中国数学会2020年学术年会报告摘要10-10

中国数学会2020年学术年会会议报告摘要目录大会邀请报告 (1)TM A 代数与数论 (3)TM B 几何与拓扑 (9)TM C1 常微动力统 (15)TM C2 偏微分方程 (19)TM C3 实分析和复分析 (27)TM D 计算数学 (32)TM E 概率和统计 (48)TM F 运筹与控制 (53)TM G 组合与计算数学 (57)TM H 数学教育 (62)大会邀请报告机器学习的数学理论鄂维南北京大学现代机器学习的核心问题是怎样有效地逼近一个高维空间的函数。

传统的逼近论方法会导致维数灾难，这是对许多领域来说困惑了我们多年的问题。

在这个演讲里，我们将介绍以下几方面的内容。

1.怎样建立起一个数学理论？这里的问题本身跟传统的数值分析基本一样。

不同的是机器学习需要处理的核心问题是维数灾难。

所以我们需要建立起一个高维数值分析理论，包括逼近论，先验和后验误差估计，优化理论等。

这个理论会帮助我们理解什么样的模型和算法没有维数灾难。

2.怎样formulate一个好的机器学习的数学模型？正确的方法是首先在连续的层面formulate好的机器学习的模型，然后采用数值分析的想法，对这些连续模型作离散化而得到所需要的机器学习算法。

我们发现许多神经网络模型，包括残差网络模型，都可以通过这种途径得到。

因为有一个好的连续模型作为背景，这样得到的机器学习模型和算法自然就有比较好的性质。

3.实际应用中有一些比较奇怪的现象，比方说double descent。

怎样解释这些现象？实际应用中人们也没有按照前面所说的套路来做，那为什么其效果也还很好呢？4.哪些问题还有待解决？数学机械化新进展高小山中国科学院数学与系统科学研究院数学机械化主要研究如何借助计算机进行定理自动证明与发现，是我国数学家吴文俊教授创立的的重要研究方向。

