Junliang Shen: Cohomology of the moduli of Higgs bundles I, II, III

Abstract: Moduli of Higgs bundles has played a key role in many directions in algebra, geometry, and physics after it was introduced by Hitchin in the 80’s. In these three lectures, I will discuss some recent progress concerning the cohomology of the moduli of Higgs bundles from various perspectives: non-abelian Hodge theory, enumerative geometry, motives and algebraic cycles, etc. We will summarize some new tools and discuss open questions. I will try to make each lecture as independent as possible.

Wei Gu: Quantum cohomology/quantum cohomology correspondence via their relationships with the Verlinde algebra

Abstract: In this talk, I will present a connection between quantum cohomologies of different target spaces with the same axial anomaly. The Verlinde algebra is the bridge to this relationship. If time permits, I will also discuss a similar statement in quantum K theory.

Maxime Cazaux: Quantum K-theory for the quintic singularity

Abstract:The Landau-Ginzburg Calabi-Yau correspondence relates the quantum  cohomology of a CY hypersurface X, with that of the associated  singularity in the affine space. More precisely, both theories are encoded in generating I-functions,  which match under analytic continuation and satisfy the Picard-Fuchs  equation. In quantum K-theory, an analogue of quantum cohomology, the I-function  of X satisfies a q-difference equation instead. In this talk, we will discuss an approach for K-theoretic invariants of  the Fermat singularity, and explain how to recover all the solutions to the q-difference equation of the quintic threefold.

Conan Nai Chung LEUNG: 3d mirror symmetry is mirror symmetry

Abstract: 3d mirror symmetry is a mysterious duality for certian pairs of hyperkahlermanifolds, or more generally complex symplectic manifolds/stacks. In this talk, I will describe its relationships with 2d mirror symmetry. This could be regarded as a 3d analog of the paper ’Mirror symmetry is T-duality’ by Strominger, Yau and Zaslow which described 2d mirror symmetry via 1d dualities.  

Andrea Brini: Refined Gromov-Witten invariants

Abstract: I will discuss a conjectural definition of refined curve counting invariants of Calabi--Yau threefolds with a C*-action in terms of stable maps on Calabi--Yau fivefolds. The corresponding disconnected generating function should conjecturally equate the Nekrasov--Okounkov K-theoretic membrane index under a refined version of the GW/PT correspondence. I'll present several acid tests validating the conjecture, both in the A and the B-model. This is based on joint work with Yannik Schuler (ETH Zurich), arXiv:2410.00118.

Patrick Lei: Higher-genus Gromov-Witten theory of smooth Calabi-Yau hypersurfaces in weighted P^4

Abstract: There are a number of conjectures regarding the all-genus Gromov-Witten theory of Calabi-Yau threefolds which have their origin in physics. While mathematical progress on these conjectures has been very slow, breakthroughs by Chang-Guo-Li and Guo-Janda-Ruan enabled proofs of some of these conjectures for the quintic. In this talk, I will describe a proof of the Yamaguchi-Yau finite generation conjecture and the BCOV holomorphic anomaly equation for smooth Calabi-Yau hypersurfaces in P(1,1,1,1,2), P(1,1,1,1,4), and P(1,1,1,2,5) and give explicit formulae for the genus one invariants.

Alexey Basalaev：Landau--Ginzburg A and B models of the simple--elliptic singularities

Abstract:It's well-known since the work of Y.Ruan and T.Milanov, that the mirror of a LG B--model defined by a simple--elliptic singularity is an elliptic orbifold. This result was further extended by Milanov and Shen who established the LG--LG type isomorphisms for the simple--elliptic singularities. In this talk we will discuss in details the A and B models of the simple--elliptic singularities with the nonmaximal/nontrivial symmetry groups.

