Welcome to Freemath!

An online geometry seminar hosted at BBB on Tuesdays at (usually) 3 pm UTC.

A link is sent to the e-mail list before each seminar.

22 June
Ian Zemke
Title:
Heegaard Floer homology and complex curves with non-cuspidal singularities
Abstract:
We will discuss joint work with B. Liu and M. Borodzik, concerning applications of Heegaard Floer d-invariants to the study of complex curves in CP^2 with non-cuspidal singularities. We focus on the simplest such singularity, which is a double point.

15 June
Johan Asplund
Title:
Chekanov-Eliashberg dg-algebras for singular Legendrians
Abstract:
The Chekanov-Eliashberg dg-algebra is a holomorphic curve invariant associated to a Legendrian submanifold of a contact manifold. In this talk we explain how to extend the definition to singular Legendrian submanifolds admitting a Weinstein neighborhood. Via the Bourgeois-Ekholm-Eliashberg surgery formula, the new definition gives direct geometric proof of the pushout diagrams and stop removal formulas in partially wrapped Floer cohomology of Ganatra-Pardon-Shende. It furthermore leads to a proof of the conjectured surgery formula relating partially wrapped Floer cohomology to Chekanov--Eliashberg dg-algebras with coefficients in chains on the based loop space. This talk is based on joint work with Tobias Ekholm. slides video

8 June
Ivan Smith
Title:
Lagrangian links on surfaces and the Calabi invariant
Abstract:
The identity component in the group of area-preserving homeomorphisms of a compact surface admits a 'mass-flow' (or flux) homomorphism to the reals. We will prove that the kernel of this homomorphism is not simple (extending earlier results of Cristofaro-Gardiner, Humilière and Seyfaddini in the genus zero case), resolving a question of Fathi from the late 1970s. The proof appeals to a new family of Lagrangian spectral invariants associated to Lagrangian links on the surface, which are used to probe the small-scale geometry of the surface; their crucial feature is that they can be used to recover the classical Calabi invariant of a Hamiltonian. The Floer cohomology theory behind these spectral invariants is a close cousin of the knot Floer homology of Ozsváth-Szabó and Rasmussen. This talk reports on joint work with Dan Cristofaro-Gardiner, Vincent Humilière, Cheuk Yu Mak and Sobhan Seyfaddini. notes video

1 June
Richard Hind
Title:
The shape invariant and path lifting
Abstract:
The shape invariant of a symplectic manifold encodes the area classes of Lagrangian submanifolds. This talk describes joint work with Jun Zhang computing the shape for some simple domains in 4-dimensional Euclidean space. We then consider the path lifting problem, which amounts to finding Lagrangian isotopies with specified flux. Finally we discuss possible relations to stabilized symplectic embeddings. video

25 May
Bernhard Keller
Title:
Singular Hochschild cohomology and reconstruction of singularities
Abstract:
We show that under a mild regularity assumption, singular Hochschild cohomology (also known as Tate-Hochschild cohomology) identifies with Hochschild cohomology of the (dg enhanced) singularity category. In joint work with Zheng Hua, we apply this to the reconstruction of a (complete isolated) compound Du Val singularity from its contraction algebra together with the additional datum of a class in its zeroth Hochschild homology. This provides some evidence towards a conjecture by Donovan-Wemyss according to which the contraction algebra alone determines such a singularity. slides video

18 May
Emre Sertöz
Title:
Separating periods of quartic surfaces
Abstract:
Kontsevich--Zagier periods form a natural number system that extends the algebraic numbers by adding constants coming from geometry and physics. Because there are countably many periods, one would expect it to be possible to compute effectively in this number system. This would require an effective height function and the ability to separate periods of bounded height, neither of which are currently possible. In this talk, we introduce an effective height function for periods of quartic surfaces defined over algebraic numbers. We also determine the minimal distance between periods of bounded height on a single surface. We use these results to prove heuristic computations of Picard groups that rely on approximations of periods. Moreover, we give explicit Liouville type numbers that can not be the ratio of two periods of a quartic surface. This is joint work with Pierre Lairez (Inria, France). slides video

11 May
Yin Li
Title:
Exact Lagrangian tori in affine varieties
Abstract:
We will discuss what is known and unknown about the existence of exact Lagrangian tori in smooth affine varieties. Based on homological mirror symmetry and computations of Hochschild cohomology, we prove the nonexistence of exact Lagrangian tori in a class of affine conic bundles over C^n, which cannot in general be embedded in the complement of ample divisors in smooth Fano varieties. This result should be regarded as evidence for the existence of dilations. slides video

