Combinatorics

The combinatorics group is one of the major groups in School of Mathematics, with William Y.C. Chen as the group leader. It is one of the major group in China and granted by NNSF via the project “Innovative group grant”(only one for Math. each year). The group has close relationship with the combinatorics group in MIT, Rutgers Univ., Linz Univ. etc.

In recent years, we made some contributions on enumerative combinatorics and discrete geometric. The main results includes: we confirmed the conjecture of Andrews-Dyson-Rhoades on spt-partitions; we showed that the seqeunce of integer partitions satisfies higher order of Turan inequality; we give the grammar for the Ramanujan-Shor polynomials. Our results on tilling appeared in Notices of AMS as a survey paper.

Algebraic Geometry

The group of Algebraic Geometry includes Xiaotao Sun, Song Yang, Xun Yu, and Mingshuo Zhou, their research interests include moduli theory, algebraic geometry in characteristic p>0, complex geometry, and automorphism group of varieties. Their contributions include: Degeneration of moduli spaces (J. of alg. geom.); Minimal rational curves in moduli spaces (Math. Ann.); Direct images of Frobenius map (Invent. Math.); Proof of a Gieseker’s conjecture completely (Ann. Sc. Norm. Super. Pisa); D-modules and representation spaces (Advances in Math.); Blow-up formulas for Dolbeault cohomology groups (J. Math. Pures Appl.). Recently, they settled completely a long-standing question of Shepherd-Barron by proof of a Miyaoka-Yau type inequality for surfaces of general type in characteristic p>0 (J. of Eur. Math. Soc.), completed the determination of  minimum positive entropy of automorphisms for complex surfaces in each Enriques-Kodaira class (Duke Math. J.), proved that moduli spaces of parabolic and generalized parabolic sheaves are of globally F-regular type (Math. Ann.), and obtained a blow-up formula for Bott-Chern cohomology groups (Trans. Amer. Math. Soc.).

Discrete Geometry

Zong Chuanming is engaged in the research of geometry of numbers and lattice theory. The geometry of numbers and lattice theory have important applications in crystallography and modern cryptography. Professor Zong systematically developed the geometric structure theory of the packing gaps, and unified several well-known problems in the field of packing and covering into the theoretical framework for in-depth research.

In the study of Hilbert’s eighteenth problem, Professor Zong obtained the first non-trivial upper bound of the maximum translational packing density of a regular tetrahedron, and proposed a computer proof scheme for the maximum packing density of a regular tetrahedron. Professor Zong’s research work determined the two-dimensional packing-covering constant, proposed the concept of Blocking number, obtained the first clear-cut upper bound for the n-dimensional shading  problem, proposed a computer proof scheme for the Hadwiger conjecture for the first time, obtained the formula for the number of classes of sublattices, and made a complete classification for the five-fold and six-fold lattice tilings of the plane.

Professor Zong published two important survey articles in Bulletin AMS and three monographs in Springer and Cambridge University Press, and published more than ten review papers in Bulletin AMS every year by appointment, introducing important directions of research and major developments in mathematics. Professor Zong's work entitled "From Deep Holes to Free Planes" is the first review article published by a Chinese mathematician in the history of the journal Bulletin AMS

In 2020 Professor Zong’s paper titled “Can You Pave the Plane with Identical Tiles” was published as a feature in Notice of the American Mathematical Society (see Volume 67, Number 5). In the paper Professor Zong provides a series of important complete research results about the classical tiling problems.

Probability

The group of Probability leaded by Professor Wang Fengyu includes Professors Shao Jinghai and Bao Jianhai, Associated Professors Li Huaiqian, Fan Xiquan, Hu Eryan, Yang Xue, and Assistant Professor Huang Xing. The group has developed the theory of Functional Inequalities and Applications, made contributions to Stochastic Analysis on Riemannian Manifolds, and focuses on the study of Stochastic PDEs, heat kernel estimates, regime-switching processes, and probability limit theory. The group has been granted by National Natural Science Foundation via a major project.

Harmonic Analysis

The group of harmonic analysis main focuses on: 1) the theory of singular integrals on manifolds and general metric measure spaces, their relations with harmonic functions and heat kernels in the study of metric geometry; 2) weighted theory of singular integrals, boundedness of weighted multilinear operators. The group has made contributions in the study of Riesz transform, regularity of harmonic functions and heat kernels, as well as (weighted) multilinear operators. These results had been published in journals including, Adv. Math.、Comm. Pure Appl. Math.、J. Math. Pures Appl.、Math. Ann., and has been widely quoted by researchers in the society.

Control Theory

The group of Control Theory is leaded by Professor Gengsheng Wang, who serves as an editor of the journal SIAM J. Control and Optimization (SICON). This group has made important contributions to the field of the mathematical control theory, especially to the directions on the controllability/observability for PDEs, stabilization for period evolution systems and time optimal control problems etc.. Recently, they found the relation between the Nazarov’s uncertainty inequality and the observability at two time points for the free Schrödinger equation, and built up the equivalence among the spectral inequality and the observability and the interpolation inequality for the free heat equation over the whole space.

Partial Differential Equations

The group of PDEs mainly works on the wave equation, elliptic and parabolic equations. The main achievements of the group include: established stability of multi-solitons for the derivative nonlinear Schrödinger equation and instability of the solitary wave solutions for the generalized derivative nonlinear Schrödinger equation in the critical frequency case, established regularity theory for the elliptic equations with highly degenerate coefficients, and proved the Landis-Oleinik conjecture. The results had been published in journals including Adv. Math.、Amer. J. Math.、J. Euro. Math. Soc.、Memoirs AMS, etc.

Data Science and Big Data

The research areas of Data Science and Big Data include data intelligence and edge intelligence.

For the research of data intelligence, more than twenty papers have been published at top conferences and journals, including IEEE TKDE, IEEE TNNLS, IEEE TAFFIC, IEEE TCYB, IEEE TMM, ACM TKDD, ACM TIST, ICCV, IJCAI, AAAI, etc. More than twenty China invention patients have been authorized. Several data intelligence algorithms have been integrated into the applications of companies such as Chinese Mobile. Researchers had won China Patent Award, First prize of Beijing Science and Technology, First Prize of Beijing Invention Patent Award, etc.

For the research of Edge Intelligence, we have conducted cutting-edge research related to cloud and edge computing, internet of things, and wireless/mobile networks, and published many papers at top conferences and journals, including IEEE TPDS、TMC、TCC、TII、and IoT-J. Research projects with Huawei and Tencent, involve novel designs and performance evaluation by modeling and analysis using analytical, numerical, and simulation methods