Mar 3
- Speaker: Wenhao Ou 欧文浩 (中科院数学所)
- Pretalk: Rational curves in complex manifolds
- We review some classical results in birational geometry, particularly on rational curves in complex manifolds.
- Research talk: A characterization of uniruled Kaehler manifolds
- We adapt Bost's algebraicity characterization to the situation of a germ in a compact Kaehler manifold. As a consequence, we extend the algebraic integrability criteria of Campana-Paun and of Druel to foliations on compact Kaehler manifolds. As an application, we prove that a compact Kaehler manifold is uniruled if and only if its canonical line bundle is not pseudoeffective.
Mar 4 (SPECIAL TIME/LOCATION: TUESDAY in the lecture room of IASM)
- Speaker: Jihao Liu 刘济豪 (北京大学)
- Pretalk: An Introduction to Foliations and the Minimal Model Program
- This pretalk will provide an overview of foliations from the perspective of birational geometry, along with an introduction to the minimal model program for foliations. We will discuss foundational concepts and set the stage for the main talk, which explores recent advances in the field.
- Research talk: Boundedness of Algebraically Integrable Fano Foliations
- In this talk, I will discuss a recent joint work with Paolo Cascini, Jingjun Han, Fanjun Meng, Calum Spicer, Roberto Svaldi, and Linyao Xie. We establish two key results: (1) the existence of good minimal models for klt algebraically integrable adjoint foliated structures of general type, and (2) the boundedness of \epsilon-lc Fano algebraically integrable adjoint foliated structures. These results are analogues of the BCHM and BAB theorems respectively in the context of algebraically integrable foliations/adjoint foliated structures..
Mar 10
- Speaker: Yalong Cao 曹亚龙 (中科院晨兴数学中心)
- Pretalk: Why critical loci are important
- We review why critical loci are important.
- Research talk: Quasimaps to critical loci
- Critical loci are fundamental objects in geometry, physics, and representation theory. In this talk, we will introduce their quantum K-theory by counting quasimaps from curves. Joint work with Yukinobu Toda and Gufang Zhao.
Mar 24
- Speaker: Shi Wang 汪湜 (上海科技大学)
- Pretalk: Basics in nonpositive curvature
- We review basics in nonpositive curvature.
- Research talk: The natural flow and the critical exponent
- For a complete Riemannian manifold of nonpositive curvature, we introduce a flow. We give an upper bound on the k-Jacobian of the flow in terms of the critical exponent of the fundamental group. We also give several applications connecting the geometry and topology of the manifold, which include the linear isoperimetric inequality, the homological vanishing theorem, and the non-existence of compact complex subvarieties in certain complex hyperbolic manifolds. This is joint work with Chris Connell and Ben McReynolds.
Mar 31
- Speaker: Lars Andersson (BIMSA)
- Pretalk: General Relativity and Geometry
- I will give a brief introduction to the Einstein equation of general relativity and mention some open problems related to it.
- Research talk: Gravitational instantons and special geometry
- Gravitational instantons are Ricci flat complete Riemannian 4-manifolds with at least quadratic curvature decay. Classical examples include the Taub-NUT and the Euclidean Kerr instanton. In this talk I will present some recent results on classification, stability, and conservation laws, for instantons with special geometry.
Apr 7
- Speaker: Zheng Hua 华诤 (香港大学)
- Pretalk: On Modular Poisson structures
- We will briefly recall Poisson structures in the algebraic geometry context. We will focus on two main classes of examples: moduli space of sheaves on del Pezzo surfaces and semi-classical limits of noncommutative deformations.
- Research talk: Symplectic foliation of modular Poisson structures
- We will give a unified construction of the algebraic Poisson structures mentioned in the first part of the talk, via the moduli space of complexes on degeneration of elliptic curves. Our construction leads to a neat description of the symplectic leaves. We will discuss several applications on deformations of Hilbert schemes and positroid varieties.
Apr 8 (SPECIAL TIME/LOCATION: TUESDAY in the lecture room of IASM)
- Speaker: Yuji Odaka 尾高悠志 (京都大学)
- Pretalk: Canonical torus action on (holomorphic) symplectic singularities
- Symplectic singularities arise from e.g., nilpotent orbit closure, compact hyperKahler varieties, quiver varieties, and so on. These have been studied actively recently, not only in algebraic geometry, but from geometric representation theoretic or string theoretic perspectives. 2-dimensional case is the classical orbifolds, the so-called Klein / ADE singularity. D. Kaledin conjectured around 2000 that they are always quasi-homogeneous, i.e., admit local good C^*-action. We prove the conjecture conditionally by a new approach, i.e., using complex differential geometry and related algebraic geometry with Poisson structures. In particular, we obtain a canonical action with DG interpretation. Joint with Yoshinori Namikawa.
