Spring 2020

Time: Thursdays 2:00pm-3:00pm (CST)
Place: Zoom Meeting Room (Virtual)
The meeting room opens at 1:45pm and closes at 3:30pm.
Organizers: Bahar Acu (baharacu[at]northwestern dot edu), Ezra Getzler, and Eric Zaslow

April 23, 2020

• Speaker: Doğancan Karabaş (Northwestern)
• Title: Fukaya categories of some rational homology balls via microlocal sheaves
• Abstract: It is shown by Kashiwara and Schapira (1980s) that for every constructible sheaf on a smooth manifold, one can construct a closed conic Lagrangian subset of its cotangent bundle, called the microsupport of the sheaf. This eventually led to the equivalence of the category of constructible sheaves on a manifold and the Fukaya category of its cotangent bundle by the work of Nadler and Zaslow (2006), and Ganatra, Pardon, and Shende (2018) for partially wrapped Fukaya categories. One can try to generalise this and conjecture that Fukaya category of a Weinstein manifold can be given by constructible (microlocal) sheaves associated with its skeleton. In this talk, I will briefly explain these concepts and confirm the conjecture for a family of Weinstein manifolds which are certain quotients of A_n-Milnor fibres. I will outline the computation of their wrapped Fukaya categories and microlocal sheaves on their skeleta, called pinwheels.
• Notes: PDF
• Recording: Zoom link

April 30, 2020

• Speaker: Honghao Gao (Michigan State University)
• Title: Infinitely many Lagrangian fillings
• Abstract: A filling is an oriented surface bounding a link. Lagrangian fillings can be constructed via local moves in finite steps, but it was unknown whether a Legendrian link could admit infinitely many Lagrangian fillings. In this talk, I will show that Legendrian torus links other than (2,m), (3,3), (3,4), (3,5) indeed have infinitely many fillings. These fillings are constructed using Legendrian loops, and proven to be distinct using the microlocal theory of sheaves and the theory of cluster algebras. This is a joint work with Roger Casals.
• Notes: PDF and Slides
• Recording: Zoom link

May 7, 2020

• Speaker: Catherine Cannizzo (Simons Center for Geometry and Physics at Stony Brook)
• Title: Towards homological mirror symmetry for genus 2 curves
• Abstract: The first part of the talk will discuss work in https://arxiv.org/abs/1908.04227 on constructing a Donaldson-Fukaya-Seidel type category for the generalized SYZ mirror of a genus 2 curve. We will explain the categorical mirror correspondence on the cohomological level. The key idea uses that a 4-torus is SYZ mirror to a 4-torus. So if we view the complex genus 2 curve as a hypersurface of a 4-torus V, a mirror can be constructed as a symplectic fibration with fiber given by the dual 4-torus V^. Hence on categories, line bundles on V are restricted to the genus 2 curve while fiber Lagrangians of V^ are parallel transported over U-shapes in the base of the mirror. Next we describe ongoing work with H. Azam, H. Lee, and C-C. M. Liu on extending the result to a global statement, namely allowing the complex and symplectic structures to vary in their real six-dimensional families. The mirror statement for this more general result relies on work of A. Kanazawa and S-C. Lau.
• Notes: PDF
• Recording: Zoom link

May 14, 2020

• Speaker: Mohammed Abouzaid (Columbia University)
• Title: Floer homotopy without spectra
• Abstract: The construction of Cohen-Jones-Segal of Floer homotopy types associated to appropriately oriented flow categories extracts from the morphisms of such a category the data required to assemble an iterated extension of free modules (in an appropriate category of spectra). I will explain a direct (geometric) way for defining the Floer homotopy groups which completely bypasses the algebra. The key point is to work on the geometric topology side of the Pontryagin-Thom construction. Time permitting, I will also explain joint work in progress with Blumberg for building a spectrum from the new point of view, as well as various generalisations which are relevant to Floer theory.
• Notes: PDF and GoodNotes
• Recording: Zoom link

