Courses: upcoming

The following courses were scheduled for the upcoming academic year:

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

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
Advanced studies in Theoretical and Computational Biology
Course Number
MATH 605E
Section Number
Section Code
Special Topics

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, please see the Western Deans' Agreement website .

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 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

Class Schedule

  • TBA

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.

Availability

This course may be open to students from universities outside of the PIMS network.

Optimal Transport: Theory and Applications

Instructor(s)

Prerequisites

  • First year graduate course in real analysis and/or probability.

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
Optimal Transport: Theory and Applications
Course Number

University of Washington Students:

  • University of Washington: Math 581

All Other WDA Students:

Section Number
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, please see the Western Deans' Agreement website .

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 modern theory of Monge-Kantorovich optimal transport is barely three decades old. Already it has established itself as one of the most happening areas in mathematics. It lies at the intersection of analysis, geometry, and probability with numerous applications to physics, economics, and serious machine learning. This two quarter long graduate topics course will serve as an introduction to this rich and useful theory. We will roughly follow the following outline. Fall: Classical theory. Analytic description of solutions. Duality. Displacement convexity. The geometry of the Wasserstein space and Otto calculus. Winter: Entropy-regularized OT. Schroedinger bridges and statistical OT. This is a continuation of the sequence of OT+X courses under the Kantorovich Initiative.

Other Information

Delivery Details

Registration

Students at Canadian PIMS Member Universities may register through the Western Deans Agreement for the “shadow course” offered at UBC (see registration details above). 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.

Class Schedule

  • TBA

Remote Participation

Online instructions over Zoom. Written on a tablet. Notes will be provided. A Slack channel will be created for answering student questions. Weekly in person office hours will be held at UW and UBC.

Lecture notes will be distributed over Slack. Recorded lectures may be viewed on our YouTube channel.

Availability

This course may be open to students from universities outside of the PIMS network, and those coming from industry/government.

Algebraic Topology I

Instructor(s)

Prerequisites

  • a first course in real analysis, and some point-set topology, including quotient topologies, connectedness, path-connectedness.

  • Homotopy of maps and homotopy equivalence of spaces will be assumed, but the necessary background here can be quickly covered by self-study.

  • Fundamental groups and covering spaces, while helpful, are not necessary.

  • the theory of abelian groups, isomorphism theorems and the classification of finitely generated abelian groups.

  • Ring theory and the theory of modules over commutative rings is extremely helpful, but not formally required.

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
Algebraic Toplogy I
Course Number
MATH 527
Section Number
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, please see the Western Deans' Agreement website .

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 is a course in homology and cohomology of topological spaces. We study spaces and continuous functions by means of abelian groups and their homomorphisms. Topics will include cellular homology of spaces, calculation techniques and applications (e.g., fixed point theorems, invariance of domain), homological algebra, and cohomology, including the cup product and Poincaré duality.

Other Information

Class Schedule

  • TBA

Remote Access

Remote access for this course will be provided via zoom. The instructor intends to lecture from handwritten notes on a tablet. Lecture notes will be provided after the lectures have been delivered.

Availability

This course may be open to students from universities outside of the PIMS network, and those coming from industry/government.

Applied Stochastic Analysis

Instructor(s)

Prerequisites

  • Good upper level undergraduate or early graduate knowledge of:

    • Probability
    • Linear Algebra
    • PDEs
    • ODEs
    • Prior experience with numerical analysis is helpful but not necessary

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
Applied Stochastic Analysis
Course Number
MATH 605F
Section Number
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, please see the Western Deans' Agreement website .

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 will introduce the major tools in stochastic analysis from an applied mathematics perspective. Topics to be covered include Markov chains (both discrete and continuous), Gaussian processes, Ito calculus, stochastic differential equations (SDEs), numerical algorithms for solving SDEs, forward and backward Kolmogorov equations and their applications. It will pay particular attention to the connection between stochastic processes and PDEs, as well as to physical principles and applications. The class will attempt to strike a balance between rigour and heuristic arguments: it will assume that students have seen a little analysis, particularly in the context of studying PDEs, but will generally avoid measure theory. The target audience is graduate students in applied mathematics or related fields, who wish to use these tools in their research for modelling or simulation. The course will be divided roughly into two parts: the first part will focus on stochastic processes, particularly Markov chains, and the second part will focus on stochastic differential equations and their associated PDEs.

Syllabus

syllabus.pdf

Other Information

Class Schedule

  • TBA

Remote Access

Remote access will be provided via zoom. The lectures will be delivered mostly on blackboards with occasional slides. PDF lecture notes will be handed out.

Availability

This course is open to students from within the PIMS network of universities.