# Courses: ongoing

The following courses were scheduled for the ongoing academic year:

The following courses were scheduled for the ongoing academic year:

Bojan Mohar : mohar@sfu.ca

Simon Fraser University

Undergraduate course in graph theory

Undergraduate course on (discrete) probability

Linear algebra

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. '

- 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

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.

- 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

- 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

- Homework assignments 30%
- Midterm 30%
- Final exam 40%

Lior Silberman : lior@math.ubc.ca

University of British Columbia

Galois Theory

Basic number theory

Introductory algebra (groups, rings, modules, polynomial rings, UFD and PID).

Commutative algebra is useful but not required.

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/

Lectures will take place every Wednesday and Friday from 10:00am-11:30am (Pacific Time).

Lectures will be shared via zoom. Students will need to have installed the zoom app on their desktop/laptop.

Anthony Quas : aquas@uvic.ca

University of Victoria

A course on measure theory.

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.

Lectures will take place every Monday and Thursday from from 8:30-9:50 (Pacific time).

Lectures will be shared via zoom. Students will need to have installed the zoom app on their desktop/laptop.

James D. Lewis : lewisjd@ualberta.ca

University of Alberta

Students should have taken a course on algebraic geometry. It is helpful to know some differential geometry, particularly how it applies to complex manifolds, de Rham and Betti (singular) cohomology. Some exposure to homological algebra will be useful.

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.

Lectures will take place on Mondays, Wednesdays and Fridays from 13:00-13:50 (Mountain Time)

These lectures will take place via zoom. Students should have zoom installed on their laptop or other device.

Thomas Hillen : thillen@ualberta.ca

University of Alberta

Some basic knowledge on partial differential equations.

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.

Lectures will take place Monday, Wednesday and Friday from 13:00-13:50 (Mountain Time).

Lectures are online on zoom.

Clifton Cunningham : ccunning@ucalgary.ca

University of Calgary

Undergraduate linear algebra, abstract algebra (groups, rings, fields)

multivariable calculus and algebraic number theory

A course in modules would be helpful, but not necessary

A course in classical commutative algebra is not required

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.

Joshua Zahl : jzahl@math.ubc.ca

University of British Columbia

This course assumes graduate-level background in measure theory, real analysis, and harmonic analysis (i.e. at the level of Math 420/507 and Math 404/541 at UBC).

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.

Lectures will take place every Monday, Wednesday and Friday 11:00am-12:00pm (Pacific Time)

Lectures will be shared via zoom. Students will need to have installed the zoom app on their desktop/laptop.