（All seminars will be held in C654, Shuangqing Complex Building on Wednesday at 15:30-16:30, unless marked in red. ）
Upcoming Talk:
Time: Wed., 15:30-16:30, Dec. 11, 2024
Place: C654, Shuangqing Complex Building
Speaker: Lian Duan (ShanghaiTech University)
Title: The Tate Conjecture for Certain Elliptic Surfaces via Irreducibility of Geometric Galois Representations
Abstract: In this talk, we discuss the irreducibility of compatible systems of certain five-dimensional geometric Galois representations. Using the modularity results of classical cases, we prove that such a compatible system of representations is either irreducible for almost all p, or decompose uniformly as p varies. Our irreducibility result will deduce an algorithm for checking the irreducibility of corresponding Galois representations. Moreover, by applying a result of Cardoret and Tamagawa on the open image theory of monodromy representation, we can spread out the irreducibility among the family of representations induced by the transcendental part of a family of elliptic surfaces, and consequently, verify the corresponding Tate conjecture for those surfaces.
Past Talks：
Title：The matrix-resolvent method to tau-functions in integrable systems
Speaker: Di Yang (中科大USTC)
Time: Wed., 15:30-16:30, Dec. 4, 2024
Online: Tencent meeting 腾讯会议：936-859-6245
Abstract：In this seminar, we will give an introduction to the matrix-resolvent method of computing logarithmic derivatives of tau-functions in integrable systems. We also discuss the recent development of this method, and present several applications.
Title: On the full Kostant-Toda lattice and the flag varieties
Speaker: Yuancheng Xie (BICMR)
Time: Wed., 15:30-16:30, Nov 27, 2024
Place: C654, Shuangqing Complex Building
Abstract: In 1967, the same year modern integrable system theory was born, Japanese physicist Morikazu Toda proposed an integrable lattice model to describe motions of a chain of particles with exponential interactions between nearest neighbors. Since then, Toda lattice and its generalizations have become the test models for various techniques and philosophies in integrable systems and wide connections are built with many other branches of mathematics. In this talk, I will mention some of these connections and then characterize singular structure of solutions of the so-called full Kostant-Toda (f-KT) lattices defined on simple Lie algebras in two different ways: through the τ -functions and through the Kowalevski- Painlev´e analysis. Fixing the spectral parameters which are invariant under the f-KT flows, we build a one to one correspondence between solutions of the f-KT lattices and points in the corresponding flag varieties. This talk is based on preprint arXiv:2212.03679.
Title: Counter-examples to Gamma conjecture I
Speaker: Huazhong Ke (Zhongshan University)
Time: Friday at 10:00, Nov 22, 2024
Place: C654, Shuangqing Complex Building
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 based on joint work with S. Galkin, J. Hu, H. Iritani, C. Li and Z. Su.
Speaker: 王成茜 (YMSC)
Title : Calabi-Yau varieties with extreme behavior
Time: Wed., 15:30-16:30, Oct. 30, 2024
Venue: Shuangqing C654
Abstract: A projective variety X is called Calabi-Yau if its canonical divisor is Q-linearly equivalent to zero. The smallest positive integer m with mK_X linearly equivalent to zero is called the index of X. Using ideas from mirror symmetry, we construct Calabi-Yau varieties with index growing doubly exponentially with dimension. We conjecture they are the largest index in each dimension based on evidence in low dimensions. We also give Calabi-Yau varieties with large orbifold Betti numbers or small minimal log discrepancy. Joint work with Louis Esser and Burt Totaro.
Speaker: Tao Su (BIMSA)
Title: Log-concavity from Hodge theory of character varieties
Time: Wed., 15:30-16:30, Oct. 30, 2024
Venue: Shuangqing C654
Abstract: We propose a conjecture on the log-concavity from E-polynomials of character varieties over Riemann surfaces. Via some 'BPS calculus', we explain an idea of reducing the conjecture to a local one: log-concavity from Severi strata of a versal deformation of planar algebraic curve singularities. In the case when the singularity link is a torus knot, we verify the local conjecture via a connection to the HOMFLY-PT polynomials. Joint work in progress with Chenglong Yu.
