Abstracts GQT Colloquium

Younghan Bae (ETH Zürich) – Surfaces on Calabi-Yau 4-folds

Abstract: A smooth projective variety X is called Calabi-Yau if the canonical line bundle of X is trivial. The goal of this talk is to study the enumerative aspect of counting surfaces inside Calabi-Yau 4-folds. By the work of Borisov-Joyce and Oh-Thomas, the Hilbert scheme (in general moduli space of stable sheaves) of X carries a pure dimensional cycle, called the virtual cycle. However, when this theory applies to the surface counting problem, the virtual class often vanishes. The reason is because the codimension of the Hodge locus of the surface class is positive in many cases. We reduce the obstruction theory of the Hilbert scheme and obtain non-trivial reduced virtual cycle. This reduced virtual cycle has deformation invariance property along the Hodge locus. As an application we derive the variational Hodge conjecture when the reduced virtual cycle is nontrivial. This is a joint work with Martijn Kool and Hyeonjun Park.

Abstract: Associated to decorated trivalent graphs I will describe a family of Laurent polynomials called graph potentials. These polynomials satisfy interesting symmetry and compatibility properties for different choices of graphs, leading to the construction of a topological quantum field theory which efficiently computes the classical periods as the partition function.

Under mirror symmetry graph potentials are related to moduli spaces of rank 2 bundles (with fixed determinant of odd degree) on a curve of genus g>1, which is a class of Fano varieties of dimension 3g-3. I will discuss how enumerative mirror symmetry relates classical periods to quantum periods in this setting. Time permitting I will touch upon aspects of homological mirror symmetry for these Fano varieties and their mirror partners.

Abstract: Hyperkähler varieties are widely studied objects in algebraic geometry; their moduli spaces are objects whose points represent hyperkähler varieties in a natural way. In a series of papers from 2007-2011, Gritsenko, Hulek and Sankaran proved that moduli spaces of hyperkähler
varieties are often of general type, meaning that their Kodaira dimension, a basic birational invariant, is maximal. In this talk I will introduce the above-mentioned notions, and present some new general type results for hyperkähler moduli spaces. If time permits, I will say some words about the strategy of the proof. This is joint work in progress with I. Barros, P. Beri and L. Flapan.

Rima Chatterjee (Cologne) – Knots in contact manifolds

Abstract: Contact manifolds and the knots in them have been extensively studied in the last few decades. A contact 3-manifold is a smooth 3-manifold endowed with a special geometric structure called the contact structure. In this talk, I’ll start by introducing contact manifolds and Legendrian and transverse knots in them. Legendrian and transverse knots lie in the intersection of knot theory and contact topology. They are not just interesting in their own right but they also provide tools in proving several important results in the contact world. Still lot of things are unknown about them which make them an interesting area of study. After giving the basic definitions, I’ll discuss some
applications of these knots. If time permits I’ll also discuss the classification and structure problems of Legendrian and transverse knots. No prior knowledge of contact geometry will be assumed.

Marton Hablicsek (Leiden) – Virtual classes of character stacks

Abstract: In this talk, I will explain how to construct a Topological Quantum Field Theory to compute virtual classes of character stacks in the Grothendieck ring of stacks. This construction extends the work of González-Prieto, Logares and Muñoz. I will also show a few features of the construction focusing on a couple of simple examples including the affine linear group of rank 1 and the semi-direct product of the multiplicative group G_m with the finite group of order 2. The work is joint with Jesse Vogel and Ángel González-Prieto.

Abstract: (joint work with Nima Rasekh) The theory of shadows is an axiomatic, bicategorical framework that generalizes topological Hochschild homology (THH) and satisfies analogous important properties, such as Morita invariance. After introducing shadows and explaining why they’re interesting, I’ll explain how to use Berman’s extension of THH to bicategories to prove that there is an equivalence between functors out of THH of a bicategory and shadows on that bicategory. As an application we provide a new, conceptual proof that shadows are Morita invariant and construct the shadow of THH of enriched infinity-categorical bimodules.

Abstract: Since 1950s, it turned out convenient to group mathematical objects into categories: at first those in topology, then permeating algebra, geometry, representation theory and many other areas. However, some objects – like paths in a space – resist this categorical thinking, as they only are “associative up to equivalence”. But what does it mean? Following this line of thought, we will enter the world of infty-categories and their relatives, discussing the notions of equivalences there. This talk is based upon joint work with Martina Rovelli.

Federica Pasquotto (Leiden) – Rabinowitz-Floer homology: definition and computations

Over the last 30 years, Floer-theoretic invariants have been responsible for many important results in symplectic and contact topology. While these invariants have all been originally defined for compact manifolds, when considering applications one is often interested in situations where the compactness assumption fails to be satisfied. This motivates several recent, interrelated efforts to extend the definition and the computation of Floer-type symplectic invariants beyond the compact setting. This talk focuses on Rabinowitz Floer homology and its definition for a class of non-compact hypersurfaces in standard symplectic space, which have received the name tentacular. I will describe the challenges arising from the lack of compactness and discuss how additional assumptions enable us to obtain the necessary bounds on moduli spaces of Floer trajectories, and thus extend the definition of the homology. The content of the talk is (partly) joint work with Rob Vandervorst and Jagna Wisniewska.