Xavier Blot: The DR-DZ equivalence I

Abstract: The Dubrovin-Zhang (DZ) hierarchies and the Double Ramification (DR) hierarchies are two families of integrable hierarchies. In 2014, it was conjectured by Buryak that these hierarchies are equivalent. This talk is the first of a series of three talks about the proof of this conjecture, its refinements and related results. In the first talk we will present these hierarchies and explain how to reformulate this equivalence as a relation A=B of classes in the tautological ring of the moduli space of curves.

Yaoxiong Wen: Mirror symmetries for parabolic Hitchin systems, from classical to global III

Abstract: This is the last of three lectures about mirror symmetries for parabolic Hitchin systems. We first recall the new moduli space (associated to nilpotent orbit) we defined in the first lecture and focus on the case of the orbit being Richardson. Then, we study the geometry of Hitchin maps of this new moduli space (associated to the Richardson orbit) and of the usual moduli space (associated to the polarization of the Richardson orbit). Finally, we prove SYZ and topological mirror symmetries. This is based on the joint work with B.Wang and X. Wen (arXiv:2403.07552).

Hyeonjun Park: Virtual Lagrangian cycles

Abstract: Donaldson-Thomas invariants of Calabi-Yau 4-folds are defined through recently developed virtual Lagrangian cycles associated to (-2)-shifted symplectic derived moduli spaces. In this talk, we discuss various properties of these virtual Lagrangian cycles in the perspective of shifted symplectic geometry. We first provide a virtual pullback formula for Lagrangian correspondences and use it to compute Hilbert scheme invariants. We then construct reduced virtual cycles for counting surfaces via (-1)-shifted closed 1-forms and show that they can detect the variational Hodge conjecture. We also revisit cosection localization via virtual Lagrangian cycles for (-2)-shifted twisted cotangent bundles. We finally explain deformation invariance in terms of the exactness of the symplectic forms.

Hu Zhao: Quantization Problem in Noncommutative Hamiltonian Setting

Abstract: Given a noncommutative Hamiltonian space A, we show that the conjecture “quantization commutes with reduction” holds on A. Also, applications and possible future directions will be discussed in this talk.

Max Hallgren: Tangent cones of Kahler-Ricci flow singularity models

Abstract: By the compactness theory of Bamler, any finite time singularity of the Kahler-Ricci flow is modeled on a singular Kahler-Ricci soliton, and such solitons are infinitesimally metric cones. In this talk, we will see that these cones are normal affine algebraic varieties, using a new method for proving Hormander-type L^2 estimates on singular shrinking solitons.

Konstantin Aleshkin: Quasimap central charges and wall crossing for GLSM

Abstract.Quasimap central charges are generating functions of GLSM invariants associated to matrix factorizations. These functions admit different kinds of integral representations: Mellin-Barnes type and Euler type. I plan to explain how the former arises and leads to relations of central charges for different phases of GLSM. If the time permits, I will explain how to generalize these ideas for   K-theoretic invariants using elliptic central charges.

Andres Fernandez Herrero: A version of stable maps into the classifying stack BGL_n

Abstract: In this talk, I will describe a version of stable maps into a quotient stack [Z/GLN], where Z is a projective variety with an action of the general linear group GLN. We will mostly focus on the case when Z is a point, so the target is the classifying stack BGL_n.I will also update on the ongoing piece of the story with marked points, which involves some surprises, such as the inclusion of a notion of orientation for the markings in order to compactify the evaluation morphisms and define reasonable gluing morphisms. This talk is based on joint work in progress with Daniel Halpern-Leistner.

