Titles and abstracts
Monday 23 March
10.00am Jonny Evans (Lancaster)
Isotopy uniqueness for pin-ellipsoids
Pin-ellipsoids are a class of symplectic domains which generalise standard symplectic ellipsoids. They arise as Milnor fibres of cyclic quotient T-singularities. There are many interesting pin-ellipsoids inside the complex projective plane, coming from the ways it can degenerate to a weighted projective space. We will show that any symplectic pin-ellipsoid in CP^2 is isotopic to one coming from an algebraic degeneration. Based on joint work with Nikolas Adaloglou, Joé Brendel, Johannes Hauber and Felix Schlenk and builds upon and illuminates earlier joint work with Ivan Smith.
11.30am Thomas Kragh (Uppsala)
Why twisted algebraic K-theory of spaces should enhance family Floer invariants
In this talk, I will discuss how twisted algebraic K-theory of spaces should lift the usual notion of family Floer homology. That is, I want to present a heuristic that should generalize the invariants we obtained from twisted generating functions in cotangent bundles (arising from joint work with Abouzaid, Courte and Guillermou). These invariants, which are related to algebraic K-theory of spaces, were seen to be much stronger than the usual family Floer homology given by the family of fibers in the cotangent bundle (at least as treated so far). I will rely heavily on specific examples for motivation. This heuristic generalization, motivates the need for a twisted version of Waldhausen's theorem, which would relate a twisted assembly map to a space of twisted h-cobordisms. If time permits, I will discuss ongoing joint work with Oldervoll trying to prove such a twisted Waldhausen theorem - relying on a category closely related to the category of flow categories.
2.30pm Abigail Ward (Cambridge)
Weinstein manifolds without arboreal skeleta
We will exhibit cohomological obstructions to the existence of an arboreal skeleton for a given Weinstein manifold and as a corollary show that many familiar affine varieties fail to be arborealisable. We will also discuss the relationship between the existence of an arboreal skeleton for a space X and the existence of a Fukaya category for X over the sphere spectrum. This is joint work with D. Alvarez-Gavela and T. Large.
4.00pm Marcelo Atallah (Sheffield)
C⁰-rigidity of the Hamiltonian diffeomorphism group of symplectic rational surfaces
A natural question bridging the celebrated Gromov–Eliashberg theorem and the C⁰-flux conjecture is whether the identity component of the group of symplectic diffeomorphisms is C⁰-closed in Symp(M,ω). Beyond surfaces and the cases in which the Torelli subgroup of Symp(M,ω) coincides with the identity component, little is known. In joint work with Cheuk Yu Mak and Wewei Wu, we show that, for all but a few positive rational surfaces, the group of Hamiltonian diffeomorphisms is the C⁰-connected component of the identity in Symp(M,ω), thereby giving a positive answer in this setting. Here, “positive rational surface” essentially means a k-point blow-up of CP² whose symplectic form evaluates positively on the first Chern class.
Tuesday 24th March
9.00 Jeff Hicks (St Andrews)
Tropical Geometry, Lagrangian Submanifolds, and Realizability
Given a tropical curve in Rn, the realizability problem asks whether it arises as the tropicalization of an honest algebraic curve in the corresponding algebraic torus. In this talk, I will report on current progress toward solving the symplectic-geometric version of this problem: the unobstructedness of Floer theory on the Lagrangian submanifold lift associated with the tropical curve.
11.00 Adrian Dawid (Cambridge)
A probabilistic perspective on the Hamiltonian diffeomorphism group
What is a random Hamiltonian diffeomorphism? In this talk, we will provide an answer to this question by constructing a family of probability measure on the group of Hamiltonian diffeomorphisms. After an outline of the construction, we show that classical invariants such as the Hofer norm and spectral invariants become well-behaved random variables with finite expectations under these measures. Furthermore, the family of measures we construct is closed under the group operations. We will outline why these measures have full support on Ham(M) and behave like Gaussian measures in certain ways. Additionally, we will see how they induce a random walk on Ham(M) that can be used to study the geometry of the Hofer metric. The talk will end with an overview of some open problems and questions.
