Mar 2, 2026
- Speaker: Fangzhou Jin 金方舟 (同济大学)
- Pretalk: Introduction to intersection theory
- Research talk: Towards an intersection theory with quadratic forms
- Intersection theory via algebraic cycles provides important tools in enumerative geometry. Recent advances in motivic homotopy theory have led to an enriched enumerative geometry with quadratic forms. I will discuss a corresponding intersection theory with applications in topology, geometry and arithmetics. Based on joint work with F. Déglise, N. Feld and A. Khan.
Mar 9, 2026
- Speaker: Chen JIANG 江辰 (复旦大学)
- Pretalk: Introduction to hyperkahler manifolds
- A hyperkahler manifold is a higher dimensional analogue of K3 surfaces. Such manifolds have many interesting geometric properties and are among one type of the building blocks of manifolds with trivial first Chern classes together with torus and Calabi-Yau manifolds. I will briefly recall basic definitions and properties of them.
- Research talk: Positivity in hyperkahler manifolds via Rozansky-Witten theory
- For a hyperkahler manifold $X$ of dimension $2n$, Huybrechts showed that there are constants $a_0, a_2, \dots, a_{2n}$ such that$$\chi(L) =\sum_{i=0}^n\frac{a_{2i}}{(2i)!}q_X(c_1(L))^{i}$$for any line bundle $L$ on $X$, where $q_X$ is the Beauville--Bogomolov--Fujiki quadratic form of $X$. Here the polynomial $\sum_{i=0}^n\frac{a_{2i}}{(2i)!}q^{i}$ is called the Riemann--Roch polynomial of $X$. In this talk, I will discuss the positivity of coefficients of the Riemann--Roch polynomial and also positivity of Todd classes. Such positivity results follows from a Lefschetz-type decomposition of the root of Todd genus via the Rozansky—Witten theory.
Mar 16, 2026
- Speaker: Junwu TU 涂君武 (上海科技大学)
- Pretalk: Introduction to categorical enumerative invariants
- We discuss the definition of categorical enumerative invariants associated with smooth proper Calabi-Yau categories.
- Research talk: B-model categorical enumerative invariants and the holomorphic anomaly equation
- In this talk, we consider categorical enumerative invariants associated with the derived category of coherent sheaves on smooth projective Calabi-Yau threefolds. We prove these invariants, when considered in a family, satisfy Bershadsky-Cecotti-Ooguri-Vafa’s holomorphic anomaly equation (HAE). We also discuss the implications of HAE for understanding the geometry of moduli spaces of Calabi-Yau threefolds.
Mar 20, 2026 (SPECIAL TIME and LOCATION: 10.30 AM in the lecture room of IASM)
- Speaker: Xiaokui Yang 杨晓奎 (清华大学)
- Research talk: Geometry and Analysis Inspired by RC-Positivity
- RC-positivity is a concept that emerged from algebraic geometry, originally introduced to characterize uniruled manifolds and rationally connected manifolds. It has since revealed profound implications for understanding geometric phenomena. In this talk, we survey recent progress in geometry and analysis, highlighting key developments inspired by RC-positivity.
Mar 23, 2026
- Speaker: Xiping Zhang 张希平 (同济大学)
- Pretalk: Chern Classes on Singular Varieties
- The Poincaré–Hopf theorem states that the topological Euler characteristic of a compact complex manifold equals the degree of the Euler class of its tangent bundle. In this talk, we will review how this theorem extends to the singular setting, following the reformulation of Grothendieck and Deligne.
- Research talk: Betti Bounds for Hypersurfaces in Projective Varieties
- In this talk we will consider a degree $d$ hypersurface $Y$ in a $n$-dimensional projective variety $X$ and discuss the upper bound for the total sum of Betti numbers of $Y$. We show that this sum is bounded by $3\deg(X)\cdot d^n+C\cdot d^{n-1}$, where $C$ is an explicit constant given by the Chern-Schwartz-MacPherson class of $X$. When $X$ is a complete intersection this bound modifies to $\deg(X)\cdot d^n+C\cdot d^{n-1}$ and isasymptotically sharp with respect to $d$. This is a joint work with Xuanyu Pan and Dingxin Zhang.
Mar 30, 2026
- Speaker: Nan Li 李楠 (City University of New York)
- Pretalk: Introduction to Alexandrov Spaces
- We will discuss some basic properties of Alexandrov spaces.