Speaker: 方博汉 (北京大学)
Title: Remodeling conjecture with descendants
Time: Wed., 15:30-16:30, Oct. 23, 2024
Venue: Shuangqing C654
Abstract: For a toric Calabi–Yau threefold, I will explain the correspondence between an equivariant line bundle supported on a toric subvariety and a relative homology cycle on the covering space of the mirror curve. The Laplace transform of the holomorphic Liouville form along this cycle gives genus-zero descendant Gromov–Witten invariants with a certain Gamma class of that bundle. Hence, the Laplace transform of the topological recursion produces all-genus descendant Gromov–Witten invariants with Gamma classes. This talk is based on ongoing joint work with Chiu-Chu Melissa Liu, Song Yu, and Zhengyu Zong.
Speaker: 张希平 (同济大学)
Title: The Characteristic Cycle of Restricted Constructible Functions
Time: Wed., 15:30-16:30, Oct. 16, 2024
Venue: Shuangqing C654
Abstract: When a constructible function is restricted to a hypersurface complement, its characteristic cycle is classically described by specializing the sharp construction of Ginzburg. When the divisor is SNC, Maxim-Rodriguez-Wang-Wu recently proved that this process is equivalent to pulling back the logarithmic characteristic cycle. In this talk we will discuss some generalizations of this result when the divisor is free and strongly Euler homogeneous. This is a joint work with Xia Liao.
Speaker: Qixiao Ma 马骐骁 (ShanghaiTech University)
Title: Torsors of the Jacobians of the generic Fermat curves
Time: 2024.10.9 (Wed.) 15:30-16:30 pm
Place: Shuangqing Complex Building A, C654
Abstract: Let C/S be the universal family of degree m (m>3) Fermat curve. Then A=Pic^0_{C/S} is an abelian scheme over S. We show that: (1) All torsors of A/S are of the form Pic^d_{C/S}. (2) Passing to the function field, there are uncountably many non-isomorphic torsors of A_k(S). Finally we discuss some partial results on extending the result to M_g.
Speaker: Yalong Cao (MCM & CAS)
Title: Towards a complexification of Donaldson-Witten TQFT
Time: 2024.9.25 (Wed.) 15:30-16:30 pm
Place: Shuangqing Complex Building A, C654
Abstract: Donaldson-Thomas theory on Calabi-Yau 4-folds (DT4) is a complexification of Donaldson theory on 4-manifolds. In this talk, we will discuss a complexification of Donaldson-Witten TQFT. This establishes a degeneration formula of DT4 invariants and a Gromov-Witten type theory for critical loci (quivers with potentials).
Speaker: Gerard van der Geer (University of Amsterdam)
Title: Constructing modular forms via geometry
Time: 2024.9.18 (Wed.) 15:30-16:30 pm
Place: Shuangqing Complex Building A, C654
Abstract: Vector-valued Siegel modular forms are a natural generalization of elliptic modular forms and find applications in algebraic geometry, number theory and mathematical physics. We indicate a number of geometric ways of constructing such forms. This is joint work with Cléry, Faber and Kouvidakis.
Speaker: Shihao Wang 王士浩 (Tsinghua 探微书院)
Title: On Bott-Samelson rings for Coxeter groups
Time: 2024.9.11 (Wed.) 15:30-16:30 pm
Place: Shuangqing Complex Building A, C654
Abstract: We study the cohomology ring of the Bott-Samelson variety. We compute an explicit presentation of this ring via Soergel's result, which implies that it is a purely combinatorial invariant. We use the presentation to introduce the Bott-Samelson ring associated with a word in an arbitrary Coxeter system by generators and relations. In general, it is a split quadratic complete intersection algebra with a triangular pattern of relations. By a result of Tate, it follows that it is a Koszul algebra and we provide a quadratic (reduced) Grobner basis. Furthermore, we prove that it satisfies the whole Kahler package, including the Poincare duality, the hard Lefschetz theorem, and the Hodge-Riemann bilinear relations. Joint with Tao Gui, Lin Sun, and Haoyu Zhu.