2.30pm Elliot Gathercole (Lancaster)
Superheavy skeleta from non-normal crossings divisors
A (smooth) complex Fano projective variety M is canonically a monotone symplectic manifold. Given an effective anticanonical divisor D, we obtain a choice of Liouville primitive on the complement of D, which gives a retraction onto the skeleton, L.
Superheaviness is a strong rigidity property of a closed subset of a symplectic manifold, which implies, in particular, non-displaceability.
In the case that D has singularities in a certain class, worse than normal crossings, we will describe a result establishing a sufficient numerical condition on D for L, or a neighbourhood thereof of a certain size, to be a superheavy subset of M. We will illustrate this with some interesting examples where the skeleton itself can be shown to be superheavy, giving examples of rigid isotropic cell complexes which are not easily detected by Lagrangian Floer homology.
4.00pm Amanda Hirschi (Sorbonne, Paris)
String-valued open Gromov--Witten invariants
I will discuss joint work in progress with Yash Deshmukh in which we construct a geometrically defined model for the homotopy involutive bi-Lie algebra structure on reduced S^1-equivariant chains on the free loop space of an arbitrary smooth closed manifold. We define a higher genus open Gromov--Witten potential of a closed Lagrangian brane valued in the involutive bi-Lie algebra of the reduced equivariant chains on the free loop space of the Lagrangian, realizing a proposal of Cieliebak--Fukaya--Latschev.
Wednesday 25th March
9.00 Alexia Corradini (Cambridge)
A Lagrangian cycle theory
In algebraic geometry, Chow groups and Griffiths groups - defined by taking quotients of cycles by rational and algebraic equivalence respectively - package rich information about the geometry of the underlying variety. Mirror symmetry relates the former to cylindrical Lagrangian cobordism groups Cob(X), as is known from work of Sheridan-Smith. I will motivate how the latter is related to algebraic Lagrangian cobordism groups algCob(X), i.e. groups of Lagrangians modulo algebraic Lagrangian cobordism, which I will define.
To study these groups, classical Abel-Jacobi theory combined with Hodge-theoretic mirror symmetry suggest that if two Lagrangians define the same class in algCob(X), their open Gromov-Witten invariants should be related. I will explain the various ingredients involved while building towards a precise conjecture. Time permitting, I will state evidence of this conjecture through a classical limit statement.
10.30 Sobhan Seyfaddini (ETH, Zurich)
On the Topological Invariance of Helicity I
Helicity is an invariant of divergence free vector fields on three-manifolds. One of its fundamental properties is invariance under volume preserving diffeomorphisms. Arnold, having derived an ergodic interpretation of helicity as an asymptotic linking number, asked whether helicity remains invariant under volume preserving homeomorphisms, and more generally, whether it admits an extension to topological volume preserving flows.
In these two lectures, we will present affirmative answers to both questions in the case of non-singular flows and will outline the main ideas behind the proofs. Our approach relies on recent advances in C^0-symplectic topology, particularly new insights into the algebraic structure of the group of area-preserving homeomorphisms, which we will also review.
Part I of a two part talk on joint work with Oliver Edtmair
11.40 Mohammed Abouzaid (Stanford)
Can machine learning teach us something about symplectic topology?
There are calls to look for applications of machine learning for mathematics. I propose that, in symplectic topology, the most fruitful potential applications will be about quantitative questions. I will focus on key challenges: (i) producing enough data, and (ii) the failure of rigidity when epsilon-errors are allowed. I will also mention some (mathematical) ideas for trying to resolve these issues, such as studying piecewise linear Lagrangians.
Thursday 26th March
9.00 Noah Porcelli (MPI, Bonn)
Grothendieck-Riemann-Roch for spectral symplectic cohomology
For a graded Liouville domain X, its symplectic cohomology (normally defined over the integers) can be lifted to a module over the complex cobordism ring using Floer homotopy theory. I'll discuss some computational aspects in the rational case, using Chern classes on Floer moduli spaces as well as X.
