Location: 507 Mathematics Building
Time: Friday 1:30pm-3:00pm
Organiser: Yoonjoo Kim, Tianqing Zhu
Seminar schedule:
Jan 30
Speaker: Tianqing Zhu
Title: Higher dimensional Hitchin fibrations
Abstract: I will give a brief introductory talk on Chen-Ngô's paper (1905.04741) pioneering work on the higher dimensional analog for the Hitchin fibration. We will also talk about some recent progress on this direction if time is allowed.
Feb 6 Yoonjoo Kim ( 1:00pm-2:30pm Note for the special time! )
Title: Tate-Shafarevich twist of abelian fibrations and sections
Abstract: I will talk about Kollár's preprint (arXiv:2504.21705) on degenerations of abelian torsors, their twists, and sections. A degeneration of abelian torsors may or may not have a (rational) section. When it doesn't have a section, one tries to "twist it" in the hope of building up a section. The paper gives a necessary and almost sufficient condition to achieve this.
Feb 13 James Hotchkiss
Title: The semiregularity theorem
Abstract: Suppose we are given a family of smooth projective varieties, and a coherent sheaf on a special fiber. The semiregularity theorem (due in increasing generality to Bloch, Buchweitz and Flenner, and Pridham) is a simple and beautiful criterion for when the sheaf may be deformed to nearby fibers. I will give a modern formulation (and maybe proof) of the result, and explain how it fits into Markman's preprint on the proof of the Hodge conjecture for abelian fourfolds.
Feb 20 Hanlin Cai
Title: Cartier duality for (gerbes of) vector bundles
Abstract: I will discuss Rodrigues Camargo’s recent preprint (arXiv:2512.24967) regarding Cartier duality for gerbes of vector bundles. I will begin by reviewing classical manifestations of Cartier duality, such as the Fourier transform and the duality of finite flat commutative group schemes. Building on this foundation, I will describe how these concepts are interpreted within the framework of vector bundles. Finally, I will explain the 'twisted' generalization: Cartier duality for gerbes of vector bundles.
Feb 27 Anh Ðúc Võ
Title: Extending holomorphic forms from the regular locus to a resolution of singularities.
Abstract: Let X be a reduced complex space and f: Y \to X a resolution of singularities of X. Which holomorphic forms on the regular locus of X extend to holomorphic forms on the complex manifold Y? This simple-looking extension theorem turns out to have many important applications, and the answer is closely related to the singularity of the complex space X. One of the earliest applications/motivations is the Zariski-Lipman conjecture, which says that if the tangent sheaf of X is locally free, then X is smooth.
In this talk, I will discuss this extension problem, and survey the key developments including recent results of Kebekus-Schnell and Park. If time permits, I will also discuss some applications including the Zariski-Lipman conjecture.
Mar 6 Yifan Wu
Title: Ngô Fibrations
Abstract: In the proof of the fundamental lemma, Ngô Bảo Châu (2010) introduced the concept of a weak abelian fibration and the support theorem. The theorem was subsequently applied for support problem of Hitchin fibrations in the G=GL_n setting through the landmark works of Chaudouard–Laumon and Maulik–Shen. Building on this, a recent preprint by de Cataldo, Fernandez Herrero, Fringuelli, and Mauri (arXiv:2502.04966) distills these conditions into the formal notion of an "Ngô fibration." This talk will review advances on the support theorem for the Hitchin fibration over the past decade and state the main results of de Cataldo et al. (2025).
Mar 13 Che Shen
Title: Zastava Spaces and Kazhdan-Lusztig Theory
Abstract: I will talk about a paper by Braverman-Finkelberg-Nakajima (arXiv:2007.09799) that uses the Zastava spaces to give an alternative proof of the Kazhdan-Lusztig conjecture. Given a reductive group G, Zastava spaces of G are partial compactifications of maps to the flag variety of G. One can construct an action of the Lie algebra of the Langlands dual group of G on the intersection cohomology of Zastava spaces using the geometric Satake equivalence. This reduces the problem of computing the multiplicities of simple modules in Verma modules, a central problem in Kazhdan-Lusztig theory, to certain geometric properties of fixed locus of Zastava spaces.
Mar 27 Gyujin Oh
Title: Cartan geometry and functional transcendence of uniformizers
Abstract: Ax showed that, given Q-linearly independent formal power series f_1(X), ..., f_n(X) in XC[[X]], C(f_1, ..., f_n, exp(f_1), ..., exp(f_n)) is of transcendence degree at least n over C(X). One can alternatively view this result as a comparison of the algebraic structures of the source and target of the uniformization map exp: C->C\{0}. This viewpoint has been vastly generalized (the "Ax--Schanuel theorems"), but the proofs require intricate arguments that use model theory and Diophantine geometry. I will give an overview of the context, culminating in a preprint of Blázquez-Sanz--Casale--Freitag--Nagloo (arXiv:2102.03384), which proves the Ax--Schanuel theorems more naturally by using the Cartan-geometric approach to differential equations (i.e., flat connections on principal bundles).
Apr 3 Shijie Dai
Title: Irrationality of cubic fourfolds and Gushel-Mukai fourfolds via quantum cohomology
Abstract: Following the work of Katzarkov-Kontsevich-Pantev-YU (arXiv:2508.05105), Guéré (arXiv:2603.04518) obtained a finer result characterizing middle primitive hodge structures of rational cubic fourfolds by utilizing Iritani's blow-up formula ( arXiv:2307.13555) to study the birational invariants coming from quantum products by euler vector fields, without necessarily relying on previous formalisms. Based on these ideas, Benedetti–Manivel–Perrin (arXiv:2603.17487) proved the irrationality of very general Gushel–Mukai fourfolds, and also established an analogue of Guéré’s result by deforming small quantum products along an additional cohomology class.
In this talk, I will discuss these recent developments of studying rationality by computing quantum cohomology.
Apr 10 Andrés Ibáñez Núñez ( 10:00am-11:00am at 407 Note for the special time and the special room! )
Title: A categorical perspective on non-abelian localization
Abstract: I will talk about Halpern-Leistner’s paper with the same title. Usually, localization formulas relate the torus equivariant geometry of a space and that of its fixed points. In this talks I will talk about the case when the group is non-abelian, or more generally a stack. Instead of the torus action, there is a Theta-stratification of the stack, and non-abelian localization relates the K-theory of the stacks and that of the centres of the stratification.
Apr 10 Jiahe Shen ( 2:00pm-3:00pm Note for the special time! )
Title: The Kakeya set conjecture in finite and p-adic fields
Abstract: The recent breakthrough on the Kakeya conjecture in R^3 is a remarkable result, but its proof is highly nontrivial and draws on techniques from several different areas of mathematics. By contrast, the Kakeya set conjecture in the finite-field and p-adic settings has much simpler proofs. In this talk, I will first introduce Dvir’s proof in the finite-field setting, a landmark application of the polynomial method. I will then discuss Arsovski’s proof in the p-adic setting, which proceeds in a similar spirit.
Apr 24 Matthew Hase-Liu
May 1 Tommaso Maria Botta