## Department of Mathematics

### Introduction

The Department of Mathematics at Nanjing University originated from the mathematics department at the National Central University and the mathematics department at Jingling University. The mathematics department at Central University can be further traced back to the mathematics department at Nanjing Higher Normal School founded in 1920 by Professor Xiong Qinglai.

In 1952, the department was formally established under the name of the Department of Mathematics and Astronomy, Nanjing University, through the readjustment of universities and colleges in China.

In 1958, the astronomy faculty was separated from the department and set up the new department of astronomy, while the Department of Mathematics and Astronomy changed its name to the Department of Mathematics.

In 1978, the faculty of computer technology was also separated from the department and established the new department of computer sciences. Since then, the structure of the Department of Mathematics has remained the same.

The department has been well-known for its long history, its strength, and its rigorous scholarship. Since its establishment, many outstanding mathematicians have worked here. It has always had a leading status at home and has been well-known abroad for its academic influences, especially in the areas of dynamic system, differential equations, number theory and K theory, and scientific and engineering computing.

The department currently has three undergraduate specialties, namely, the Mathematics and Applied Mathematics (a Characteristic Specialty Construction Point of the State Ministry of Education), Information and Computing Sciences, and Statistics.

At present, the department is a national training base for basic science research and teaching talents and is a Jiangsu advantaged subject with two grade-one subjects (mathematics is the national grade-one key subject and statistics is the Jiangsu grade-one key subject), and has ascended into the highest 1% in the ranking of ESI global subjects for its mathematics.

### Features and Highlights

The Department of Mathematics has a strong academic faculty of 76 members, including 34 full professors, 31 associate professors, 26 doctoral supervisors, 2 persons elected into the “1,000-Talent Program,” 3 Cheung Kong professors, 7 winners of the National Science Fund for Distinguished Young Scholars, 1 winner of the National Science Fund for Excellent Young Scholars, 2 persons elected into the National Talent Program, 2 winners of the Fund for Rewarding Young Teachers of the Ministry of Education, 8 (Cross-Century) New-Century Excellent Talents of the Ministry of Education, and 1 Innovation Team of the Ministry of Education.

Altogether, 96.8% of the basic and specialized courses are taught by professors and associate professors.

The faculty members have obtained more than 30 various awards in recent years, including the high-grade awards such as the second prize of the National Natural Science, Chenxing Prize in Mathematics, Qiushi Outstanding Youth Scholars Award, first prize of Natural Sciences of Universities and Colleges of China, and first prize of Scientific and Technological Advancement of Jiangsu Province.

More than two thirds of the faculty members are leaders of national-level scientific research projects, and at present, they take charge of nearly 70 on-going national key projects including 973 Program as well as outstanding, key and general programs of the National Natural Science Foundation of China, equivalent to one project per person, ranking among the top mathematics departments in universities and colleges in China.

The faculty members devote themselves to teaching, original research and making high-quality achievements. In recent years, they have published large quantities of important papers in influential international journals on mathematics. Especially, in 2011 and 2012, they published three papers in internationally top-ranking mathematics journal Inventiones Mathematicae. The research of the faculty has produced significant influence in the international mathematical society. For example, Professor Cheng Chongqing was the invited speaker at the 2010 International Congress of Mathematicians (ICM 2010).

The department has established highly efficient mechanisms for academic exchange and international cooperation and has strengthened talent training and academic exchange by means of “inviting in and dispatching out.” Every year, it invites nearly 100 preeminent scholars from at home and abroad (including winners of Fields Medal, lifetime professors of world top-ranking universities, and academicians of Chinese Academy of Sciences) to visit and give lectures at this department, and dispatches dozens of excellent undergraduate students as exchange students to study at partner schools overseas.

The department uses diversified methods of talent training. It recruits new undergraduate students by the general category of mathematics and does not stipulate their specialty in the first two years, but strengthens the basic training for them. In the third and fourth years, it divides them into different specialties according to the different types and characteristics of different students. In recent years, the department has vigorously carried out the reform of the graduate programs and changed the recruitment of doctoral students from the unified examination to an independently implemented “application-assessment” system.

The department has won 1 first prize of Jiangsu Award for Teaching Achievements and 1 second prize of Jiangsu Award for Excellent Teaching Achievements. It has been titled the Excellent Teaching Team of Universities and Colleges of Jiangsu once and has 1 course titled the National High-Quality Course. It has compiled 5 “oriented at the 21st century” teaching materials, 2 teaching materials under the 11th Five-Year National Planning for Common Higher Education, 1 key teaching material for higher education schools of Jiangsu Province during the 12th Five-Year Plan period, and it has compiled a total of more than 40 teaching materials.