Xueqing Wen: Mirror symmetries for parabolic Hitchin systems, from classical to global II

Abstract: This is the second lecture about SYZ and topological mirror symmetries for parabolic Hitchin systems in type B/C. One of the key step to establish mirror symmetries for Hitchin systems is to understand the generic fibers of the Hitchin map, and the most powerful tool is the so called BNR correspondence which relates Higgs bundles to rank one sheaves on the spectral curves. However, in parabolic case, the spectral curves usually have singularities, which may cause many difficulties but also many interesting phenomena. In this talk, I will focus on the formal disk around the singular point, and define the notion of local Higgs bundles, which can be seem as not only the restriction of parabolic Higgs bundle to the formal disk but also an infinitesimal enhancement of nilpotent Lie algebra elements. The main results in this part are some decomposition theorems of the local Higgs bundles in type B/C. As a byproduct, we found a new geometric interpretation of the Lusztig’s canonical quotient(arXiv:2403.07552).

Yaoxiong Wen: Mirror symmetries for parabolic Hitchin systems, from classical to global, 1

Abstract: This is the first of three lectures about Strominger-Yau-Zaslow and topological mirror symmetries for parabolic Hitchin systems. In this talk, we will explain the significance of nilpotent orbit closures in parabolic Hitchin systems, i.e., the classical level. Furthermore, we establish a new perspective of Springer duality for special nilpotent orbits of types B and C. This is based on the joint work with B.Wang and X. Wen (arXiv:2403.07552).

Omar Kidwai: Donaldson-Thomas invariants for the Bridgeland-Smith correspondence

Abstract: We describe the calculation of (refined) Donaldson-Thomas invariants for a certain class of 3-Calabi-Yau triangulated categories whose space of stability conditions can be interpreted as a space of quadratic differentials on a Riemann surface. This category is slightly different from the usual one discussed by Bridgeland and Smith, which in particular allows us to recover a nonzero invariant in the case where the quadratic differential has a second-order pole, in agreement with predictions from the physics literature. Based on joint work with N. Williams.

Changzheng Li: A PLU ̈CKER COORDINATE MIRROR FOR PARTIAL FLAG VARIETIES AND QUANTUM SCHUBERT CALCULUS

Abstract.In this talk, we will review the current study of mirror symmetry for flag varieties. We will focus more on the construction of Landau-Ginzburg model, and discuss a folklore mirror symmetry expectation on the eigenvalues of the first Chern class, using concrete examples of flag varieties of Lie type A. This is based on a work in progress joint with Konstanze Rietsch, Mingzhi Yang and Chi Zhang.

Mauro Porta: Homotopy theory of Stokes data and derived moduli

Abstract: Stokes data are the combinatorial counterpart of irregular meromorphic connections in the Riemann-Hilbert correspondence. In dimension 1, they have been studied by Deligne and Malgrange. In higher dimensions the situation is intrinsically more complicated, but the solution of Sabbah's resolution conjecture by Mochizuki and Kedlaya unlocked a much deeper understanding of the Stokes phenomenon. In recent work with Jean-Baptiste Teyssier, we develop a framework to define and study Stokes data with coefficients in any presentable stable infinity category. As an application, we construct a derived moduli stack parametrizing Stokes data in arbitrary dimensions, extending all previously known representability results. One fundamental input is a finiteness theorem for the stratified homotopy types of algebraic and compact real analytic varieties that we obtained in collaboration with Peter Haine and that extends finiteness theorems for the underlying homotopy types of Lefschetz–Whitehead, Łojasiewicz and Hironaka.

Xin Wang: Universal structures for Gromov-Witten invariants

Abstract: In this talk, we discuss two kinds of universal structures for higher genus Gromov-Witten invariants of any target varieties. One is certain explicit universal equations from Hodge integrals. Another one is certain basic structures from topological recursion relations on the moduli space of curves. This is based on joint work with Felix Janda.

Hiroshi Iritani: Decompositions of quantum cohomology under blowups

Abstract. It is an interesting question how (analytic continuation of) quantum cohomology is related to birational geometry. In particular, we expect that quantum cohomology decomposes in a certain way for extremal contractions. In this talk, I will explain such a decomposition for blowups: quantum cohomology of the blowup of X along Z will decompose into QH(X) and (codim Z-1) copies of QH(Z). The idea of the proof comes from Teleman's conjecture on quantum cohomology of symplectic reductions.