Tag - Symplectic geometry

Russell Avdek: CH, GW and the geography of tight convex hypersurfaces

20th October, 2023

While convex hypersurfaces are well understood in 3d contact topology, we are just starting to explore their basic properties in high dimensions. I will describe how to compute contact homologies (CH) of their neighborhoods, which can be used to infer tightness in any dimension. Then I’ll give a general construction of high-dimensional convex hypersurfaces in the style of Gompf’s fiber sum. For these convex hypersurfaces, relative Gromov-Witten can often compute CH in the style of Diogo-Lisi. We’ll work through some interesting examples.

Bulent Tosun: Contact Surgeries and Symplectic Fillability

9th October, 2023

It is well known that all contact 3-manifolds can be obtained from the standard contact structure on the 3-sphere by contact surgery on a Legendrian link. Hence, an interesting and much studied question asks what properties are preserved under various types of contact surgeries. The case for the negative contact surgeries is fairly well understood. In this talk, extending an earlier work of the speaker with Conway and Etnyre, we will discuss some new results about symplectic fillability of positive contact surgeries, and in particular we will provide a necessary and sufficient condition for contact (n) surgery along a Legendrian knot to yield a weakly fillable contact manifold, for some integer n > 0. When specialized to knots in the three sphere with its standard tight structure, this result can be effectively used to find many examples of fillable surgeries along with various obstructions and surprising topological applications. For example, we prove that a knot admitting lens space surgery must have slice genus equal to its 4-dimensional clasp number.

Mohammed Abouzaid: Bordism and Floer theory

5th October, 2023

In the late 1980s Andreas Floer revolutionized low-dimensional and symplectic topology by discovering the existence an extension of Morse theory to an infinite-dimensional setting where the standard methods of variational calculus fail. While he foresaw that his theory should be able to encompass generalized homology theory (bordism, K-theory, ...), severe foundational difficulties prevented any significant progress on this question until two years ago. I will explain the advances that have been made on two fronts: (I) defining concrete models, in terms of equivariant vector bundles, for the moduli spaces that appear in Floer theory, and (II) understanding the geometric consequences of lifting Floer homology to generalized homology theories. I will end by formulating how the notion of derived orbifold bordism provides a universal receptacle for Floer's invariants, and its descendants.

Christian Lehn: Lagrangian Fibrations of Holomorphic Symplectic Varieties

11th September, 2023

By Matsushita's fundamental results, Lagrangian fibrations are essentially the only morphisms on irreducible holomorphic symplectic varieties with positive fibre dimension. We will start by reviewing these results and discuss their validity also for singular symplectic varieties. We will study singular fibres and some of the fundamental conjectures. Towards the end of the course, we will turn to some of the fascinating recent developments in the Hodge theory of Lagrangian fibrations.

Ana Balibanu: Moment maps, multiplicative reduction, and Dirac geometry

31st August, 2023

We develop a general approach to reduction along strong Dirac maps, which are a broad generalization of Poisson moment maps. We recover a number of familiar constructions and we give several new reduction procedures, including a multiplicative analogue of Whittaker reduction.

Joé Brendel: Local exotic tori

26th May, 2023

We discuss exotic Lagrangian tori in dimension greater than or equal to six. First, we give another approach to Auroux's result that there are infinitely many tori in ℝ6 which are distinct up to symplectomorphisms of the ambient space. The exotic tori we construct naturally appear in a two-​parameter family, some of which are not monotone. Small enough tori in this family can be embedded by a Darboux chart into any tame symplectic manifold and one can show that they are still distinct up to symplectomorphisms.

Vinicius Gripp Barros Ramos: The Toda Lattice, Billiards and the Viterbo Conjecture

19th May, 2023

The Toda lattice is one of the earliest examples of non-linear completely integrable systems. Under a large deformation, the Hamiltonian flow can be seen to converge to a billiard flow in a simplex. In the 1970s, action-angle coordinates were computed for the standard system using a non-canonical transformation and some spectral theory. In this talk, I will explain how to adapt these coordinates to the situation to a large deformation and how this leads to new examples of symplectomorphisms of Lagrangian products with toric domains. In particular, we find a sequence of Lagrangian products whose symplectic systolic ratio is one and we prove that they are symplectic balls. This is joint work with Y. Ostrover and D. Sepe.

Michael Entov: Kähler-type and Tame Embeddings of Balls into Symplectic Manifolds

5th May, 2023

A symplectic embedding of a disjoint union of domains into a symplectic manifold M is said to be of Kähler type (respectively tame) if it is holomorphic with respect to some (not a priori fixed) integrable complex structure on M which is compatible with (respectively tamed by) the symplectic form. I'll discuss when Kähler-type embeddings of disjoint unions of balls into a closed symplectic manifold exist and when two such embeddings can be mapped into each other by a symplectomorphism. If time permits, I'll also discuss the existence of tame embeddings of balls, polydisks and parallelepipeds into tori and K3 surfaces.

Michael Entov: Kähler-type and Tame Embeddings of Balls into Symplectic Manifolds

5th May, 2023

A symplectic embedding of a disjoint union of domains into a symplectic manifold M is said to be of Kähler type (respectively tame) if it is holomorphic with respect to some (not a priori fixed) integrable complex structure on M which is compatible with (respectively tamed by) the symplectic form. I'll discuss when Kähler-type embeddings of disjoint unions of balls into a closed symplectic manifold exist and when two such embeddings can be mapped into each other by a symplectomorphism. If time permits, I'll also discuss the existence of tame embeddings of balls, polydisks and parallelepipeds into tori and K3 surfaces.

Michael Bialy: Locally Maximizing Orbits and Rigidity for Convex Billiards

28th April, 2023

Given a convex billiard table, one defines the set ℳ swept by locally maximizing orbits for convex billiard. This is a remarkable closed invariant set which does not depend (under certain assumptions) on the choice of the generating function. I shall show how to get sharp estimates on the measure of this set, recovering as a corollary rigidity result for centrally symmetric convex billiards. Also I shall discuss rigidity of Mather β-function.