About jGMRT:
jGMRT (junior Geometric Methods in Representation Theory) is a grad student seminar created in the image of the GMRT seminar, broadly covering all algebra/geometry/topology (AGT) topics, and featuring presentations by the grad students of UNC for the grad students of UNC. This seminar is a space for grad students to practice giving talks, keep tabs on what their peers are thinking about, and participate in the community of grad students working in AGT. This seminar was founded by me at UNC in Fall 2022, but the idea was stolen from the University of Texas, where I attended similar seminars as an undergrad. There they have many “junior” counterparts to “senior/faculty” seminars. As a result, ambitious undergrads are also welcome to attend, with the understanding that they may not be the main audience for the talks.
Past Seminars
This semester’s (Fall 2024) jGMRT takes places in the Chapman Hall conference room, Thursdays 2-3PM.
9/19: Jeff Ayers
Title: Feynman diagrams in K theoretic Enumerative Geometry
Abstract: Enumerative Geometry is an old field which attempts to count various geometric objects. In the last 30 years there has been a major subfield development into counting certain curves and studying their K-theories. This study has proven to be remarkable in its connections to physics, in particular with QFT. Moreover the study “generalizes” the already popular area of Groton-Witten theory to the K-theoretic versions. The objects that arise are difficult to understand on their own, and thus mathematicians have invented so called “Feynman diagrams” to help simplify the notations.
In this talk we’ll discuss exactly what kinds of curves we wish to count, and where the curves are living. Our main goal will be to consider the various interesting marked points on these curves, discuss what sorts of objects arise, and display their diagrammatic representations. Time permitting we’ll discuss how these objects connect to Representation Theory and the famous Bethe Ansatz.
9/26: Tiger Cheng
Title: Some words about vector bundles in algebraic geometry
Abstract: Vector bundles are some of the main tools we have to understand geometric objects. In this talk, I will go over some of the basic concepts of vector bundles in algebraic geometry, with an eye towards what happens in the case of algebraic curves. Some topics include basic words about the relation between vector bundles and sheaves, Serre’s GAGA, basics about divisors and line bundles, and topics in moduli theory. Audience participation is highly encouraged, as I will happily talk about whichever concept interests the audience the most.
10/3: Xiangjia Kong
Title: Automorphisms on K3 surfaces
Abstract: Irreducible Holomorphic Symplectic (IHS) manifolds, also known as Hyperkähler manifolds, are important objects of study due to the structure theorem on complex manifolds with trivial first Chern class. In particular, 2-dimensional IHS manifolds are called K3 surfaces, and in this talk we will look at some properties and examples of these things. I will also talk about a generalisation by Markman and Verbitsky of the Torelli theorem, which is used to study automorphism groups of K3 surfaces, and some known properties of automorphism groups. Perhaps lattice theory will be mentioned.
10/24: Isaac Weiss
Title: Gröbner Methods for Representations of Combinatorial Categories
Abstract: Suppose one has a category C of combinatorial interest. Much like any category, this too has representations, defined as functors from C to Mod_k, the category of left k-modules. The category of these representations, denoted Rep_k(C), is of particular interest to representation theorists. One such property of interest is whether said category is Noetherian. How does one determine such a thing? In this talk, we will assess Gröbner methods to determine if our desired category is Noetherian. If time permits, we may even have the pleasure of computing an example of the consequences of this work.
10/31: Isaac Weiss
Title: Gröbner Methods for Representations of Combinatorial Categories (PART 2)
11/7: Matt Crawford
Title: An Introduction to p-adic numbers
Abstract: Following a mixture of algebraic, analytic, and topological perspectives, we will explore the world of p-adic numbers. This will be an audience-led survey of (mainly) Gouvêa’s “p-adic Numbers: An Introduction.” We will cover some classic results and proofs of more general nonarchimedean fields, with plenty of examples. Come see what strange world these numbers live in!
11/21: Matt Crawford
Title: An introduction to p-adic numbers (PART 2)