- Research talk: Tangent cone, bubbling, compactness and algebraic geometry
- In the latter half, we talk about more general or technical aspects. First main aspect of our approach to the Kaledin’s conjecture to interpret the predicted conical structure as metric tangent cone, after the Donaldson-Sun theory which revealed general algebro-geometric structure underlying the general metric tangent cone of klt singularities with local singular Kahler-Einstein metrics (the second aspect is understanding of Poisson deformations). Their method is extended to understanding the bubbling of the metric family with euclidean volume growths e.g., Kronheimer ALE space. As in the case of (metric) tangent cones, these bubblings, or "older" Gromov-Hausdorff limit of compact Kahler-Einstein manifolds (singular Q-Fano var, by Donaldson-Sun I) etc are all some differential geometric limits which turn out to be varieties. I plan to explain key ideas and methods of how their underlying limit varieties can be algebro-geometrically re-constructed or recovered in a somewhat unified deformation theoretic manner (without any technical prerequisites in algebraic geometry).
Apr 14
- Speaker: Yifei Zhu 朱一飞 (南方科技大学)
- Pretalk: Quantum materials, Higgs bundles, and geometric Langlands correspondence
- With motivation from condensed matter physics and materials science in collaboration with physicists, I’ll discuss explicit examples of eigenbundles associated to both gapped and gapless quantum mechanical systems, that is, to families of matrices with prescribed symmetry, as well as their homotopical classifications. These include Hopf bundles and, more generally, certain rank-2 and rank-3 Higgs bundles. The geometry and topology of such mathematical objects and their moduli spaces shed light on questions in physics, namely, hyperbolic band theory (after A. J. Kollár et al.) and bulk–edge correspondence, which I will at least indicate.
- Research talk: Spheres, spectral algebraic geometry, and Jacquet–Langlands correspondence
- The ring spectrum of spheres plays a similar role in spectral algebraic geometry that the ring of integers does for classical algebraic geometry. It recovers the latter in a natural way while encoding additional higher-homotopical information. With motivation from structure and computation centered around the homotopy groups of spheres, I’ll explain joint work with Xuecai Ma of Westlake University. We define relative effective Cartier divisors for a spectral Deligne–Mumford stack. We then apply the representability of their moduli to questions at the crossroad of algebraic topology and arithmetic geometry.
Apr 18 (SPECIAL TIME/LOCATION: FRIDAY in the lecture room of IASM)
- Speaker: Jian Xiao 肖建 (清华大学)
- Pretalk: Examples on log-concavity
- We present several examples on log-concavity in geometry and combinatorics, behind which is the Hodge index theorem.
- Research talk: Extremals in log-concavity: an algebro-geometric viewpoint
- We show how to formulate the extremal problem in log-concavity in the framework of algebraic geometry, and discuss some of our progress towards the conjectural picture. Based on joint work with Jiajun Hu.
Apr 21
- Speaker: Guchuan Li 李谷川 (北京大学)
- Pretalk: Bott Periodicity and topological K-theory
- We will assume Bott Periodicity and use it to define a generalized cohomology theory: topological K-theory.
- Research talk: Real Bott periodicity at higher chromatic heights
- Real Bott periodicity shows that the homotopy groups of real topological K-theory is 8-periodic. From chromatic homotopy theory perspective, Real topological K-theory is a height 1 theory, and for each natural number n, there are height n periodic cohomology theories. Important examples include 192 periodic topological modular forms at height 2 and Hill-Hopkins-Ravenel’s 256 periodic Kervarie detection spectrum at height 4. In this talk, we give a general formula for the periodicity for all heights. This is joint work with Zhipeng Duan, Mike Hill, Yutao Liu, Danny XiaoLin Shi, Guozhen Wang, and Zhouli Xu.
Apr 28
- Speaker: Rixin Fang 方日鑫 (复旦大学)
- Pretalk: Introduction to K-theory and Hochschild Homology
- We introduce low dimensional K-theory first, and briefly explain Quillen's plus construction. Then we recall the Hochschild homology of an algebra, and its relation with K-theory. If time permits, we give a briefly introduction on stable homotopy theory, and $v_n$-periodicity.
- Research talk: Chromatic redshift and Segal conjecture
- The Lichtenbaum--Quillen property comes with an arithmetic background, Waldhausen reformulated this property as a telescopic homotopy problem. The chromatic redshift introduced by Rognes generalized this idea to higher height ring spectra. And algebraic K-theory for ring spectra can be well understood by trace method, we will briefly recall the trace method. By the work of J. Hahn, D. Wilson, et al., the redshift problem can be reduced to Segal conjecture and (weak) canonical vanishing problem. We recall the techniques to prove Segal conjecture, and we present examples that Segal conjecture holds. We use the cyclic decomposition to demonstrate some examples that Segal conjecture fails, and thus Lichtenbaum--Quillen property fails also.
