2019年京区博士后交流研讨会

言（北京大学）

杰（中科院晨兴数学中心）

笛（北京大学）

斌（中科院晨兴数学中心）

（中科院数学院）

Conference Schedule

 December 6, 2019 (MCM 110) 主持人：章志飞 08:00-08:50 签到：晨兴楼一楼前台 08:50-09:00 欢迎词：杨乐 09:00-09:40 刘彦麟 Global solutions of $3$-D Navier-Stokes system with small unidirectional derivative 09:50-10:30 史鹏帅 The index of a perturbed Dirac operator with local boundary condition 10:30-10:50 茶歇 (合影) 10:50-11:30 石云峰 Lattice Green's functions estimates and applications 11:40-12:20 吴笛 A homogenized limit for the 2D Euler equations in a perforated domain 12:20-14:00 午餐（盒饭） 主持人：万昕 14:00-14:40 邓太旺 Characteristic cycles and semi-canonical basis 14:50-15:30 胡创强 A new modular interpretation of BBGS Towers 15:30-15:50 茶歇 15:50-16:30 刘诗南 Local model of Shimura varieties 16:40-17:20 赵斌 Slopes of modular forms 17:50-20:00 晚宴 December 7, 2019 (MCM 110) 主持人：付保华 09:30-10:10 刘杰 Positivity of tangent bundle and its subsheaves 10:10-10:30 茶歇 10:30-11:10 王振建 Monotonic invariants under blowups 11:20-12:00 张科伟 Recent progress on Tian's partial C^0 estimate 12:00-14:00 午餐（盒饭） 主持人：杨晓奎 14:00-14:40 胡鹰翔 Harmonic mean curvature flow and geometric inequalities 14:50-15:30 黄瑞芝 Stringc structures and modular invariants 15:30-15:50 茶歇 15:50-16:30 郦言 Modified Ding functional and Mabuchi metric 16:40-17:20 罗思捷 On characterizations of order preserving mappings in convex analysis and its applications to operator theory

# Characteristic cycles and semi-canonical basis

Twenty years ago Lusztig introduced the semi-canonical basis for the enveloping algebra $U(n)$, where n is a maximal unipotent sub-Lie algebra of some simple Lie algebra of type A, D, E. Later on B. Leclerc found a counter-example to some conjecture of Bernstein-Zelevinsky and related it to the difference between dual canonical basis and dual semi-canonical basis. He further introduced a condition (open orbit conjecture of Geiss-Leclerc-Schoer) under which dual canonical basis and dual semi-canonical basis coincide. In this talk we explain in detail the above relations and show a relation between the two bases above through micro-local analysis.

# A new modular interpretation of BBGS Towers

In 2000, Elkies had given explicit equations of rank-2 Drinfeld modular curves which coincide with the asymptotically optimal towers of curves constructed by Garcia and Stichtenoth. To generalize this result, we develop a new procedure to obtain equations of rank-m Drinfeld modular curves, with $m \geq 2$. The resulting modular curves coincide with the celebrate (recursive) towers of curves constructed by Bassa, Beelen, Garcia, and Stichtenoth.

# Harmonic mean curvature flow and geometric inequalities

In this talk, we use the harmonic mean curvature flow to prove Alexandrov-Fenchel type inequalities for strictly convex hypersurfaces in hyperbolic space. Using the new Alexandrov-Fenchel type inequalities and the inverse mean curvature flow, we show that the Alexandrov-Fenchel inequality for the total curvature in terms of the area for strictly convex hypersurfaces.

# Stringc structures and modular invariants

Spin structure and its higher analogies play important roles in index theory and mathematical physics. In particular, Witten genera for String manifolds have nice geometric implications. As a generalization of the work of Chen-Han-Zhang (2011), we introduce the general Stringc structures based on the algebraic topology of Spinc groups. It turns out that there are infinitely many distinct universal Stringc structures indexed by the infinite cyclic group. Furthermore, we can also construct a family of the so-called generalized Witten genera for Spinc manifolds, the geometric implications of which can be exploited in the presence of Stringc structures. As in the un-twisted case studied by Witten, Liu, etc, in our context there are also integrality, modularity, and vanishing theorems for effective non-abelian group actions. We will also give some applications.

This a joint work with Haibao Duan and Fei Han.

# Modified Ding functional and Mabuchi metric

In this talk, I will first review a joint work with Zhou Bin, which concerns the properness of modified Ding functional and existence of Mabuchi metric. In particular we will review the existence criterion on Fano group compactifications. Also I will construct some new examples and talk about the generalization of this work in singular cases.

# Local model of Shimura varieties

To calculate the Hasse-Weil zeta function of a Shimura variety (and relate it to automorphic forms) is a central subject in the Langlands program. Local models are certain explicitly-defined closed subschemes of grassmannian varieties. In this talk we explain how to use local model to calculate the zeta function of a Shimura variety at a bad prime, i.e. where the Shimura variety does not have a smooth model. We will use the modular curve as our guiding example.