4 May
Julie Rana
Title:
T-singular surfaces of general type
Abstract:
We explore the moduli space of stable surfaces, where the simplest of questions continue to remain open for almost all invariants. A few such questions: Of the allowable singularities, which ones actually occur on a stable surface? Which of these deform to smooth surfaces? How can we use this knowledge to find divisors in the moduli spaces? Can we develop a stratification of these moduli spaces by singularity type? Our focus will be on cyclic quotient singularities, with an emphasis on discussing concrete visual examples built out of rational, K3, and elliptic surfaces.

27 April
Nicolo Sibilla
Title:
Fukaya category of surfaces and pants decomposition
Abstract:
In this talk I will explain some results joint with James Pascaleff on the Fukaya category of Riemann surfaces. I will explain a local-to-global principle which allows us to reduce the calculation of the Fukaya category of surfaces of genus g greater than one to the case of the pair-of-pants, and which holds both in the punctured and in the compact case. The starting point are the sheaf-theoretic methods which are available in the exact setting, and which I will review at the beginning of the talk. This result has several interesting consequences for HMS and geometrization of objects in the Fukaya category. The talk is based on 1604.06448 and 2103.03366. video

20 April
Dan Pomerleano
Title:
Intrinsic Mirror Symmetry and Categorical Crepant Resolutions
Abstract:
A general expectation in mirror symmetry is that the mirror partner to an affine log Calabi-Yau variety is "semi-affine" (meaning it is proper over its affinization). We will discuss how the semi-affineness of the mirror can be seen directly as certain finiteness properties of Floer theoretic invariants of X (the symplectic cohomology and wrapped Fukaya category). One interesting consequence of these finiteness results is that, under fairly general circumstances, the wrapped Fukaya of X gives an ("intrinsic") categorical crepant resolution of the affine variety Spec(SH^0(X)). This is based on 2103.01200. slides video

13 April (at 10:00 BST)
Ian Le
Title:
Mirror Symmetry for Truncated Cluster Varieties
Abstract:
Gross, Hacking and Keel gave an algebro-geometric construction of cluster varieties: take a toric variety, blow up appropriate subvarieties in the boundary, and then remove the strict transform of the boundary. We work with a modification of this construction, which we call a truncated cluster variety--roughly, this comes from performing the same procedure on the toric variety with all the codimension 2 strata removed. The resulting variety differs from the cluster variety in codimension 2. I will describe a construction of a Weinstein manifold mirror to a truncated cluster variety and explain how to prove a mirror symmetry via Lagrangian skeleta. We hope that this is a first step towards understanding mirror symmetry for the entire cluster variety. This is joint work with Benjamin Gammage. video

30 March
Conan Leung
Title:
Quantum cohomology of flag varieties via wonderful compactifications
Abstract:
Peterson conjectured that quantum cohomlogy ring of G/T is isomorphic to the homology of the based loop space of G after localization. Lam and Shimozono proved the conjecture by combinatorial method. We studied the wrapped Floer theory of the complexification of G and used the geometry of its wonderful compactification to give a geometric proof of this result. slides video

23 March
Charlotte Kirchhoff-Lukat
Title:
Towards Floer theory and Fukaya categories for Generalized Complex Manifolds: Some first ideas
Abstract:
Generalized complex (GC) manifolds encompass both symplectic and complex manifolds as examples. From the inception of the field of GC geometry in the early 2000s, questions have thus been raised about its relation to mirror symmetry: Can mirror symmetry be understood as a generalized complex duality, and if so, how? An answer to this general question currently seems out of reach both from the point of view of mirror symmetry, as well as GC geometry -- general GC manifolds are so far relatively poorly understood. However, I have identified a number specific initial questions and approaches which I hope will ultimately help a more general understanding. These ideas -- currently still in their infancy -- are what I would like to outline in this talk. video

16 March
Egor Shelukhin
Title:
Lagrangian configurations and Hamiltonian maps
Abstract:
We study configurations of disjoint Lagrangian submanifolds in certain low-dimensional symplectic manifolds from the perspective of the geometry of Hamiltonian maps. We detect infinite-dimensional flats in the Hamiltonian group of the two-sphere equipped with Hofer's metric, showing in particular that this group is not quasi-isometric to a line. This answers a well-known question of Kapovich-Polterovich from 2006. We show that these flats in Ham(\(S^2\)) stabilize to certain product four-manifolds, prove constraints on Lagrangian packing, find new instances of Lagrangian Poincare recurrence, and present a new hierarchy of normal subgroups of area-preserving homeomorphisms of the two-sphere. The technology involves Lagrangian spectral invariants with Hamiltonian term in symmetric product orbifolds. This is joint work with Leonid Polterovich.

