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 semester Fall 2023, so please reach out to me with any questions or suggestions about the seminar, or if you would like to give a presentation. I will relinquish my tyrannical hold on organizing responsibilities next semester (Spring 2024), so if you’re interested in organizing later, also get in contact with me.
Fall 2023 Schedule:
Sept 11: Reese Lance
Title: Cohomology ring of the full flag variety
Abstract: We compute the cohomology ring of the full flag variety, Fl(n). This cohomology ring has a well known “geometric” basis of Schubert classes, arising from an affine paving of Fl(n). Here we take a different route to present an “algebraic basis” in terms of Chern classes of certain tautological bundles over Fl(n). Along the way we will learn (then heavily utilize) how to compute cohomology of the total space of projective bundles in terms of cohomology of the base space. I doubt we will get to this but the so-called Schubert polynomials are essentially “change of basis formulas” between the geometric and algebraic basis.
Sept 18: Tiger Cheng
Title: Proof of the Kontsevich Formula
Abstract: In this talk I will prove Kontsevich’s formula for the number of degree d rational plane curves passing through 3d-1 general points in P^2. In order to get there, we will develop some understanding of the moduli space of stable maps and the definition of Gromov-Witten numbers. Time permitting, we will also explore how these notions relate to quantum cohomology, and various enumerative geometry questions on homogeneous spaces.
Sept 25: UNIVERSITY WELLNESS DAY
Title: YOU HAVE BEEN DECLARED WELL
Oct 2: Luke Conners
Title: An Invitation to Derived Categories and Derived Functors
Abstract: Classical derived functors form the technical backbone of a vast swath of homology and cohomology theories. Consideration of these functors in the more modern framework of derived categories often both expands and simplifies the theory considerably and has become mainstream in fields as disparate as knot theory and microlocal analysis. Despite its elegance, the language of derived categories can be intimidating to the uninitiated. In this talk, we hope both to make derived categories more accessible to the working mathematician and demonstrate their utility. Our main technical goal will be to generalize first examples of classical derived functors (Ext and Tor) as functors between derived categories. We will be motivated throughout by examples from commutative algebra, algebraic geometry, and differential geometry.
Oct 9: Alex Foster
Title: Equivariant Cohomology
Abstract: Equivariant cohomology is a cohomology theory that captures information about a group action on a space. After a short discussion of its definition and properties, we’ll use fixed points and fixed curves to compute the equivariant cohomology for a simple class of spaces called GKM spaces. Then we will use more advanced tools to compute it for a non-GKM space.
Oct 16: Joe Compton
Title: A deformed product in cohomology
Abstract: We will begin by stating a theorem from 2004 (conjectured by Fulton) regarding the tensor decomposition problem for representations of GL(n, C). The direct generalization of this theorem for arbitrary reductive groups is false. We will use this as motivation for defining a “new” product in cohomology (due to Belkale and Kumar) and explain how this provides an appropriate setting for the generalization of the theorem.
Oct 30: Luke Conners
Title: An Invitation to Derived Categories and Derived Functors II
Nov 13: Reese Lance
Title: Cohomology ring of the full flag variety II