Time:26 March 2024 (16:00-17:00)

Speaker: Robert Crumplin (Imperial College London)

Venue: Room 233A,Taizhou Hall,SICM Seminar Room,SUSTech (Zoom ID: 931 4365 2875　 Password: 935689)

Abstract:For (X|D) a smooth pair, the Gromov—Witten theory of the root stacks of X along D with parameter r is a well-studied way of counting maps to X with given tangency to D. By involved calculations, Tseng and You have shown that the resulting invariants are polynomial in r of degree bounded in terms of the genus g of the domain curves. I will show a geometric way to understand this polynomiality by an explicit description of the moduli space of twisted maps to the ''universal pair"- a space which controls the enumerative geometry of any smooth pair (X|D). It turns out this space is built from a distinguished main component, along with abelian variety fibrations over boundary strata of this component arising from Jacobians of curves. The irreducible components of this space then correspond to certain combinatorial/tropical data, and each contributes a single monomial to the polynomial discovered by Tseng and You.

Time:20 March 2024 10:00 - 22 March 2024 12:00

Speaker: Mohammed Abouzaid (Stanford University)

Venue: 5/F, Academic Building No. 1 , The Chinese University of Hong Kong

Abstract:Each semester we will invite some guest speaker(s) to give us a mini-course. The main guest speaker for this semester is Prof. Mohammed Abouzaid from Stanford University. He will give a series of lectures at the Chinese University of Hong Kong. More information on the lectures can be found via the hyperlink of the venue. All are welcome.

Time:12 March 2024 (16:00-17:00)

Speaker: Yi Hu (University of Arizona)

Venue: M1001, College of Science Building, SUSTech (Zoom ID: 889 8176 7749　 Password: 666666)

Abstract:I will explain the matrix local equations defining the moduli spaces of stable maps of arbitrary genus, found jointly by Jun Li and the speaker. These equations already guided us to find explicit global resolutions for these moduli spaces in the cases when the genera are one and two. By Murphy’s law, stable map moduli possess arbitrary singularities. Turning to this, I will explain Lafforgue’s version of Mnev’s universality, how it leads to standard local equations for arbitrary singularity types, and how it should guide to resolve arbitrary singularities.

Time:28 November 2023 (16:00-17:15)

Speaker: Junwu Tu (ShanghaiTech University)

Venue: M1001, College of Science Building, SUSTech (Zoom ID: 983 7636 8911　 Password: 111431)

Abstract:Assuming certain comparison between non-commutative Hodge structures and classical Hodge structures, we show the categorical enumerative invariants associated with a smooth projective family of Calabi-Yau 3-folds satisfy the holomorphic anomaly equations. This naturally leads to the study of geometric structures on moduli spaces of smooth projective Calabi-Yau 3-folds.

Time:21 November 2023 10:00 - 24 November 2023 12:00

Speaker: Hiroshi Iritani (Kyoto University)

Venue: 5/F, Academic Building No. 1, The Chinese University of Hong Kong

Abstract:Each semester we will invite some guest speaker(s) to give us a mini-course. The main guest speaker for this semester is Prof. Iritani from Kyoto Univeristy. He will give a series of lectures at the Chinese University of Hong Kong. More information on the lectures can be found via the hyperlink of the venue. All are welcome.

Time:19 September 2023 (16:00-17:00)

Speaker: Michel van Garrel (University of Birmingham)

Venue: M1001, College of Science Building, SUSTech (Zoom ID: 980 1095 8704　 Password: 522124) 　

Abstract:Gromov-Witten theory is the study of counts of curves in algebraic varieties. These counts may be formulated in several ways including log, open, orbifold and local variants. Recent years have seen the establishment of correspondences linking various curve counting invariants, often associated with different geometries. In this talk, my focus will be on the all genus correspondence linking log and local Gromov-Witten invariants of surfaces, and on specific examples. This is based on joint works with Andrea Brini and Pierrick Bousseau, and Navid Nabijou and Yannik Schüler.

Time:01 June 2023 (16:00-17:00)

Speaker: Tyler Kelly (University of Birmingham)

Venue: Zoom ID: 396 431 7007 　　　　　Password: 000222

Abstract:Landau-Ginzburg (LG) models consist of the data of a quotient stack X and a regular complex-valued function W on X. Here, geometry is encapsulated in the singularity theory of W. One can find that LG models are deformations of many Calabi-Yau varieties in some sense. For example, if the Calabi-Yau is a hypersurface in a smooth projective variety Z cut out by a polynomial f, then one can take X to be the canonical bundle of Z with function W=uf, where u is the bundle coordinate—when the hypersurface is smooth, the critical locus of uf will indeed just be the hypersurface. Exoflops were introduced by Aspinwall as a way to effectively find new birational models of the quotient stack to get new geometries. They effectively create new GIT problems of partial compactifications of X, expanding the tractable birational geometries related to Z. We will explain this technique, provide some foundational results about this, and then provide some new applications proven recently for Calabi-Yau varieties with nontrivial scaling symmetry groups. This talk contains results from a series of joint works (some in progress) with D. Favero (UMinn), C. Doran (Bard/Alberta), A. Malter (Birmingham).