At each session of China Undergraduate Mathematical Contest in Modeling and Mathematical Contest in Modeling (MCM), the department makes outstanding achievements.

The department has many doctoral dissertations that won the title of Provincial Excellent Doctoral Disseretations of the Province, and 1 of them won the title as one of the 100 Doctoral Disseretations of the Country.

Over the years, around 30% of the master’s degree candidates from the department have been admitted to the doctoral program in advance, around 10% of them have been admitted to doctoral programs in other universities at home and abroad, and around 60% of them have earned their degree and obtained employment with satisfaction. The department has obtained many achievements in the talent training, and one of the benchmarks is the 100% employments rate of its graduates. These graduates have been making important contributions to China’s economic and cultural development in almost all academic, business, and industrial fields.

These graduates have found favor with employers for their solid professional foundation, good observation ability, abstract thinking ability and creative and development potentials. Meanwhile, more than 70% of the graduates continue their study at the world’s top-ranking universities abroad such as Harvard, Stanford, Berkeley, Columbia, Cambridge, as well as well-known universities within China.

### Important research directions

#### Dynamic system

The team of Hamiltonian Dynamics is internationally well-recognized. In recent years, some breakthrough progress has been made by faculty members towards the solution of some notable problems such as Arnold diffusion which, because of its importance, was intensively studied by many top-level mathematicians. The faculty shall continue its efforts to develop the team of dynamical system, not only for Hamiltonian dynamics, but also for differentiable and complex dynamics, so that it will become one of the international centers for dynamical systems.

#### Algebraic number theory and K-theory

A central subject of number theory is the study of L-functions of number fields. The famous Riemann Hypothesis is about the distribution of non-trivial zeros of the zeta-functions. The values at the integers of an L-function of a number field contain information of many arithmetic invariants of the field, some of which are closely related to the K-groups of the ring of algebraic integers of the field. Hence there exists deep connection between algebraic number theory and K-theory. In recent years, K-theory has also played an essential role in the proof of the BSD conjecture for function fields.

The faculty will continue its in-depth research in related topics, including representations of quadratic forms, elliptic curves, arithmetic algebraic geometry, dynamical system, Euler system, Mahler measure, and structures of various K-groups. It will focus on the mainstream of the international mathematical developments and major problems, such as the Lang-Trotter conjecture, the dynamical Mordell-Lang conjecture of quasi-projective varieties, the Beilinson conjecture, the Coleman-Oort conjecture, and the relation between the algebraic K-groups of an elliptic curve over a number field and the values at the integers of the L-function of the curve.

#### Theory of Partial Differential Equations and their numerical methods.

The research at the department has been focusing on the following (but not limited to) areas and some progress has been made.

The phenomena of transonic flows and transonic shocks in fluid dynamics;

Theory of mixed type partial differential equations and degenerate elliptic equations;

Efficient numerical methods for scattering problems with high wave numbers and theirs theoretical analysis, which has been an open problem for several decades;

The combined multi-scale finite element methods (MFEM) for multi-scale problems, which raise the simulation efficiency for some multi-scale problems such as underground water;

Stability, error, and super-convergence analysis for the local discontinuous Galerkin methods with various time discretizations, which has important applications in the fields of computational fluid dynamics; and

The inverse problems of anomalous diffusion equations of fractional order, which has important applications in the fields of material mechanics, biochemistry, medicine, and image processing.

#### Mathematical statistics

The main research directions of the theory and application of modern statistical analysis include the statistical analysis of time series models and related fields, the research on stochastic mathematics [statistics for stochastic processes, stochastic process and network, stochastic analysis and stochastic (ordinary/partial) differential equations, optimization on stochastic process and stochastic optimal control] and its cross applications in information science and financial management, spatial statistical analysis, the statistical analysis of Bayesian econometric models, and small value probability of branching processes.

Some research achievements have been published in international top-ranking journals and have produced great significances and influences in the international arena. Such research has important theoretical and application values in the fields of biology, computer science, medicine and health, hydrogeology, environmental science, ecology, forestry science, telecommunications, economy, finance, electronic commerce, and physics and control science.

Along with the development of economy and science and technology, high-dimensional big data have emerged gradually in these fields, their statistical analysis has attracted increasingly close attention, and the research prospect is very broad. Related research achievements may be applied to the analysis of socio-economic data, such as futures, foreign exchange rate, stock market returns, the unemployment rate, electronic commerce, forestry resources, remote sensing monitoring, and earthquake center distribution. By establishing reasonable statistical models, the faculty may adjust the distribution of fishery resources, control the spreading scope of infectious diseases, predict or control the risk and development of financial markets.

