【10.21-10.22】Singularity Theory and Combinatorics 2023

Time: October 21-22, 2023

            2023年10月21-22日

Room: C1124, Material Science Research Building (Section C), USTC 

               中国科学技术大学东校区物质科研楼C座 C1124

Organizer

刘永强 Yongqiang Liu (University of Science and Technology of China)  liuyq@ustc.edu.cn

王振建 Zhenjian Wang (University of Science and Technology of China) wzhj@ustc.edu.cn

Secretary

谭纯 nanbei0104@ustc.edu.cn

Sponsor

Institute of Geometry and Physics, USTC 

中国科学技术大学几何与物理研究中心

Conference Schedule

Title and abstract

陈颖 Ying Chen（华中科技大学 ) 

Title: Fibrations of tamely composable maps
Abstract: J. Milnor proved in 1968 that a holomorphic function induces a locally trivial fibration in the neighbourhood of a singular point, which is named after him called Milnor fibration. However, the existence of fibrations sets new challenges in the real setting. By introducing a general condition, called tameness, we have a local singular tube fibration structure in stratified sense. In this talk, we show that under some specific condition called tamely composable, composed map germs have singular tube fibrations, and the topological type of their fibers can be explicitly determined. Our results provide a natural characterization of Némethi's generalized Thom-Sebastiani theorem and extend the relevant results of R.N.A dos Santos and N. Dutertre for non-isolated singularities.
Reference:
[AD] R.N. Araújo dos Santos, N. Dutertre, Topology of real Milnor fibrations for non-isolated singularities, Int. Math. Res. Not. IMRN 2016, no. 16, 4849–4866.
[ART] R.N. Araújo dos Santos, M. Ribeiro, M. Tibar, Fibrations of highly singular map germs, Bull. Sci. Math. 155 (2019), 92-111.
[Ne] A. Némethi, Generalized local and global Sebastiani-Thom type theorems, Compositio Math.80 (1991), 1-14.

郝峰 Feng Hao（山东大学）

Title: Holomorphic 1-forms without Zeros and Smooth Morphism to Abelian varieties

Abstract: A celebrated result of Popa and Schnell shows that any holomorphic 1-form on a smooth complex projective variety of general type admits zeros. More generally, they show that the Kodaira dimension κ(X) of a smooth complex projective variety X satisfies the following inequality κ(X)≤dimX−g,where g is the maximal number of pointwise linearly independent holomorphic 1-forms. In this talk I will give a classification of (minimal) varieties satisfying the equality conditions κ(X)=dimX−g. Roughly speaking, they arise as diagonal quotients of the product of an abelian variety with a variety of general type. This is a joint work with Nathan Chen and Benjamin Church.

郭威力 Weili Guo（北京化工大学）

Title  Holonomy Lie algebre of a geometric lattice

Abstract: Motivated by Kohno’s result on the holonomy Lie algebra of a hyperplane arrangement, we define the holonomy Lie algebra of a finite geometric lattice in a combinatorial way. For a solvable pair of lattices, we show that the holonomy Lie algebra is an almost-direct product of the holonomy Lie algebra of the sublattice and a free Lie subalgebra. This yields the structure of the holonomy Lie algebra of a finite hypersolvable (including supersolvable) lattice. As applications, we obtain the structure of the holonomy Lie algebra of (the Salvetti complex of) a supersolvable oriented matroid, and that of a hypersolvable arrangement, as well as their lower central series formulae.

李彦霖 Yanlin Li (杭州师范大学)

Title: Singularities and dualities of evolving developable frontals

Abstract: In the field of differential geometry from viewpoint of singularity theory, cuspidal edge，swallowtail and cuspidal cross cap, etc. are classical and fundamental singularity types. We can often find interesting results from those singularities. On the other hand, surfaces with singularities, named singular surfaces are hot research object in the recent years. Wave fronts and frontals are particularly interesting singular surfaces which may appear interesting singularity types. In this presentation, I will introduce a deformation on frontals, where we call the frontal homotopy. Then, I will discuss the local and global properties of evolving swallowtail and cuspidal cross cap. Furthermore, I talk about the evolving cuspidal edge, and show a classification of it and a duality property between cuspidal edge and swallowtail. This is a joint work with Zhichao Yang.

王博潼 Botong Wang（University of Wisconsin, Madison）Minicourse (Two one-hour lectures)

Title: An introduction to the work of Adiprasito-Huh-Katz

Abstract: Starting with Huh’s paper resolving Rota’s log-concave conjecture for matroids realizable in characteristic zero, matroid theory entered a new era using ideas from algebraic geometry. The goal of the lectures is to give an overview of the ideas of the proof of Rota’s conjecture. In the first lecture, we will introduce the wonderful model of a hyperplane arrangement, and the proof in the realizable case. In the second lecture, we will introduce Bergman fans and the proof in the nonrealizable case.

王岁杰 Suijie Wang ( 湖南大学 ) 

Title: Partition of Grassmannian induced by hyperplane arrangements

Abstract：In this talk, I will introduce a new approach for constructing new hyperplane arrangements from the intersection lattice of a given hyperplane arrangement, called k-adjoint of a hyperplane arrangement,  which has connections or applications with the following objects.

1) Serve as a combinatorial decomposition of Grassmannian, equivalent to both matroid stratification and permuted Schubert decomposition;

2) Provide a combinatorial classification of all k-restrictions of a fixed hyperplane arrangement;

3) Extend the concept of adjoint of a hyperplane arrangement by Bixby and Coulard;

4) Present the anti-continuity property of some combinatorial invariants for all k-restrictions of a fixed hyperplane arrangement.

吴磊 Lei Wu（浙江大学）

Title: Log geometry and log D-modules

Abstract: The theory of D-modules provides very powerful tools and solved many important problems. In this talk, I will introduce a natural way to generalize the D-module theory in logarithmic geometry. I will explain log Bernstein inequality and define log holonomic D-modules on smooth log schemes. Then I will explain log constructibility by using Kato-Nakayama spaces associated to log schemes. If time allowed, I will also explain how the theory is related to the classical b-function theory as well as log Riemann-Hilbert correspondence. This is based on an ongoing project with Andreas Hohl.

肖建 Jian Xiao （清华大学）

Title: Intersection theoretic inequalities via Lorentzian polynomials 

Abstract: The theory of Lorentzian polynomials was recently introduced and systematically developed by Braden-Huh and independently (with part overlap) by Anari-Liu-Gharan-Vinzant. It has many important applications in combinatorics, including a resolution of  the strongest version of Mason conjecture and new proofs of the Heron-Rota-Welsh conjecture. In this talk, we explore its applications to geometry. In particular, we establish a series of intersection theoretic inequalities, which we call rKT property.  We will discuss the origin of the rKT property in analytic geometry, and its connections with the submodularity for numerical dimension type functions and the sumset estimates for volume type functions. Joint work with J. Hu.

赵以庚 Yigeng Zhao（西湖大学）

Title: Vanishing cycles and the Milnor formula for constructible etale sheaves

Abstract: As a generalization from topological to etale settings, Deligne's Milnor formula conjecture links the Milnor number at an isolated singularity point to the total dimension of the vanishing cycles of the constant sheaf. We briefly review this conjecture and its known cases. Using the six-functor formalism, we present a cohomological analog of the Milnor formula in terms of non-acyclicity classes. This is a joint work with Enlin Yang.