Tom A Bridgeland: Geometric structures defined by Donaldson-Thomas invariants

Abstract. In recent work the author has explored the idea that the DT invariants of a three-dimensional Calabi-Yau category can be used to define a geometric structure on the space of stability conditions. The relevant structure was christened a Joyce structure and can be viewed as a kind of non-linear Frobenius structure. There are close links with hyperkahler structures and the work of Gaiotto, Moore and Neitzke, and also with recent work in physics on non-perturbative completions of partition functions. The main examples of Joyce structures considered so far involve moduli spaces of Higgs bundles and flat connections on Riemann surfaces, and are a kind of complexification of the Hitchin system. The rough plan for the three talks is (1) Introduction, and definition of Joyce structures, (2) Moduli-theoretic construction of Joyce structures on spaces of quadratic differentials, (3) Tau function associated to a Joyce structure.

Changzheng Li：Automorphisms of the quantum cohomology of the Springer resolution and applications

Abstract.In this talk, we will introduce quantum Demazure--Lusztig operators acting by ring automorphisms on the equivariant quantum cohomology of the Springer resolution. Our main application is a presentation of the torus-equivariant quantum cohomology in terms of generators and relations. We will discuss explicit descriptions for the classical types. We also recover Kim's earlier results for the complete flag varieties by taking the Toda limit. This is based on my joint work with Changjian Su and Rui Xiong.

Todor Milanov：Reflection vectors and quantum cohomology of blowups

Abstract: Let X be a smooth projective variety with a semi-simple quantum cohomology. The monodromy group of the quantum cohomology is by definition the monodromy group of the so-called second structure connection. There is a very interesting conjecture, more or less equivalent to Dubrovin's conjecture, that gives an explicit description of the monodromy group in terms of the exceptional objects of the derived category of X. In a joint work with my student Xiaokun Xia, we made an interesting progress in proving that the conjectural description of the monodromy group is compatible with the operation of blowing up at finitely many points. More precisely, we proved that the exceptional objects in Orlov's full exceptional collection of the blowup Bl(X) that are supported on the exceptional divisor are reflection vectors. 

Weiqiang He: Equivariant Hikita conjecture for minimal nilpotent orbit

Abtract: The theory of symplectic duality is a kind of mirror symmetry in mathematical physics. Suppose two (possibly singular) manifolds are symplectic dual to each other, then there are some highly nontrivial identities between the geometry and topology of them. One of them is the equivariant Hikita conjecture. Suppose we are given a pair of symplectic dual conical symplectic singularities, then Hikita’s conjecture is a relation of the quantized coordinate ring of one conical symplectic singularity to the equivariant cohomology ring of the symplectic resolution of the other dual conical symplectic singularity. In this talk, I will focus on this case: the minimal nilpotent orbit and the slodowy slice of the subregular orbit. This is a joint work with XIaojun Chen and Sirui Yu.

Yefeng Shen: Quantum spectrum and Gamma structures for quasi-homogeneous polynomials of general type

Abstract. In this talk, I will explain quantum spectrum and asymptotic expansions in FJRW theory of quasi-homogenous singularities of general type. Inspired by Galkin-Golyshev-Iritani's Gamma conjectures for quantum cohomology of Fano manifolds, we propose Gamma conjectures for FJRW theory of general type. Here the Gamma structures are essential to understand the connection between algebraic structures of the singularities (such as Orlov's semiorthogonal decompositions of matrix factorizations) and the analytic structures in FJRW theory. The talk is based on the work joint with Ming Zhang

Cheng Shu: Mirror of Orbifold Singularities in the Hitchin Fibration

Abstract. For the group SLn, we study the geometry of singular Hitchin fibres over the elliptic locus of the Hitchin base. This gives interesting information on the Fourier-Mukai transform of a skyscraper sheaf supported at an orbifold singularity of the Hitchin moduli space for PGLn. The main results prove a classical version of the conjectures of Frenkel-Witten concerning the mirror of orbifold singularities.

Longting Wu: All-genus WDVV recursion, quivers, and BPS invariants

Abstract: Let D be a smooth rational ample divisor in a smooth projective surface X. In this talk, we will present a simple uniform recursive formula for (primary) Gromov-Witten invariants of O_X(-D). The recursive formula can be used to determine such invariants for all genera once some initial data is known. The proof relies on a correspondence between all-genus Gromov–Witten invariants and refined Donaldson–Thomas invariants of acyclic quivers. In particular, the corresponding BPS invariants are expressed in terms of Betti numbers of moduli spaces of quiver representations. This is a joint work with Pierrick Bousseau.

Junwu Tu: CEI and HAE

Abstract. Assuming certain comparison between non-commutative Hodge structures with classical Hodge structures, we prove the CEI associated with a smooth projective family of Calabi-Yau's satisfy the holomorphic anomaly equation. This is based on a work in progress with Yefeng Shen.

