Courses: past

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.

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 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.

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.

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

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