报告人简介：桂弢，北京大学北京国际数学研究中心博士后。2018年本科毕业于四川大学数学学院，2023年于中国科学院数学与系统科学研究院获得博士学位，师从席南华研究员。主要研究方向是李理论，表示论，组合代数几何与组合霍奇理论。
时间地点：
2025年4月16日、23日9:50-11:25（第3-4节），腾讯会议号717-9005-3856
4月18日、25日15:50-17:25（第8-9节），腾讯会议号628-5140-2494
Classical Hodge theory is a crucial part of a parcel of remarkable statements (Hodge decomposition, the weak and hard Lefschetz theorem, the Hodge--Riemann relations) concerning the cohomology of projective complex algebraic varieties (or more generally Kähler manifolds). Over the last decades, it has been discovered that there are several settings where structures similar to the cohomology groups of classical Hodge theory show up, but where the underlying variety is "missing".
Typically, the existence of Hodge like structures on such spaces provides deep results in combinatorics. Examples include the g-conjecture on the face numbers of polytopes, the positivity of Kazhdan-Lusztig polynomials coming from the theory of Soergel modules, the Heron--Rota--Welsh conjecture and Dowling and Wilson's "top-heavy" conjecture on matroids. There is a remarkable parallel between the theory of polytopes, Coxeter groups, and matroids, based on a combinatorial cohomology theory. The central theorems in Hodge theory continue to hold in a realm that goes far beyond that
of Kahler geometry. This cohomology theory gives strong restrictions ("shadows of Hodge theory") on numerical invariants of these combinatorial objects.
I will try to introduce this web of ideas and give an overview of the similarity, present some work on these classical conjectures in combinatorics, and also present some open problems and conjectures. We hope that these lecture series can build bridges---providing intricate, combinatorially inspired spaces to the topologist and geometer, and versatile geometric tools to the combinatorialist.
The four lectures in this series are designed to be accessible (hopefully!) to a broad audience and appropriate for a Department Colloquium. One of the intentions is
to setup a stage for after-lecture discussions with and between participants.
1. Overview of classical cohomology and Hodge theory
I will give a broad overview of the cohomology and Hodge theory for smooth complex projective varieties, especially the so-called Kahler package: the Poincare duality, the hard Lefschetz theorem and the Hodge--Riemann relations. Then I will explain the restrictions ("shadows of Hodge theory") on numerical invariants of complex projective
varieties with an affine paving. I will also present a conjecture of myself on certain numbers coming from symmetric groups (or more generally, finite Coxeter groups) that "look like" a Hodge diamond!
References:
Alexander Beilinson, Joseph Bernstein, Pierre Deligne, and Ofer Gabber, Faisceaux pervers (French) [Perverse sheaves]
Mark de Cataldo and Luca Migliorini, The decomposition theorem, perverse sheaves and the topology of algebraic maps
Kyle Peterson, A two-sided analogue of the Coxeter complex
Claire Voisin, Hodge theory and complex algebraic geometry. I & II
2. Cohomology and Hodge theory for polytopes
How many vertices, edges, faces, ... can a polytope have? This is a very difficult question in general. For simplicial (or dually, simple) polytopes, there is beautiful sufficient and necessary condition conjectured by Peter McMullen known as the "g-conjecture"and proved by Richard Stanley (for the necessity) using Hodge theory of toric varieties. I will also present our work on the Weyl group symmetries of the toric variety associated to Weyl chambers.
References:
David Cox, John Litttle, and Henry Schenck, Toric Varieties
Tao Gui, Hongsheng Hu, and Minhua Liu, Weyl group symmetries of the toric variety associated to Weyl chambers
Ezra Miller and Bernd Sturmfels, Combinatorial Commutative Algebra
Richard Stanley, Combinatorial applications of the hard Lefschetz theorem
Richard Stanley, The number of faces of a simplicial convex polytopes
Gunter Ziegler, Lectures on Polytopes
3. Cohomology and Hodge theory for Coxeter groups
Kazhdan and Lusztig in 1979 defined certain polynomials for every pair of elements in a Coxeter groups. These polynomials have remarkable properties and appear everywhere in the representation theory of Lie theoretic objects. I will explain the proof of Kazhdan--Lusztig positivity conjecture by Elias--Willamson. I will also present the combinatorial invariance conjecture of Lusztig and Dyer (independently) and our idea to attack it. However, these polynomials are still very mysterious and only time will tell what they are trying to tell us!
References:
Ben Elias and Geordie Williamson, Kazhdan--Lusztig conjectures and shadows of Hodge theory
Ben Elias and Geordie Williamson, The Hodge theory of Soergel bimodules
Tao Gui, Lin Sun, Shihao Wang, and Haoyu Zhu, On Bott--Samelson rings for Coxeter groups
David Kazhdan and George Lusztig, Representation of Coxeter groups and Hecke algebras
David Kazhdan and George Lusztig, Schubert varieties and Poincare duality
Geordie Williamson, The Hodge theory of the Hecke category
4. Cohomology and Hodge theory for matroids
Matroids are abstractions of (in)dependence structures in mathematics. I will explain the proof of Heron--Rota--Welsh conjecture by Adiprosito--Huh--Katz and the proofs of Dowling and Wilson's "top-heavy" conjecture by Huh--Wang and Braden--Huh--Matherne--Proudfoot--Wang. I will also present the equivariant log-concavity conjecture of myself and of Nick Proudfoot.
References:
Karim Adiprasito, June Huh and Eric Katz, Hodge theory for combinatorial geometries
Tom Braden, June Huh, Jacob Matherne, Nicholas Proudfoot and Botong Wang, Singular Hodge theory for combinatorial geometries
Tao Gui, On the equivariant log-concavity for the cohomology of the flag varieties
Tao Gui and Rui Xiong, Equivariant log-concavity and equivariant Kahler packages
June Huh and Eric Katz, Log-concavity characteristic polynomials and the Bergman fan of matroids
June Huh, Combinatorial applications of the Hodge--Riemann relations
June Huh, Combinatorics and Hodge theory
June Huh, Milnor numbers of projective hypersurfaces and the chromatic polynomials of graph
