Algebraic geometry：

Introduction：

Algebraic geometry is a core mathematical research subject, originating from studying geometric problems via algebraic method. Its primary research objects are algebraic varieties.

Current algebraic geometry has three main branches, classifications of varieties, their algebraic sub-objects, and the moduli of algebraic structures.

Algebraic geometry faculties in algebraic geometry affiliated to Shanghai Center for Mathematical Sciences are working on the research directions:

Birational geometry of high dimensional varieties;

Boundedness of Fano/Calabi-Yau varieties;

Degeneration method in algebraic geometry;

Geography of high dimensional varieties;

Geometry of Abelian varieties;

Geometry of hyperkahler manifolds;

Geometry of Shimura varieties;

Gromov-Witten invariants and mirror symmetry conjecture;

Hodge theory and its geometric applications;

Minimal model program;

Moduli space of algebraic varieties;

Moduli space of vector bundles over algebraic surfaces;

Faculty:

韩京俊（博导）Birational geometry, minimal model program

江辰（博导）Birational geometry, Minimal model theory, Fano varieties, Calabi-Yau varieties, singularities

江智（博导）Birational Geometry, Generic Vanishing, Syzygies, Abelian Varieties, Hodge theory

李骏（博导）Algebraic Geometry

李志远（博导）Algebraic geometry, arithmetic geometry

周杨 （博导）moduli spaces, enumerative geometry, Gromov-Witten theory

Algebra, Number Theory, Representation Theory：

Introduction：

Noncommutative Algebra

Noncommutative algebras are associative algebras which are not necessarily commutative. Examples are algebras from operator point view (say, rings of differential operators), from representation theory (say, group algebras), cohomology rings from topology, Hopf algebras/quantum groups.

Noncommutative algebra faculties in SCMS are working in the following areas:

Geometrically motivated homological properties;

Number Theory

Number theory is the study of the integers and related objects. This topic has advanced rapidly in the recent years and has the feature of unifying numerous main branches of mathematics. Especially the so-called Langlands program is a series of conjectures that connects number theory with representation theory and algebraic geometry.

Number theory faculties in SCMS are working in the following areas:

Geometry of Shimura varieties and Rapoport-Zink spaces;

Geometry of p-adic period domains;

The Birch-Swinnerton-Dyer conjecture and its generalizations;

Representation Theory

Representation theory is a core mathematical research subject, originating from studying the symmetry of spaces. Its primary research objects are representations, group homomorphisms into spaces of linear operators.

Current representation theory centrals around the Langlands program.

Representation theory faculties in representation theory working on those research directions:

Representations of reductive groups;

Representations of Hecke algebras;

Gan-Gross-Prasad conjectures;

Branching laws and restriction problems;

Representations of general linear groups;

Faculty:

王海宁（硕导）Number theory

吴泉水（博导）Algebra

陈苗芬 Number theory

Computational Systems Biology

Introduction：

Computational systems biology is an interdisciplinary field that combines different approaches from mathematics, physics, biology and other disciplines to understand how biological systems function. Computational systems biology uses tools including mathematical modeling, stochastic analysis, and dynamical systems to build models of complex biological systems. There are two major branches in this field, quantitative data-driven approaches and dynamical model-driven approaches, both can be applied to study the underlying mechanism of complex biological processes, such as development, tumor and brain disease.

Faculties in computational systems biology affiliated to Shanghai Center for Mathematical Sciences are working on the research directions:

Mathematical biology;

Cell fate decisions;

Computational neuroscience;

Computational biophysics;

Brain science;

Data mining;

Gene regulatory network modeling;

Stochastic and nonlinear dynamics;

Single cell data analysis;

Computational psychiatry;

Biophysical models of single neuron and neuronal networks;

Faculty:

冯建峰（博导）Computational Biology, Stochastic and Nonlinear Dynamics, Data Mining and Mathematical Physics

李春贺（博导）Computational Systems Biology; Biophysics; Computational Neuroscience

Combinatorics, Graph Theory

Introduction：

Combinatorics is an area of mathematics concerns the general study of discrete objects. It grows rapidly in the past decades due to the broad connection with information technology and other modern sciences.

There are several main branches in Combinatorics, including Enumeration Combinatorics, Graph Theory, Extremal combinatorics, Algebraic Combinatorics, Probablistic Combinatorics, etc.

