Place: Zoom Meeting Room (Virtual)
Time: Central Time (CT)
Organizers: Ezra Getzler, Dogancan Karabas (dogancan.karabas[at]northwestern dot edu), and Eric Zaslow
Tuesday, January 12, 2021 at 1:00pm
- Speaker: Tony Yue Yu (Paris-Sud)
- Title: Non-archimedean mirror symmetry and its applications
- Abstract: Mirror symmetry is one of the most mysterious dualities in mathematics. My research explores a new approach to mirror symmetry, via non-archimedean geometry. I will give an overview of the recent progress in this direction. I will talk about non-archimedean enumerative geometry, the Frobenius structure conjecture, and derived non-archimedean geometry. I will also describe an application towards cluster algebras in representation theory, and another application towards the moduli spaces of Calabi-Yau pairs and their compactifications.
- Zoom Meeting ID: 924 2487 2281 (email Dogancan Karabas to access the meeting password.)
Monday, January 25, 2021 at 3:00pm
- Speaker: Junliang Shen (MIT)
- Title: The P=W conjecture and hyper-Kähler geometry
- Abstract: Topology of Hitchin’s integrable systems and character varieties play important roles in many branches of mathematics. In 2010, de Cataldo, Hausel, and Migliorini discovered a surprising phenomenon which relates these two very different geometric objects in an unexpected way. More precisely, they predict that the topology of Hitchin systems is tightly connected to Hodge theory of character varieties, which is now called the “P=W” conjecture. In this talk, we will discuss recent progress of this conjecture. In particular, we focus on general interactions between topology of Lagrangian fibrations and Hodge theory in hyper-Kähler geometries. This hyper-Kähler viewpoint sheds new light on both the P=W conjecture for Hitchin systems and the Lagrangian base conjecture for compact hyper-Kähler manifolds.
- Zoom Meeting ID: 938 2310 1668 (email Dogancan Karabas to access the meeting password.)
Monday, February 01, 2021 at 3:00pm
- Speaker: Gus Schrader (Columbia University)
- Title: Modular functors and higher Teichmuller theory
- Abstract: In the approach to the construction of invariants of links in 3-manifolds pioneered by Witten, a crucial role is played by the notion of a modular functor. Such a functor assigns to a surface a finite dimensional representation of its mapping class group in a way that is compatible with gluing surfaces together along a boundary circle. I’ll report on joint work with A. Shapiro in which we prove the conjecture of Fock and Goncharov that the quantization of moduli spaces of local systems on hyperbolic surfaces, a.k.a. higher Teichmuller theory, delivers an infinite dimensional analog of such a modular functor. I’ll conclude by describing some applications of our construction to the representation theory of non-compact quantum groups.
- Zoom Meeting ID: 942 1636 3027 (email Dogancan Karabas to access the meeting password.)
Thursday, February 25, 2021 at 1:00pm
- Speaker: Lara Simone Suarez Lopez (Ruhr-Universität Bochum)
- Title: On the rigidity of Legendrian cobordisms
- Abstract: In symplectic and contact topology, there are different notions of cobordisms. One of them due to Arnold concerns Lagrangian and Legendrian cobordisms between Lagrangian/Legendrian submanifolds. For Lagrangian cobordisms Biran-Cornea showed that monotone ones preserve Floer homology. In the first part of this talk I will show a similar statement for Legendrian cobordisms that are hypertight. Then I will talk about positive Legendrian cobordism. This last part is based on a joint project with Maÿlis Limouzineau.
- Zoom Meeting ID: 936 2594 0833 (email Dogancan Karabas to access the meeting password.)
Thursday, March 04, 2021 at 3:00pm
- Speaker: Nick Rozenblyum (University of Chicago)
- Title: Langlands duality and categorical traces
- Abstract: By analogy with characteristic p geometry, Beilinson and Drinfeld formulated a categorical analogue of the Langlands program for Riemann surfaces over the complex numbers. One of the remarkable features of this conjecture is its relation to conformal field theory and higher dimensional quantum field theory. However, this formulation is very specific to complex algebraic geometry. I will describe a general categorical conjecture suitable to arbitrary geometric settings, including l-adic sheaves on algebraic curves over finite fields. One remarkable application of these ideas is a description of the space of automorphic forms as the categorical trace (aka Hochschild homology) of Frobenius. This is joint work with Arinkin, Gaitsgory, Kazhdan, Raskin, and Varshavsky.
- Zoom Meeting ID: 997 6959 0131 (email Dogancan Karabas to access the meeting password.)
- Notes: PDF
- Recording: Zoom
Thursday, March 11, 2021 at 3:00pm
- Speaker: Renato Vianna (Universidade Federal do Rio de Janeiro)
- Title: Sharp ellipsoid embeddings and almost-toric mutations
- Abstract: We will show how to construct volume filling ellipsoid embeddings in some 4-dimensional toric domain using mutation of almost toric compactification of those. In particular we recover the results of McDuff-Schlenk for the ball, Fenkel-Müller for product of symplectic disks and Cristofaro-Gardiner for E(2,3), giving a more explicit geometric perspective for these results. To be able to represent certain divisors, we develop the idea of symplectic tropical curves in almost toric fibrations, inspired by Mikhalkin’s work for tropical curves. This is joint work with Roger Casals.
Obs: The same result appears in “On infinite staircases in toric symplectic four-manifolds”, by Cristofaro-Gardiner — Holm — Mandini — Pires. Both papers were posted simultaneously on arXiv.
- Zoom Meeting ID: 997 6959 0131 (email Dogancan Karabas to access the meeting password.)
- Recording: Zoom