9 March
Abigail Ward
Title:
Mirror symmetry for certain non-Kähler elliptic surfaces
Abstract:
The logarithmic transformation is an operation on complex elliptic surfaces which can be used to produce interesting spaces from more familiar ones. I will first give homological mirror symmetry results for surfaces which are constructed by performing two logarithmic transformations to the product of \(P^1\) with an elliptic curve, a class of surfaces which includes the classical Hopf surface (\(S^1 \times S^3\)). I will then use this work, along with work of Auroux, Efimov and Katzarkov on the Fukaya category of singular curves, to describe some work in progress on a potential mirror operation to the logarithmic transformation and some applications. video

2 March
Nicholas Wilkins
Title:
Quantum Steenrod operations are covariant constant
Abstract:
We explore the quantum Steenrod operations (which are quantum cohomology operations that utilise a symmetry under the cyclic group of order p), and observe that these operations are covariant constant with respect to the quantum connection. In particular, they can be partially calculated in a variety of cases (and fully calculated in a subset). This work is joint with Paul Seidel. slides video

23 February (at 14:00 UTC)
Aleksander Doan
Title:
Counting pseudo-holomorphic curves in symplectic six-manifolds
Abstract:
The signed count of embedded pseudo-holomorphic curves in a symplectic manifold typically depends on the choice of an almost complex structure on the manifold and so does not lead to a symplectic invariant. However, I will discuss two instances in which such naive counting does define a symplectic invariant. The proof of invariance combines methods of symplectic geometry with results of geometric measure theory, especially regularity theory for calibrated currents. The talk is based on joint work with Thomas Walpuski. Time permitting, I will also discuss a related project, joint with Eleny Ionel and Thomas Walpuski, whose goal is to use geometric measure theory to prove the Gopakumar-Vafa finiteness conjecture. slides video

16 February
Matthew Habermann
Title:
Homological mirror symmetry for nodal stacky curves
Abstract:
In this talk I will explain the proof of homological mirror symmetry where the B-side is a ring or chain of stacky projective lines joined nodally, and where each irreducible component is allowed to have a non-trivial generic stabiliser, generalising the work of Lekili and Polishchuk. The key ingredient is to match categorical resolutions on the A- and B-sides with an intermediary category given by the derived category of modules of a gentle algebra. I will begin by explaining how to construct this category from the data of the A- and B-models before moving on to applications. In particular, one can show homological mirror symmetry where the B-model is taken to be an invertible polynomial in two variables, but where the grading group is not necessarily maximal. In the maximally graded case the mirror is shown to be graded symplectomorphic to the Milnor fibre of the transpose invertible polynomial, thus establishing the Lekili-Ueda conjecture in dimension one. slides video

9 February
Junwu Tu
Title:
On the categorical enumerative invariants of a point
Abstract:
We briefly recall the definition of categorical enumerative invariants (CEI) first introduced by Costello around 2005. Costello's construction relies fundamentally on Sen-Zwiebach's notion of string vertices V_{g,n}'s which are chains on moduli space of smooth curves M_{g,n}'s. In this talk, we explain the relationship between string vertices and the fundamental classes of the Deligne-Mumford compactification of M_{g,n}. More precisely, we obtain a Feynman sum formula expressing the fundamental classes in terms of string vertices. As an immediate application, we prove a comparison result that the CEI of the field \mathbb{Q} is the same as the Gromov-Witten invariants of a point. slides video

2 February
Maxim Jeffs
Title:
Mirror symmetry and Fukaya categories of singular varieties
Abstract:
In this talk I will explain Auroux' definition of the Fukaya category of a singular hypersurface and two results about this definition, illustrated with some examples. The first result is that Auroux' category is equivalent to the Fukaya-Seidel category of a Landau-Ginzburg model on a smooth variety; the second result is a homological mirror symmetry equivalence at certain large complex structure limits. I will also discuss ongoing work on generalizations. slides video