- Research talk: A Canonical Proof of Perelman's Stability Theorem
- Perelman's remarkable stability theorem states that if $X$ is a compact $n$-dimensional Alexandrov space with curvature $\ge k$, then for any $\epsilon>0$, there exists $\delta=\delta(X,\epsilon)>0$ such that for any $n$-dimensional Alexandrov space $Y$ with curvature $\ge k$, if $Y$ is Gromov-Hausdorff close to $X$ by a $\delta$-approximation $f: X\to Y$, then there is a homeomorphism $\tilde f: X\to Y$ which is $\epsilon$-close to $f$. We present a canonical proof of this theorem, which is distinct from Perelman's original approach. In fact, the homeomorphism $f$ is constructed purely inline with the metrics of $X$ and $Y$.
Apr 3, 2026 (SPECIAL TIME and LOCATION: 4.00-5.30 PM in the lecture room of IASM)
- Speaker: Cristiano Spotti (Aarhus University)
- Research talk: On degenerations and singularities of Kähler-Einstein metrics
- Thanks to their deep relations with algebraic geometry Kähler-Einstein metrics provide a very powerful example to study in detail degenerations and collapsing phenomena at the boundary of Einstein moduli spaces. In the pre-talk, after reviewing some of the basics, I will discuss the toy case of moduli of conical (with cone angle less than 2pi) metrics with constant curvature on the sphere, as an illustrative example of the type of phenomena we are interested in. In the main talk, instead, I will present some results and conjectures in higher dimensions.
Apr 13, 2026
- Speaker: Jonathan Trejos (南方科技大学 Southern University of Science and Technology)
- Pretalk: Embeddings Problems in Symplectic Geometry
- We briefly discuss the problem of symplectically embedding one manifold into another and explain its significance in symplectic geometry. We then introduce toric domains and singular toric domains, and describe several embedding problems arising in this setting.
- Research talk: ECH capacities of Singular Concave Toric Domains
- We describe the main theorem, which provides a method to compute the ECH capacities of singular concave toric domains via a suitable ball-packing construction. We then outline the proof, which relies on understanding an equivalence between the embedded contact complex of the concave toric domain and a combinatorial model.
Apr 20, 2026
- Speaker: Gerard Freixas i Montplet (CNRS-Centre de Mathématiques Laurent Schwartz École Polytechnique)
- Pretalk: Determinants of Laplacians and the geometry they encode
- Given a well-behaved Laplace type operator, say on a compact Riemannian manifold, it is possible to define its determinant as a regularized product of its eigenvalues. The regularization is obtained via zeta functions and heat kernel theory. Determinants of Laplacians encode some geometric features of the manifold. In this talk, I will explain this construction and provide some simple examples of the geometric properties they reflect.
- Research talk: Analytic torsion and degnerations of Calabi-Yau varieties
- Given a compact Kähler manifold and a Hermitian vector bundle E on it, the holomorphic analytic torsion is defined as a suitable combination of determinants of the Laplacians acting on (0,p)-forms with coefficients on E. This is a deep invariant involved in a differential form version of the Grothendieck-Riemann-Roch formula. For a Calabi-Yau manifold, one can consider another suitable combination of analytic torsions of sheaves of holomorphic differentials. This quantity, called the BCOV torsion, can then be normalized in order to produce a real invariant which does not depend on any choice of Kähler metric. It is called the BCOV invariant. In joint work with Dennis Eriksson, we obtain general expressions for the degeneration of the BCOV invariant for some degenerations of Calabi-Yau varieties. As an application, we obtain a necessary numerical criterion for the existence of a smooth filling of the degeneration. For some simple degenerations, such as those acquiring ADE singularities, this criterion entails a non-smooth filling result. This generalizes results by Voisin and others for A_1 and A_2 singularities. In the talk I will give an overview of these facts.
Apr 27, 2026
- Speaker:Tong Zhang, 张通 (华东师范大学)
- Pretalk: Geography of algebraic varieties: an introduction
- In this pretalk, I will give a brief introduction to the geography of algebraic varieties, starting from the surface case.
- Research talk: Moduli spaces of threefolds on the Noether line
- In this talk, I will discuss some recent results on classifying threefolds of general type with p_g >= 5 and with smallest possible canonical volume. This is a joint work with S. Coughlan, Y. Hu, and R. Pignatelli.
