Tag - Symplectic geometry

Cheol-Hyun Cho: Fukaya category for Landau-Ginzburg orbifolds and Berglund-Hübsch homological mirror symmetry for curve singularities

21st September, 2020

For a weighted homogeneous polynomial and a choice of a diagonal symmetry group, we define a new Fukaya category based on wrapped Fukaya category of its Milnor fiber together with monodromy information. It is analogous to the variation operator in singularity theory. As an application, we formulate a complete version of Berglund-Hübsch homological mirror symmetry and prove it for two variable cases. Namely, given one of the polynomials f=xp+yq, xp+xyq, xpy+xyq and a symmetry group G, we use a Floer-theoretic construction to obtain the transpose polynomial ft with the transpose symmetry group Gt as well as an explicit A∞-equivalence between the new Fukaya category of (f,G) to the matrix factorization category of (ft,Gt). In this case, monodromy is mirror to the restriction of LG model to a hypersurface. For ADE singularities, Auslander-Reiten quiver for indecomposable matrix factorizations were known from the 1980s, and we find the corresponding Lagrangians as well as surgery exact sequences.

Vincent Colin: Reeb dynamics in dimension 3 and broken book decompositions

11th September, 2020

In joint work with Pierre Dehornoy and Ana Rechtman, we prove that on a closed 3-manifold, every non-degenerate Reeb vector field is supported by a broken book decomposition. From this property, we deduce that in dimension 3 every non-degenerate Reeb vector field has either 2 or infinitely periodic orbits and that on a closed 3-manifold that is not graphed, every non-degenerate Reeb vector field has positive topological entropy.

Georgios Dimitroglou Rizell: Hamiltonian classification and unlinkedness of fibres in cotangent bundles of Riemann surfaces

4th September, 2020

In a joint work with Laurent Côté we show the following result. Any Lagrangian plane in the cotangent bundle of an open Riemann surface which coincides with a cotangent fibre outside of some compact subset, is compactly supported Hamiltonian isotopic to that fibre. This result implies Hamiltonian unlinkedness for Lagrangian links in the cotangent bundle of a (possibly closed Riemann surface whose components are Hamiltonian isotopic to fibres.

John Pardon: Pontryagin-Thom for orbifold bordism

24th July, 2020

The classical Pontryagin-Thom isomorphism equates manifold bordism groups with corresponding stable homotopy groups. This construction moreover generalizes to the equivariant context. I will discuss work which establishes a Pontryagin-Thom isomorphism for orbispaces (an orbispace is a 'space' that is locally modelled on Y/G for Y a space and G a finite group; examples of orbispaces include orbifolds and moduli spaces of pseudo-holomorphic curves). This involves defining a category of orbispectra and an involution of this category extending Spanier-Whitehead duality. Global homotopy theory also plays a key role.

Patrick Le Meur: Equivariant models for graded algebras and application to Calabi-Yau duality

16th July, 2020

Skew Calabi-Yau algebras are generalizations of Calabi-Yau algebras due to Reyes, Rogalski, and Zhang. Within the graded (associative and unital) algebras over a field k, they form the non-commutative analogues of the regular algebras. As a special feature, such an algebra A is equipped with its so-called Nakayama automorphism φ. The talk will present ongoing investigations on the presentations of these algebras by generators and relations taking into account their homological specificities. Such presentations are well-known for Calabi-Yau algebras (after Ginzburg, Bocklandt and van den Bergh) and also for Koszul skew Calabi-Yau algebras (after Bocklandt, Wemyss and Schedler). The general situation involves the interaction of the A∞-Yoneda algebra E(A) := ExtA(k,k) with the Nakayama automorphism φ, and also the A∞-Yoneda algebra E(A[x,φ]) of the Ore extension A[x,φ] of A by φ. More precisely, one is particularly intereseted in minimal models of these A∞-algebras. After having presented all these concepts, I will discuss the relationship between these minimal models as well as consequences in terms of presentations of A.

Peter Ozsváth: Knot Floer homology and bordered algebras

10th July, 2020

Knot Floer homology is an invariant for knots in three-space, defined as a Lagrangian Floer homology in a symmetric product. It has the form of a bigraded vector space, encoding topological information about the knot. I will discuss an algebraic approach to computing knot Floer homology, and a corresponding version for links, based on decomposing knot diagrams.

Ana Rita Pires: Infinite staircases and reflexive polygons

3rd July, 2020

A classic result, due to McDuff and Schlenk, asserts that the function that encodes when a four-dimensional symplectic ellipsoid can be embedded into a four-dimensional ball has a remarkable structure: the function has infinitely many corners, determined by the odd-index Fibonacci numbers, that fit together to form an infinite staircase. The work of McDuff and Schlenk has recently led to considerable interest in understanding when the ellipsoid embedding function for other symplectic 4-manifolds is partly described by an infinite staircase. In this talk we will discuss a general framework for analysing this question for a large family of targets, and in particular give an obstruction to the existence of an infinite staircase that experimentally seems strong.

We will then look at the special case of rational convex toric domains / closed symplectic toric manifolds, for which we prove the existence of six families of targets with infinite staircases that are distinguished by the fact that their moment polygon is reflexive. The proof uses, among other tools, almost toric fibrations -- see also the second of the ellipsoid day talks.

Finally, we conjecture that these six families constitute a complete answer to the question of existence of infinite staircase. This conjecture has been verified in the case when the target is an ellipsoid.

Ailsa Keating: Distinguishing monotone Lagrangians via holomorphic annuli

26th June, 2020

We present techniques for constructing families of compact, monotone (including exact) Lagrangians in certain affine varieties, starting with Brieskorn-Pham hypersurfaces. We will focus on dimensions 2 and 3. In particular, we'll explain how to set up well-defined counts of holomorphic annuli for a range of these families. Time allowing, we will give a number of applications.

Igor Uljarevic: Exotic symplectomorphisms and contact circle action

19th June, 2020

An exotic symplectomorphism is a symplectomorphism that is not isotopic to the identity through compactly supported symplectomorphisms.Using Floer-theoretic methods, we prove that the non-existence of an exotic symplectomorphism on the standard symplectic ball, 𝔹2n, implies a rather strict topological condition on the free contact circle actions on the standard contact sphere, S2n−1. We also prove an analogue for a Liouville domain and contact circle actions on its boundary. Applications include results on the symplectic mapping class group, the fundamental group of the group of contactomorphisms, and exotic contact structures on S3.

Mark McLean: Floer Cohomology and Arc Spaces

12th June, 2020

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.