# Positivity of tangent bundle and its subsheaves

It is a very classical problem in algebraic geometry to derive geometric information of algebraic varieties from the properties of tangent bundles. In recent years, it has been lots of interest in understanding the positivity of tangent bundle via its subsheaves. I will talk about examples, known results and present some ideas lying behind our recent work towards this problem.

# Global solutions of $3$-D Navier-Stokes system with small unidirectional derivative

Given initial data $u_0=(u_0^h,u_0^3)\in H^{\frac 12}(R^3)\cap B^{0,\frac 12}_{2,1}(R^3)$ with $u^h_0$ belonging to $L^2(R^3)\cap L^\infty(R_v; H^{-\delta}(R^2_h))\cap L^\infty(R_v; H^3(R^2_h))$ for some $\delta\in ]0,1[,$ if  in addition $\partial_3u_0$ belonging to the homogeneous anisotropic Sobolev space, $H^{-\frac12,0},$ we prove that the classical $3$-D Navier-Stokes system has a unique global Fujita-Kato solution  provided that the $H^{-\frac 12,0}$ norm of  $\partial_3u_0$ is sufficiently small compared to $\exp\left(-C\big(A_\delta(u^h_0)+B_\delta(u_0)\big)\right)$ with $A_\delta(u^h_0)$ and $B_\delta(u_0)$ being scaling invariant quantities of the initial data, and which is scaling invariant  with respect to the variable $x_3$. This result provides some classes of large initial data which are large in Besov space $B^{-1}_{\infty,\infty}$ and which generate unique global solutions to  3-D Navier-Stokes system. In particular, we extend the previous results in a series of works by Chemin, I. Gallagher et al for initial data with a slow variable to multi-scales slow variable initial data.

# On characterizations of order preserving mappings in convex analysis and its applications to operator theory

In this talk, we will recall some historical background of the study of order preserving (resp. reversing) mappings in convex analysis, such as the classical “Artstein-Avidan-Milman representation theorems” and their generalization-s. Secondly, we will provide a localized version of the “Artstein-Avindan-Milman representation theorems” and the characterizations of order preserving (resp. re-versing) mappings defined on certain classes of convex functions will also be discussed. Finally, we conclude our talk with some applications to operator theory.

# The index of a perturbed Dirac operator with local boundary condition

We first introduce the boundary value problems for Dirac-type operators on Riemannian manifolds with compact boundary. This was studied in full generality by B？r and Ballmann, through a hybrid Sobolev space on the boundary. We talk about its extension to manifolds with non-compact boundary. We then consider a special class of perturbed Dirac operators on a complete Riemannian manifold whose boundary is a disjoint union of finitely many complete connected manifolds. Under a local boundary condition, the index of the operator can be computed in terms of the indexes on the boundary. This generalizes a result of D. Freed. Based on joint work with Maxim Braverman.

# Lattice Green's functions estimates and applications

In this talk, we consider estimates of quasi-periodic Green's functions on higher-dimensional lattice. The main tools employed here include sub-harmonic estimates and semi-algebraic geometry arguments. As applications, we will introduce some recent results on Anderson localization and KAM theory. This talk is mainly based on a joint work with Svetlana Jitomirskaya and Wencai Liu.

# Monotonic invariants under blowups

We discuss a conjecture on the quotient of Milnor number $\mu$ and Tjurina number $\tau$ for a singular reduced plane curve stating that $\mu/\tau<4/3$. This conjecture was posed of A. Dimca and G.M. Greuel in 2018, and it aroused a great deal of interest in 2019. It has already been completely proved in October 2019. In this talk, we will show that $3\mu-4\tau$ has some monotonicity under blowups. This monotonicity leads to a solution to the conjecture for an irreducible curve. Since the conjecture has been proved in the general case, we will not discuss the details of proof of our results; instead, we discuss the idea of seeking monotonic quantities under blowups, which is inspired by the theory of geometric flows.

# A homogenized limit for the 2D Euler equations in a perforated domain

We study the motion of an ideal incompressible fluid in a perforated domain. The porous medium is composed of inclusions of size a separated by distances d and the fluid fills the exterior. We focus on the asymptotic behavior of the fluid when $a$ and $d$ both tend to zero. If the inclusions are distributed on the unit square, this issue is studied recently when $a/d$ tends to zero or infinity, leaving aside the critical case where the volume fraction of the porous medium is bounded and non-zero. In this paper, we provide the first result in this regime. In contrast with former results, we obtain an Euler type equation where a homogenized term appears in the elliptic problem relating the velocity and the vorticity. Our analysis is based on the so-called method of reflections whose convergence provides novel estimates on the solutions to the div-curl problem which is involved in the 2D-Euler equations.

# Recent progress on Tian's partial C^0 estimate.

Partial C^0 estimate plays crucial roles in the proof of Yau-Tian-Donaldson conjecture, which measures the very ampleness of a line bundle in a quantitative way. I will report some recent progress on the partial C^0 estimate. For instance, I will show that, along the normalized Kaehler-Ricci flow on a Fano manifold, the partial $C^0$ estimate holds uniformly.

# Slopes of modular forms

We will explain the motivation to study the slopes of modular forms and mention several conjectures towards their computation. Then we will explain the relation between these conjectures with the geometric properties of eigencurves and some more general eigenvarieties.