May 12
- Speaker: Siqi He 何思奇 (中科院晨兴数学中心)
- Pretalk: The Analytic Compactification of SL(2,C) Character Varieties
- In the pretalk, we will introduce Taubes’ work on the analytic compactification of the SL(2,C) character variety of 3- and 4-manifolds from a gauge theory perspective. We will discuss the construction of the compactification, and understanding the boundary of the compactified space.
- Research talk: The Deformation Problem for Z/2 Harmonic 1-Forms over Kähler Manifolds
- Z/2 harmonic 1-forms, introduced by Taubes, describe the boundary behavior of moduli spaces arising from gauge-theoretic equations. The Hitchin–Simpson equations on a Kähler manifold are first-order nonlinear equations for a pair consisting of a connection on a Hermitian vector bundle and a 1-form valued in the endomorphism bundle. We study the behavior of solutions to the Hitchin–Simpson equations as the norm of the 1-form becomes unbounded, and explore its relationship with Z/2 harmonic 1-forms. In addition, we will discuss the deformation problem of Z/2 harmonic 1-forms in the Kähler setting.
May 19
- Speaker: Honghao Gao 高鸿灏 (清华大学丘成桐数学中心)
- Pretalk: Legendrian knots and its invariants
- Knots are loops in Euclidean 3 space. On top of that, Legendrian knots additionally satisfy a particular differential equation, imposing constraints to the tangent direction at a certain point. Understanding Legendrian knots is one of the central topics in a subject called contact topology. In this pretalk, we will explain how to describe Legendrian knots through (a lot of) pictures, and introduce some algebraic invariants that are used to study Legendrian knots.
- Research talk: Legendrian knots and Lagrangian fillings
- Lagrangian fillings are surfaces bounding Legendrian knots, which live in a 4-dimensional space called symplectization. Different Lagrangian fillings can be distinguished by cluster charts in a moduli space associated to the Legendrian knots. In this talk, I will explain the joint work with Roger Casals, on how to construct a Lagrangian filling from a given cluster chart.
May 26
- Speaker: Yu Zhang 张宇 (天津大学)
- Pretalk: Gröbner Bases: The Linear Algebra of Polynomial Ideals
- This talk explores Gröbner bases as the nonlinear analog of Gaussian elimination and row echelon form in linear algebra. By comparing polynomial ideals to linear subspaces, we’ll see how Gröbner bases enable systematic solutions to systems of polynomial equations. Using intuitive examples and analogies, we demystify the core principles of Gröbner basis theory.
- Research talk: Extensive Computations of the Adams Spectral Sequence E_2-Page at Odd Primes
- The Adams spectral sequence of the sphere is a cornerstone of stable homotopy theory, yet explicit computations remain notoriously difficult due to prohibitive computational complexity. In this talk, we focus on the case of odd primes, which exhibits distinct structural features compared to the p=2 setting. By leveraging insights from computational algebra, we (1) construct a tailored extension of Gröbner basis theory for the noncommutative Steenrod algebra at odd primes, and (2) develop a novel algorithmic framework using this generalized theory to compute the E_2-page of the Adams spectral sequence in significantly higher degrees than previously achievable. These advances not only expand the algorithmic toolkit for spectral sequence computations but also open new avenues for exploring stable homotopy groups of spheres. This is joint work with Weinan Lin.
Jun 2
- 端午节
Jun 9
- Speaker: Zhipeng Duan 段志鹏 (南京师范大学)
- Pretalk: Atiyah’s Real K-theory and Slice Spectral Sequences
- In this pre-talk, we will introduce Atiyah’s Real K-theory and explain its connections to complex and real K-theory. We will then apply the slice spectral sequence to compute its homotopy groups.
- Research talk: Vanishing lines and periodicities of higher real K-theories
- In this talk, we generalize computations in Atiyah’s Real K-theory to broader classes of groups and to higher chromatic heights. We will present two key structural results arising from these developments: a vanishing line result and a periodicity result. These results enable a more effective approach to computing homotopy fixed point spectral sequences and slice spectral sequences. We will also discuss an interesting geometric application that follows from these structural insights. This talk is based on a combination of several joint works with Mike Hill, Hana Jia Kong, Guchuan Li, Yutao Liu, Yunze Lu, Xiaolin Danny Shi, Guozhen Wang, and Zhouli Xu.
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