May 21, 2020

• Speaker: Maÿlis Limouzineau (Mathematical Institute of the University of Cologne)
• Title: About reversing surgery for Lagrangian fillings of Legendrian knots.
• Abstract: Consider Sigma an immersed Lagrangian filling of a Legendrian knot Lambda. Polterovich surgery allows to solve double points to get an embedded Lagrangian filling of Lambda, each solved point increasing the genus by one. We wonder if the surgery procedure is reversible: Can any Lagrangian filling Sigma with genus g(Sigma)>0 and p(Sigma) double points can be obtain from surgery on a Lagrangian filling Sigma’ with g(Sigma’)=g(\Sigma)-1 and p(\Sigma’)=p(\Sigma’)+1? We will see that the answer is no and give a family of counter-examples. This is work in progress with Orsola Capovilla-Searle, Noémie Legout, Emmy Murphy, Yu Pan and Lisa Traynor.
• Notes: PDF and GoodNotes
• Recording: Zoom link

May 28, 2020

• Speaker: Jesse Wolfson (University of California, Irvine)
• Title: The Geometry of Hilbert’s 13th Problem
• Abstract: The goal of this talk is to explain how enumerative geometry can be used to simplify the solution of polynomials in one variable. Given a polynomial in one variable, what is the simplest formula for the roots in terms of the coefficients? Hilbert conjectured that for polynomials of degree 6,7 and 8, any formula must involve functions of at least 2, 3 and 4 variables respectively (such formulas were first constructed by Hamilton). In a little-known paper, Hilbert sketched how the 27 lines on a cubic surface should give a 4-variable solution of the general degree 9 polynomial. In this talk I’ll recall Klein and Hilbert’s geometric reformulation of solving polynomials, explain the gaps in Hilbert’s sketch and how we can fill these using modern methods. As a result, we obtain best-to-date upper bounds on the number of variables needed to solve a general degree n polynomial for all n, improving results of Segre and Brauer.
• Notes: Slides
• Recording: Zoom link

June 4, 2020

• Speaker: Xin Jin (Boston College)
• Title: Homological mirror symmetry for the universal centralizers
• Abstract: I will present work (partly in progress) on the homological mirror symmetry for the universal centralizer $J_G$ associated to a complex semisimple Lie group G. The A-side will be a partially wrapped Fukaya category of $J_G$ and the B-side is the category of coherent sheaves on the categorical quotient of a dual maximal torus by the Weyl group action (with some modification if $G$ has a nontrivial center).
• Notes: Notes
• Recording: Zoom link

June 11, 2020

• Speaker: Haniya Azam (Lahore University of Management Sciences)
• Title: Topological Fukaya category of Riemann surfaces
• Abstract: Introduced by Fukaya in his work on Morse theory, A-infinity categories and Floer homology, the Fukaya category constitutes one side of the homological mirror symmetry conjecture of Kontsevich. In this talk, I will present a topological variant of Floer homology and the Fukaya category of a Riemann surface of genus greater than one. We will introduce an admissibility condition borrowed from Heegard Floer theory which ensures invariance under isotopy and finiteness and compute the Grothendieck group of the derived Fukaya category in this setup. If time permits, we will also discuss the induced action of the Mapping class group on the topological Fukaya category. This talk is based on joint work with Christian Blanchet.
• Notes: OneNote, GoodNotes
• Recording: Zoom link

June 18, 2020

• Speaker: Eric Zaslow (Northwestern)
• Title: A Diagrammatic Calculus For Legendrian Surfaces
• Abstract: I will describe work with Roger Casals. We show how planar diagrams called N-graphs encode Legendrian surfaces which cover the plane N-to-1. These N-graphs can be used to express Reidemeister moves, surgeries, and connect sums; to describe a Markov move a` la braids; to construct large classes of examples of any genus; to define moduli spaces which can be used to distinguish surfaces up to Legendrian isotopy; to construct exact Lagrangian fillings; and to define a (hopefully interesting) planar algebra.
• Notes: PDF
• Recording: Zoom link