26 January
Benjamin Gammage
Title:
Mirror symmetry for Berglund-Hübsch Milnor fibers
Abstract:
After recalling some joint work with Jack Smith proving homological Berglund-Hübsch mirror symmetry, we explain the calculation of the Fukaya category of a Berglund-Hübsch Milnor fiber, proving a conjecture of Yankı Lekili and Kazushi Ueda; the main technical trick is the reduction of the calculation to a certain extension of perverse schobers, essentially already computed by David Nadler. video

19 January
Sylvain Courte
Title:
Twisted generating functions and the nearby Lagrangian conjecture
Abstract:
I will explain the notion of twisted generating function and show that a closed exact Lagrangian submanifold L in the cotangent bundle of M admits such a thing. The type of function arising in our construction is related to Waldhausen's tube space from his manifold approach to algebraic K-theory of spaces. Using the rational equivalence of this space with BO, as proved by Bökstedt, we conclude that the stable Lagrangian Gauss map of L vanishes on all homotopy groups. In particular when M is a homotopy sphere, we obtain the triviality of the stable Lagrangian Gauss map and a genuine generating function for L. This is a joint work with M. Abouzaid, S. Guillermou and T. Kragh. slides video

12 January
Daniel Halpern-Leistner
Title:
Derived Theta-stratifications and the D-equivalence conjecture
Abstract:
Every vector bundle on a smooth curve has a canonical filtration, called the Harder-Narasimhan filtration, and the moduli of all vector bundles admits a stratification based on the properties of the Harder-Narasimhan filtration at each point. The theory of Theta-stratifications formulates this structure on a general algebraic stack. I will discuss how to characterize stratifications of this kind, and why their local cohomology is particularly well-behaved. I will then explain how Theta-stratifications are part of a recent proof of a case of the D-equivalence conjecture: for any projective Calabi-Yau manifold X that is birationally equivalent to a moduli space of semistable coherent sheaves on a K3 surface, the derived category of coherent sheaves on X is equivalent to the derived category of this moduli space. This confirms a prediction from homological mirror symmetry for this class of compact Calabi-Yau manifolds. video

15 December
Daniel Álvarez-Gavela
Title:
Polarized Weinstein manifolds and their positive arboreal skeleta
Abstract:
The goal of this talk is to give a geometric introduction to arboreal singularities, as well as to the distinguished subclass of *positive* arboreal singularities, and to state precisely the theorem joint with Y. Eliashberg and D. Nadler that a Weinstein manifold admits a global field of Lagrangian planes if and only if the Weinstein structure can be deformed so that the skeleton becomes positive arboreal. In particular it follows that complete intersections in complex affine space can be arborealized. video

8 December
Kenji Fukaya
Title:
Atiyah-Floer type conjecture and Virtual fundamental chain
Abstract:
This is a report on my work in progress with Aliakbar Daemi We are studying an SO(3) version of Atiyah-Floer conjecture relating Instanton Floer homology to Lagrangian Floer homology, via cobordism method. In the case when the moduli space of flat connections on 3 manifold is an {\it embedded} Lagrangian submanifold of the space of flat connections on 2 manifold, we can perturb Lipyanskiy type mixed moduli space using geometric perturbation. In the case it is an immersed Lagrangian submanifold, we need abstract perturbation and virtual technique. The singularity of the instanton moduli space is wilder than the case of pseudo-holomorphic curve and we need certain `stratified' version of Kuranishi structure. I will explain how we can define such a notion, show the existence of such structure and use it to obtain virtual fundamental chain. slides video

1 December
Francesco Lin
Title:
Floer homology and closed geodesics of hyperbolic three-manifolds
Abstract:
Floer homology and hyperbolic geometry are fundamental tools in the study of three-dimensional topology. Despite this, it remains an outstanding problem to understand whether there is any relationship between them. I will discuss some results in this direction that use as stepping stone the spectral geometry of coexact 1-forms. This is joint work with M. Lipnowski. video

24 November
Mark Mclean
Title:
Floer Cohomology and Arc Spaces
Abstract:
Let f be a polynomial over the complex numbers with an isolated singular point at the origin and let d be a positive integer. To such a polynomial we can assign a variety called the dth contact locus of f. Morally, this corresponds to the space of d-jets of holomorphic disks in complex affine space whose boundary `wraps' around the singularity d times. We show that Floer cohomology of the dth power of the Milnor monodromy map is isomorphic to compactly supported cohomology of the dth contact locus. This answers a question of Paul Seidel and it also proves a conjecture of Nero Budur, Javier Fernandez de Bobadilla, Quy Thuong Le and Hong Duc Nguyen. The key idea of the proof is to use a jet space version of the PSS map together with a filtration argument. slides video

