Instructor: Cailan Li
E-mail: ccl2166@columbia.edu
Classes:Tuesdays 6:10 PM - 8:10 PM in Math 507
Office hours:Please email me to make an appointment.
I will usually be in my office on Tuesdays 5-6 PM if you have any last minute questions/concerns.
Coxeter groups appear at the intersection between combinatorics, geometry, representation theory and topology. Examples of Coxeter groups include Weyl groups which are the combinatorial backbone of semisimple lie algebras and their representation theory. A slight generalization of Weyl groups are reflection groups (which are also Coxeter groups), which arise as the symmetries of regular polytopes (loosely speaking, these are polytopes with very high degrees of symmetry). In three dimensions these are exactly the Platonic solids. Below is an image of the hyperdodecahedron, or “120-cell”, a regular polytope in 4 dimensions.
The only formal requirement will be MATH UN2000: Intro to Higher Mathematics. MATH UN2010: Linear Algebra might be useful for one or two lectures. Similarly, MATH GU4041: Intro Modern Algebra I, might be useful for material related to groups, but we will only need the definition of a group for the seminar, and I will review/explain this in the introductory/background talk(s).
Grading will be based on participation, attendance, and effort. Specifically,
Audience members are expected to actively engaged during the talk. In particular I want to emphasize that if you are confused or need an additional explanation at any point during a talk, PLEASE ASK A QUESTION . There will be no dumb questions in this seminar, so please feel free to ask anything you want (pertaining to the lecture).
Speakers are expected to put effort into making their lecture as clear, cohesive and engaging as possible. In particular, please put in more effort than just copying what is written in the references on the blackboard. There should be a logical flow to your talk and you should explain how the different things you are writing down relate to each other. Here are some guidelines/suggestions for your talk
Dates | Speaker | Topic | Supplementary Images |
---|---|---|---|
9/6 | Organizational Meeting | ||
9/13 | Cailan Li | Crash Course on Groups and Linear Algebra: Notes | |
9/20 | Elizabeth Marks | Definitions and Examples: Notes | Figure 1.1, Figure 1.2, Figure 1.3 |
9/20 | Tuan Dolmen | Finite Coxeter Groups and Root Systems: Notes | E_8 projection |
9/27 | Nirvaan Iyer | Lengths, Descents and Exchange Conditions: Notes | |
9/27 | Hengzhi Zhang | The Longest element and Matsumoto’s theorem: Notes | Tetrahedron Axis of Symmetries |
10/4 | Olivia Solanot | Principle of Inclusion Exclusion: Notes | |
10/4 | Talia Fine | Posets and Lattices: Notes | |
10/11 | Savik Kinger | The Bruhat Order (of Sn): Notes | |
10/18 | Ahkeel Timothy | Properties of the Bruhat Order: Notes | |
10/18 | Cailan Li | Review | |
10/25 | CJ Magleby | Parabolic Subgroups and the Tableau Criterion: Notes | |
10/25 | Dawson Franz | Generating Functions: Notes | |
11/1 | Gabi D'Agostino | q−analogs: Notes | |
11/1 | Charlotte Coats | The Mobius Function: Notes | |
11/15 | Sarah Kuriyama | (Semi)Standard Young Tableaux: Notes | |
11/15 | Eli Baucom-Hays | Catalan Numbers and Fully Commutative Elements: Notes | |
11/22 | Param Gujral | Young’s Lattice and Differential Posets: Notes | |
11/22 | Jacob Daum | Poincare Series: Notes | |
11/29 | Polina Zakharov | Eulerian Polynomials: Notes | |
11/29 | Berkley Fang | Log Concavity, Unimodality and Matroids: Notes | |
12/6 | Zara Hall | The Work of June Huh: Notes | |
12/6 | Cailan Li | Closing Remarks and Q&A |