Junwu Tu: An introduction to categorical enumerative invariants (CEI)

Abstract. In this talk, we shall present the definition of CEI associated with smooth and proper Calabi-Yau categories. We also sketch an explicit combinatorial formula of CEI. In the end, we discuss about concrete computations as well as some interesting questions. The talk is based on joint works with Andrei Caldararu and Lino Amorim.

Yalong Cao: From curve counting on Calabi-Yau 4-folds to quasimaps for quivers with potentials

Abstract: I will start by reviewing an old joint work with Davesh Maulik and Yukinobu Toda on relating Gromov-Witten, Gopakumar-Vafa and stable pair invariants on compact Calabi-Yau 4-folds. For non-compact CY4 like local curves, similar invariants can be studied via the perspective of quasimaps to quivers with potentials. In a joint work in progress with Gufang Zhao, we define a virtual count for such quasimaps and prove a gluing formula. Computations of examples will also be discussed.

Denis Nesterov：Enumerative mirror symmetry for moduli spaces of Higgs bundles

Abstract: We discuss conjectures, called genus 1 Enumerative mirror symmetry for moduli spaces of Higgs bundles, which relate curve-counting invariants of moduli spaces of Higgs SL(r)-bundles to curve-counting invariants of moduli spaces of Higgs PGL(r)-bundles.

Jie Gu：Resurgent structure in topological strings

Abstract. Topological string theory has (spacetime) instanton sectors, which the resurgence theory predicts to be completely controlled by the perturbative free energy via Stokes transformations. Recent results also suggest the Stokes constants are related to BPS invariants/DT invariants. To make this picture complete, one needs to first solve the instanton amplitudes and then calculate the Stokes constants. We demonstrate that the first problem can be solved exactly and completely through a transseries extension of the BCOV holomorphic anomaly equations. We also make progress in the second problem. We focus on examples in local Calabi-Yau threefold and comment that similar results can be obtained for quintic as well.

Albrecht Klemm: Symplectic Invariants on Calabi-Yau 3 folds, Modularity and Stability (Lecture I & II & III)

We discuss techniques to calculate symplectic invariants of CY 3-folds $M$, namely Gromov-Witten (GW) invariants, Pandharipande-Thomas (PT) invariants, and Donaldson-Thomas (DT)  invariants. Physicallythe latter are closely related to BPS brane bound states in type IIA string compactifications on $M$. We focus on the rank $r_{\bar 6}=1$  DT invariants  that count $\bar D6-D2-D0$ brane bound states related to PT- and  high genus GW invariants, which are calculable by mirror symmetry and topological string B-model methods modulo certain boundary conditions, and the rank zero DT invariants that count rank $r_4=1$   $D4-D2-D0$ brane bound states. It has been conjectured  by Maldacena, Strominger, Witten and  Yin that  the latter are governed by an index that has modularity properties due to $S$-duality in string theory and extends to a mock modularity index of higher depth for $r_4>1$. Again the modularity allows to fix at least the $r_4=1$ index up to boundary conditions fixing their polar terms.  Using Wall crossing formulas obtained by Feyzbakhsh certain PT invariants or Katz-Klemm-Vafa (KVV) invariants close to the Castelnuovo bound can be related to the $r_4=1$  $D4-D2-D0$ invariants. This provides further boundary conditions for the topological string B-model approach as well as for the $D4-D2-D0$ brane indices.  The approach allows to prove the Castelnuovo bound and  calculate the  $r_{\bar 6}=1$  DT- invariants or the GW invariants to higher genus than hitherto possible, as was pointed out in https://arxiv.org/abs/2301.08066 by Alexandrov, Feyzbakhsh, Pioline, Schimannek and me. See also  http://www.th.physik.uni-bonn.de/Groups/Klemm/data.php for concrete evaluations. Lecture I: ``Recursive solution of the perturbative topological String''In this lecture we explain how to solve the topological string recursively in terms of non holomorphic modular objects and discuss the integer invariants that it calculates.Lecture II:  S-duality and the index of $D4-D2-D0$ bound states''We review the approach of  Maldacena, Strominger, Witten and Yin  to the calculate the abelian ($r_4=1$)   $D4-D2-D0$ bound state degneracies from a modular index and the relation of these invariants to the PT  and DT invariants discussed above ( if time permits we comment on the mock  modularity of higher depth for $r_4>1$).             Lecture III: Castelnuovo bound and Wall crossing'' In this lecture we define the above  mentioned  symplectic invariants  more mathematically  and discuss their stability and their Wall crossing behaviour and summarize the implication on concrete  calculations of them. 

Gregory W. Moore: Supersymmetric QFT & Invariants Of Smooth Four-Manifolds

Abstract. After describing briefly some aspects of nonabelian Yang-Mills theory and Donaldson invariants  the physicists' derivation of the "Witten conjecture" expressing Donaldson invariants in terms of Seiberg-Witten invariants will be sketched. Time permitting, more recent developments related to this physical approach will be sketched.