After introducing Hamiltonian homeomorphisms and recalling some of their properties, I will focus on fixed point theory for this class of homeomorphisms. The main goal of this talk is to present the outlines of a C0 counterexample to the Arnold conjecture in dimensions four and higher.
We shall give a construction of the quantized sheaf of a Lagrangian submanifold in T∗N and explain a number of features and applications.
I will recall the construction of the space of states in a gauged topological A-model. Conjecturally, this gives the quantum cohomology of Fano symplectic quotients: in the toric case, this is Batyrev's presentation of quantum cohomology of toric varieties. Time permitting, I will discuss the role of 'Coulomb branches' in gauge theory in relation to equivariant quantum and symplectic cohomology.
Let n > 1. Given two maps of an n-dimensional sphere into Euclidean 2n-space with disjoint images, there is a ℤ/2 valued linking number given by the homotopy class of the corresponding Gauss map. We prove, under some restrictions on n, that this vanishes when the components are immersed Lagrangian spheres each with exactly one double point of high Maslov index.
Singular algebraic (sub)varieties are fundamental to the theory of smooth projective manifolds. In parallel with his introduction of pseudo-holomorphic curve techniques into symplectic topology 30 years ago, Gromov asked about the feasibility of introducing notions of singular (sub)varieties suitable for this field. I will describe a new perspective on this question and motivate its appropriateness in the case of normal crossings singularities. It leads to multifold versions of symplectic sum and cut constructions expected by Gromov and notions of one-parameter families of degenerations of symplectic manifolds and logarithmic tangent bundles in the spirit of the Gross-Siebert program. In our category, the standard triple point condition of algebraic geometry is the only obstruction to the smoothability of NC singularities.
In this talk, I will discuss two measurements of Lagrangian cobordisms between Legendrian submanifolds in symplectizations: their length and their relative Gromov width. The Gromov width, in particular, is a fundamental global invariant of symplectic manifolds, and a relative version of that width helps understand the geometry of Lagrangian submanifolds of a symplectic manifold. Lower bounds on both the length and the width may be produced by explicit constructions; this talk will concentrate on upper bounds that arise from a filtered version of Legendrian contact homology, a Floer-type invariant.
We show that there is a 1-parameter family of monotone Lagrangian tori in the cotangent bundle of the 3-sphere with the following property: every compact orientable monotone Lagrangian with non-trivial Floer cohomology is not Hamiltonian-displaceable from either the zero-section or one of the tori in the family. The proof involves studying a version of the wrapped Fukaya category of the cotangent bundle which includes monotone Lagrangians. Time permitting, we may also discuss an extension to other cotangent bundles.
This is a part of my joint work with Oh-Ohta-Ono and is a part of project to rewrite the whole story of virtual fundamental chain in a way easier to use. In general we can construct virtual fundamental chain on (basically all) the moduli space of pseudo-holomorphic curve. It depends on the choices. In this talk I want to provide a statement to clarify which is the data we need to start with and in which sense the resulting structure is well defined. A purpose of writing such statement is then it can be a black box and can be used without looking the proof. Also it is useful to see some properties of it such as its relation to the (target space) group action or compatibility with forgetful map.
In this talk, I would like to explain my joint work with Weiwei Wu about understanding projective Dehn twist using Lagrangian cobordism.
Joint work with Ivan Smith. Let p be a positive integer. Take the quotient of a 2-disc by the equivalence relation which identifies two boundary points if the boundary arc connecting them subtends an angle which is an integer multiple of (2π/p). We call the resulting cell complex a 'p-pinwheel'. We will discuss constraints on Lagrangian embeddings of pinwheels. In particular, we will see that a p-pinwheel admits a Lagrangian embedding in ℂP2 if and only if p is a Markov number. Time permitting, I will discuss nondisplaceability results, which are a purely symplectic analogue of the Hacking-Prokhorov classification of Q-Gorenstein degenerations of ℂP2.
You must be logged in to post a comment.