May 4, 2026 （HOLIDAY, NO TALK!）
- Speaker: TBA
- Pretalk: TBA
- TBA
- Research talk: TBA
- TBA
May 11, 2026
- Speaker: Yigen Zhao 赵以庚 (西湖大学)
- Pretalk: On characteristic classes for étale sheaves
- In the Pretalk, we will first present a historical overview of characteristic classes for constructible étale sheaves, tracing the theory from its classical cohomological construction to modern microlocal approaches.
- Research talk: On characteristic classes for étale sheaves
- In the Research talk, we will then examine the functorial properties and uniqueness of these classes. Finally, we will conclude by discussing their geometric and arithmetic applications. This talk is based on joint work with Enlin Yang.
May 12, 2026 (SPECIAL TIME and LOCATION: 4:30pm Tuesday in Room 203)
- Speaker: Disheng Xu 许地生 (大湾区大学)
- Pretalk: Dynamical systems and group actions
- We will discuss dynamical systems and group actions.
- Research talk: The Zimmer program for hyperbolic actions
- Zimmer’s superrigidity theorems on higher rank Lie groups and their lattices launched a program of study aiming to classify actions of semisimple Lie groups and their lattices, known as the Zimmer program. When the group is too large relative to the dimension of the phase space, the Zimmer conjecture predicts that the actions are all virtually trivial. At the other extreme, when the actions exhibit enough regular behavior, the actions should all be of algebraic origin. I will talk about recent progress on the Zimmer program, joint work with D. Damjanovic, R. Spatzier and K. Vinhage. (Will be self-contained.)
May 18, 2026
- Speaker: Song-Yan Xie 谢松晏 (中国科学院大学)
- Pretalk: Introduction to Nevanlinna theory and complex hyperbolicity
- How many values can a nonconstant entire function miss? In 1879, Picard gave a stunning answer: at most one. This seemingly simple result opened a century-long journey. We trace the milestones: Hadamard's famous regret over Jensen's formula, Nevanlinna's quantitative First and Second Main Theorems (1925), Cartan's generalization to higher dimensions (1933), and the geometric synthesis achieved by Kobayashi (1967/70), whose hyperbolicity conjectures now guide the search for entire curves in projective varieties. Along the way, we encounter a rich interplay between complex analysis, algebraic geometry, and number theory — culminating in Lang's conjecture and the Vojta dictionary.
- Research talk: Vanishing of Invariant 2-Jet Differentials and Applications
- We present two recent advances in Nevanlinna theory and complex hyperbolicity, both powered by a new method of proving vanishing for invariant \(2\)-jet differentials via algebraic reduction and computer algebra. First, we establish an effective Second Main Theorem in Nevanlinna theory for three generic smooth conics in \(\mathbb{P}^2\). For any algebraically nondegenerate entire curve, we obtain \[ T_f(r) \;\leqslant\; 5 \sum_{i=1}^3 N_f^{[1]}(r,\mathcal{C}_i) + o(T_f(r)) \quad\parallel, \] with the optimal truncated counting function. This settles a long-standing open problem. Second, we prove new \emph{key vanishing lemmas} for negatively twisted invariant logarithmic and compact \(2\)-jet differentials. Together with the P\u{a}un--Rousseau strategy, these yield improved degree bounds for the Kobayashi hyperbolicity conjecture in dimension two: \[ d \geqslant 17 \quad\text{(very generic surfaces in }\mathbb{P}^3\text{)}, \qquad d \geqslant 12 \quad\text{(generic curve complements in }\mathbb{P}^2\text{)}. \] Both results rely on the same core framework: an algebraic reduction that turns the geometric problem into a finite linear system, which is then solved by optimized Maple computations. The method also illuminates the path toward the 2-jet approach thresholds \(d=15\) and \(d=11\).
May 25, 2026
- Speaker: Yu-Wei Fan 范佑维 (SIMIS)
- Pretalk: Introduction to stability conditions
- In the pretalk, we will discuss the notion of slope stability of vector bundles, and generalize this concept to stability conditions on abelian and triangulated categories.
- Research talk: Stable implies special Lagrangian: a higher-dimensional example
- The profound link between special Lagrangians and stability conditions, proposed by Thomas and Yau around 2000, has inspired significant developments in symplectic and algebraic geometry. The one-dimensional case was established by the seminal work of Haiden, Katzarkov, and Kontsevich in 2017. In this talk, after reviewing the necessary background, we will discuss a higher-dimensional example where "stable implies special Lagrangian" can be rigorously proved.
May 26, 2026 (SPECIAL TIME and LOCATION: 10.30 AM in the lecture room of IASM)
- Speaker: Nero Budur (KU Leuven)
- Pretalk: NONE
- NONE
- Research talk: An update on some zeta functions
- In this talk we present an overview of recents results on archimedean, birational, and motivic zeta functions related to singularities of hypersurfaces.