17 November
Kyler Siegel
Title:
On the embedding complexity of Liouville manifolds
Abstract:
I will describe a new framework for obstructing exact symplectic embeddings between Liouville manifolds, based on L-infinity structures in symplectic field theory. As a main application, we study embeddings between normal crossing divisor complements in complex projective space, giving a complete characterization in many cases. This is based on joint work in preparation with S. Ganatra. video

10 November
Tom Bridgeland
Title:
Donaldson-Thomas invariants and a non-perturbative topological string partition function
Abstract:
I will introduce a class of Riemann-Hilbert problems which (I claim) arise naturally in Donaldson-Thomas theory. I will start with the simplest example (corresponding to the DT theory of the A1 quiver) which leads via undergraduate mathematics to the gamma function. Then I will explain how the same procedure applied to the DT theory of coherent sheaves on the resolved conifold leads to a non-perturbative version of the Gromov-Witten generating series, i.e. a particular choice of holomorphic function having this series as its asymptotic expansion (in fact the same result holds for any non-compact CY threefold having no compact divisors). If there is time left at the end (which there never is) I will discuss recent attempts to go beyond these results. slides video

3 November
Laurent Côté
Title:
Homological invariants of codimension 2 contact embeddings
Abstract:
There is a rich theory of transverse knots in 3-dimensional contact manifolds. It was a major open question in contact topology whether non-trivial transverse knots (i.e. codimension 2 contact embeddings) also exist in higher dimensions. This question was recently settled in the affirmative by Casals and Etnyre. Motivated by their result, I will talk about recent work with Francois-Simon Fauteux-Chapleau in which we develop invariants of codimension 2 contact embeddings using the machinery of symplectic field theory. video

27 October
Francois Greer
Title:
Cycle-valued quasi-modular forms on Kontsevich space
Abstract:
On a general rational elliptic surface (fibered over \(\mathbb{P}^1\)), the number of sections of height \(n\) is equal to the coefficient of the Eisenstein series \(E_4(q)\) at order \(n+1\). I will describe a conjectural generalization of this fact, which associates to any smooth projective variety a quasi-modular form valued in the Chow group of its Kontsevich moduli space. The proof is in progress. video

20 October
Andrew Manion (USC)
Title:
Higher representations and cornered Floer homology
Abstract:
I will discuss recent work with Raphael Rouquier, focusing on a higher tensor product operation for 2-representations of Khovanov's categorification of U(gl(1|1)^+), examples of such 2-representations that arise as strands algebras in bordered and cornered Heegaard Floer homology, and a tensor-product-based gluing formula for these 2-representations expanding on work of Douglas-Manolescu. video

13 October (at 10 am UTC+01:00)
Hansol Hong (Yonsei, Korea)
Title:
Maurer-Cartan deformation of a Lagrangian
Abstract:
The Maurer-Cartan algebra of a Lagrangian is the algebra that encodes the deformation of its Floer complex as an A-infinity algebra. I will give a convenient description of the Maurer-Cartan algebra through a natural homological algebra language, and relate it with (a version of) Koszul duality for the Floer complex. It helps us to obtain the mirror-symmetry interpretation for the Maurer-Cartan deformation and its locality in SYZ situation. Namely, the Maurer-Cartan algebra provides a neighborhood of the point mirror to the Lagrangian, which varies in size depending on geometric types of Floer generators involved in the deformation. video

6 October
Pieter Belmans (Bonn)
Title:
Graph potentials as mirrors to moduli of vector bundles on curves
Abstract:
In a joint work with Sergey Galkin and Swarnava Mukhopadhyay we have a class of Laurent polynomials associated to decorated trivalent graphs which we called graph potentials. These Laurent polynomials satisfy interesting symmetry and compatibility properties. Under mirror symmetry they are related to moduli spaces of rank 2 bundles (with fixed determinant of odd degree) on a curve of genus $g\geq 2$, which is a class of Fano varieties of dimension $3g-3$. I will discuss (parts of) the (enumerative / homological) mirror symmetry picture for Fano varieties, and then explain what we understand for this class of varieties and what we can say about the (conjectural) semiorthogonal decomposition of the derived category. notes

