# Courses: ongoing

The following courses were scheduled for the ongoing academic year:

The following courses were scheduled for the ongoing academic year:

Jonathan Noel : noelj@uvic.ca

University of Victoria

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

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.

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.

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.

Manish Patnaik : patnaik@ualberta.ca

University of Alberta

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 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.

**Dates**: Sep. 3 - Dec. 9**Class Time**: Tuesday/Thursday, 16-17:20 (Mountain Time)

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.

Khanh Dao Duc : kdd@math.ubc.ca

University of British Columbia

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.

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.