Mohammed Abouzaid (Columbia/Stanford), Nate Bottman (Max Planck),
Catherine Cannizzo (UC Riverside), Sheel Ganatra (USC), Kyler Siegel (USC)
In Spring 2023, our seminar meets online Fridays at 12 pm ET.
Zoom Link: https://columbiauniversity.zoom.us/j/96547665252?pwd=T0pOb28vM2JJSnRYM2RjeFBhWU96QT09
Meeting ID: 965 4766 5252 Passcode: 310229
One tap mobile: +16694449171, 96547665252#,*310229# US
Zoom meetings open 15 minutes before the talk begins.
Talks are 50 minutes, followed by a 10 min Q&A and an informal discussion session.
Preceding the talk, we will host an informal symplectic tea each Friday on gather.town from 11:45 am - 12pm ET at the following URL: https://gather.town/app/abBT3vhSL9vD0zbL/virtualsymplectictea
To receive announcements of talks, please join our email listerv.
https://forms.gle/fnBvwohDHFPK5Qfc6
Abstract: Hedden, Herald, and Kirk initiated a program to study knots using Lagrangian Floer Theory. Given a certain decomposition of the knot and appropriate perturbations, one can use character varieties to obtain a pair of Lagrangians in a space called the pillowcase. The intersections of these Lagrangians can be used to give bounds on the singular instanton homology of the knot. My work shows how to find these Lagrangians for tangle sums. Using this to compute new examples, I show that the Lagrangian Floer homology alone is not a knot invariant, suggesting that deriving a knot invariant from these Lagrangians will require more sophisticated methods.
Abstract: We discuss the map on Chekanov-Eliashberg DGAs induced by an exact Lagrangian cobordism, and an analytic strategy to lift the map to integer coefficients, introduced by Fukaya, Oh, Ohta and Ono and further adapted by Ekholm, Etnyre, and Sullivan and Karlsson respectively to the current setting. We then explain how this strategy can be applied to find a concrete combinatorial formula for a mini-dipped pinch move, thereby completely determining the integral DGA maps for all decomposable, orientable Lagrangian cobordisms. If time permits, we will show how to obtain this formula in a model case. We will also go into future potential work, including applications to Heegaard Floer Homology and nonorientable cobordisms.
Abstract: We will introduce a mathematical formulation of the open/closed correspondence originally proposed by Mayr, which is a correspondence in genus zero between the open Gromov-Witten theory of toric Calabi-Yau 3-folds and the closed Gromov-Witten theory of toric Calabi-Yau 4-folds. We will discuss the correspondence at the numerical level for individual invariants, as well as the level of generating functions. We will also discuss its compatibility with open and closed mirror symmetry. This is based on joint work with Chiu-Chu Melissa Liu.
Abstract: I will motivate and explain the construction of a global Kuranishi chart for moduli spaces of stable pseudoholomorphic maps of arbitrary genus. Using this, I will give a proof of the product formula for symplectic Gromov-Witten invariants. This joint work with Mohan Swaminathan.
Abstract: In its simplest incarnation, bordered Floer homology associates to a 3-manifold Y with connected boundary an algebraic object CFD(Y) which can be regarded as a dg-module. A pairing theorem of Lipshitz–Ozsváth–Thurston tells us that the complex Mor(CFD(Y_1),CFD(Y_2)) of module homomorphisms between two of these modules is homotopy equivalent to the Heegaard Floer complex of the manifold Y obtained by gluing -Y_1 and Y_2 along their common boundary. In this talk, we will discuss a topological interpretation of composition of such module homomorphisms as the map induced on Heegaard Floer complexes by a pair of pants cobordism and some consequences of this interpretation.
There will be 4 lectures on recent developments in the study of Legendrian Fillings and cluster algebras, which will take place at the following times (Pacific time, so the first lecture will start at 13:30 Eastern):
10:30-11:30: First lecture by Roger Casals
11:45-12:45: Lecture by Daping Weng
13:00-14:00: Second lecture by Roger Casals
14:30-15:30: Lecture by José Simental
Titles and abstracts follow.
Title Casals #1: An introduction to Lagrangian fillings of Legendrian links
Abstract Casals #1: In this first talk, we will introduce a geometric context to study Legendrian links and their Lagrangian fillings. In particular, we will discuss constructions of Lagrangian fillings as well as techniques available to distinguish their Hamiltonian isotopy classes. The perspective that we employ will naturally lead to cluster algebras. We will state the main theorem relating cluster algebras and Lagrangian fillings and start presenting the key ingredients needed for the proof.