22 September
Damien Gayet (Institut Fourier, Grenoble)
Title:
Lagrangians of (random) projective hypersurfaces.
Abstract:
I will explain that any smooth compact hypersurface in Rn appears (up to diffeomorphism) a very large number of times as disjoint Lagrangians in any complex hypersurface of CPn, if the degree of the hypersurface is high enough. Suprisingly, the proof holds on probabilistic arguments. slides video

15 September
Marco Golla (Nantes)
Title:
Symplectic hats
Abstract:
A hat for a transverse knot in a symplectic cap of a contact 3-manifold is a symplectic surface in the cap whose boundary is the knot. I will talk about existence, obstructions, and properties of hats, with an emphasis on the standard 3-sphere, and about an application to Stein fillings. This is joint work with John Etnyre. slides video

8 September (at 10 am UTC+01:00)
Andrew Macpherson
Title:
A bivariant Yoneda lemma and (infinity, 2)-categories of correspondences.
Abstract:
The notion of the *category of correspondences* of a category D with a specified, base change stable, class of morphisms S --- whose objects are those of D and whose morphisms are "spans" in D, one side of which belongs to S --- will be familiar to practitioners of Grothendieck's theory of motives. Perhaps less familiar is the fact that an obvious 2-categorical upgrade of correspondences has a universal property: it is the universal way to attach right adjoints to members of S subject to a base change formula. I will explain a little about the state of the art on enriched and iterated higher categories and show that they can be used to provide a conceptual (that is, no explicit homotopy- or simplex-chasing) proof of this phenomenon for (infinity, 2)-categories. This enhancement opens the door to direct constructions of bivariant homology theories in motivic homotopy theory and beyond. video

1 September
Wai Kit Yeung (IPMU)
Title:
Pre-Calabi-Yau categories
Abstract:
Pre-Calabi-Yau categories are algebraic structures first studied by Kontsevich and Vlassopoulos. They can be viewed as a noncommutative analogue of Poisson structures, just like Calabi-Yau structures are a noncommutative analogue of symplectic structures. It is expected that disk-counting with more than one output endows Fukaya categories with pre-Calabi-Yau structures. In this talk, we discuss several aspects of this notion. slides video

25 August
Inbar Klang (Columbia)
Title:
Twisted Calabi-Yau algebras and categories
Abstract:
This talk will begin with a discussion of the string topology category of a manifold \(M\); this was shown by Cohen and Ganatra to be equivalent as a Calabi-Yau category to the wrapped Fukaya category of \(T^*M\). In joint work with Ralph Cohen, we generalize the Calabi-Yau condition from chain complexes to spectra. I'll talk about these Calabi-Yau ring spectra and discuss examples of interest. video

20 August
Giancarlo Urzúa (Pontificia Universidad Católica de Chile)
Title:
On the geography of complex surfaces of general type with an arbitrary fundamental group
Abstract:
Surfaces of general type are lovely unclassifiable objects in algebraic geometry. Geography refers to the problem of construction of such surfaces for a given set of invariants, classically the Chern numbers \(c_1^2\) (self-intersection of canonical class) and \(c_2\) (topological Euler characteristic). In this talk, we treat the question: What can be said about the distribution of Chern slopes \(c_1^2/c_2\) of surfaces of general type when we fix the fundamental group? It turns out that there are various well-known constraints, which will be pointed out during the talk, but at least we can prove the following theorem (joint with Sergio Troncoso): "Let \(G\) be the fundamental group of some nonsingular complex projective variety. Then Chern slopes of surfaces of general type with fundamental group isomorphic to \(G\) are dense in the interval \([1,3]\).". Remember that for complex surfaces of general type we have that \(c_1^2/c_2\) is a rational number in \([1/5,3]\), and so most open questions now refer to slopes in \([1/5,1]\). On the other hand, it is known that every finite group is the fundamental group of some nonsingular projective variety, and so a lot is going on for high slopes. video

