The junior geometry seminar meets every Tuesday at 3:45-5pm central. What is the junior geometry seminar?
The organizational meeting is on January 19th at 3:45 pm central.
Zoom links and email reminders will be sent to the mailing list. Click here to be added to the mailing list.
Please email the spring 2021 Organizer: Jackson Van Dyke with any questions.
List of past organizers (aka JG ORGANIZER HALL OF FAME )
The participants of the jr. geometry seminar are encouraged to raise awareness about a social justice issue in mathematics that they personally care about.
Click here for more information about the jr. geometry SJA format.
The titles and abstracts of the talks are also available here. at the official math department seminar calendar.
Click here for a schedule of the senior geometry seminar for Spring 2021.
Title: The Moment Map and Equivariant Cohomology
Abstract: I will review the classic paper by Atiyah and Bott titled "The Moment Map and Equivariant Cohomology". The motivational statement of that paper is that if \(f\) is a function on a symplectic manifold \(M\) whose hamiltonian flow generates a circle action, the integral of \(e^{if}\) localizes to a certain integral over the critical locus of \(f\). First, I will briefly tell you why I'm personally interested in this theorem, (it tells you stuff about complex Chern Simons theory,) then I will talk about how to prove it. The proof will be easy once we learn about equivariant cohomology, and how to do equivariant cohomology in a de Rahm way.
Title: Topological field theory and algebraic geometry
Abstract: In the first part of the talk I will go over the basics of topological field theory (TFT) and algebraic geometry. Then, given a variety M, we will attach a TFT called the B-model. We can think of this TFT as forming a ``package'' of algebraic invariants/linearizations of M. We will see that various conditions and facts about general TFTs correspond to conditions and facts about the geometry of M.
Title: An Introduction to \(\ell\)-adic Sheaves
Abstract: We will introduce the definition of \(\ell\)-adic sheaves in algebraic geometry. The talk will begin by motivating why we might define such things and end with the rigorous definition of \(\mathbb{Q}_\ell\)-sheaves.
Title: Monodromy on the stack of formal groups
Abstract:
The stack of formal groups is known to possess profound (and still quite mysterious!) connections to stable homotopy theory.
In this talk, we will discuss a derived variant of this due to Lurie: the moduli stack of oriented formal groups. On a formal level, this realizes the conjectural connection perfectly. We will explain how algebro-geometric study of deformations of oriented formal groups allow us to recover some well-known theorems in chromatic homotopy theory.
Title: Quivers and their Representations
Abstract: Quivers are another name for directed graphs. In this talk we will define what a quiver representation is and explain some basic results, constructions, and examples of them. One major result is Gabriel's theorem, which says that a quiver has finitely many indecomposable representations if and only if it is a simply laced Dynkin diagram. We will explain how these indecomposable representations are in bijection with the positive roots of the corresponding simple lie algebra. Finally we will construct quiver varieties as a way to describe the space of representations of a quiver and show how the Springer resolution in type A appears as an example.
Title: Gaitsgory's construction of central sheaves on the affine flag variety
Abstract: Let G be a (split) connected reductive group over a finite field \(F_q\), and let \(K=F_q((t))\), \(O=F_q[[t]]\). Then a theorem of Bernstein realizes the bi-\(G(O)\)-invariant compactly supported functions on \(G(K)\) as the center of the algebra of bi-\(I\)-invariant compactly supported functions on \(G(K)\), where \(I\) denotes the Iwahori subgroup of \(G(O)\). In this talk we are going to talk about Gaitsgory's geometric version of this result. More precisely, using nearby cycles, Gaitsgory constructed a tensor functor from the symmetric monoidal category of \(G(O)\)-invariant perverse sheaves on the affine grassmannian to the center of the monoidal category of Iwahori-invariant perverse sheaves on the affine flag variety. Moreover, this functor satisfies a myriad of nice properties, some of which are invisible at the level of Hecke algebras.
Title: How can we count surfaces representing a given class in \(H_2(X)\)?
Abstract: As the title suggests, I'll try to address this innocent-looking question but more importantly, I'll focus on what can possibly go wrong in trying to answering it. Spoiler alert: a lot. Incredibly enough working in a symplectic setting will give us a lot of mileage thanks to a strong connection between topology, analysis and algebra that we can exploit here. To my knowledge, the question is still not fully answered but the techniques involved have far reaching (spin\(^c\) ) connections in the realm of 4-manifolds theory. Anyone interested in knowing more about what symplectic geometry is about, the wild world of 4-manifolds or to hear a more geometric viewpoint on classic algebraic geometry topics is more than welcomed to join. The first half of the talk is going to be pretty elementary (a.k.a. (co)homology), while in the second part I'll focus on the main ideas in order to keep the necessary background to a minimum.
Title: Randomly sampling half-translation surfaces
Abstract: We will discuss a recent theorem of Francisco Arana-Herrera which states that horoball segment measures on the bundle of unit-length measured geodesic laminations over the moduli space of hyperbolic surfaces equidistribute with respect to Mirzakhani measure. After a brief overview of the proof, we will discuss an application of this theorem, which gives a concrete recipe to randomly sample hyperbolic surfaces. Once we have a random hyperbolic surface, we can use train tracks to randomly sample a quadratic differential using the Thurston homeomorphism, thus giving us a random half-translation surface.
Title: Chern classes in complex algebraic geometry
Abstract: Any smooth projective variety X over the complex numbers may alternatively be viewed as a compact Kähler complex manifold by passing to its analytification. As such, several cohomology theories are available to us when setting out to study this class of varieties, depending on whether we choose to work in the settings of algebraic topology (singular/de Rham cohomology), algebraic geometry (Chow cohomology), or complex geometry (Dolbeault cohomology). With that in mind, we will explore some of the various ways in which we can go about defining a theory of Chern classes on these varieties (a certain type of cohomological invariants associated to vector bundles, or more generally to coherent sheaves on X), and give some hints as to how these constructions relate to one another.
Title: The almost-Fuchsian moduli space
Abstract: A construction of Donaldson assigns to a closed, oriented surface C a hyperkähler manifold M, that can be interpreted in several different ways: 1) as an open neighborhood of the zero section of Teichmüller space for C, 2) as the space of almost-Fuchsian 3-manifolds, 3) as an open subspace of the SL(2,C) character variety of C. I will explain what most of these words mean, and how they are connected. This will be accessible to anyone who likes hyperbolic or Riemann surfaces.
Abstract: [Cancelled]
Title: Tropical Cycles and Rational Equivalence
Abstract: Algebraic cycles on a variety tell us a lot about the structure and behavior of that variety. They give rise to many important invariants such as Chow groups, Picard groups, and cohomology. As usual, there is a tropical analogue to algebraic cycles in tropical geometry, which are called tropical cycles. Unlike in the classical world, tropical cycles are less understood. This talk will discuss work by Allermann and Rau who laid the groundwork for tropical cycles as well as outline their proof that the bounded tropical Chow group of \(\mathbb{R}^n\) is the group of fan cycles.
Abstract: [Cancelled]
Title: Topological Field Theory and Serre Duality
Abstract: In this talk I will discuss my attempts to interpret Serre duality using 2d topological field theory. To a smooth projective variety we will assign a framed 2d TFT and observe that the Serre functor arises as the image of a certain one dimensional bordism. We will interpret Serre duality using boundary theories for this 2d TFT.