Time:11 May 2023 (16:00-17:00)

Speaker: Qingyuan Jiang (University of Edinburgh)

Venue: Zoom ID: 396 431 7007 　　　　　Password: 000222

Abstract:The framework of Derived Algebraic Geometry (DAG), developed by Toen-Vezzosi, Lurie and many others, allows us to extend Grothendieck’s theory of projectivizations and Grassmannians of sheaves to the cases of complexes. This derived extension is useful for constructing and studying moduli spaces, especially when the spaces are singular and difficult to analyze in the classical framework. We will discuss the constructions of derived projectivizations and Grassmannians as well as their properties, with a focus on applications to Abel maps for singular curves and Hecke correspondences for surfaces. The talk will be based on papers arXiv:2202.11636 and arXiv:2212.10488 and works in preparation.

Time:13 April 2023 (17:15-18:15)

Speaker: Fenglong You (ETH Zurich)

Venue: Zoom ID: 396 431 7007 　　　　　Password: 000222

Abstract:I will talk about mirror symmetry for smooth log Calabi–Yau pairs. I will explain how to use the intrinsic mirror symmetry construction from the Gross—Siebert program to define the proper Landau–Ginzburg potential and show that it is the inverse of the relative mirror map in a relative version of Givental mirror theorem proved by Fan-Tseng-You.

Time:13 April 2023 (16:00-17:00)

Speaker: Shuai Guo (Peking University)

Venue: Zoom ID: 396 431 7007 　　　　　Password: 000222

Abstract:The Virasoro conjecture is a concept in enumerative geometry. It states that the generating function for the Gromov–Witten invariants of a smooth projective variety is annihilated by an action of half of the Virasoro algebra. In this talk, we will first introduce a wall-crossing formula that converts heavy markings to light markings. Then, we will prove that the Virasoro conjecture for Fano complete intersections with only ambient insertions is equivalent to the Virasoro conjecture with only one ambient insertion. In the end, we will prove the Virasoro conjecture for one ambient insertion using wall crossing formula and the twisted theory. This is a work in progress with Qingsheng Zhang and Yang Zhou.

Time:06 December 2022 (17:15-18:15)

Speaker: Yang Zhou (Fudan University)

Venue: Zoom ID: 920 7313 5240 　　　　　Password: 973824

Abstract:The theory of Mixed-Spin-P fields was introduced by Chang-Li-Li-Liu for the quintic threefold. Chang-Guo-Li has successfully applied it to prove important conjectures on the higher-genus Gromov-Witten invariants. In this talk I will explain a generalization of the construction to more spaces. The key is the stability condition which guarantees the separatedness and properness of certain moduli spaces. It also generalizes the construction of the mathematical Gauged Linear Sigma Model by Fan-Jarvis-Ruan, removing their technical assumption about "good liftings".
This is a joint work with Huai-Liang Chang, Shuai Guo, Jun Li and Wei-Ping Li.

Time:06 December 2022 (16:00-17:00)

Speaker: Huailiang Chang (HKUST)

Venue: Zoom ID: 920 7313 5240 　　　　　Password: 973824

Abstract:Gromov Witten invariants Fg encodes the numbers of genus g curves in Calabi Yau threefolds and play an important role in enumerative geometry. In 1993, Bershadsky, Cecotti, Ooguri, Vafa exhibited a hidden "Feynman structure" governing all Fg’s at once, using path integral methods. The counterpart in mathematics has been missing for many years.
After a decades of search, in 2018, a mathematical approach: Mixed Spin P field (MSP) moduli, is finally developed to provide the wanted “Feynman structure”, for quintic CY 3-fold. Instead of enumerating curves in the quintic 3-fold, MSP enumerate curves in a large N dimensional singular space with quintic-3-fold boundary. The “P fields” and “cosections” are used to formulate counting in the singular space via a Landau Ginzburg type construction.
In this talk, I shall focus on geometric ideas behind the MSP moduli. Some update will be provided. The results follow from a decade of joint works with Jun Li, Shuai Guo, Young Hoon Kiem, Weiping Li, Melissa C.C. Liu, Jie Zhou, and Yang Zhou.

Time:29 November 2022 (16:00-17:00)

Speaker: Caucher Birkar (Tsinghua University)

Venue:

Abstract:In this talk I will define stable varieties of non-negative Kodaira dimension and then describe results on existence of moduli spaces and their compactifications for such varieties in any dimension.

