Northwestern Geometry&Physics (Virtual) Seminar Schedule

****See this website for the latest schedule****

Spring 2020

Time: Thursdays 2:00pm-3:00pm (CST)

Place: Zoom Meeting Room

Zoom Meeting ID: 922-4351-6463 (E-mail Bahar Acu to access the meeting password.)

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

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Northwestern Geometry&Physics Seminar Schedule - Winter 2019

Place and Time

Pretalk: Thursdays 1:00pm - 1:50pm in Lunt 105 (aimed at graduate students)

Research talk: Thursdays 4:10pm - 5:00pm in Lunt 107

Organizers: Bahar Acu and Ezra Getzler

January 24, 2019

- Speaker: Kyler Siegel (Columbia)
- Pretalk title: Introduction to symplectic embedding problems and symplectic capacities
- Abstract: A basic problem in symplectic geometry is to determine what types of symplectic embeddings are possible. The first nontrivial result is Gromov's celebrated nonsqueezing theorem, which puts fundamental constraints on symplectic transformations beyond volume considerations. Since then, various finer obstructions have been discovered and formalized into the notion of a symplectic capacity. In this talk I will discuss some toy problems (still largely open), survey key results, and highlight the significance of Floer homology and symplectic field theory.

- Research talk at 3pm in room Lunt 105 (Note the unusual time and room.)
- Title: Higher symplectic capacities
- Abstract: I will describe a new family of symplectic capacities based on the equivariant L-infinity structure on symplectic cohomology. These give state of the art obstructions for certain embedding problems such as one symplectic ellipsoid into another. Using symplectic field theory, these capacities can be interpreted in terms of holomorphic curves with tangency constraints. I will also give some structural results which are useful for computations.

February 7, 2019

- Speaker: Emmy Murphy (Northwestern)

- Pretalk title: Weinstein manifolds and wrapped Fukaya categories
- Abstract: We discuss the basic geometry of Weinstein manifolds, particularly their relationship with contact geometry, complex geometry, and handlebody theory. We will also discuss the wrapped Fukaya category, a powerful algebraic object associated to any Weinstein manifold. Besides the basic definition and properties, we will also discuss the essential generation results, and surgery formulas relating it to Legendrian contact homology.

- Research talk
- Title: Inductively collapsing wrapped Fukaya categories and flexibility
- Abstract: A Weinstein manifold is an exact symplectic manifold which has a Lagrangian skeleton: this includes all cotangent bundles and all smooth affine varieties. Associated to any Weinstein manifold is the wrapped Fukaya category of that manifold, an algebraic invariant of the manifold. An important example is that the wrapped category of C^n is trivial. We discuss a partial converse to this statement. This will take us through the worlds of arboreal singularities, partially wrapped categories of Weinstein sectors, and Legendrian h-principles.

February 14, 2019

- Speaker: Sara Venkatesh (IAS)
- Pretalk title: Hamiltonian Floer theory and symplectic cohomology
- Abstract: Starting with closed symplectic manifolds, we introduce Hamiltonian Floer homology and discuss the dynamical information it encodes. We then translate this story to open symplectic manifolds, on which symplectic cohomology is defined.

- Research talk
- Title: Symplectic cohomology of subdomains
- Abstract: Mirror symmetry predicts the existence of Floer invariants that yield “local” information. Guided by this, we construct a quantitative symplectic cohomology theory that detects Floer-essential Lagrangians within subdomains. We illustrate the quantitative behavior of this theory by examining negative line bundles over toric symplectic manifolds.

February 21, 2019

- Speaker: Oleg Lazarev (Columbia)
- Pretalk title: Flexibility and rigidity for Weinstein domains
- Abstract: Weinstein domains are the symplectic analogs of smooth handle-bodies and can be understood explicitly via Legendrian knot theory. I will give examples of Weinstein domains, discuss recent flexibility results, including the high dimensional existence theorem, and give an overview of wrapped Fukaya category, an algebraic invariant built using Lagrangians and J-holomorphic curves.

