Mini Courses
Hyperbolic manifolds and their topological invariants
Friday, December 11 | 9am - 12pm
Professor Mauricio Bustamante
Universidad Católica de Chile
The mini-course will introduce the topology of closed hyperbolic manifolds. We will begin with the hyperboloid model of hyperbolic space, its isometry group, and the description of closed hyperbolic manifolds as quotients of hyperbolic space by discrete groups of isometries. We will discuss a theorem of Sullivan stating that every closed hyperbolic manifold admits a stably parallelizable finite cover, together with some of its consequences.
We will then introduce Charney--Davis strict hyperbolization, a procedure for replacing cubical complexes with negatively curved spaces while preserving part of their topological and combinatorial information. We will discuss applications to the construction of closed negatively curved and hyperbolic manifolds with nontrivial characteristic classes.
Learning Outcomes:
The mini-course will introduce participants to the topology of closed hyperbolic manifolds, starting from the basic geometry of hyperbolic space and progressing to the Charney--Davis strict hyperbolization construction and its applications. It addresses the first and second objectives by providing an accessible introduction to the subject, and introducing participants to an active area of research at the intersection of geometric topology, differential geometry, and geometric group theory.
Chromatic Quasisymmetric Functions: Computations, Connections, and Applications
Friday, December 11 | 1pm - 4pm
Mohammad Bardestani
John Abbott College
.jpg)
Chromatic quasisymmetric functions are a natural generalization of chromatic symmetric functions, with additional information coming from statistics on proper graph colorings. They provide a way to study graph colorings using tools from the theory of quasisymmetric functions and have interesting connections with other topics in algebraic combinatorics.
The goal of this mini-course is to introduce chromatic quasisymmetric functions, explain their basic properties, and discuss some of their connections and applications. I will start with the definitions and work through examples involving small graphs. I will then discuss some of the main results and show how these functions connect graph theory with symmetric-function theory and other areas of algebraic combinatorics.
The course will combine basic theory, explicit calculations, and applications. My aim is to give participants a good understanding of the subject and of the questions that have motivated its development, while also providing some directions for further study.
Learning Outcomes:
-Introduce the foundational theory.
Introduce participants to chromatic quasisymmetric functions and their role in algebraic combinatorics. Participants will learn the basic definitions, understand how these functions refine chromatic symmetric functions, and become familiar with their fundamental properties through illustrative examples.
-Develop computational skills and mathematical intuition. Enable participants to compute chromatic quasisymmetric
functions for small graphs and to explore their expansions in natural bases of quasisymmetric functions. Through explicit calculations and examples, participants will develop intuition for how graph structure influences the resulting functions and how computational evidence can suggest mathematical patterns and conjectures.
-Explore connections and applications. Demonstrate how chromatic quasisymmetric functions connect graph coloring with symmetric-function theory and other areas of algebraic combinatorics. Participants will encounter selected applications and significant results, gaining an understanding of the broader mathematical questions that motivate the subject and of possible directions for further investigation.
Einstein 4-manifolds and their degenerations
Friday, December 11 | 1pm - 4pm
Tristan Ozuch (Massachusetts Institute of Technology)
An introduction to Einstein metrics in dimension four. These metrics are known as gravitational instantons in physics, and are closely tied to the topology of the underlying 4-manifold. The course covers classical examples and topological obstructions, how Einstein 4-manifolds degenerate into orbifolds with gravitational instantons bubbling off, and how this picture leads to new constructions.
Matroids 101
Friday, December 11 | 9am - 12pm
Jonathan Boretsky
McGill University / LACIM
Matroids are a fundamental combinatorial object with a remarkably wide range of applications, simultaneous generalizing linear independence of vectors, spanning trees of graphs, and partial matchings in bipartite graphs. In this course we will approach them from several different directions, each with its own distinct, but equivalent, definition, highlighting the broad usefulness of these combinatorial objects. As an application, we will see how certain (realizable) matroids relate to the geometry of the Grassmannian, in a setting where combinatorics and algebraic geometry interact heavily.
Learning Objective: This course is meant as an introduction to matroid theory. Matroids are a key tool in algebraic combinatorics, algebraic geometry, discrete geometry and optimization, and representation theory. They are also a natural generalization of graph theory, and this is one of the key perspectives from which I want to approach matroids. Accordingly, they relate to a number of sessions at the CRM meeting. More importantly, to my view, matroids are a tool that is actively used in all these areas and so this minicourse will introduce attendees to a tool that will enable them to participate in modern research across a number of research areas.