- 18 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.

- 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.

- 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.