Speaker: 张诗卓 (Simons Laufer Mathematical Sciences Institute/复旦大学上海数学中心)
Title: Recent advances in categorical Torelli theorems
Time: 2024.06.12 (Wed.) 2:00-3:00 pm
Place: Shuangqing Complex Building A, C654
Abstract:
Let $X$ be a not necessarily smooth Fano variety and denote by \Ku(X) the non-trivial semi-orthogonal component. The Categorical Torelli problem asks if \Ku(X) determines the isomorphism class of $X$. In my talk, I will briefly talk about the history of this topic including the known results and popular strategies to prove these results. Then I will survey the recent advances for (weighted) hypersurfaces, a cubic threefold with a geometric involution, del Pezzo threefold of Picard
rank one, and a class of nodal prime Fano threefolds. Meanwhile, I will talk about some new approaches to solving these problems. If time permits, I will also talk about categorical Torelli problems for a class of index one prime Fano threefold as the double cover of del Pezzo threefolds. This talk is based on a series of work joint with Xun Lin, Daniele Faenzi, Zhiyu Liu, Soheyla Feyzbakhsh, Jorgen Renneomo, Xianyu Hu, Sabastian-Casalaina Martin, and Zheng Zhang.
Speaker: 杜衡 (YMSC)
Title: Bridgeland Stability Conditions Applied to the Fargues–Fontaine Curve
Time: 2024.06.05 (Wed.) 2:00-3:00 pm
Place: Shuangqing Complex Building A, C654
Abstract:
The Fargues–Fontaine curve has become a fundamental geometric object in the study of p-adic Hodge theory. One of the key theories developed by Fargues and Fontaine concerns the stability condition for vector bundles on this curve. In this talk, we will apply Bridgeland stability conditions to the derived category of coherent sheaves on the Fargues–Fontaine curve. We will see that the Fargues–Fontaine curve presents a strong similarity to elliptic curves. Additionally, we will explore how the hearts defined by slicings of stability conditions generalize the notion of Banach–Colmez spaces. This talk is based on joint work with Qingyuan Jiang and Yucheng Liu.
Title: Hitchin morphism for projective varieties
Speaker: Siqi He (何思奇)，中科院晨兴数学中心
Time: 2024.05.22 (Wed.) 2:00-3:00 pm
Place: Shuangqing Complex Building A, C654
Abstract: The Hitchin morphism is a map from the moduli space of Higgs bundles to the Hitchin base, which is generally not surjective when the dimension of the variety is greater than one. Chen-Ngo introduced the concept of the spectral base, which is a closed subscheme of the Hitchin base. They conjectured that the Hitchin morphism is surjective to the spectral base and also proved that the surjectivity is equivalent to the existence of finite Cohen-Macaulayfications of the spectral varieties. For rank two Higgs bundles, we will discuss an explicit construction of the Cohen-Macaulayfication of the spectral variety. In addition, we will discuss several applications using the spectral base to the topology of projective variety. This talk is based on some collaborative work with J. Liu and N. Mok.
Title: "Counting" pseudo-holomorphic disks via loop space
Speaker: Yi Wang (王怡), Purdue University, Postdoc
Time: 2024.05.15 (Wed.) 2:00-3:00 pm
Place: Shuangqing Complex Building A, C654
Abstract:
Let M be a symplectic manifold and L be a Lagrangian submanifold. Celebrated work of Fukaya-Oh-Ohta-Ono constructs a filtered cyclic A-infinity algebra structure on the cochain (cohomology) group of L by studying moduli spaces of pseudo-holomorphic disks to (M,L) with boundary mark points. In this talk, I will explain how this A-infinity algebra lifts to the free loop space of L in a way compatible with the string topology of loop spaces, as well as some applications of this formalism.