11 August (at 10 am UTC+01:00)
Dougal Davis (Edinburgh)
Title:
Surface singularities and their deformations via principal bundles on elliptic curves
Abstract:
It is well known that du Val (aka simple, Kleinian, ADE, ...) singularities of algebraic surfaces are classified by Dynkin diagrams of type ADE. A geometric link between the singularity and the Lie algebra of the same type was given by Brieskorn in the 70s, who showed that the singularity can be recovered by intersecting the nilpotent cone inside the Lie algebra with a transversal slice through a subregular nilpotent element. Brieskorn's construction also realises the entire transversal slice as the total space of a miniversal deformation of the singularity. In this talk, I will discuss an elliptic version of this story, where the Lie algebra is replaced with the stack of principal bundles on an elliptic curve. There is still a notion of subregular slice in this stack, and one gets a singular surface by intersecting such a thing with the locus of unstable bundles. I will explain which surfaces arise in this way, and in what sense the subregular slice is still the total space of a miniversal deformation. Time permitting, I will also touch on how the BCFG types are related to the ADE ones (in a different way to the story for Lie algebras!), and on some questions about Poisson structures and their quantisations. slides

4 August
Barış Kartal (Princeton)
Title:
p-adic analytic actions on the Fukaya category and iterates of symplectomorphisms
Abstract:
A theorem of J. Bell states that given a complex affine variety \(X\) with an automorphism \(\phi\), and a subvariety \(Y\subset X\), the set of numbers \(k\) such that \(\phi^k(x)\in Y\) is a union of finitely many arithmetic progressions and finitely many numbers. Motivated by this statement, Seidel asked whether there is a symplectic analogue of this theorem. In this talk, we give an answer to a version of this question in the case \(M\) is monotone, non-degenerate and \(\phi\) is symplectically isotopic to identity. The main tool is analogous to the main tool in Bell's proof: namely we interpolate the powers of \(\phi\) by a p-adic arc, constructing an analytic action of \(\mathbb{Z}_p\) on the Fukaya category. slides video

28 July
Catherine Cannizzo (Stony Brook)
Title:
Towards global 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. video

21 July
Alastair Craw (Bath)
Title:
Hilbert schemes of ADE singularities as quiver varieties
Abstract:
The nth symmetric product of a ADE surface singularity is well known to be a Nakajima quiver variety. I will describe recent work with Gammelgaard, Gyenge and Szendroi in which the Hilbert scheme of n points on the ADE singularity is constructed as a Nakajima quiver variety. This result provided the catalyst for the description of the generating function of Euler numbers on punctual Hilbert schemes of an ADE surface singularity by Nakajima earlier this year. slides video

14 July
Sheel Ganatra (USC)
Title:
On Rabinowitz wrapped Fukaya categories
Abstract:
This talk develops the open-string categorical analogue of Rabinowitz Floer homology, which we term the Rabinowitz (wrapped) Fukaya category. Following a conjecture of Abouzaid, we relate the Rabinowitz Fukaya category to the usual wrapped Fukaya category by way of a general categorical construction introduced by Efimov, the "categorical formal punctured neighborhood of infinity". Using this result, we show how Rabinowitz Fukaya categories can be fit into - and therefore computed in terms of - mirror symmetry. Joint work (in progress) with Yuan Gao and Sara Venkatesh. notes video (audio gets better after minute 10).

7 July
Mohammed Abouzaid (Columbia)
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 stable homotopy theory. 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. video

30 June
Nick Sheridan (Edinburgh)
Title:
Symplectic mapping class groups and homological mirror symmetry
Abstract:
I will explain how one can get new information about symplectic mapping class groups by combining two recent results: a proof of homological mirror symmetry for a new collection of K3 surfaces (joint work with Ivan Smith), together with the computation of the derived autoequivalence group of a K3 surface of Picard rank one (Bayer--Bridgeland). For example, it is possible to give an example of a symplectic K3 whose smoothly trivial symplectic mapping class group (the group of isotopy classes of symplectic automorphisms which are smoothly isotopic to the identity) is infinitely-generated. This is joint work with Ivan Smith. notes

23 June (at 4 pm UTC+01:00)
Alexander Kuznetsov (Steklov)
Title:
Residual categories and quantum cohomology
Abstract:
Dubrovin's conjecture predicts that a smooth projective variety has a full exceptional collection in the derived category of coherent sheaves if and only if its big quantum cohomology ring is generically semisimple. However, the big quantum cohomology is very hard to compute. We suggest a conjecture, where the big quantum cohomology is replaced by the small quantum cohomology (which is much more easy to compute), and a full exceptional collection is replaced by a semiorthogonal decomposition of a special form. We support this conjecture by a number of examples provided by homogeneous varieties of simple algebraic groups. This is a joint work with Maxim Smirnov.

