Courses: ongoing

Registration for this course is not currently available.

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.

Registration for this course is not currently available.

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.

Registration for this course is not currently available.

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

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.

Registration for this course is not currently available.

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.

Lecture Schedule

• Wednesday/Friday 2:30-4:20pm Pacific Time See the SFU Calendar for more details.

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. Zoom links will be available on the course webpage (via Canvas, which will be available to enrolled students).

Other Information

Please see the SFU Calendar for more details about this course.

Registration for this course is not currently available.

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.

Registration for this course is not currently available.

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.

Course Website

Full information about this course is available on the course website.

Lecture Schedule

• Tuesday/Thursday 11:00am-12:30pm Pacific Time

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.

Registration for this course is not currently available.

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.

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 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.

Registration for this course is not currently available.

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