[Introduction to Mathematics Department]
The Department of Mathematics at Nanjing University is sourced from the Department of Mathematics at Central University and the Department of Mathematics at Jingling University. The predecessor of the Department of Mathematics at Central University was the Department of Mathematics at Nanjing Higher Normal School, which was founded in 1920 by Professor Qinglai Xiong. In 1952, the department was formally established with the name of the Department of Mathematics and Astronomy at Nanjing University through readjustment of universities and colleges in China. In 1958, the astronomy group was separated from the department, and a new department, the Department of Astronomy, was established while the Department of Mathematics and Astronomy changed its name to the Department of Mathematics. In 1978, the computer technology group was also separated from the department, and another new department, the Department of Computer Sciences, was established. Since then, the basic pattern of the Department of Mathematics has been kept the same. The department has been famous for its long history, powerful strength, and rigorous scholarship. Since the establishment of the department, many outstanding mathematicians have worked here, establishing the subject architecture and features of this department, and forming a relatively perfect teaching system. This department has always ranked a leading status at home and been famous at abroad for its academic influences.
The Department of Mathematics currently has three undergraduate specialties, namely, the Mathematics and Applied Mathematics (a Characteristic Specialty Construction Point of the Ministry of Education of China), Information and Computing Sciences, and Statistics, and formed a completed cultivation system of undergraduate and postgraduate students. The department has made a great effort to enhance the national grade-one key subjects and connotation of national basic science research and teaching talents training base with the aims to consolidate and strengthen the existing advantaged subject area, and to drive the development of other areas. The department has formed many research areas with significant influences and distinctive features in China. In particular, it has had relatively great international influences in the areas of dynamic system, differential equations, number theory and K theory, scientific and engineering computing, etc.
At present, the department is a national basic science research and teaching talents training base and 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 first 1% in the ranking of ESI global subjects for its mathematics.
[Information on Features and Highlights]
The Department of Mathematics has a strong academic staff with 76 faculty members, including 34 full professors, 31 associate professors, 26 doctoral supervisors, 2 persons elected into the “1,000-Elite 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. 96.8% of professional basic courses and professional specialized courses are taught by professors and associate professors. The faculty members of the department 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, etc. More than 2/3 of the faculty members are the leaders of national-level scientific research projects, and at present, they take charge of nearly 70 in-research national key projects including 973 Program as well as outstanding, key and general programs of the National Natural Science Foundation of China, equivalent to per capita one project, ranking among the top leading level of the departments in mathematics of domestic universities and colleges.
The faculty members of the department devote themselves to teach, engage in original research, and produce high-quality achievements. In recent years, we have published a large quantity of important research achievements in influential international periodicals related to mathematics. Especially, in 2011 and 2012, we published three pieces of papers in international top-ranking mathematics periodicals Inventiones Mathematicae, and in this aspect, we rank a leading place in China. The research works contributed by our faculty members have produced significant influences in international mathematical society. For example, Professor Chongqing Cheng in dynamic systems was the 45-min invited speaker at the 2010 International Congress of Mathematicians (ICM 2010).
The department has established highly efficient mechanism for academic exchange and international cooperation, and strengthened talent training and academic exchange by means of “inviting in and dispatching out”. Every year, the department invites nearly 100 domestic and overseas preeminent scholars (including winners of Fields Medal, lifetime professors of world top-ranking universities, academicians of Chinese Academy of Sciences, etc.) to visit and give lectures at this department, and dispatches dozens of excellent undergraduate students as exchange students to study at foreign contract-signed schools.
The department executes diversified talent training modes. It recruits undergraduate students by general category of mathematics, does not distinguish specialty in the first two years, but strengthens basic training; in the third and fourth years, it executes split training by mathematics mode and mathematics application mode aiming at the different characteristics and demands of different types of students. In recent years, the department has vigorously implemented reform of postgraduate training, and changed the recruitment of doctors 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 as Excellent Teaching Team of Universities and Colleges of Jiangsu once and has 1 course titled as National High-Quality Course. It has compiled 5 teaching materials oriented at the 21st century, 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 total more than 40 teaching materials. At each China Undergraduate Mathematical Contest in Modeling and Mathematical Contest in Modeling (MCM), the department has obtained outstanding achievements. In recent years, the Department of Mathematics has many doctoral papers titled as Provincial Excellent Papers for Doctor’s Degree, and 1 doctoral paper titled as National 100 Excellent Papers for Doctor’s Degree.
