Date	Speaker	Title	Abstract
July 15	UMS Officers	Introduction	We will first introduce Summer UMS and then decide on the logistics for the next weeks. We will go over some potential textbooks we could cover and then pick one by vote. Every member will then have the opportunity to sign up to give a talk.
July 22	Lisa Faulkner Valiente	Review of Linear Algebra and Group Theory	We review key definitions and results in linear algebra and group theory that will be relevant when studying representation theory, including eigenvalues and eigenvectors, the definition of a module, normal subgroups and quotient groups, and the first isomorphism theorem. If time permits, we will also give a precise definition of a representation of a group, and an example.
July 29	Yiming Song	Properties of Representations	We go over some fundamentals of linear algebra before introducing the notion of representations, their basic properties, equivalent representations, kernels, and faithfulness. Then we introduce FG-modules, a concept closely related to representations which will be useful for further study.
August 5th	Joseph McGill	FG-Modules, Irreducibility, and Regular Representations	After learning about FG modules last week, we want to explore their structure more in depth. This week we will discuss the idea of reducibility, along with specific useful modules such as the permutation modules. Finally we will tie these all together and discuss the group algebra, which gives way to the regular representation.
August 12	Maschke's Theorem, Schur's Lemma	Zachary Lihn	We'll continue our study of FG-modules by quickly defining FG-module homomorphisms and their basic properties. We'll then discuss the important result known as Maschke's Theorem, which reduces our study of FG-modules to the irreducible ones. Finally, we'll prove Schur's Lemma, which (time permitting) will allow us to classify the representations of finite abelian groups over the complex numbers.
August 19	Characters, Orthogonality, and Some Physics	Andrew Navruzyan	We introduce and prove fundamental facts about characters and their orthogonality for groups, highlighting the general story on Lie groups. We also introduce Weyl's unitary trick in addition to giving connections of representation theory to general relativity and quantum mechanics. TBD
August 26	TBD	TBD

Date

Speaker

Title

Abstract

July 15

UMS Officers

Introduction

We will first introduce Summer UMS and then decide on the logistics for the next weeks. We will go over some potential textbooks we could cover and then pick one by vote. Every member will then have the opportunity to sign up to give a talk.

July 22

Lisa Faulkner Valiente

Review of Linear Algebra and Group Theory

We review key definitions and results in linear algebra and group theory that will be relevant when studying representation theory, including eigenvalues and eigenvectors, the definition of a module, normal subgroups and quotient groups, and the first isomorphism theorem. If time permits, we will also give a precise definition of a representation of a group, and an example.

July 29

Yiming Song

Properties of Representations

We go over some fundamentals of linear algebra before introducing the notion of representations, their basic properties, equivalent representations, kernels, and faithfulness. Then we introduce FG-modules, a concept closely related to representations which will be useful for further study.

August 5th

Joseph McGill

FG-Modules, Irreducibility, and Regular Representations

After learning about FG modules last week, we want to explore their structure more in depth. This week we will discuss the idea of reducibility, along with specific useful modules such as the permutation modules. Finally we will tie these all together and discuss the group algebra, which gives way to the regular representation.

August 12

Maschke's Theorem, Schur's Lemma

Zachary Lihn

We'll continue our study of FG-modules by quickly defining FG-module homomorphisms and their basic properties. We'll then discuss the important result known as Maschke's Theorem, which reduces our study of FG-modules to the irreducible ones. Finally, we'll prove Schur's Lemma, which (time permitting) will allow us to classify the representations of finite abelian groups over the complex numbers.

August 19

Characters, Orthogonality, and Some Physics

Andrew Navruzyan

We introduce and prove fundamental facts about characters and their orthogonality for groups, highlighting the general story on Lie groups. We also introduce Weyl's unitary trick in addition to giving connections of representation theory to general relativity and quantum mechanics. TBD

August 26

TBD

Columbia Undergraduate Math Society

Spring 2024 << Summer 2024 Learning Seminar >> Fall 2024