Speaker: 张旭成 (YMSC)
Title: A geometric approach to identifying the stability condition
Time: 2024.4.24 (Wed) 14:00--15:00
Place: Shuangqing Complex Building A, C654
Abstract:
For any reductive group we find a geometric interpretation of the stability condition for principal bundles over a curve: it is the unique maximal open locus that admits a schematic moduli space. Some applications and further progress will be discussed. This is a joint work with Dario Weissmann.
Title: Universal holomorphic maps, conflict between fully hypercyclicity and slow growth
Speaker: Zhangchi Chen (陈张弛，中科院晨兴数学中心)
Time: first talk 2024.4.17 (Wed) 2:00-3:00 pm, second talk 2024.4.18 (Thurs) 10:00-11:00 am (Remark: the first talk will be general, and the second talk will contain more details)
Place: Shuangqing Complex Building A, C654
Abstract:
In the space O(C,C) of entire functions, equipped with the open-compact topology, an element is called universal if its translation orbit is dense. It is hypercyclic w.r.t some translation operator if its orbit under this operator is dense. It is fully hypercyclic if it is simultaneously hypercyclic to all non-trivial translations in all directions.
Universal entire functions are transcendental, hence their Nevanlinna characteristic functions grows faster than O(log r). Dinh-Sibony asked what the slowest Nevanlinna growth of universal entire curves is. In a joint work with Dinh Tuan Huynh and Song-Yan Xie, we solved their question completely, by constructing universal entire curves in CP^n whose Nevanlinna growth is slower than any given transcendental entire function.
Bin Guo and Song-Yan Xie discovered the conflict between fully hypercyclicity and slow growth. They proved that if the growth is too slow then the hypercyclic directions in [0,2pi) has Hausdorff dimension 0.
Replace C by the unit disc D, and translations by Aut(D), one can also talk about universal holomorphic discs. Transcendental functions defined on D with bounded Nevanlinna characteristic functions are called of bounded type, which is the analogous property of having slow growth. In a joint work with Bin Guo and Song-Yan Xie, we constructed universal discs in CP^n of bounded type. We also discovered a weak-conflict between fully hypercyclicity and slow growth. If the disc is of bounded type, then the hypercyclic directions in [0,2pi) has Lebesgue measure 0.
Title: A Noether-Lefschetz type theorem for spectral varieties with applications
Speaker: Xiaoyu Su （苏晓羽，北京邮电大学）
Time: 2024.4.10 (Wed) 2-3 pm
Place: Shuangqing Complex Building A, C654
Abstract:
In this talk, we will discuss spectral variety and related moduli spaces of Higgs pairs on surfaces. We will first introduce the geometry of moduli of Higgs pairs and the spectral varieties. Indeed, we will talk about the Noether-Lefschetz type problems and the Picard schemes of the spectral varieties. If time permits, the Picard group of generic (very general) spectral varieties and its geometric applications will also be discussed. This is a joint work with Bin Wang.
Title: Hodge properties of confluent hypergeometric connections
Speaker: Yichen Qin (Humboldt Universität zu Berlin)
Time: Wednesday, 14:00-15:00, March 27, 2024
Place: C654, Shuangqing Complex Building A
Abstract:
Sabbah and Yu computed the irregular Hodge numbers associated with hypergeometric connections. In this talk, we introduce a new approach for hypergeometric connections whose defining parameters are rational numbers. Our method relies on a geometric interpretation of hypergeometric connections, which enables us to describe the irregular Hodge filtrations explicitly and derive several arithmetic applications on hypergeometric sums. This research is conducted in collaboration with Daxin Xu.