- Research talk
- Title: Weinstein presentations and K-theory
- Abstract: Weinstein domains are the symplectic analogs of smooth handle-bodies and their presentations can be modified by handle-slides and handle creation/cancellation as in the smooth setting. I will show how to construct minimal presentations for Weinstein domains and give examples of different Legendrian knots producing the same Weinstein domain. I will also explain how these presentations are related to the algebra of the wrapped Fukaya category and its Grothendieck group. In particular, the rank of the Grothendieck group is bounded by the rank of singular cohomology and there is a symplectic interpretation of a theorem of Thomason about subgroups of the Grothendieck group and split-generating subcategories.

March 7, 2019

- Speaker: Baptiste Chantraine (Nantes)
- Pretalk title: Front projections and Lagrangian cobordisms
- Abstract: I will explain how we can use front projections to represent some cobordisms between Legendrian submanifolds. This will allow use to describe cobordisms associated to Lagrangian surgeries and we will see why this type of cobordisms are a priori not symmetric.

- Research talk at 3pm in room Lunt 105 (Note the unusual time and room.)
- Title: Lagrangian cobordisms between Legendrian submanifolds and Lagrangian surgeries
- Abstract: In this talk I will study Lagrangian cobordisms between Legendrian submanifolds arising from some Lagrangian surgeries. From the Floer theory of those cobordisms we can deduce some geometrical descriptions of certain iterated cones in the Fukaya category. I will then explain how those considerations lead to a proof of the fact that Lagrangian cocores generates the wrapped Fukaya category of a Weinstein manifold. This is joint work with G. Dimitroglou Rizell, P. Ghiggini and R. Golovko.

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Northwestern Geometry&Physics Seminar Schedule - Fall 2018

Place and Time

Pretalk: Thursdays 1:00pm - 1:50pm in Lunt 107 (aimed at graduate students)

Research talk: Thursdays 4:10pm - 5:10pm in Lunt 107

Organizers: Bahar Acu and Ezra Getzler

October 4, 2018

- Speaker: Ezra Getzler (Northwestern)
- Title: Variational calculus in the Batalin-Vilkovisky formalism and general covariance
- Abstract: Motivated by supersymmetry, Batalin and Vilkovisky reformulated the equations governing Lagrangian mechanics in the variational calculus as a Maurer-Cartan equation (vanishing of curvature). This allowed them to arrive at a new understanding of symmetries off-shell (i.e. where the Euler-Lagrange equation does not hold). In this talk, I will show how a modification of their Maurer-Cartan equation can handle the action of the diffeomorphism group on the world-sheet (i.e. general covariance of the theory). Our approach involves the introduction of a curvature to Maurer-Cartan equation. This curvature is central (a scalar multiple of the identity matrix): this is analogous to the Berry phase in the Hamiltonian approach to quantum theory (though this is really nothing more than a formal analogy).

October 11, 2018

- Speaker: Tim Large (MIT)
- Title: Steenrod operations and the Floer homotopy type
- Abstract: A natural question in symplectic geometry and gauge theory is whether one can construct a homotopy type underlying the Floer homology groups. There are usually topological obstructions to this, but if one could, the Z/p coefficient versions of the Floer groups would come with natural Steenrod operations. On the other hand, one could try construct these operations as equivariant refinements of the pair-of-pants product, analogously to the story for singular homology. In this talk, we will explain how these two approaches relate to each other.

October 25, 2018

- Speaker: James Pascaleff (UIUC)
- Pretalk title: Monotone Lagrangians and disk counting
- Research talk title: Wall-crossing formulas for Lagrangian mutations
- Abstract: In this talk I will discuss several versions of the wall-crossing phenomenon that arise in Floer theory. The first is the interpretation of the wall-crossing formula as a coordinate change between charts on the moduli space of compact exact Lagrangian objects in the Fukaya category of an exact symplectic manifold M, and the second is the behavior of superpotentials of those same Lagrangians when M is replaced by a partial compactification X. By relating the Fukaya categories of M and X, Dmitry Tonkonog and I showed how the latter is determined by the former in a general context. This allows us to derive new wall-crossing formulas in complex dimension greater than two. A third aspect is the way that the same algebra governs also the ring structure on the wrapped Floer cohomology of certain non-compact Lagrangians.

