemiology, and pattern formation. Theory of such models formulated as difference equations, ordinary differential equations, and partial differential equations.

Other Information

Remote Access

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

Spectral Methods for PDEs

Instructor(s)

Prerequisites

  • Undergraduate analysis and PDEs

  • Some exposure to numerical analysis desirable, but not necessary

  • Some homework questions will require computer programming (MATLAB or Julia, etc.)

Registration

Registration for this course is not currently available.

Abstract

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.

Syllabus

syllabus.pdf

Other Information

Remote Access

Lectures will be delivered via Zoom using iPad with GoodNotes.

Stochastic Analysis-Stochastic Differential Equations

Instructor(s)

Prerequisites

  • Some knowledge on Differential equations and Probability Theory

Registration

Registration for this course is not currently available.

Abstract

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.

Syllabus

syllabus.pdf

Other Information

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

Topics in Mathematical Biology: biological image data and shape analysis

Instructor(s)

Prerequisites

    Registration

    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:

    Course Name
    Topics in Mathematical Biology: biological image data and shape analysis
    Course Number
    MATH 612
    Section Number
    Math
    Section Code
    612

    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.

    Abstract

    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 analysismanifold 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.

    Other Information

    Lecture Schedule

    Remote Access

    Remote access will be via zoom. A combination of pre-pared 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.

    Translation Surfaces

    Instructor(s)

    Prerequisites

    • Complex Analysis

    • Manifolds

    Registration

    Registration for this course is not currently available.

    Abstract

    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).

    Syllabus

    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.

    Other Information

    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.