The Institute of Pure Mathematics has made significant academic contributions in many aspects of the frontier fields of pure mathematics, thus having some international influence. The researches of the Institute include:
关于具有复杂边界条件、非光滑系数的非线性椭圆、抛物方程的正则性和渐近性问题的研究，成果发表在Arch. Ration. Mech. Anal.,Adv. Math. SIAM J. Numer. Anal., CVPDE等期刊。
The researches on the regularity and asymptotic behavior of nonlinear elliptic and parabolic equations with complex boundary conditions and non-smooth coefficients, the results of which have been published in Arch. Ration. Mech. Anal., Adv. Math., SIAM J. Numer. Anal. and CVPDE.
关于极大曲面外区域问题及无穷Laplace方程外区域问题和边界正则性等方面的研究，成果发表在Comm. Pure Appl. Math.，IMRN，Manuscripta Math. 和Nonlinear Anal.等期刊。
The researches on the exterior domain problems for the maximal surfaces and on the exterior domain problems and boundary regularity problems for the infinity Laplace equations, the results of which have been published in Comm. Pure Appl. Math., IMRN, Manuscripta Math., and Nonlinear Anal.
关于带正质量标量场爱因斯坦方程组的全局稳定性的研究，成果发表在Comm. Math. Phys.，并有专著两部由World Scientific出版社出版。
The research on stability of Einstein-massive scalar field equations, the results of which have been published in Comm. Math. Phys. and two monographs published by World Scientific Publishing Co Pte Ltd.
对Camassa-Holm类可积系统非光滑孤立子的稳定性的系统研究，成果发表在Adv. Math.，Comm. Math. Phys.，Arch. Ration. Mech. Anal.，J. Math. Pure Appl.，Nonlinearity等期刊。
The systematic researches on the stability of non-smooth solitons in Camassa-Holm-like integrable systems, the results of which have been published in Adv. Math., Comm. Math. Phys., Arch. Ration. Mech. Anal., J. Math. Pure Appl., and Nonlinearity.
开展了解析数论、代数数论与代数几何的交叉性研究。系统研究了Kloosterman和的Sato-Tate猜想：一方面，从殆素数角度否定回答了模结构问题，这是Nicholas Katz提出问题40年来的第一个理论性进展；另一方面，对殆素数模Kloosterman和的符号变化给出了目前最好的定量刻画。发展了代数迹函数的算术型指数对理论，并用于Pell方程的Hooley猜想、二次多项式的Schinzel猜想等问题的研究。相关成果发表在Invent. math.，Compos. Math.，IMRN等期刊。
The Institute works on the interface of analytic number theory, algebraic number theory and algebraic geometry. A systematic study on the Sato-Tate conjecture for Kloosterman sums has been conducted. On one hand, a negative answer to the problem on modular structures was given in the sense of almost prime moduli, and this is the first theoretical progress since the problem was proposed in 1980 by Nicholas Katz. On the other hand, the best record up to now is kept for the quantitative characterization on sign changes of Kloosterman sums to almost prime moduli. Moreover, the theory of arithmetic exponent pairs for algebraic trace functions has been developed and also been applied to the Hooley conjecture for Pell equations and Schinzel Hypothesis for quadratic polynomials. The relevant results have been published in Invent. math., Compos. Math., and IMRN.
在低维拓扑方面，对三维Seifert流形、双曲流形的不动点指数及不动子群的秩给出了界定，证明了曲面群任意一族自同态的不动子群之秩不超过曲面群的秩，且在自同构时为惯性的，并对不动子群可压缩、惯性的乘积群进行了分类。相关成果发表在Algebraic & Geometric Topology，J. Algebra等期刊。
In terms of low-dimensional topology, the Institute has defined the fixed point index of the three-dimensional Seifert manifold and hyperbolic manifold and the rank of the fixed subgroup, proved that the rank of the fixed subgroup of any family endomorphism of a surface group does exceed the rank of the surface group, and classifies the compressible and inertial product groups of the fixed subgroups when the automorphism is inertial. Related results were published in Algebraic & Geometric Topology, J. Algebra, etc.
在格上拓扑方面，对Domain和Quantale理论中的若干前沿问题做了深入研究，特别对偏序集上的下极限收敛和序收敛做了系统研究，给出了它们可拓扑化的充要条件，并将这些研究延拓到了T0 拓扑空间上。相关成果发表在Topol. Appl.，Rocky MT J Math.，Houston J Math.等期刊。
In terms of topology on lattice, the Institute has conducted in-depth researches on several frontier issues in the Domain and Quantale theories, especially the systematic researches on the lim-inf convergence and order convergence on partially ordered sets, giving necessary and sufficient conditions for their topologicalization and extending these researches to the T0 topological space. Related results were published in Topol. Appl., Rocky MT J Math., and Houston J Math.
在非线性泛函分析方面，对无穷维动力系统和最优控制理论研究中的若干前沿问题进行了深入研究，特别对吸引子的存在性及维数理论、最优控制中的Pontryagin 极大值原理进行了系统研究。相关研究成果发表在Z. Angew. Math. Phys., Nonlinear Anal.等期刊。
Concerning nonlinear functional analysis, the Institute has conducted deep researches in several frontier issues of the infinite-dimensional dynamical systems and optimal control theory. Especially, the existence of attractors and its fractal dimension have been systematically investigated, and the Pontryagin maximum principle in optimal control was established as well. Many related results were published in Z. Angew. Math. Phys. and Nonlinear Anal.
With the focus on major topics in several frontier fields of pure mathematics and in response to the needs of other research centers of the Institute for basic mathematical theories in the process of applying mathematical technologies, the main research directions of the Institute of Pure Mathematics include:
Regularity and stability theories of partial differential equations;
Geometric topology, topology on lattice and non-classical logic;
Soliton theory and nonlinear integrable system;
Non-linear functional analysis;