Time:09 November 2022 (10:15-11:15)

Speaker: Baohua Fu (Chinese Academy of Sciences)

Venue: Zoom ID: 971 1062 5774　　　　 Password: 474205

Abstract:Motivated by geometric Langlands, we initiate a program to study the mirror symmetry between nilpotent orbit closures of a semisimple Lie algebra and those of its Langlands dual.
The most interesting case is Bn via Cn. Classically, there is a famous Springer duality between special orbits. Therefore, it is natural to speculate that the mirror symmetry we seek may coincide with Springer duality in the context of special orbits. Unfortunately, such a naive statement fails. To remedy the situation, we propose a conjecture which asserts the mirror symmetry for certain parabolic/induced covers of special orbits. Then, we prove the conjecture for Richardson orbits and obtain certain partial results in general. In the process, we reveal some very interesting and yet subtle structures of these finite covers, which are related to Lusztig’s canonical quotients of special nilpotent orbits. For example, there is a mysterious asymmetry in the footprint or range of degrees of these finite covers. Finally, we provide two examples to show that the mirror symmetry fails outside the footprint.
This is a joint work with Yongbin Ruan and Yaoxiong Wen.

Time:09 November 2022 (09:00-10:00)

Speaker: Yu-Shen Lin (Boston University)

Venue: Zoom ID: 971 1062 5774　　　　 Password: 474205

Abstract:Gravitational instantons are the building blocks of Hawking's quantum gravity theory. They are non-compact hyperK\"ahler 4-manifolds with L^2 curvature and can be viewed as the local pieces of K3 surfaces. Similar to K3 surfaces, the cohomology classes of the hyperK\"ahler triples of gravitational instantons induce a period map. In this talk, we will restrict to a particular type of gravitational instanton of type ALH^*. As a side product of studying SYZ mirror symmetry of log Calabi-Yau surfaces, we will prove that the period map is both injective and surjective. As a consequence, the period domain is the moduli space of ALH^*-gravitational instantons. The talk is based on joint works with T. Collins, A. Jacob and the joint work with T.-J. Lee.

Time:02 November 2022 (10:15-11:15)

Speaker: Si Li (Tsinghua University)

Venue: online

Abstract:We present an effective BV quantization theory for chiral deformation of two dimensional conformal field theories. We explain a connection between the quantum master equation and the chiral homology for vertex operator algebras. As an application, we construct correlation functions of the curved beta-gamma/b-c system and establish a coupled equation relating to chiral homology groups of chiral differential operators. This can be viewed as the vertex algebra analogue of the trace map in algebraic index theory. The talk is based on the recent work arXiv:2112.14572 [math.QA].

Time:02 November 2022 (09:00-10:00)

Speaker: Junliang Shen (Yale University)

Venue: online

Abstract:Hitchin’s integrable systems lie in the crossroads of geometry, representation theory, and mathematical physics. I will discuss two central conjectures raised in the last two decades which greatly influenced the development for the algebraic geometry of Hitchin moduli spaces. The first is the P=W conjecture, which concerns the interaction of the topology of the Hitchin system and the non-abelian Hodge correspondence. The second is the topological mirror symmetry conjecture which connects the Langlands duality of groups and the mirror symmetry for Hitchin systems. I will explain that both conjectures can be proved in a uniform way, via vanishing cycles techniques and support theorem. Based on joint work with Davesh Maulik.

Time:26 October 2022 (10:15-11:15)

Speaker: Bohan Fang (Peking University)

Venue: online

Abstract:I will discuss the oscillatory integrals on the mirror Landau-Ginzburg models for toric mirrors. They are related to genus zero Gromov-Witten invariants and are compatible with homological mirror symmetry. The asymptotic expansion of such integrals leads to the Gamma II conjecture for toric varieties. I will also report some recent progress in some non-toric situation inspired by the Gross-Siebert program.

Time:26 October 2022 (09:00-10:00)

Speaker: Eric Zaslow (Northwestern University)

Venue: online

Abstract:Inside a symplectic leaf of the cluster Poisson variety of Borel-decorated $PGL_2$ local systems on a punctured surface is an isotropic subvariety we will call the “chromatic Lagrangian.” Local charts for the quantized cluster variety are quantum tori defined by cubic planar graphs, and can be put in standard form after some additional markings giving the notion of a “framed seed.”
The mutation structure is encoded as a groupoid. The local description of the chromatic Lagrangian defines a “wavefunction” which, we conjecture, encodes open Gromov-Witten invariants of a Lagrangian threefold in threespace defined by the cubic graph and the other data of the framed seed.
We also find a relationship we call “framing duality”: for a family of “canoe” graphs, wavefunctions for different framings encode DT invariants of symmetric quivers.
This talk is based on joint work with Gus Schrader and Linhui Shen.

Time:18 October 2022 (16:00-17:00)

Speaker: Rahul Pandharipande (ETH Zurich)

Venue: Online

Abstract:I will explain some developments in the study of the log Chow ring of the moduli space of curves: tautological classes, piecewis polynomials, and the log DR cycle.