Title Weng: Cluster algebras associated to Bott-Samelson braids
Abstract Weng: In this second talk, we will focus on Legendrian links that are rainbow closures of positive braids. The main result we present is that the coordinate ring of functions of an algebraic variety associated to such a Legendrian link, which informally acts as the moduli of Lagrangian fillings for these links, is a cluster algebra. In fact, we show that the ingredients needed to construct such cluster algebra - including the quiver and cluster variables - can all be named through symplectic topology. We will also introduce weaves, an important piece of the proof of the main result, and explain how to perform explicit computations with them.
Title Casals #2: Applications of the relation between cluster algebras and symplectic topology
Abstract Casals #2: In this third talk, we will explore some consequences of the fact that these moduli of Lagrangian fillings have coordinate rings that are cluster algebras. Applications to symplectic topology include the detection of infinitely many Lagrangian fillings (and closed Lagrangian surfaces in simple Weinstein 4-folds) and the existence of holomorphic symplectic structures on these moduli. Applications to cluster algebras, using symplectic topology, include the construction of cluster algebra structures on the coordinate ring of any Richardson variety, the existence of Donaldson-Thomas transformations, and a geometric source of examples that naturally explain quasi-cluster structures.
Title Simental: Cluster algebras for general braid varieties
Abstract Simental: In this last talk, we will present the most general construction of cluster algebras on braid varieties. Weaves are again a key ingredient. We introduce in detail Lusztig cycles and the cluster variables associated with weaves and provide explicit examples and computations that illustrate the main result. Two crucial steps in the proof will be a localization process and the proof that A=U. Many steps and operations with weaves needed for this final talk are motivated by (and were originally conceived from) symplectic topology, but the proof we present can be understood entirely in Lie-theoretic and combinatorial terms.
Abstract: A compact hyper-Kähler (HK) manifold X is a higher dimensional generalization of a K3 surface. It is also called a holomorphic symplectic manifold as it carries a holomorphic symplectic form. A holomorphic Lagrangian fibration of a HK manifold is a higher dimensional analogue of an elliptic fibration of a K3 surface. Given a Lagrangian fibration X -> B of a known HK manifold X, we propose an explicit construction of its dual fibration Y -> B where Y is a new HK orbifold associated to X. This dual fibration is a flat compactification of the dual torus fibration and is Lagrangian with respect to the holomorphic symplectic structure on Y. Our construction behaves similar to the duality of Hitchin systmes. In certain cases, we can check the topological mirror symmetry holds between them: the (orbifold) Hodge numbers of X and Y coincide. The last part about the topological mirror symmetry is joint work with Mirko Mauri and Justin Sawon.
Abstract : Pascaleff-Tonkonog defined higher mutations for monotone toric fibers and proved an invariance of disc potential under a change of local system. In this talk, I will define a local version of higher mutations for locally mutable Lagrangians and use neck-stretching to show the invariance of Lagrangian intersection cohomology under a change of local system which agrees with the mutation formula in Pascaleff-Tonkonog.
Abstract: We describe a toric domain in $\mathbb R^4$ with smooth boundary; it is unbounded as a toric domain but is symplectomorphic to a bounded set. However the domain is not the interior of any compact symplectic manifold with smooth boundary, and packing stability fails badly; in particular it does not admit a volume filling symplectic embedding from any finite disjoint union of bounded domains in $\mathbb R^4$ with piecewise smooth boundaries. Our obstructions come from the subleading asymptotics of the ECH capacities. This is joint work with Dan Cristofaro-Gardiner.
Abstract: Given a planar oval, consider the maximal area of inscribed n-gons resp. the minimal area of circumscribed n-gons. One obtains two sequences indexed by n, and one of Dowker’s theorems states that the first sequence is concave and the second is convex. In total, there are four such classic results, concerning areas resp. perimeters of inscribed resp. circumscribed polygons, due to Dowker, Moln ́ar, and Eggleston. We show that these four results are all incarnations of the convexity property of Mather’s β-function (the minimal average action function) of the respective billiard- type systems. We then derive new geometric inequalities of similar type for various other billiard system. Some of these billiards have been thoroughly studied, and some are novel. This is joint work with Sergei Tabachnikov.
Abstract: A compact complex manifold is Kobayashi non-hyperbolic if there exists an entire curve on it. Using mirror symmetry we establish that there are (possibly singular) elliptic or rational curves on any Calabi-Yau manifold X, whose mirror dual exists and is not "Hodge degenerate", therefore proving that X is Kobayashi non-hyperbolic. We are not aware of any higher dimensional simply connected Calabi-Yau manifolds that satisfy the "Hodge degenerate" condition. This paper is joint with Cumrun Vafa.
Preceding the talk, we will host an informal symplectic tea each Friday on gather.town from 11:45 am - 12 pm ET at the following URL:
https://gather.town/app/abBT3vhSL9vD0zbL/virtualsymplectictea
For those of you who haven't used it, gather.town simulates the experience of a large group gathering by allowing you to "walk up to people" in its interface and video chat with the people you are close to.
We are looking forward to seeing you there!