本报告将简要介绍数学机械化的基本想法与若干新进展：微分Chow坐标与稀疏微分结式。

1Weierstrass 型函数图像的分形性质沈维孝复旦大学Weierstrass 处处不可微连续函数的图像是分形几何的经典研究对象之一.我们将回顾这些函数图像的研究历史,并介绍近期如下二分性定理(与任浩杰合作)的证明:假设b 是大于1的整数,f 是周期的解析函数,0<t <1.那么要么函数W (x )=∑︀∞n =0b−nt f (b n x )是解析函数,要么其图像的Hausdorff 维数等于2−t .2分组报告TM A代数与数论Some results about finite p-groups安立坚山西师范大学In Shanxi Normal University,there is a team of theorists who are interested in finite p-groups mainly.This team began from2003.It has8members now.In this talk,we present some results and progresses about finite p-groups obtained by this team.This talk contains four aspects:commutativity,normality,anzahl and lattice of subgroups.On the aspects of commutativity and normality,we classified some finite p-groups with“stronger”commutativity or normality.On the aspect of anzahl,we investigate a conjecture proposed by Chinese late mathematicians Loo-Keng Hua and Hsin-Fu Tuan.On the aspect of lattice of subgroups,we concentrate on Chermak-Delgado lattice.Twisted quantum affinizations陈福林厦门大学In this talk,I will introduce a notion of twisted quantum affinization of an arbitrary quantum Kac-Moody algebra and discuss some of its algebraic properties,including the triangular decomposition,the(topological)Hopf algebra structure,the specialization and etc.Constructions and applications of recollements of derivedmodule categories陈红星首都师范大学3In1982Beilinson,Bernstein and Deligne introduced recollements of triangulated categories in contexts of the derived categories of perverse sheaves over singular spaces, which provide a derived version of Grothendieck’s six functors on abelian categories. Since then,they have been used in algebraic geometry and topology and recently in representation theory.For instance,Happel used recollements of bounded derived mod-ule categories to establish reduction techniques for homological conjectures.To produce recollements of triangulated categories,there are many methods available.For instance, one may take stable categories of Frobenius categories or quotient categories by smash-ing subcategories.But,to construct recollements of derived module categories of rings seems to be a difficult question and little was known about it before2010.A known ex-ample is the Cline-Parshall-Scott construction from hereditary ideals,or more generally stratifying ideals of rings.In this talk,I will report two general methods to construct recollements of derived module categories of rings from infinitely generated tilting modules and homological exact contexts,based on a series of joint work with Changchang Xi.These recollements have many applications in algebraic representation theory and algebraic K-theory.For example,they were applied to establish estimation inequalities of finitistic dimensions of rings and also infinite Mayer-Vietoris sequences of higher algebraic K-groups of rings.Gorenstein flat representations of left rooted quivers狄振兴西北师范大学We study Gorenstein flat objects in the category Rep(Q,R)of representations of a left rooted quiver Q with values in Mod(R),the category of all left R-modules,where R is an arbitrary associative ring.We show that a representation X in Rep(Q,R)is Goren-stein flat if and only if for each vertex i the canonical homomorphismϕXi :⊕a:j→i X(j)→X(i)is injective,and the left R-modules X(i)and CokerϕXi are Gorenstein flat.As anapplication of this result,we show that there is a hereditary abelian model structure on Rep(Q,R)whose cofibrant objects are precisely the Gorenstein flat representations, fibrant objects are precisely the cotorsion representations,and trivial objects are pre-cisely the representations with values in the right orthogonal category of all projectively coresolved Gorenstein flat left R-modules.The integral form of U(̂︀gl n)and U(gl n[t])4付强同济大学In the seminal work,Beilinson–Lusztig–MacPherson gave a beautiful realization for quantum gl n via a geometric setting of quantum Schur algebras.In particular they give a geometric construction of canonical bases for the integral form of modified quantum gl n.We will investigate the integral form of U(̂︀gl n)and U(gl n[t]).In particular we will construct canonical bases for the integral form of modified quantum̂︀gl n and modifiedquantum gl n[t].Basepoint-freeness of primitive line bundles on abelian varieties江智上海数学中心/复旦大学Given an ample line bundle L on abelian varieties,it is well-known that L⊗2is basepoint free and L⊗3is very ample.In general,it is difficult to tell when L is basepoint free.In this talk we will present some recent progress on this topic.V-topologies and beyondWill Johnson复旦大学A V-topology is a field topology induced by a valuation ring or an absolute value. V-topologies play an important role in number theory and algebra.A more general class of“finite weight”field topologies naturally arises from problems in mathematical logic. Weight-1topologies are equivalent to V-topologies.The picture in higher weights is unexpectedly complicated,but a classification