Title: Curves on K3 surfaces and Mukai’s program
Speaker: Haoyu Wu (Fudan University)
Time: Wednesday, 14:00-15:00, March 20, 2024
Place: C654, Shuangqing Complex Building A
Abstract:
The Mukai’s program seeks to recover a K3 surface X from any curve C on it by exhibiting it as a Fourier-Mukai partner to a Brill--Noether locus of vector bundles on the curve. In this talk, I will give an introduction to work of Feyzbakhsh for Picard number one K3 and primitive curve C. We extend the results to the case of non-primitive curves by introducing the tools of destabilizing regions. As an application, we show that there are hyper-K\"{a}hler varieties as Brill-Noether loci of curves in every dimension. This is a joint work with Yiran Chen and Zhiyuan Li.
Title: On the moduli space of certain plane sextic curves
Speaker: Yiming Zhong (BICMR)
Time: Wednesday, 14:00-15:00, March 13, 2024
Place: C654, Shuangqing Complex Building A
Abstract:
We study moduli spaces of certain sextic curves with a singularity of multiplicity 3 from both perspectives of Deligne–Mostow theory and periods of K3 surfaces. In both ways we can describe the moduli spaces via arithmetic quotients of complex hyperbolic balls. We show that the two ball-quotient constructions can be unified in a geometric way. This is a joint work with Zhiwei Zheng.
Title: Higher dimensional Heegaard Floer homology and Hecke algebras
Speaker: Yin Tian (BNU)
Time:Wednesday at 11:00, March 6, 2024
Place: C654, Shuangqing Complex Building
Abstract:
Higher dimensional Heegaard Floer homology (HDHF) is a higher dimensional analogue of Heegaard Floer homology in dimension three. It's partly used to study contact topology in higher dimensions. In a special case, it's related to symplectic Khovanov homology. In this talk, we discuss HDHF of cotangent fibers of the cotangent bundle of an oriented surface and show that it is isomorphic to various Hecke algebras. This is a joint work with Ko Honda and Tianyu Yuan.
Title: On Hurwitz-Severi numbers
Speaker: Boris Shapiro, Stockholm University
Time: Dec 20, 2023, Wednesday at 10:00am
Abstract: For a point p in CP2 and a triple (g, d, l) of non-negative integers we define a Hurwitz–Severi number Hg,d,l as the number of generic irreducible plane curves of genus g and degree d+l having an l-fold node at p and at most ordinary nodes as singularities at the other points, such that the projection of the curve from p has a prescribed set of local and remote tangents and lines passing through nodes. Under certain conditions we express the above Hurwitz-Severi numbers via appropriate Hurwitz numbers. Several questions will be posed.
Title：Mixed-Spin-P fields for GIT quotients
Speaker：Zhou Yang 周杨 （复旦大学上海数学中心）
Time: Dec 13, 2023, Wednesday at 10:00am
Venue：C654, Shuangqing Complex Building
Abstract：
The theory of Mixed-Spin fields was introduced by Chang-Li-Li-Liu for the quintic threefold, aiming at studying its higher genus Gromov-Witten invariants. Chang-Guo-Li has successfully applied it to prove famous conjectures on the higher-genus Gromov-Witten invariants proposed by physicists. In this talk I will explain a generalization of the construction to more spaces. The generalization usually depends on some choices and I will give some concrete examples in the talk.
The key is a 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 technique assumption about "good lifitings".
This is a joint work with Huai-Liang Chang, Shuai Guo, Jun Li and Wei-Ping Li.
Title: Springer correspondence and mirror symmetry for Sp/SO Hitchin Systems
Speaker: Bin Wang 王彬 (Postdoc, 香港中文大学)
Time：Wednesday at 10:00, Dec 6, 2023
Place: C654, Shuangqing Complex Building
Abstract:
Starting from special nilpotent orbits in Sp_{2n}/SO_{2n+1} which are related by Springer correspondence, we construct various Hitchin systems on curves with marked points. We resolve singularities of generic spectral curves. We then apply it to analyze the corresponding affine Spaltenstein fibers，which can be treated as the local version of (parabolic) Hitchin fibers. As a result, we obtain the (Strominger-Yau-Zaslow) mirror symmetry for these Hitchin systems. This is a joint work with X. Su, X. Wen and Y. Wen.
