August 7, 2020
Speaker: Terence Coelho
Title: Affine Weyl groups and untwisted affine Lie algebras
Abstract: One of the most beautiful properties of simple Lie algebras is their correspondence to root systems (whose reflection groups are known as Weyl groups). Given a Weyl group, one can form the corresponding affine Weyl group by adding in translations by the coroot lattice. In this talk, I will talk about these groups, along with how they relate to Untwisted Affine Lie algebras.
September 2, 2020
Speaker: Tamar Lichter Blanks
Title: Why study schemes?
Abstract: Let's face it: schemes are weird. As algebraists, we like rings, and we might even like topological spaces, but the definition of a scheme can seem like it comes out of nowhere. In this talk, we'll start with a review of varieties, and then slowly build up to the idea of schemes. We'll see how schemes can help us answer questions from algebra, geometry, and number theory, and how scheme theory extends beyond the theory of varieties to deal with things like SL_n, "generic" points, and intersections of curves.
September 9, 2020
Speaker: Yael Davidov
Title: Representation theory of finite groups
Abstract: In this talk we will be covering some of the basics in representation theory of finite groups, looking forward to Jason’s talk next week. We will go through some definitions, study some examples, understand the connection to and use of character theory in relation to representations, and really just try to have a good time. There is so much material to cover but I will assume no knowledge beyond first year algebra and we’ll see where we end up!
September 16, 2020
Speaker: Jason Saied
Title: Representation theory and symmetric functions
Abstract: This is a follow-up to Yael's talk. First, we will discuss some more finite group representation theory (orthogonality of characters, induced representations, and Frobenius reciprocity). We will then study the connection between representations of symmetric groups and the ring of symmetric functions: it turns out that you can learn a lot about representation theory by studying the combinatorial theory of symmetric functions. We will look at a few applications of this connection. I won't assume any outside knowledge of rep theory or symmetric functions, but I WILL assume that you went to Yael's talk or watched the recording, which is available on the GARTS website: https://sites.math.rutgers.edu/~jjs435/garts/
September 23, 2020
Speaker: Terence Coelho
Title: Introduction to Lie algebras
Abstract: We'll be covering definitions, basic concepts, and examples of Lie algebras as well as the classification of finite dimensional modules of sl(2). No background will be necessary at all.
September 30, 2020
Speaker: Johnny Fonseca
Title: Where the Hopf are the Hopf algebras?
Abstract: I will focus on the categorical significance of U_q(sl2)-modules, which I hope to discuss more explicitly, and its applications, in future talks. The present talk will be motivated by the representation theory of Lie algebras and of groups. Along the way I will discuss the Hopf algebras working behind the scenes in each of these studies, and present a couple of interesting results about Hopf algebras. From here, I will then emphasize the important categorical differences between U_q(sl2)-modules versus modules of a Lie algebra or a group.
November 11, 2020
Speaker: Edna Jones
Title: Möbius Transformations and the Bends and Centers of Generalized Circles, Spheres, and Hyperspheres
Abstract: We know from complex analysis that Möbius transformations map generalized circles to generalized circles. However, given a generalized circle C and a Möbius transformation f, can we easily compute the radius and the center of the generalized circle f(C)? Can we do a similar thing with generalized spheres and Möbius transformations in higher dimensions? The answer for both of these questions is yes. To show this, we will use inversive coordinates and Clifford algebras. This will be similar to (but not exactly the same as) the talk I gave in the Graduate Number Theory Seminar/Learning Seminar this semester.
November 18, 2020
Speaker: Brian Pinsky
Title: Birkoff's HSP theorem
Abstract: Have you ever wondered why fields are terrible? Have you always thought fields were pretty okay and you're reading this like "hey what's wrong with fields"? Have you ever wished I would write abstracts that were just like "we're gonna talk about this thing" instead of starting with a bunch of questions, each less related to the topic than the last?
If you answered yes, no, maybe, or "I don't know" to any of these questions, then you should learn universal algebra. Universal algebra studies structures like groups and rings defined by operations and axioms. Fields are bad because they are not defined by operations and axioms, since division by 0 is impossible. This is why the product of rings is a ring, but the product of fields is useless garbage.
The HSP theorem says that a class of structures is definable by equations iff it is closed under Products, Subspaces, and Quotients ("homomorphic images"). Like most good theorems, the forwards direction is trivial, and the backwards direction is also trivial, but only after you build enough theory to see that it is trivial.
P.S. I promise this isn't a category theory talk. I haven't drawn any commutative diagrams and I'm only using the word adjoint once. There is real, hopefully intelligible mathematical content.
P.P.S. unless you want it to be a category theory talk; I can totally make that happen.
December 2, 2020
Speaker: Johnny Fonseca
Title: Not Really a Knot Talk (Part 1)
Abstract: The goal of my sequence of talks is to showcase how useful quantum groups can be through one of my favorite applications: (quantum) knot invariants! These are an instance of something more general called Reshetikhin-Turaev invariants, or RT invariants for short. In order to define and discuss these objects, we will need some background on a couple of topics; this is why I will give a couple of talks!
The first installment of my talks will predominantly be the necessary algebraic background to construct RT invariants. This will feature plenty of category theory, but all motivated and exemplified from familiar categories e.g. kVect, Rmod, Rep(G), Rep(g), and Rep(U_q(g)). Specifically, we will work toward the definition of ribbon categories and a notion of trace therein. I will then use the remaining time to lay the foundation for next week's geometric part of these presentations by introducing a nice pictorial tool to work with ribbon categories.
December 9, 2020
Speaker: Johnny Fonseca
Title: Not Really a Knot Talk (Part 2)
Abstract: This talk is a continuation of my talk last week in which we defined ribbon categories. We will proceed to establish graphical notation for ribbon categories, then define the geometric objects of interest: colored ribbon graphs! The definition of RT-invariants will then be natural to define, and I'll make a final appeal to why quantum groups are worth studying with some "examples" of RT-invariants and closing remarks. Also, I will briefly recap what a ribbon category is at the start, so aside from this there really is no expectation of prior knowledge!