16 June (at 10 am UTC+01:00)
Tatsuki Kuwagaki (IPMU, Japan)
Title:
Symplectic geometry of exact WKB analysis
Abstract:
A sheaf quantization is a sheaf associated to a Lagrangian brane. This sheaf conjecturally has information as much as Floer theory of the Lagrangian. On the other hand, exact WKB analysis is an analysis of differential equations containing \(\hbar\) (the Planck constant). In this talk, I will explain how to construct a sheaf quantization over the Novikov ring of the spectral curve of an \(\hbar\)-differential equation by using exact WKB method. In the construction, one can see how (conjecturally) the convergence in WKB analysis is related to the convergence in Fukaya category. In degree 2, there is an application to cluster theory: the sheaf quantization associates a cluster coordinate which is the same as the Voros-Iwaki-Nakanishi-Fock-Goncharov coordinate. I will also mention about some relationships to Riemann-Hilbert correspondence of D'Agnolo-Kashiwara and Kontsevich-Soibelman.

9 June
Yuhan Sun (Stony Brook)
Title:
Displacement energy of Lagrangian 3-spheres
Abstract:
We study local and global Hamiltonian dynamical behaviours of some Lagrangian submanifolds near a Lagrangian sphere S in a symplectic manifold X. When dim S = 2, we show that there is a one-parameter family of Lagrangian tori near S, which are nondisplaceable in X. When dim S = 3, we obtain a new estimate of the displacement energy of S, by estimating the displacement energy of a one-parameter family of Lagrangian tori near S.

2 June
Octav Cornea (Univ. of Montreal)
Title:
Lagrangians, surgery and rigidity
Abstract:
I will discuss a framework for analyzing classes of Lagrangian submanifolds that aims to endow them with a metric structure. The tools involve certain Floer type machinery for immersed Lagrangians. Part of the picture is a correspondence between certain cobordism categories endowed with surgery models and derived Fukaya categories. The talk is based on joint work with Paul Biran.

26 May (at 10 am UTC+01:00)
Kazushi Ueda (Univ. of Tokyo, Japan)
Title:
Noncommutative del Pezzo surfaces
Abstract:
It is known after the works of Artin-Tate-Van den Bergh and Bondal-Polishchuk that noncommutative deformations of the projective plane are classified by triples consisting of a cubic curve and two line bundles. Similarly, Van den Bergh gave a classification of noncommutative quadric surfaces in terms of quadruples consisting of (a degeneration of) an elliptic curve and three line bundles. In the talk, I will discuss a joint work in progress with Tarig Abdelgadir and Shinnosuke Okawa on classifications of noncommutative del Pezzo surfaces.

19 May
Gleb Smirnov (ETH, Zürich)
Title:
Isotopy problem for symplectic forms in the presence of an anti-holomorphic involution
Abstract:
Suppose we are given an algebraic surface equipped with an anti-holomorphic involution. From the symplectic viewpoint, a natural question to ask is: are there cohomologous anti-invariant symplectic forms on this manifold which are not isotopic within anti-invariant forms? And, if so, how many? During the talk, we will look at a particularly simple case of complex quadrics and do some explicit computations.

12 May
Jenny August (MPIM, Bonn)
Title:
Stability for Contraction Algebras
Abstract:
For a finite dimensional algebra, Bridgeland stability conditions can be viewed as a continuous generalisation of tilting theory, providing a geometric way to study the derived category. Describing this stability manifold is often very challenging but in this talk, I'll look at a special class of symmetric algebras whose tilting theory is very well behaved, allowing us to describe the entire stability manifold of such an algebra. This is joint work with Michael Wemyss.

5 May
Alexandru Oancea (Sorbonne, Paris)
Title:
Duality for Rabinowitz-Floer homology
Abstract:
I will explain a duality theorem with products in Rabinowitz-Floer homology. This has a bearing on string topology and explains a number of dualities that have been observed in that setting. Joint work in progress with Kai Cieliebak and Nancy Hingston.

28 April
Pierrick Bousseau (ETH, Zürich)
Title:
Holomorphic curves, Lagrangians, and coherent sheaves
Abstract:
I will describe a new correspondence between coherent sheaves on the projective plane and holomorphic curves in the projective plane with tangency condition along a smooth cubic curve. This correspondence is motivated by a combined application of hyperkaehler rotation and mirror symmetry. The actual proof uses tropical geometry as a connecting bridge.