I will give an introduction to sheaves and microlocal sheaves, as pioneered by Kashiwara-Schapira. The goal will be to explain recent work with Shende establishing that microlocal sheaves on a Weinstein manifold are a symplectic invariant.
The discovery of the Jones polynomial in the early 80's was the beginning of ``quantum topology'': the introduction of various invariants which, in one sense or another, arise from quantum mechanics and quantum field theory. There are many mathematical constructions of these invariants, but they all share the defect of being first defined in terms of a knot diagram, and only subsequently shown by calculation to be independent of the presentation. As a consequence, the geometric meaning has been somewhat opaque.
I will discuss results relating different partially wrapped Fukaya categories. These include a K\"unneth formula, a `stop removal' result relating partially wrapped Fukaya categories relative to different stops, and a gluing formula for wrapped Fukaya categories. The techniques also lead to generation results for Weinstein manifolds and for Lefschetz fibrations. The methods are mainly geometric, and the key underlying Floer theoretic fact is an exact triangle in the Fukaya category associated to Lagrangian surgery along a short Reeb chord at infinity.
As it was observed a few years ago, there exists a certain signed count of real lines on real projective hypersurfaces of degree 2n+1 and dimension n that, contrary to the honest "cardinal" count, is independent of the choice of a hypersurface, and by this reason provides, as a consequence, a strong lower bound on the honest count. Originally, in this invariant signed count the input of a line was given by its local contribution to the Euler number of an appropriate auxiliary universal vector bundle.
We consider the problem of extending a representation of the fundamental group of 3-manifolds from part of the boundary surfaces. Applications to links will be discussed. Combining this with some cohomology classes of Atiyah and Bott leads to new multivariable polynomial invariants of 3-manifolds with boundary.
This is joint work with Edward Miller.
No background in 3-dimensional topology will be assumed in this survey and research talk.
Given two Hamiltonian isotopic curves in a surface, one would like to tell whether they are "close" or "far apart". A natural way to do that is to consider Hofer's metric which computes mechanical energy needed to deform one curve into the other. However due to lack of tools the large-scale Hofer geometry is only partially understood. On some surfaces (e.g. S^2) literally nothing is known.
I will review homological mirror symmetry for the torus, which describes Lagrangian Floer theory on T^2 in terms of vector bundles on the Tate elliptic curve --- a version of Lekili and Perutz's works "over Z", where t is the Novikov parameter. Then I will describe a modified form of this story, joint with Lekili, where the Floer theory is altered by a locally constant sheaf of rings on T^2 (an "F-field").
In the last 35 years, geometric flows have proven to be a powerful tool in geometry and topology. The Mean Curvature Flow is, in many ways, the most natural flow for surfaces in Euclidean space. In this talk, which will assume no prior knowledge, I will illustrate how mean curvature flow could be used to address geometric questions.
Cohomology jump loci are secondary cohomological invariants of discrete groups and topological spaces. I will describe some recent work on the cohomology jump loci of complements of unions of smooth complex hypersurfaces, and I will motivate the notion of abelian duality spaces that I introduced in joint work with Alex Suciu and Sergey Yuzvinsky. The study of such hypersurface arrangements involves a mix of combinatorics and complex geometry.
This talk is a survey on results concerning the Teichmuller space of negatively curved Riemannian metrics on M. It is defined as the quotient space of the space of all negatively curved Riemannian metrics on M modulo the space of all isotopies of M that are homotopic to the identity. This space was shown to have highly non-trivial homotopy when M is real hyperbolic by Tom Farrell and Pedro Ontaneda in 2009.
The Whitehead conjecture asks whether a subcomplex of an aspherical 2-complex is always
aspherical. This question is open since 1941. Howie has shown that the existence of a finite counterexample implies (up to the Andrews-Curtis conjecture) the existence of a counterexample within the class of labelled oriented trees. Labelled oriented trees are algebraic generalisations of Wirtinger presentations of knot groups.
We shall discuss several topics regarding symplectic measurements in the classical phase space. In particular: Viterbo's volume-capacity conjecture and its relation with Mahler conjecture, the symplectic size of random convex bodies, the EHZ capacity of convex polytopes (following the work of Pazit Haim-Kislev), and (if time permits) also computational complexity aspects of estimating symplectic capacities.
Open Gromov-Witten (OGW) invariants count pseudoholomorphic maps from a Riemann surface with boundary to a symplectic manifold, with constraints that make sure the moduli space of solutions is zero dimensional. In joint work with J. Solomon (2016-2017), we defined OGW invariants in genus zero under cohomological conditions. In this talk, also based on joint work with J. Solomon, I will describe a family of PDEs satisfied by the generating function of our invariants. We call this family the open WDVV equations.