may be possible.On the algebraic side, there is a class of“weight n”integral domains generalizing valuation rings.We will discuss the basic theory,examples,and open problems.Geometry of moduli space of hyper-K¨a hler varieties李志远上海数学中心/复旦大学5The study of moduli space of hyper-K¨a hler varieties have received a lot of attentions from different aspects in the past decades.In this talk,I will survey the recent progress in this area,especially for K3surfaces.I will explain an important interaction between the geometry of such moduli spaces and arithmetic properties of automorphic forms.There are a lot of useful consequences and I will focus on some fundamental problems,such as Noether-Lefchetz conjectures,Tautological conjectures and Effective cone problems.Cells in affine q-Schur algebra罗栗华东师范大学We introduced the notion of affine q-Schur algebras of arbitrary type and devel-oped their algebraic and geometrical approaches toward canonical bases.The cells and asymptotic forms for these q-Schur algebras are studied.These results are generaliza-tions of those for affine Hecke algebras due to Lusztig.It is joint work with Weideng Cui and Weiqiang Wang.丢番图逼近中上极限集的分形维数王保伟华中科技大学由一列球或矩形确定的上极限集是丢番图逼近中两类基本的集合,一个源自于Dirichlet定理另一个源自于Minkowski定理.由球确定的上极限集的度量理论已经非常丰富/完备,然而由矩形确定的上极限集的研究却非常滞后,甚至一些基本问题都尚未完全解决.在此报告中,通过引入矩形的无处不在性/满测性,我们确定了由矩形生成的上极限集的Hausdorff理论的一般原理.函数域上的一个Siegel-Weil公式熊玮湖南大学6Siegel-Weil公式是一个等式,它说的是在很多情况下Eisenstein级数和Theta函数的积分是相等的.在这个报告中,我们将介绍函数域上的一个Siegel-Weil公式.这个公式是A.Weil在1964年建立的数域上的Siegel-Weil公式在函数域上的一个类比.我们将首先介绍下Siegel-Weil公式的雏形及一些例子,然后介绍下C.L.Siegel的经典结果,再介绍下Weil用表示论及Adele的语言对Siegel工作的推广,最后介绍下函数域上的结果.Periods,global packets,and L-functions许宾四川大学The central values of L-functions play an important role in the study of automorphic representations.For example,it is closely related to the periods of automorphic forms via the Gan-Gross-Prasad conjectures,and is crucial for the non-vanishing of global theta lifts.In this talk,we will recall some relations among global Arthur(Vogan)packets, automorphic periods,and automorphic L-functions.Then we introduce a new approach to show that,for a cuspidal representation of PGL2having a quadratic twist with root number+1,there exist infinitely many quadratic twists with non-zero central L-values.The new approach is based on concrete constructions of automorphic representa-tions.A bound of permutation resolutions of modules for some blocks徐行忠湖北大学Recently,Balmer and Benson introduce a kind of resolution whose objects are permutation modules,and those resolutions are finite.In this talk,we recall these resolutions and try to give a bound of these permutation resolutions for some module in some given blocks.This is a jointed work with H.Liu and J.Zhang.Free idempotent generated semigroups:maximal subgroups, general structures and word problems杨丹丹西安电子科技大学7The set of idempotents of any semigroup carries the structure of a biordered set, which contains a great deal of information concerning the idempotent generated subsemi-group of the semigroup in question.This leads to the construction of a free idempotent generated semigroup IG(E)–the‘free-est’semigroup with a given biordered set E of idempotents.It was thought from the1970s that all maximal subgroups of IG(E)would be free,but this conjecture was false.In this talk,I will present my joint work with my collaborators on this topic,including maximal subgroups,general structures and word problems of IG(E).Construction of Gelfand-Tsetlin modules for gl n张健华中师范大学A classical paper of Gelfand and Tsetlin describes a basis of irreducible finite di-mensional modules over the Lie algebra gl n.This is one of the most remarkable results of the representation theory of Lie algebras which initiated a development of the theory of Gelfand-Tsetlin modules.The Gelfand-Tsetlin modules form the largest subcategory of gl n-modules where there is some understanding of irreducible modules.The main remaining problem is how to construct explicitly these modules.We propose a new effective method of constructing explicitly Gelfand-Tsetlin modules for gl n and obtain a large family of irreducible modules that have a basis consisting of Gelfand-Tsetlin tableaux and the action of the Lie algebra is given by the Gelfand-Tsetlin formulas.As an application of our construction we prove necessary and sufficient condition for the Gelfand and Graev’s continuation construction to define a module which was conjec-tured by Lemire and Patera.The talk is based on joint results with Vyacheslav Futorny and Luis Enrique Ramirez.8TM B几何与拓扑Regularity for Dirac-harmonic maps into Lorentzian manifolds艾万君西南大学Dirac-harmonic maps are motivated by the supersymmetric nonlinear sigma model in quantum field theory,which generalize the classical harmonic maps and harmonic spinors.In recent decades,the existence,regularity and blow-up analysis of Dirac-harmonic maps from a Riemann surface to another compact Riemannian manifold have