#### Mathematical programming and optimization methods

Mathematical programming is a branch of operational research, and its research object is the arrangement and dispatching of related work in plan management, namely, searching for an optimal scheme according to certain measuring index under some given conditions. Mathematically, it can be expressed as calculating the maximum or minimum value of a real-valued function under certain constraints. It mainly concerns the mathematical nature, solution methods and computer execution of these problems. Mathematical programming includes many branches such as linear programming, nonlinear programming, multi-objective programming, dynamic programming, parametric programming, integer programming, stochastic programming, and variational inequality and complementarity problem. It has important applications in industry, commerce, agriculture, traffic and transportation, and governmental. It is a powerful instrument in the fields of economic programming, system engineering, and modern management.

#### Mathematical logic and theoretical computer

The main research direction of mathematical logic is recursion theory (also called as computability theory), set theory and their crossing fields. In terms of recursion theory, the faculty focuses on the global structure research on the degree of unsolvability and its application to theoretical computer science. In terms of degree theory, it mainly focuses on model theory properties of the degree of unsolvability. In terms of applications, it focuses on the research on the recursion theory to the algorithmic randomness theory. In terms of set theory, it focuses on effective descriptive set theory and inner model, as well as the research of applications of recursion theory method to set theory.

Over the past 50 years, explosive development has been achieved in recursion theory, and it is embodied not only at the constant expansion of intrinsic contents, but also at the plentiful applications in other fields. The global structure of degree theory has become clear gradually. The automorphism problem is still pending, but elementary substructure problem has been solved basically, and the faculty has made important contributions to the tackling of this problem. In terms of the algorithmic randomness theory, the faculty has realized the most successful application of recursion theory to theoretical computer, and the problem of lowness for randomness has nearly been solved thoroughly.

Now, the faculty is gradually turning to the applications to analysis and dynamic system, especially the ergodic theory in the algorithmic theory. In terms of set theory, it has formed a relatively characteristic research field, which is mainly about the research of recursion theory and algorithmic randomness by applying forcing method and constructibility. For example, higher randomness, chains and antichains are all the new research fields it has gradually found out, and it has attracted the following of many international young prominent scholars.

#### Geometry and topology

Geometry and Topology are two important branches which are mutually independent and closely related in modern mathematics, and they take differential manifolds and topological invariants as research objects. Many problems in geometry and topology attract generations of mathematicians. For example, the famous problems include Poincare conjecture and Borel conjecture. Over the past half a century, much great progress has been made in research on modern geometry and topology, and the progress has aroused research on plentiful new important theoretical problems.

The faculty plans to enrich the academic force of each branch of geometry and topology through many years of efforts, combine analysis, geometry and topology, and solve important problems in inter-disciplines, and we plan to conduct research on curvature and topology, global Riemannian geometry, complex geometry, symplectic geometry, toric topology, topological group and general topology theory.

#### Algebraic combination and additive combinatorics

Along with the rise and vigorous development of discrete mathematics, the cross agglomeration of combinatorics with algebra and number theory has given a strong impetus to the settlement of some significant mathematical problems. The famous Szemeredi Theorem and Green-Tao Theorem are just the outcomes of the cross penetration of combinatorics, number theory and analysis. The latest significant breakthrough on the Twin Prime Conjecture by Zhang Yitang, Maynard and Tao are also benefited from the glossy combination of number-theoretic tools and combinatorial arguments, and combinatorics also plays a key role in the important progress made by Tao, et al. recently on the problems concerning big gaps between successive prime numbers. Algebraic combinatorics and additive combinatorics is a popular modern branch in combinatorics, and it involves researching combinatorial properties of algebraic structures as well as combinatorial problems with algebraic tools. The important topics in this field include the combinatorial properties of primes, Ramsey-type problems, cyclic permutations, restricted sum sets over fields, and zero-sum problems in abelian groups. Our work in this aspect has been quoted by some famous mathematicians like Tao and Alon. On the basis of the existing work, the faculty plans to further study the related front-edge problems with advanced tools from number theory, combination, algebra, analysis, probability, and ergodic and strive to obtain influential significant results.

### Contact

Department of Mathematics, Gulou Campus, No. 22, Hankou Road, Gulou District, Nanjing, Jiangsu Province

Tel: +86-25-83593301

Email: ndmaths@nju.edu.cn

For more information visit the school's website.