Speaker: Cristina Manolache (Imperial College London)

Title: Degenerate contributions to enumerative invariants
Abstract: Enumerative questions have a very long history in Mathematics and have been revolutionised in the nineties with the construction of the moduli space of stable maps and the machinery allowing us to integrate on these very singular spaces. However, moduli spaces of stable maps have many "unwanted" components which are reflected in the intersection numbers.

In my talks I will first give examples of "enumerative invariants" which are not enumerative, then I will shortly explain virtual classes and why it is difficult to split virtual classes on components of a moduli space. In the last talk I will give examples of contributions of the components of the moduli space of stable maps to Gromov--Witten invariants. More precisely, I will discuss the relationship between Gromov--Witten invariants and reduced invariants (or Gopakumar--Vafa invariants) and if time permits the relationship between Gromov--Witten invariants and quasi-map invariants.

Speaker: Ben Davison (École Polytechnique Fédérale de Lausanne)

Title: Cohomological Donaldson--Thomas theory

Abstract: I will talk about recent work with Sven Meinhardt, in which we have proved the integrality conjecture for refined Donaldson-Thomas invariants, and also provided a mathematically rigorous definition of the cohomology of the space of primitive BPS states from Physics. This rigorous definition lies at the heart of "categorification" of DT theory, in which we promote the integrality conjecture and the wall crossing formula from their original formulation, as equalities, to the category of mixed Hodge modules, where they become isomorphisms in cohomology.  I will present a couple of recent applications to this categorification in the context of positivity conjectures. But have no fear, I will in addition motivate the theory throughout with plenty of down-to-earth (but still fun!) examples.

Speaker: Sheldon Katz (University of Illinois at Urbana-Champaign)

Title: Moduli and mirror symmetry: motivations from physics
Abstract: My goal is to explain a little bit of physics (using mathematical language and assuming nothing) to try to explain why we should expect things like BPS counts, mirror symmetry, and Gromov-Witten invariants.

 
Speaker: Atsushi Kanazawa (Harvard University)
Title: Geometry of branes and stability conditions
Abstract: I will talk about two kinds of D-branes on Calabi-Yau manifolds, namely A-branes and B-branes. Roughly, A-branes are Lagrangian submanifolds and B-branes are coherent sheaves, and they are interchanged under mirror symmetry. I will explain this correspondence and how stability conditions of the branes come into the play. The goal is to show how the space of stability conditions of the B-branes is related to various fields of mathematics, such as MMP and Weil-Petersson geometry.

Speaker: Helge Ruddat (Johannes Gutenberg-Universität Mainz)
Title: Logarithmic Gromov-Witten invariants
Abstract: Logarithmic Gromov-Witten invariants generalize usual and relative Gromov-Witten invariants and were first suggested by Siebert and then introduced by Gross-Siebert and Abramovich-Chen.
The key results are stable reduction and the existence of a natural virtual fundamental class as well as by means of the notion of "basicness" the algebraicity and quasi-compactness of the moduli space.
I will give a comprehensive introduction including several examples that demonstrate their main features and use.

Research talks

Speaker: Jinwon Choi (Sookmyung Women's University) and Michel van Garrel (Korea Institute for Advanced Study)

Title: Local and log BPS numbers
Abstract: BPS numbers were introduced by physicists and have only recently been defined rigorously through the work of Kiem-Li. We consider the local BPS numbers for a del Pezzo surface S. In joint work with S. Katz and N. Takahashi, we introduce log BPS numbers of S, which are weighted counts of rational curves fully tangent to a smooth anticanonical divisor. In this talk, we start by reviewing local BPS numbers and then introduce log BPS numbers. We state and motivate their conjectural relationship and explain how to prove it for all curve classes of arithmetic genus at most 2.

Speaker: Sukmoon Huh (Sungkyunkwan University)

Title: A simple construction of nilpotent co-Higgs bundles
Abstract: A co-Higgs bundle on a smooth complex projective variety X is a pair of a holomorphic vector bundle E and a morphism E to E tensored by the tangent bundle of X (called a co-Higgs field), satisfying the integrability condition. It is introduced by Hitchin, as a generalized vector bundle on X, considered as a generalized complex manifold. Adapting the notion of semistability of vector bundles, we may define semistable co-Higgs bundle and it is expected to exists at the lower end of Enriques-Kodaira spectrum.

Motivated by the work of Correa to classify the surfaces admitting a semistable nilpotent co-Higgs bundle of rank two, we investigate several aspects of nilpotent co-Higgs bundles. In this talk, we suggest a simple way of constructing nilpotent co-Higgs bundle, using the Hartshorne-Serre correspondence and apply it to several cases.

This is a joint work with Edoardo Ballico.

Speaker: Kiryong Chung (Kyungpook National University)

Title: Mori program for the moduli space of conics in Grassmannian
Abstract: Rational curves in Grassmannian (or its sections) has been studied in various view points: Birational and complex geometry. Motivated by the results of Iliev-Manivel and Hosono-Takagi, we complete Mori's program for Kontsevich's moduli space of degree $2$ stable maps to Grassmannian. One of key ingredient is to provide some moduli theoretic meaning for the double symmetroid which directly connects the Kontsevich space and Hilbert scheme. We also discuss about its birational relation with the moduli space of one dimensional sheaves on $mathbb{P}^2$. This is a joint work with Han-Bom Moon.

Speaker: Bumsig Kim (Korea Institute for Advanced Study)
Title: Residue Mirror Symmetry for Grassmannians
Abstract: The talk is based on joint work with Jeongseok Oh, Kazushi Ueda and Yutaka Yoshida.

Motivated by recent works on localizations in A-twisted gauged linear sigma models, we discuss a generalization of toric residue mirror symmetry to complete intersections in Grassmannians.