Spring 2023 Math UN3952 Undergraduate Seminars: Symmetric Functions and Macdonald Polynomials

Instructor: Cailan Li
E-mail: ccl2166@columbia.edu
Classes: Thursdays 6:05 PM - 8:05 PM in Math 507
Office hours:Please email me to make an appointment.
I will usually be in my office on Thursdays 5-6 PM if you have any last minute questions/concerns.

Description

The study of symmetric functions has a rich history, dating back to at least the mid-nineteenth century and remains an active field of research today. In part this is because it provides a tool to investigate various phenomenon occuring in representation theory and algebraic geometry. To provide a quick definition, a symmetric function (or rather polynomial in this case) on n variables x₁, ... , x_n is a polynomial that is unchanged by any permutation of its n variables. Because of this symmetry, it turns out symmetric functions (polynomials) have remarkable algebraic properties, much of which can be expressed using the language of combinatorics. A large portion of the seminar will be about investigating these properties and the combinatorics that arises.

Prerequisites

The formal requirements will be MATH UN2010: Linear Algebra and one course demonstrating mathematical maturity such as MATH UN2000: Intro to Higher Mathematics. MATH BC2006: Combinatorics might be useful for familiarity with some topics and MATH GU4044: Representations of Finite Groups might be useful as motivation but both aren't strict requirements.

Grading

Grading will be based on participation, attendance, and effort. Specifically,

Attendance:

• At the end of each class, on a piece of paper please submit to me (with your name and date) two "things" that you got from the talk. For more information on what a "thing" is, please see Ravi Vakil's description: Three Things. This will also serve as your attendance record for the lecture. (Note, you can write your two things down during the lecture.)
• During the week you are speaking, you don't need to write two things. However please your name on a piece of paper.
• Each absence will result in the deduction of a half a letter grade (e.g. A to A-). In order to have an absence be excused, please ask an intelligent question in any of the following class sessions.
• On the day you ask your intelligent question, send me an email with the question you asked during class. If it is sufficient I will inform you and your absence will be excused. Otherwise keep trying to ask questions until it's passable.
• 3 absences will result in a pernament half a letter grade deduction.

• Put effort into actually understanding the material you are presenting. Understanding the material at a deep level isn't necessary, but simply copying the references onto the chalkboard is a bad idea.
• Submit to me your notes for the talk via email after the talk. If you have good handwriting, then you can handwrite and submit them. Otherwise I expect a pdf (you can write your notes on a tablet and convert to pdf for instance) or word document.
• I will be grading your notes for effort, so do not try any shortcuts.
• Per Professor Bayer's suggestion, the audience CANNOT USE LAPTOPS OR TABLETS during the lectures.
• Any student that puts in a nontrivial amount of effort into learning the material in the seminar will get an A+, regardless of whether they understand the material well.

Expectations

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

• You should present the information in the order of logical flow (if Theorem B is needed in the proof of Theorem A, then present Theorem A first) which does not always coincide with the order that the information is presented in the references.
• You should present information that helped you understand the material. For example, you came up with example C that helped you understand concept D, so present example C.
• Some things you should just say instead of writing the entire thing on the board such as a block of text. In this case you should write 1-3 key words relating to the block of text and say the rest.
• REMINDER that instead of presenting the proof of a theorem, you can always do examples demonstrating the theorem.
• Print out the notes you use for the lecture in a large font/enlarge them. Otherwise it will be hard for you to read when presenting.

Main References

An Introduction to Symmetric Functions and their Combinatorics
Algebraic Combinatorics and Coinvariant Spaces

Seminar Schedule