Over the past years, around 30% of the master’s degree candidates in the department have been enrolled in advance for pursuing their doctor’s degrees, around 10% masters studied for doctor’s degree at abroad, foreign or domestic other universities, and around 60% masters have obtained their jobs with satisfactions. The department has obtained many achievements in the talent training. In recent years, the employment ratios of our graduated students have been kept the 100% with the employments in almost all academic, business, and industry sectors, making non-under-estimable contributions in the constructions of material and spiritual civilizations of China. The graduates of the department have found favor with employers for their solid professional foundation, good observation ability, abstract thinking ability and creation & development potentials. Meanwhile, more than 70% of graduates continue to study at world top-ranking universities abroad including Harvard, Stanford, Berkeley, Columbia, Cambridge, as well as domestic well-known universities.
[Introduction to the important subject directions of the Department of Mathematics]
(1) Dynamic system
The team of Hamiltonian Dynamics is internationally well-reconganized. In recent years, some breakthrough progresses have been made by our faculty members towards the solution of some notable problems, such as Arnold diffusion which, because of its importance, was intensively studied by many mathematicians of top-level. We shall continue our efforts to develop the team of dynamical system, not only for Hamiltonian dynamics, but also for differentiable and complex dynamics, so that our department becomes one of the international centers for dynamical systems.
(2) 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. We will continue to conduct in-depth research in related topics, including representations of quadratic forms, elliptic curves, arithmetic algebraic geometry, dynamical system, Euler system, Mahler measure, structures of various K-groups, etc. We will focus on the mainstream of 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, 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, etc.
(3) Theory of Partial Differential Equations and their numerical methods.
Our research has been focus on the following (but not limited to) areas and some progresses have been made. 1) The phenomena of transonic flows and transonic shocks in fluid dynamics; 2) Theory of mixed type partial differential equations and degenerate elliptic equations; 3) Efficient numerical methods for scattering problems with high wave numbers and theirs theoretical analysis, which has been an open problem for several decades; 4) 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, etc; 5) Stability, error, and superconvergence analysis for the local discontinuous Galerkin methods with various time discretizations, which has important applications in the fields of computational fluid dynamics, etc.; 6) The inverse problems of anomalous diffusion equations of fractional order, which has important applications in the fields of material mechanics, biochemistry, medicine, image processing, etc.
(4) 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, etc. Some research achievements have been published in international top-ranking and influential magazines, and produced great significances and influences in international arena. These researches have 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, physics and control science, etc. Along with the development of economy and science & technology, high-dimensional big data have emerged gradually in these fields, their statistical analysis has aroused increasing 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, earthquake center distribution, etc. By establishing reasonable statistical models, we may adjust the distribution of fishery resources, control the spreading scope of infectious diseases, predict or control the risk and development of financial markets.
(5) 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 optimum 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, variational inequality and complementarity problem, etc. It has important applications in industry, commerce, agriculture, traffic and transportation, governmental department, etc. It is a powerful instrument in the fields of economic programming, system engineering, modern management, etc.
(6) Mathematical logic and theoretical computer
The main research direction of mathematical logic in Nanjing University is recursion theory (also called as computability theory), set theory and their crossing fields. In terms of recursion theory, we mainly focus on the global structure research on the degree of unsolvability and its application to theoretical computer science. In terms of degree theory, we mainly focus on model theory properties of the degree of unsolvability. In terms of applications, we focus on the research on the recursion theory to the algorithmic randomness theory. In terms of set theory, we mainly focus on effective descriptive set theory and inner model, as well as the research of applications of recursion theory method to set theory.
In the latest 50 years, explosive development has been achieved in recursion theory, and it’s embodied at not only the constant expansion of intrinsic contents, but also 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 we have made important contributions to it. In terms of the algorithmic randomness theory, we have realized the most successful application of recursion theory to theoretical computer, and the problem of lowness for randomness has nearly been solved thoroughly. Now, we gradually turn to the applications to analysis and dynamic system, especially the ergodic theory in the algorithmic theory. In terms of set theory, we have formed a relatively characteristic research field, which is mainly about the research of recursion theory and algorithmic randomness by applying forcing method and constructibility, etc. For example, higher randomness, chains and antichains are all the new research fields we have gradually explored out, and we have attracted the following of many international young prominent scholars.
(7) 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, Borel conjecture, etc. Over the past nearly half a century, Many great progresses have been made in the research on modern geometry and topology, and they have aroused the research on plentiful new important theoretical problems. We plan to enrich the academic force of each branch in the direction 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 do influential researches on curvature and topology, global Riemannian geometry, complex geometry, symplectic geometry, toric topology, topological group and general topology theory, etc.
(8) 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 sumsets 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, we plan to further study related front-edge problems with advanced tools from number theory, combination, algebra, analysis, probability, ergodic, etc., and strive to obtain influential significant results.