Online Seminar

05292024
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).

05222024
Hyeonjun Park: Virtual Lagrangian cycles
Abstract: DonaldsonThomas invariants of CalabiYau 4folds 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 1forms 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.

05172024
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.

05162024
Max Hallgren: Tangent cones of KahlerRicci flow singularity models
Abstract: By the compactness theory of Bamler, any finite time singularity of the KahlerRicci flow is modeled on a singular KahlerRicci 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 Hormandertype L^2 estimates on singular shrinking solitons.

05132024
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: MellinBarnes 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 Ktheoretic invariants using elliptic central charges.

05062024
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 HalpernLeistner.

04182024
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).

04102024
Yaoxiong Wen: Mirror symmetries for parabolic Hitchin systems, from classical to global, 1
Abstract: This is the first of three lectures about StromingerYauZaslow 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).

04072024
Omar Kidwai: DonaldsonThomas invariants for the BridgelandSmith correspondence
Abstract: We describe the calculation of (refined) DonaldsonThomas invariants for a certain class of 3CalabiYau 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 secondorder pole, in agreement with predictions from the physics literature. Based on joint work with N. Williams.

03152024
Abstract.In this talk, we will review the current study of mirror symmetry for flag varieties. We will focus more on the construction of LandauGinzburg 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.

03082024
Mauro Porta: Homotopy theory of Stokes data and derived moduli
Abstract: Stokes data are the combinatorial counterpart of irregular meromorphic connections in the RiemannHilbert 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 JeanBaptiste 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.

02222024
Xin Wang: Universal structures for GromovWitten invariants
Abstract: In this talk, we discuss two kinds of universal structures for higher genus GromovWitten 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.

02152024
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 Z1) copies of QH(Z). The idea of the proof comes from Teleman's conjecture on quantum cohomology of symplectic reductions.

01112024
MinhTam Trinh：TriplyGraded Link Homology and the HilbvsQuot Conjecture
Abstract: Any complex algebraic plane curve singularity defines a link up to isotopy, via its intersection with a surrounding 3sphere. The Oblomkov–Rasmussen–Shende conjecture describes the a, q, tgraded Khovanov–Rozansky homology of the link in terms of the punctual Hilbert schemes of the singularity. The t = –1 limit is a theorem of Maulik. However, no prior results managed to incorporate all three gradings, except in very simple examples. Oscar Kivinen and I introduce a different kind of Quot scheme, and prove that an analogue of the ORS conjecture for these Quot schemes does hold, with all three gradings, for many singularities of the form y^n = x^d. This motivates a further conjecture about how these new Quot schemes are related to the Hilbert schemes by a motivic substitution of variables. I have proven it for singularities y^3 = x^d with d coprime to 3. This, combined with the previous result, establishes the ORS conjecture for the same class of singularities. If time permits, I will mention how ORS relates to Gromov–Witten theory.