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.

8 December
Kenji Fukaya
Title:
Abstract:

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.

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 Fernández de Bobadilla, Quy Thuong Lê 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.

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.