JGMRT

JGMRT (pronounced "jug-mart"), short for Junior Geometric Methods in Representation Theory, is a grad student seminar created in the image of the GMRT seminar at UNC, broadly covering all algebra/geometry/topology (AGT) topics, and featuring presentations by the grad students for the grad students. 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 Reese Lance at UNC in Fall 2022. Ambitious undergrads are also welcome to attend, with the understanding that they may not be the main audience for the talks.

This semester's JGMRT is headed by Isaac Weiss.

Spring 2024

We will be holding JGMRT this semester on Fridays from 14:00-15:00.

2/16: Tiger Cheng
Title: A gentle introduction to stacks

Abstract: This talk will be a very gentle introduction to some of the ideas of stacks. I will not claim to be comprehensive, nor super precise, but my goal is to introduce some of the ideas behind stacks and why people thought of these things. Stacks can be thought of as spaces where instead of having an underlying set with a topology, we replace the set with a site \(\mathcal{C}\) which is a category equipped with what's called a Grothendieck topology. In Part I of the talk, I will talk about some of the motivation behind stacks coming from moduli theory. In Part II I will set up some technology regarding Grothendieck topologies, and in Part III, I will define what a stack is and show off some examples.
2/23: Empty
3/1: Empty
3/8: Tiger Cheng
Title: Gentle introduction to stacks part 2

Abstract: We will continue from our talk on 2/16. This time I will actually define what a stack is along with discussing some example as well as some further topics.
3/15: Spring Break
3/22: Luke Conners
Title: Six Functor Formalisms

Abstract: Many cohomology theories (e.g. singular cohomology of topological spaces, de Rham cohomology of smooth manifolds, Betti/l-adic/etale cohomology of schemes, etc.) share similar structural enhancements generalizing the usual functoriality, cup product, Kunneth formula, and Poincare duality of singular cohomology. In many cases, these structures can be seen as consequences of the existence of six 'distinguished' functors on categories giving rise to these theories, together with various relationships among these functors and properties they satisfy. We'll spend a bit of time discussing six functor formalisms in general, then specialize our attention to the case of singular cohomology of sufficiently nice topological spaces. Here the relevant category is sheaves of abelian groups on a space; we'll construct each of the six functors in this case in detail and sketch the passage from the properties of these functors to familiar structures in singular cohomology.
3/29: Spring Holiday
4/5: Luke Conners: Six functors part 2
4/12: Alex's defense
4/19 No JGMRT today
4/26 Tiger Cheng
Title: Mori's Bend and Break

Abstract: The purpose of this talk is to give a glimpse at a very interesting piece of mathematics. Mori's Bend and Break technique incorporate some of the most interesting features of algebraic geometry, namely deformation theory techniques and passing to characteristic \(p\). The main goal of this exposition is to use the bend and break technique to prove that on every point on a (smooth) Fano variety there is a rational curve passing through it.