Title: Gromov--Witten/Pandharipande--Thomas correspondence via conifold transitions
Speaker: Lin Yinbang 林胤榜（同济大学）
Time: Monday at 10:00 am, Nov 27, 2023
Venue: B626, Shuangqing Complex Building
Abstract:
Given a (projective) conifold transition of smooth projective threefolds from $X$ to $Y$, we show that if the Gromov--Witten/Pandharipande--Thomas descendent correspondence holds for the resolution $Y$, then it also holds for the smoothing $X$ with stationary descendent insertions. As applications, we show the correspondence in new cases. This is joint work with Sz-Sheng Wang.
Title: Rigidity problems on moduli spaces of polarized manifolds
Speaker: Ruiran Sun (McGill University)
Time: Wednesday at 10:00, Nov 22, 2023
Place: C654, Shuangqing Complex Building
Abstract: I will survey the recent progress on the rigidity problems on moduli spaces of polarized manifolds. This talk is based on the joint works with Ariyan Javenpaykar, Steven Lu and Kang Zuo, and with Chenglong Yu and Kang Zuo.
Title：Kodaira-type and Bott-type vanishings via Hodge theory
Speaker: Wei Chuanhao, Westlake University （魏传豪，西湖大学)
Time: Nov 8 ，10:00-11:00 am
Venue: C654, Shuangqing Complex Building
Abstract：
I will first give a brief introduction to T. Mochizuki's Theory of twistor D-modules. Then, we use it to study Kodaira-type vanishings. In particular, we will generalize Saito vanishing, and give a Kawamata-Viehweg type statement. As an application, we will also prove a Bott-type Vanishing using M. Saito's mixed Hodge module.
Title: Simpson's correspondence and the P=W conjecture
Speaker：Zhang Zili, Tongji University (张子立, 同济大学）
Time： Nov 1 ，10:00-11:00 am
Venue：C654, Shuangqing Complex Building
Abstract:
For a complex projective curve C and a reductive group G, the character variety M_B and the moduli of Higgs bundles M_Dol are canonically homeomorphic via the Simpson's correspondence and hence the cohomology groups of them are naturally identified. The geometric structures of the moduli spaces induce various filtrations in the cohomology groups. De Cataldo-Hausel-Migliorini conjectured in 2012 that the Perverse filtration (P) of M_Dol is identical to the Hodge-theoretic Weight filtration (W) of M_B; the P=W conjecture. We will introduce some background and recent progress of the P=W conjecture.
Title: Tropicalizations of Riemann surfaces and their moduli
Speaker: Dali Shen, BIMSA
Time: Oct 27 (Fri.) 1：30pm-2：30pm
Place: Shuangqing Complex Building A513
Abstract: The tropical methods have already been used to study the moduli theory of algebraic curves during the past decade. In this talk, I will first discuss about the tropicalization of a smooth pointed Riemann surface via its (hyperbolic) pair of pants decomposition, and then about how to compactify the moduli space of tropicalizations in a geometrically meaningful way.
Title: L^2 type invariants of hyperplane arrangement complement
Speaker: Liu Yongqiang, USTC （刘永强，中国科学技术大学）
Time: Wednesday at 10:00, Oct 18, 2023
Place: C654, Shuangqing Complex Building
Abstract:
We first give an brief introduction on the topic of hyperplane arrangement. Then we give concrete formulas for these L^2 type invariants at degree 1 and study their connections with combinatorics. If time allows, some similar results for smooth complex quasi-projective variety will be discussed.