June 1, 2026
- Speaker: Ping Xu 徐平 (Penn State University)
- Pretalk: Introduction to DG Manifolds
- I will give a brief introduction to dg manifolds, a useful framework for studying geometric objects that may have singularities. Roughly speaking, dg manifolds extend the notion of smooth manifolds by incorporating homological (chain complex) data into their structure.
DG manifolds of amplitude [−n,−1] are equivalent to Lie n-algebroids, which can be viewed as infinitesimal models of higher groupoids. On the other hand, DG manifolds of amplitude [1,n] are closely related to derived manifolds, which arise in derived geometry and allow one to systematically handle "derived" or hidden intersections.
More generally, DG manifolds with amplitude [−m,n] provide a unified language that can encode both stacky and derived types of singularities, making them a flexible tool in modern differential geometry. - Research talk: Duflo--Kontsevich Type Theorem for DG Manifolds
- It is a classical result that for any dg algebra A, the pair of its Hochschild (co)homologies (H•(A, A), H•(A, A)) carries rich algebraic structures resembling the usual Cartan calculus, often referred to as the Tamarkin--Tsygan calculus.
DG manifolds provide a useful geometric framework for describing spaces with singularities. In this talk, I will discuss the Tamarkin--Tsygan calculus associated with the dg algebra of a dg manifold and present a Duflo--Kontsevich type theorem in this setting.
As applications to several important examples, we recover the Duflo theorem on the center of the universal enveloping algebra of a Lie algebra and Kontsevich's theorem on the Hochschild cohomology of complex manifolds, placing them within a unified framework.
This is joint work with Hsuan-Yi Liao and Mathieu Stiénon.
Jun 4, 2026 (SPECIAL TIME and LOCATION: 4.30pm Thursday in Room 204)
- Speaker:Ningchuan Zhang 张凝川 (Indiana University)
- Research talk: Profinite transfers in $K(n)$-local homotopy theory
- The classical $J$-homomorphism is a map from the orthogonal group $O(n)$ to the iterated loop space $\Omega^nS^n$ of the sphere $S^n$. After $K(1)$-localization, the stable $J$-homomorphism can be interpreted as a profinite transfer map. More precisely, it is a transfer map $\Sigma^{-1}KO^\wedge_2 \to S_{K(1)}$ from the $C_2$-homotopy fixed points (with a twist) to the $\mathbb{Z}_2^\times$-homotopy fixed points of the $2$-complete complex topological $K$-theory. In joint work in progress with Guchuan Li, we extend this idea to define and study profinite transfers between homotopy fixed points of the Morava $E$-theory by closed subgroups of the Morava stabilizer group in $K(n)$-local homotopy theory. We introduce two definitions of the profinite transfer maps. One working definition is as duals to the profinite restriction maps in the appropriate category. At large primes, we show that the image of the transfer map $\Sigma^{-n^2}E_n \to S_{K(n)}$ on homotopy groups is the filtration $n^2$-line in the homotopy fixed point spectral sequence. A second definition of the profinite transfer maps is based on the 6-functor formalism for smooth representations of $p$-adic Lie groups by Heyer—Mann. We prove that the two definitions are equivalent.
Jun 8, 2026 (SPECIAL TIME:4.00-5:00pm)
- Speaker:Wanchun Shen (Harvard University)
- Research talk: Hodge Theory and algebraic K-theory
- In this talk, we will discuss how to study the algebraic K-groups of a singular variety from the perspective of Hodge theory
Jun 15, 2026
- Speaker: TBA
- Pretalk: TBA
- TBA
- Research talk: TBA
- TBA
Jun 22, 2026
- Speaker: TBA
- Pretalk: TBA
- TBA
- Research talk: TBA
- TBA
Jun 29, 2026 (SPECIAL LOCATION:the lecture room of IASM)
- Speaker:Xiaodong Wang (Michigan State University)
- Pretalk: The Yamabe Problem and the Yamabe constant
- I will discuss the Yamabe problem and its solution as well as the ongoing efforts to understand the Yamabe constant.
- Research talk: Problems and Results on a nonlinear elliptic PDE on Riemannian manifolds
- The nonlinear elliptic PDE in question is closely related to the Yamabe problem. I will discuss uniqueness results and problems in various settings and their geometric applications.
Collapse Spring 2026 Schedule