# Past jGMRT Seminars

jumpto:

## 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

**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.

Abstract:

**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

Abstract: TBD

**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**

Title: A Visual Approach to the Representation Theory of sl(3,C)

Abstract: In this talk, we will study the representation theory of the semisimple Lie algebra sl(3,C). We will start with an introduction to the algebra, and then study several representations of the algebra via their weight diagrams. Ultimately this visual approach points to the general form of the irreducible representations of sl(3,C), leading to two “major theorems” that we will introduce and discuss (but not prove).

**Fall 2022:**

This semester was our first test run. We started close to the end of the semester, thus only 3 talks featured here:

#### November 8 – Reese Lance

Title: The Derived Category of Coherent Sheaves. Lecture Notes

Abstract: The object of study is a certain category which one can associate to a certain class of topological spaces which ends up being a useful tool in many areas of algebraic geometry and representation theory. It can be used as an invariant to detect isomorphisms as well as encode properties of the geometry/topology of the space.

The talk will feature 3 sections: The category of coherent sheaves, the derived category of an abelian category, and what happens when you combine the two/why someone may care. Familiarity with basics of sheaves and categories will ensure you get the most out of this talk. I’m presenting a streamlined path to D^bCoh(X), so we will try not to introduce anything unnecessary. In particular, we will not discuss derived functors for the sake of brevity. If there ends up being a part II, it will be all about the role of derived functors in this story, which are necessary to prove the interesting results which we will state in the third section of this talk.

Many advanced topics in AG require understanding of the derived category of coherent sheaves as a starting off point, so we will be in a good position to discuss a large number of topics afterwards, some of which I will have listed in the notes, to be posted after the talk. These would all make great talk topics for anyone interested.

#### November 15 – Reese Lance

Title: The Derived Category of Coherent Sheaves Part II

#### November 22 – Paul Teszler

Title: Introduction to Intersection Homology

Abstract: Nice cohomological properties like Poincare Duality can fail in spaces with singularities. Intersection Homology aims to recover these luxuries by ignoring certain chains that intersect singularities “too much.” In this talk, we aim to introduce this theory with motivating examples and briefly discuss some historical uses and modern applications.