Title: Graded character sheaves, HOMFLY-PT homology, and Hilbert schemes of points on C^2
Speaker: Penghui Li (YMSC)
Time: Wednesday at 10:00, Oct 11, 2023
Place: C654, Shuangqing Complex Building
Abstract:
Using a geometric argument building on our new theory of graded sheaves, we compute the categorical trace and Drinfel'd center of the (graded) finite Hecke category H_W in terms of the category of (graded) unipotent character sheaves, upgrading results of Ben-Zvi-Nadler and Bezrukavninov-Finkelberg-Ostrik. In type A, we relate the categorical trace to the category of 2-periodic coherent sheaves on the Hilbert schemes of points on C^2 (equivariant with respect to the natural C*×C* action), yielding a proof of a conjecture of Gorsky-Negut-Rasmussen which relates HOMFLY-PT link homology and the spaces of global sections of certain coherent sheaves on Hilbert schemes. As an important computational input, we also establish a conjecture of Gorsky-Hogancamp-Wedrich on the formality of the Hochschild homology of H_W. This is a joint work with Quoc P. Ho.
Title: Degeneration of Hodge structures and cubic hypersurfaces
Speaker: Renjie Lyu 吕人杰 (中科院AMSS, Postdoc)
Time: June 22 (Thur.) 3:30-5:00 pm
Place: Jinchunyuan West Building, Floor 3
Abstract:
The degeneration of Hodge structures is related to how a smooth projective variety degenerate. And it provides a Hodge-theoretic perspective to compactify moduli spaces. In this talk, I will focus on a particular degeneration of cubic hypersurfaces and study the associated limiting mixed Hodge structure. It generalizes some results in Radu Laza’s and Brendan Hassett’s works on cubic fourfolds. This is a joint work with Zhiwei Zheng.
Title: Moduli spaces of modules over even Clifford algebra and Prym varieties
Speaker: Jia Choon Lee (BICMR)
Time: Wednesday at 15:30, June 14, 2023
Place: Conference Room 3, Jinchunyuan West Building
Abstract:
A conic fibration has an associated sheaf of even Clifford algebra on the base. In this talk, I will discuss the relation between the moduli spaces of modules over the sheaf of even Clifford algebra and the Prym variety associated to the conic fibration. I will begin by motivating the connection between them from the viewpoint of the rationality problem of cubic hypersurfaces. Then I will explain the construction of a rational map from the moduli space of modules over the sheaf of even Clifford algebra to the special subvarieties in Prym varieties. As an application, we get an explicit correspondence between instanton bundles of minimal charge on cubic threefolds and twisted Higgs bundles on curves.
Title: Sheaves on non-reduced curves in a projective surface
Speaker: Yao Yuan (Capital Normal University)
Time: Tuesday 10:00am, June 06, 2023
Place: Third floor, Jinchunyuan West Building
Abstract:
Sheaves on non-reduced curves can appear in moduli spaces of 1-dimensional semistable sheaves over a surface, and moduli spaces of Higgs bundles as well. We estimate the dimension of the stack M_X(nC, \chi) of pure sheaves supported at the non-reduced curve nC (n ≥ 2) with C an integral curve on X. We prove that the Hilbert-Chow morphism h_{L,\chi} : M_X^H(L, \chi) -> |L| sending each semistable 1-dimensional sheaf to its support have all its fibers of the same dimension for X Fano or with trivial canonical line bundle and |L| contains integral curves. The strategy is to firstly deal with the case with C smooth and then do induction on the arithmetic genus of C which once can decrease by a blow-up given C singular. As an application, we generalize the result of Maulik-Shen on the cohomology \chi-independence of M_X^H(L,\chi) to X any del Pezzo surface not necessarily toric.
Title: Cubic threefolds with an involution and their intermediate Jacobians
Speaker: Zheng Zhang 张正 (ShanghaiTech University)
Time: Wednesday at 15:30, May 31, 2023
Place: Conference Room 3, Jinchunyuan West Building
Abstract:
We study the moduli space of cubic threefolds admitting an involution via the period map sending such a cubic threefold to the invariant/anti-invariant part of the intermediate Jacobian. Our main result is global Torelli holds for the period map. Key ingredients of the proof include a description of the invariant/anti-invariant part of the intermediate Jacobian as a Prym variety and a detailed study of certain positive dimensional fibers of the corresponding Prym map. The proof also relies on the results of Donagi-Smith, Ikeda and Naranjo-Ortega on related Prym maps. This is joint work with S. Casalaina-Martin and L. Marquand.