November 1, 2018

- Speaker: Harold Williams (UC Davis)
- Title: Kasteleyn operators from mirror symmetry
- Abstract: Given a consistent bipartite graph Γ in T^2 with a complex-valued edge weighting E, we show the following two constructions are the same. The first is to form the Kasteleyn operator of ( Γ, E) and pass to its spectral transform, a coherent sheaf supported on a spectral curve in C*^2. The second is to form the conjugate Lagrangian L in T* T^2 of Γ, equip it with a brane structure prescribed by E, and pass to its mirror coherent sheaf. This lives on a stacky toric compactification of C*^2 determined by the Legendrian link which lifts the zig-zag paths of Γ (and to which the noncompact Lagrangian L is asymptotic). We work in the setting of the coherent-constructible correspondence, a sheaf-theoretic model of toric mirror symmetry. This is joint work with David Treumann and Eric Zaslow.

November 8, 2018

- Speaker: Michele Schiavina (Max-Planck Institut, Bonn)
- Title: Equivalence of gauge theories in the presence of boundaries: insights from General Relativity
- Abstract: The standard notion of classical equivalence between field theories establishes an arguably simple relation between the critical loci of two variational problem, with little to no mention on how to properly treat the data coming from their symmetries. Although for some applications this is enough, we argue that modern "cohomological" approaches to field theories, like those that stem from the works of Batalin, Fradkin and Vilkovisky, might help in roadmapping beyond this standard notion. In this talk I will present one possible approach to the problem and show how natural examples coming from General Relativity in different space-time dimensions might be a rich class of theories for which a refined notion of equivalence becomes relevant.

November 15, 2018

- Speaker: Kevin Sackel (MIT)
- Pretalk title: Convex surfaces in contact 3-manifolds
- Abstract: We discuss contact geometry in three dimensions, focusing in particular on the power of convex surfaces.

- Research talk title: Convex contact handle decompositions
- Abstract: Analogous to Weinstein structures in symplectic geometry, there is a notion of convex structures in contact geometry. We discuss an explicit surgery theory for these contact manifolds with convex structures, showing that they decompose naturally into handles. We also explore the relationships between these handle decompositions and Weinstein open books, proving as a corollary that every closed contact manifold has a convex structure.

November 22, 2018

- No seminar (Thanksgiving)

November 29, 2018

- Speaker: Honghao Gao (Institut Fourier)
- Pretalk title: An Odyssey of Augmentations
- Abstract: In this talk, I will introduced four types of augmentations of a knot or a link. They are knot/link invariants defined over either a framed cord algebra or a Legendrian differential graded algebra associated to the knot/link. I will explain the relations among these augmentations.

- Research talk at 3pm (Note the unusual time.)
- Title: Augmentations and sheaves for knot conormals
- Abstract: Knot invariants can be defined using Legendrian isotopy invariants of the knot conormal. There are two types of invariants raised in this way: one is the knot contact differential graded algebra together with augmentations associated to this dga, and the other one is the category of simple sheaves microsupported along the knot conormal. The SFT and Nadler-Zaslow correspondence suggest a connection between the two types of invariants. In this talk, I will manifest an explicit bijection between augmentations and simple sheaves for knot conormals.

December 6, 2018

- Speaker: Dmitry Tonkonog (UC Berkeley)
- Pretalk title: Lagrangian tori in Fano manifolds
- Abstract: I will recall a beautiful construction, due to Vianna, that associates to each Markov triple a monotone Lagrangian torus in the complex projective plane. I will explain how these tori can be distinguished using holomorphic disk counts, and the wall-crossing formula that arises in this question.

- Research talk at 3pm (Note the unusual time.)
- Title: Disk potentials and mirror symmetry
- Abstract: Fix a Fano manifold and a monotone Lagrangian torus inside it. The simplest enumerative invariant of the torus is its disk potential, which is a certain Laurent polynomial. It can be seen as a piece of the mirror to the Fano, according to the SYZ conjecture. I will explain how to prove some classical mirror symmetry predictions from this point of view. I will focus on two theorems: a formula for the quantum periods of a Fano manifold in terms of period integrals, and the quantum Lefschetz formula.

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This page is maintained by Bahar Acu.

For the departmental seminar webpage, click here.