The Faculties in Combinatorics affiliated to Shanghai Center for Mathematical Sciences are mostly working on Structural Graph Theory, Extremal Combinatorics, Probalistic Combinatorics, in particularly interest in the following subjects:

Graph Coloring, vertex partition and its variance;

Longest cycle problems in Graphs;

Graph Embedding Problem;

Graph Packing Problem;

Extremal problems in Combinatorics, including Ramsey type or Turan type problems on graphs or set systems;

Application of probabilistic, algebraic and topology method in Combinatorics;

Combinatorial Optimization;

Faculty:

吴河辉（博导）Structural Graph Theory & Extremal Combianatorics

Data Science and Machine Learning

Introduction：

Data Science and Machine Learning is a growing applied math subject rooted in statistics, information theory, optimization, probability theory, control theory, stochastic processes, stochastic geometry, combinatorics, etc. Currently focuses on the following topics:

Optimization with an emphasis on stochastic zeroth-order optimization and stochastic gradient/Hessian estimation.

Stochastic decision making with an emphasis on Multi-Armed Bandits in metric spaces.

Real-world applications of data science principles and machine learning algorithms.

Faculty:

王天宇（硕导） Machine learning，mathematics of data science

Dynamical Systems：

Introduction：

Dynamical systems theory is an active area of pure mathematics. There are intimate interactions between dynamics and other branches of mathematics such as number theory, mathematical physics and geometry, etc. In the study of dynamical systems people try to understand the orbits of points, with emphasis on long term behaviours, under a map, a flow, a semigroup or a group. The directions of dynamical systems include ergodic theory, topological dynamics, complex dynamics, differentiable dynamics, Hamiltonian dynamics, homogeneous dynamics, Teichmüller dynamics, etc. Faculty members in dynamical systems affiliated to the Shanghai Center for Mathematical Sciences are working on the research directions:

Complex dynamics;

Low dimensional dynamics;

Homogeneous dynamics;

Dynamics related to fractal geometry;

Dynamics related to number theory;

Dynamics of group actions;

Faculty:

沈维孝（博导）Dynamical systems (in particular, real and complex one-dimensional dynamics)

石荣刚（博导）Homogneous Dynamical, Ergodic Theory, Number Theory, fractal geometry

Geometry, topology and geometric group theory

Introduction：

Differential geometry is a core mathematical research subject, originating from studying geometric problems via analytic method.

Current differential geometry has three main branches, classifications of Riemannian manifolds, complex manifolds, and submanifolds.

Faculties in differential geometry affiliated to Shanghai Center for Mathematical Sciences are

working on the research directions:

Riemannian geometry;

Kähler geometry and non-Kähler geometry;

Minimal submanifolds;

Geometric flows;

Geometric measure theory;

Topology is a core mathematical research subject,  studying properties of spaces invariant under continues deformations. Its primary research objects are manifolds and other related spaces. Current topoligy has three main branches, classifications of manifolds, classification of homotopy types, and the foundation of mathematics with higher structures.

Faculties in topology affiliated to Shanghai Center for Mathematical Sciences are working on classification of higher dimensional manifolds, computation of homotopy groups, motivic homotopy theory, topological cyclic homology, algebraic K theory.

Geometric group theory is a fast-growing area of mathematics. It situates at the intersection of low-dimensional topology, hyperbolic geometry, algebraic topology,  group theory, differential geometry and dynamical systems. It analyzes connections between the algebraic properties of the groups and their geometric and topology properties. Faculty in geometric group theory at Shanghai Center for Mathematical Sciences work on finiteness properties of groups, Artin groups and big mapping class groups.

Faculty:

丁琪（博导）Differential geometry, Geometric analysis, Partial differential equations, Geometric measure theory

傅吉祥（博导）Differential geometry, Complex geometry

卿于兰（硕导）Geometric Group Theory, Low Dimensional Topology

王国祯（博导）Algebraic topology

伍晓磊（博导）Geometric group theory, geometric topology, algebraic K-theory

王志超（博导）Geometric analysis, Minimal surfaces

Noncommutative Geometry：

Introduction：

Noncommutative geometry (NCG) studies noncommutative algebras as geometric objects, often employing techniques from functional analysis, algebraic topology, group theory, dynamical systems, and many other areas of mathematics. A noncommutative algebra, i.e., one where xy=yx may not hold, may be viewed as the algebra of functions on a "noncommutative space". NCG has been successfully applied to solve major problems in differential topology and differential geometry and is also closely related to geometric group theory, dynamical systems, number theory, and mathematical physics, among others.

