We present some obstructions to the existence of Lagrangian cobordisms in ℝ4. The obstructions arise from studying moduli spaces of holomorphic disks with corners with boundaries on immersed objects called Lagrangian tangles. The obstructions boil down to area relations and sign conditions on disks bound by knot diagrams of the boundaries of the Lagrangian. We present examples of pairs of knots that cannot be Lagrangian cobordant and knots that cannot bound Lagrangian disks.
The question in the title was one of the founding questions in symplectic topology 40 years ago, and despite a lot of progress since that time, it remains widely open. In the talk I will discuss the initial questions, the progress, and the remaining challenges.
We will discuss the existence of rational (multi)sections and unirulings for projective families f: X→ℙ1 with at most two singular fibres. Specifically, we will discuss two ingredients for constructing the above rational curves. The first is local symplectic cohomology groups associated to compact subsets of convex symplectic domains. The second is a degeneration to the normal cone argument that allows one to produce closed curves in X from open curves (which are produced using local symplectic cohomology) in the complement of X by a singular fibre.
In this talk, I will discuss recent joint work with D. Cristofaro-Gardiner and B. Zhang showing that a generic area-preserving diffeomorphism of a closed surface has a dense set of periodic points. This follows from a result called a 'smooth closing lemma' for area-preserving surface diffeomorphisms; this answers in the affirmative Smale’s 10th problem in the setting of area-preserving surface diffeomorphisms. The proof uses quantitative analysis of spectral invariants from periodic Floer homology via various estimates in Seiberg-Witten theory.
We will discuss a complete computation of Savelyev's homomorphism associated to any coadjoint orbit of a compact Lie group G, where the domain is restricted to the based loop homology of G. This gives at the same time some applications to the Hamiltonian groups of these spaces and a geometric proof of an unpublished theorem of Peterson. This theorem tells us explicitly how the multiplicative structure constants of the based loop homology of G determine those of the quantum cohomology of its coadjoint orbits.
In this talk, as a continuation of my talk in the Members' Colloquium but with a specialized audience in mind, I will discuss in more detail some of the general geometric and dynamical structures underlying the theoretical aspects of the restricted 3-body problem, and outline new research directions.
This talk is based on a joint work with Thomas Kragh. Using the generating function theory we split inject homotopy groups of pseudo-isotopy and/or h-cobordism spaces into various spaces of Legendrian manifolds, e.g. the space of Legendrian unknots in ℝ2n+1 for a sufficiently large n. For instance, there is a non-trivial element in π2 of the space of Legendrian unknots in ℝ2n+1 for n ≥ 12.
Despite the fact that the 3-body problem is an ancient conundrum that goes back to Newton, it is remarkably poorly understood, and is still a benchmark for modern developments. In this talk, I will give a (very) biased account of this classical problem, both from a modern theoretical perspective, i.e. outlining possible lines of attack coming from symplectic geometry, holomorphic curves and Floer theory; as well as comment on practical and numerical aspects, within the context of finding orbits for space mission design and ocean worlds exploration.
proof of homological mirror symmetry between the complex and symplectic manifold associated to local pieces of the combinatorial data.
This is part of a programme with Vivek Shende to prove homological mirror symmetry over a global SYZ base.
In various areas of mathematics there exist 'big fibre theorems', these are theorems of the following type: 'For any map in a certain class, there exists a 'big' fibre', where the class of maps and the notion of size changes from case to case.
We will discuss three examples of such theorems, coming from combinatorics, topology and symplectic topology from a unified viewpoint provided by Gromov's notion of ideal-valued measures.
We adapt the latter notion to the realm of symplectic topology, using an enhancement of Varolgunes’ relative symplectic cohomology to include cohomology of pairs. This allows us to prove symplectic analogues for the first two theorems, yielding new symplectic rigidity results.
Necessary preliminaries will be explained.
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