been extensively studied.In this presentation,we will start from classical regularity results of spherical harmonic maps and Lorentz spherical harmonic maps,and then show some generalizations of these results for general target manifolds,both in Riemannian and Lorentzian cases.Finally,we present a new result about the regularity of Dirac-harmonic maps from a Riemann surfaces into a stationary Lorentzian manifolds.It turns out the regularity of weakly Dirac-harmonic maps depends on a general regularity theorem of critical elliptic systems without an L2-antisymmetric structure.Our results generalize the corresponding regularity results of H´e lein,Rivi`e re and Rivi`e re–Struwe for harmonic maps.This is joint work with Zhu,Miaomiao.Shifted derived Poisson manifolds associated with Lie pairs陈酌清华大学I will talk about the shifted analogue of the“Lie-Poisson”construction between L∞algebroids and shifted derived Poisson manifolds via the example of a Lie algebroid pair(L,A).We show that the pullback of the normal bundle L/A over the dg manifold (A[1],d A)is an L∞algebroid,thus the space totΩA(∧L/A)admits a canonical degree (+1)derived Poisson algebra structure with the wedge product as associative multi-plication and the Chevalley-Eilenberg differential d BottA as the unary L∞bracket.As aconsequence,its Chevalley-Eilenberg hypercohomology admits a canonical Gerstenhaber algebra structure.If time were permitted,I will explain that this degree(+1)derived Poisson algebra structure can also be recovered from Fedosov dg Lie algebroid of this Lie pair and from the Dirac deformation of the associated Courant algebroid,respectively. This is a joint work with R.Bandiera,M.Sti´e non,and P.Xu.Recent progress in symplectic birational geometry杜承勇四川师范大学In this talk,we first introduce the backgrounds of and basic notions in symplectic birational geometry,and then we review some recent progresses of the symplectic bira-tional invariance of symplectic uniruledness and some results on symplectically rational connectedness.Part of the results are based on joint work with Bohui Chen,Jianxun Hu and Rui Wang.Maxwell-Einstein度量关庄丹河南大学In every K¨a hler class of a compact almost homogeneous manifold with two ends we found an unique Calabi extremal metric earlier.Moreover,we found an unique extremal metric in a given K¨a hler class on certain C P1bundle if certain function there is positive. We realized later on that this is equivalent to the geodesic stability of the K¨a hler class. In this talk,we shall prove that for these kind of manifolds,there is another natural K¨a hler metric in the given K¨a hler class,which is called the Maxwell-Einstein metrics. This kind of metrics came from physics and in the efforts of finding Hermitian-Einstein metrics.Recently,it became a hot topic in the K¨a hler geometry.It is worked by several famous mathematicians as Le Brun and Futaki.Our solution gave an answer to a central problem people tried to obtain in series of many papers.曲率有下界球面区域的刚性来米加上海交通大学在本报告中,我将汇报两个有关球面区域的几何刚性定理:一个是在里奇曲率有下界时,由Clifford环面界定的球面区域的刚性;另一个是在Q曲率有下界时半球面的刚性.后者是和上海数学中心韦韡合作的.Recent progress on eta invariants and eta forms刘博华东师范大学In1975,Atiyah-Patodi-Singer developed an index theory for the Dirac operator on compact manifolds with boundary.Their index formula involves a contribution of the boundary,called the eta invariant.In1989,Bismut-Cheeger extended the eta invariant to the family case,called eta form,which is the boundary contribution of the family index theorem with boundary.In this talk,we will discuss the recent progress on eta invariants and eta forms.Some results in this talk are based on works jointed with Xiaonan Ma.凸几何中的赋值理论马丹上海师范大学在1872年德国著名数学家Klein提出了Erlangen纲领之后,研究和刻画在变换群作用下不变的几何量一直是几何学研究的核心.赋值则是几何量通常满足的另一性质.经典的赋值Z是定义在K n(n维欧氏空间中的凸体空间)上取值于阿贝尔半群的一类满足以下性质的映射Z(K)+Z(L)=Z(K∪L)+Z(K∩L),其中,K,L,K∪L∈K n.凸几何中的赋值理论在积分几何、泛函分析、微分几何、随机几何、信息论、体视学、图像分析和物理学中均有广泛的应用.本报告将介绍一些列凸几何中的赋值刻画结果,包括Sobolev空间上的L p范数,凸体空间上的Laplace 变换,凸多胞形空间上的矩向量、矩矩阵和LYZ矩阵.Projective embedding of pairs and K-stability孙京洲汕头大学Given a smooth polarized Riemann surface(X;L)endowed with a hyperbolic metric with cusp singularities along a divisor D,we show the L2projective embedding of(X;D) defined by L k is asymptotically almost balanced in a weighted sense.We also show its generalization to the higher dimensional case of projective completion of a positive line bundle over a smooth projective manifold.Topological data analysis and topological approaches to drugdesign and discovery吴杰河北师范大学In this talk,we will report our current research on topological data analysis.The talk will consist of three sections.In the first section,we give a brief introduction to topological data analysis.Then,in the second section,we will give a report on the current progress of topological approach to drug design and discovery in the world.In the last section,we give a report on our works.度量黎曼几何近年来的新发展胥世成首都师范大学从上世纪七、八十年代以来，几何分析与度量黎曼几何得到了蓬勃的发展。