The noncommutative geometers at SCMS are working on the following research directions: K-theory, higher index theory, the Novikov conjecture, topological and geometric rigidity, large-scale geometry, positive scalar curvature, C*-algebras, topological and C*-dynamical systems, etc.

Faculty:

吴健超（博导）Operator Algebras, Noncommutative Geometry, K-theory, Coarse Geometry, and Topological Dynamics

郁国樑（博导）

Operations Research and Control Theory

Introduction：

Operations research and control theory is an important subject of mathematical sciences, deals with the development and application of advanced analytical methods to improve decision-making. Control theory deals with the control of dynamical systems in engineered processes and machines. Nowadays, operations research and control theory is increasingly applied to high-dimensional, strongly nonlinear, and multilevel problems, such as machine learning, neuroscience, finance, epidemiology, intelligent manufacturing, autonomous robots, and self-driving cars.

Faculties in operations research and control theory algebraic geometry affiliated to Shanghai Center for Mathematical Sciences are working on the research directions:

Dynamical programming;

Partially observable Markov decision process;

Stochastic optimal control;

Controllability and observability;

Data-driven modelling and control;

Intelligent control;

Reinforcement learning;

Multi-agent systems;

Mean-field control and games;

Financial mathematics;

Faculty:

杜恺（博导）Optimal control, intelligent control, reinforcement learning, financial mathematics

PDE, Inverse Problems

Introduction：

PDEs are ubiquitous across modern science and technology. The research of PDEs is the cornerstone of classical mechanics, electrodynamics, quantum mechanics, thermodynamics, fluid dynamics, general relativity, quantum field theory, financial mathematics, acoustics, optics,

materials science and engineering etc.

From the viewpoint of pure mathematics, primary research objects of PDEs are qualitative and quantitative properties of solutions, including existence, uniqueness, regularity, well-posedness, stability, asymptotic, dynamics, and related a priori estimates.

Inverse problems concern the reconstruction of background information of physical models

from measurable local data. It is the research that makes invisible objects observable from

measurable information. There are applications in physical

sciences and engineering, such as medical imaging, probing the universe and Earth.

The research of inverse problems is at the interface of PDEs, geometry, computational mathematics, and mathematical physics.

The PDE & inverse problems group of the SCMS explores the following topics:

• PDE

Microlocal analysis of variable-coefficient PDEs;

Hyperbolic PDEs and propagation of singularities;

Spectral theory of differential operators;

• Inverse problems

Inverse problems of PDEs arising in mathematical physics;

Regularization theory;

Bayesian inverse problems;

Faculty:

陈曦 Microlocal analysis, Inverse Problems, Spectral Theory, Harmonic Analysis

Probability Theory and Statistics

Introduction：

Probability theory and statistics are important branches of mathematics with major interests in the understanding of random events. Central research objects in probability theory include random variables, probability distributions, and stochastic processes. It lays the mathematical foundation for statistics, wherever later is a scientific discipline that focuses on the collection, organization, analysis, interpretation, and presentation of numerical data. In the era of big data, probability theory and statistics have played crucial roles in various fields which involve quantitative analysis of data, including biology, chemistry, physics, engineering, as well as social sciences.

Probability theory and statistics faculty members in the research group of probability theory and statistics at the Shanghai Center for Mathematical Sciences are working on the following directions:

Stochastic processes;

Extreme value theory;

Random networks;

Time series;

Statistical learning;

Survival analysis;

Non-parametric and semi-parametric statistics;

Longitunal data analysis;

Risk management;

Functional data analysis;

Biostatistics;

Faculty:

王天栋 （博导）Complex Networks, Extreme Value Theory

应志良（博导）Mathematical Statistics

双聘导师

陈猛*（博导）代数几何

袁小平*（博导）动力系统

郭坤宇*（博导）泛函分析

李洪全*（博导）调和分析

吴宗敏*（博导）计算几何

林伟*（博导）神经网络的数学方法与应用、非线性科学、计算生物学

卢文联*（博导）神经网络的数学方法与应用、计算生物学

雷震*（博导）应用偏微分方程

严军*（博导）应用偏微分方程

周忆*（博导）应用偏微分方程

陆帅*（博导）数学物理反问题

苏仰锋*（博导）新型算法、数值线性代数

汤善健*（博导）随机控制理论与数学金融