Based on joint work-in-progress with Kenny Blakey.
11.00 Luya Wang (IAS)
Complements of submanifolds in 4-dimensional Weinstein domains
I will introduce the notion of nearly Lefschetz fibrations, which can be viewed as the complements of neighborhoods of positive multisections in bordered Lefschetz fibrations. We show that these objects are supported by canonical Stein structures and provide a tool to understand Lagrangian submanifolds in Weinstein domains. In particular, we show that in 4-dimensional Weinstein domains, all exact Lagrangian disks with Legendrian boundaries are regular, in the sense of Eliashberg-Ganatra-Lazarev. This is joint work in progress with Agniva Roy and Joseph Breen.
2.30 Oliver Edtmair (ETH Zurich)
On the Topological Invariance of Helicity II
Helicity is an invariant of divergence free vector fields on three-manifolds. One of its fundamental properties is invariance under volume preserving diffeomorphisms. Arnold, having derived an ergodic interpretation of helicity as an asymptotic linking number, asked whether helicity remains invariant under volume preserving homeomorphisms, and more generally, whether it admits an extension to topological volume preserving flows.
In these two lectures, we will present affirmative answers to both questions in the case of non-singular flows and will outline the main ideas behind the proofs. Our approach relies on recent advances in C^0-symplectic topology, particularly new insights into the algebraic structure of the group of area-preserving homeomorphisms, which we will also review.
Part II of a two part talk on joint work with Sobhan Seyfaddini
4.00-5.00 Nick Sheridan (Edinburgh)
On homological mirror symmetry for Kuznetsov components
Abstract: The derived category of coherent sheaves on a variety can often be decomposed into simpler pieces, by a semi-orthogonal decomposition. In some settings there are a number of simple pieces, and one complicated one called the Kuznetsov component. The mirror to this is the decomposition of a Fukaya-Seidel category into pieces corresponding to its singular fibres. I will report on an ongoing project joint with Di Dedda, Gugiatti, and Koževnikov, where we study how features of the Kuznetsov component are reflected in the geometry of the corresponding critical fibre of the mirror. I will focus on the particular example of the Kuznetsov component of a cyclic branched cover, as studied by Kuznetsov-Perry.
Friday 27th March
9.00am Maxim Jeffs (Cambridge)
A splitting formula for fixed point Floer cohomology of Dehn twists
In previous joint work with Yuan Yao and Ziwen Zhao, we showed how certain algebras built from fixed point Floer cohomology groups can be considered as substitutes for 'quantum cohomology' for singular varieties, and verified an instance of mirror symmetry by explicitly computing these algebras for curves. In the present work, we give tools for computing this quantum cohomology in the case of nodal singularities in higher dimensions: for iterated Dehn twists, we show that the fixed point Floer cohomology splits into local and Morse-theoretic contributions, using a confinement principle for holomorphic curves; the local contributions can then be explicitly computed using algebraic methods. I will finish by discussing applications of these computational tools to mirror symmetry.
11.00am Cheuk Yu Mak (Sheffield)
Lagrangian torus fibration on Calabi-Yau hypersurfaces
In joint work with Matessi-Ruddat-Zharkov, we prove the existence of Lagrangian torus fibrations on Calabi-Yau hypersurfaces in toric Fano manifolds given by a reflexive polytope. The result is motivated by the Strominger-Yau-Zaslow conjecture, which predicts the existence of these fibrations on Calabi-Yau manifolds near large complex structure limits. In this talk, I will outline the main setup of the construction. The idea is to replace the ordinary algebraic equation with a new one involving "ironing coefficients" and a convex (but not strictly convex!) potential which has the effect of breaking the manifold into local models. Over these models, we apply the Liouville flow technique in the style of Evans-Mauri.