Title: Universal meromorphic functions with slow growth
Speaker: Songyan Xie (CAS)
Time: Wednesday at 15:30, May 24, 2023
Place: Conference Room 3, Jinchunyuan West Building
Abstract:
I will show a solution to a problem asked by Dinh and Sibony in their open problem list, about minimal growth of universal meromorphic functions. This is joint work with Dinh Tuan Huynh and Zhangchi Chen. If time permits, I will also discuss my recent joint work with my Ph.D. student Bin Guo, about the existence of universal holomorphic functions in several variables with slow growth.
Title: On attractor points on the moduli space of Calabi-Yau threefolds
Speaker: Emanuel Scheidegger (BICMR)
Time: Wednesday at 15:30, May 17, 2023
Place: Conference Room 3, Jinchunyuan West Building
Abstract:
We briefly review the origin in physics of attractor points on the moduli space of Calabi-Yau threefolds. We turn to their mathematical interpretation as special cases of Hodge loci. This leads to fascinating conjectures on the modularity of the Calabi-Yau threefolds at these points in terms of their periods and L-functions. For hypergeometric one-parameter families of Calabi-Yau threefolds, these conjectures can be verified at least numerically to very high precision.
Title: stable (parabolic) holomorphic vector bundles over complex curves and instanton Floer homology
Speaker: 谢羿（北京大学）
Time: Wednesday at 15:30, Apr 26, 2023
Place: Conference Room 3, Jinchunyuan West Building
Abstract:
Stable holomorphic bundles are objects in algebraic geometry which have been studied by many people. Instanton Floer homology is an invariant of 3-manifolds, which has been used to solve many problems in the low dimensional topology. It turns out the two things are closely related: knowledge on the moduli space of stable bundles can help the calculation of Instanton Floer homology. In this talk, I will explain this relationship and its generalization to stable parabolic bundles. This is joint work with Boyu Zhang.
Title: On the cohomology of BG
Speaker: 李时璋（中科院晨兴数学中心）
Time: Wednesday at 15:30, Apr 19, 2023
Place: Conference Room 3, Jinchunyuan West Building
Abstract:
In this talk, if time permits, we will discuss:
(1) classify order p group schemes over Spec(char p alg closed field) using Dieudonne modules;
(2) a new way of understanding Dieudonne modules in terms of cohomology of BG (due to Mondal);
(3) an attempt of using BG to construct a counterexample that Deligne--Illusie asked for (work of Antieau--Bhatt--Mathew);
(4) why this attempt cannot succeed (joint work in progress with Kubrak--Mondal), and how this attempt can be made successful (due to Petrov).
Title: Tropicalization of curves and applications
Speaker: Xiang He (YMSC, Tsinghua)
Time: Wednesday at 15:30, Apr 12, 2023
Place: Conference Room 3, Jinchunyuan West Building
Abstract:
The tropicalization process assigns to an algbraic variety a polyhedral complex with extra structure that records certain degeneration data. In this talk, I will introduce the tropicalization of (a family of) algebraic curves and explain the connection between the geometry of the algebro-geometric side and the tropical side. I will then discuss the application of this construction to the irreducibility of Severi varieties and the moduli space of curves. This is joint work with Karl Christ and Ilya Tyomkin.
Title: Moduli spaces of hyperKahler manifolds and cubics
Speaker: Zhiwei Zheng, YMSC, Tsinghua
Time: Mar. 29 (Wed.) 3:30-5:05 pm
Place: Jinchunyuan West Building, Second Floor, Conference Room 3
Abstract:
The study of moduli spaces of hyperKahler manifolds and low dimensional cubic hypersurfaces is an active direction in algebraic geometry. Thanks to kinds of Torelli theorem, many moduli spaces can be realized as locally symmetric varieties of unitary type or orthogonal type. Hodge theory, birational geometry and arithmetic geometry converge in this topic. In this talk I will give a general introduction to the theory and examples, and discuss the future directions.