Victor Przyjalkowski：Fibers of Landau--Ginzburg models: the Fano threefold case

Abstract：Mirror Symmetry relates a Fano variety to its dual object --- Landau--Ginzburg model. Such models are one-dimensional Calabi--Yau varieties that are dual to anticanonical sections of Fano varieties. The most studied higher-dimensional case of smooth Fano varieties is the threefold one: due to Iskovskikh and Mori--Mukai they are classified, and their geometry is well studied. On the other hand Landau--Ginzburg models for them are constructed. We shortly observe the construction.We also discuss properties of Landau--Ginzburg models that reflect the geometry of their corresponding Fano varieties: rationality, Hodge numbers, etc., and algebro-geometric conjectures on Fano varieties based on them. Finally, we discuss derived categories of Fano varieties, their relations to Landau--Ginzburg models,and some recent predictions based on the observations.

Zhiyu Liu: A new deformation type of irreducible symplectic varieties

Abstract: Irreducible symplectic varieties are one of three building blocks of varieties with Kodaira dimension zero, which are higher-dimensional analogs of K3 surfaces. Despite their rich geometry, there have been only a limited number of approaches to construct irreducible symplectic varieties. In this talk, I will introduce a general criterion for the existence of irreducible symplectic compactifications of non-compact Lagrangian fibrations, based on the minimal model program and the geometry of general fibers. As an application, I will explain how to get a 42-dimensional irreducible symplectic variety with the second Betti number at least 24, which belongs to a new deformation type. This is a joint work with Yuchen Liu and Chenyang Xu.

Di Yang: Quantum KdV hierarchy and quasi-modularity

Abstract: In this seminar, we give a review of the recent development on the quantum Hopf hierarchy and the quantum KdV hierarchy. The quantum Hopf hierarchy (aka the quantum dispersionless KdV hierarchy) was introduced by Eliashberg, and the quantum KdV hierarchy that we consider was constructed by Buryak and Rossi using double ramification cycles on the moduli space of curves. We will mainly focus on the spectral problems of the quantum integrable hierarchies. In particular, we will explain how quasi-modularity naturally appears in this study and Zagier's conjecture.

Hua-Zhong Ke: Counter-examples to Gamma conjecture I (2)

Abstract: For quantum cohomology of a Fano manifold X, Gamma conjectures try to describe the asymptotic behavior of Dubrovin connection in terms of derived category of coherent sheaves on X, via the Gamma-integral structure of the quantum cohomology. In particular, Gamma conjecture I expects that the structure sheaf corresponds to a flat section with the smallest asymptotics. Recently, we discovered that certain toric Fano manifolds do not satisfy this conjecture. In this talk, we will report our results on these counter-examples, and propose modifications for Gamma conjecture I. This talk is self-contained, and is based on joint work with S. Galkin, J. Hu, H. Iritani, C. Li and Z. Su.

Changzheng Li: Counter-examples to Gamma conjecture I (1)

Abstract: Gamma conjectures I, II and the underlying Conjecture O for Fano manifolds were proposed by Galkin, Golyshev and Iritani. In this talk, we will briefly review these conjectures, with an emphasis on counter-examples of Conjecture O and interesting questions arising from the counter-examples. This talk is based on joint work with S. Galkin, J. Hu, H. Iritani, H. Ke and Z. Su.

Danilo Lewanski: The DR-DZ equivalence III

Abstract: Building on the result from the second talk, where we established the relation A = B, we will introduce a third class, defined as a sum over trees decorated with Omega classes (also known as Chiodo classes), and demonstrate that this new class is equal to the A-class and therefore the B-class. This class originates from a formula we conjectured representing quantum tau functions in the framework of quantum DR hierarchies. We will discuss this formula, and if time permits, present its recently discovered proof.

Xavier Blot: The DR-DZ equivalence II

Abstract: This talk is the second of a series of three talks about the DR-DZ equivalence. In the first talk, we explained how the equality of the so-called A-class and B-class in the cohomology of the moduli space of curves implies the equivalence of the DR and DZ hierarchies. In this talk, we explain how to prove the relation A=B using a virtual localization formula.