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.
I am organizing for the fall semester of 2022 and spring 2023, so please reach out to me with any questions about the seminar. Meetings for spring 2023 are in Phillips 334 from 12-1.
Spring 2023 Schedule:
Feb 8: Tiger Cheng
Title: What’s in a scheme? Notes
Abstract: Schemes are one of the fundamental objects of algebraic geometry. However, they have a reputation of being abstract and therefore difficult to understand as well as being difficult to motivate. In this talk, we will go on a journey from manifolds to varieties and ultimately to schemes, highlighting the natural connection and progression of ideas which led to the development of schemes. After highlighting some basic notions of schemes, I want to talk about the phenomenology of schemes, paying particular attention to illuminating examples.
Feb 15: Tiger Cheng
Title: What’s in a scheme? Pt II.
Feb 22: Reese Lance
Title: Introduction to Kahler Geometry
Abstract: Kahler manifolds are a unification of three important structures one encounters in differential geometry: complex, symplectic, and Riemannian. It turns out that with a little extra condition, any two of these structures on a smooth manifold determines a third, and with another extra condition, we will call such a manifold Kahler. These manifolds enjoy many nice properties which we will discuss, but also are important because they seem to arise naturally in many areas of mathematical physics/physically inspired mathematics. Time permitting we will discuss a little bit of Hodge theory and Dolbeault groups, but I will have to skip proofs for these.
March 1: Reese Lance (Pushed back to Later in the Semester)
Title: Hyperkahler Quotients and Singularities
March 8: Luke Conners
Title: Link polynomials arising from representation theory
Abstract: The category of finite-dimensional representations of (the quantized enveloping algebra of) a Lie algebra has a rigid algebraic structure, including a tensor product, an interesting braiding operation, and dual representations. In this talk, I’ll describe how these algebraic structures can be encoded diagramatically in the form of an oriented tangle diagram. We’ll also see how to obtain link polynomials, including the Jones polynomial, from this procedure.
March 15: No talk
Title: Spring break
Abstract: Go outside and touch grass
March 22: Jeff Ayers
Title: What is Equivariant Cohomology
Abstract: Cohomology theories provide us with many interesting invariants of topological and geometric objects. Sometimes our geometric objects have group actions associated to them, and we can ask: “Is there a Cohomology theory that takes this space and the action into consideration?”. This is the role of Equivariant Cohomology. In this talk we’ll explore what Equivariant Cohomology is, compute many examples, and if time permits, explore some applications to Mathematical Physics.
March 29: Tiger Cheng
Title: Moduli problems in GIT
Abstract: One of the central themes of mathematics is the problem of classification. This is the central idea of moduli problems: we are given some notion of equivalence and we would like to classify the different equivalence classes that we have, and preferably in a way compatible with some underlying geometry. One way of doing such classifications is through taking quotients of certain ‘universal’ spaces by the action of some algebraic group. This is the idea of geometric invariant theory (GIT). In this talk, I will go through the general idea and notions behind moduli problems, and then move on to describe how quotients by group actions work in the algebraic geometry setting
April 5: Luke Conners
Title: Link Invariants Arising from Tangle Functors
Abstract: Last time, we saw how the Jones polynomial can naturally be realized as the result of applying a functor from the diagrammatic category of tangles to the category of vector spaces over rational functions. Today, we’ll consider other possible targets for this functor. This will naturally lead to the notion of a (ribbon) Hopf algebra and, more generally, a ribbon category. We’ll see how an arbitrary instance of these structures gives rise to a tangle functor, which itself gives rise to an invariant of (framed, oriented) links.
April 12: Kaitlyn Hohmeier
April 19: Joe Compton
April 26: Reese Lance
Title: Hyperkahler Quotients and Singularities