1, functional analysis
Research content: Functional analysis is developed on the basis of variational method, differential equation, integral equation, function theory and quantum physics. It uses the viewpoints and methods of geometry and algebra to study and analyze problems. The main research direction is: (1) Banach space geometry theory, such as convexity and adjacency. (2) fixed point theory; (3) Critical point theory.
Preparatory knowledge: mathematical analysis, topology, functional analysis.
Application fields: differential equations, wavelet theory, etc.
Research results: the yoke property of strong convexity in Banach spaces is solved; The concept of strongly flat space is introduced to study the properties of weakly convex Banach spaces. The approximation properties of Banach empty tips are studied. More than 50 academic papers have been published in the English edition of J.Math.Anal. Application, calculation. mathmatics Application and nonlinear analysis.
2. Geometric analysis
Research content: Based on the theory of partial differential equations, the geometry, topology and analytical structure of differential manifolds are studied.
Preparatory knowledge: partial differential equations, differential geometry.
Research achievements: 199 1 won the second prize of natural science of China Academy of Sciences; 1998 won the National Outstanding Youth Fund; 200 1 distinguished professor was hired as the Changjiang Scholar Award Program of the Ministry of Education, and won the Morningside Mathematics Award, the highest award of the World Chinese Mathematicians Congress, in 2004.
3. Symplectic topology and mathematical physics
Research contents: The main problems studied are the blow-up formula of gromov-Witten invariants of symplectic manifolds, the change of quantum homology groups under double rational operations, the relationship between gromov-Witten invariants and integrable systems, and mirror symmetry.
Preparatory knowledge: functional analysis, partial differential equations, abstract algebra, differential geometry, topology.
Research results: The blow-up formula of gromov-Witten invariants of symplectic manifolds is given, and the hypothesis of quantum minimum model of cohomology groups is verified for Mu Kai flop.
4. Dynamic systems, fractal geometry and dynamic equations on time scales.
Research contents: This paper mainly studies the calculation and estimation of Hausdorff measure of self-similar sets, the stability and oscillation of dynamic equation solutions on time scales, etc.
Preparatory knowledge: real variable function theory, measure theory, ordinary differential equation, difference equation, etc.
Research results:
Bound of Hausdorff measure of 1. Jia, Koch curve, applied mathematics and calculation. 182(2007).
2. Jia, Sierpinski carpet Hausdorff measure bounds, theory and application analysis, 22:4, 2006.
3. Generalization of Jia, Ostrowski Inequality, Inequalities in Pure Mathematics and Applied Mathematics, Vol.7, No.5, 2006.
4. Jia, Notes on an Inequality of Gamma Function, Inequalities in Pure Mathematics and Applied Mathematics, Vol.7, No.5, 2006.
5. Bound of Hausdorff Measure of Sierpinski Gasket, Journal of Mathematics. Anal. Application (2006), doi:10.1016/j.jmaa.2006.08.026.
6. A New Lower Bound of Hausdorff Measure of Sierpinski Gasket, Theory and Application Analysis, 22: 1, 2006,8-19.
7. Zhou, Jia, Hausdorff measure and convex density of a kind of self-set on the plane, Journal of Mathematics, 48 (3), 2005, 535-540.
8. Qu, Zhou, Jia, Upper Density of Symmetric Perfect Sets, Journal of Mathematics. Anal. Page 292 (2004): 23-32.
9. A Lower Bound of Hausdorff Measure between Jia, Zhou, Zhou and the cartoon product of the middle third detector set and itself, China Contemporary Mathematics, 2003, Vol.24, No.4, 34 1-350.
10. the filling measure of the third-order cantor set and its own cartesian product in Jia, Zhou,,,, Journal of Mathematics. Anal. Application, 288(2003) 424-44 1.
1 1. Jia, Zhou,, Hausdorff measure of self-product of three-point cantor set, Journal of Mathematics, Vol.46, No.4, 2003,747-752.
12. Lower bound of Hausdorff measure of self-product of Cantor set, Annual Journal of Mathematics, 24a: 5 (2003), 575-582.
A lower bound of Hausdorff measure of 13. Jia, Zhou, Sierpinski gasket, nonlinear 15(2002) 393-404.
5. algebra
Research contents: Galois theory includes Galois extension theory of fields, algebras and rings with Galois groups, which is the extension and generalization of classical Galois theory on fields, and studies the structure and group action of extensions; When a Hopf algebra has Galois effect in fields, algebras and rings, Hopf-Galois theory studies the Galois extension structure and the structure of Hopf algebra itself.
Preparatory knowledge: the mathematical foundation of undergraduate students in the department of mathematics of the university, with a good foundation in modern algebra.
Application field: The interaction between group and algebra provides a method to discuss algebraic structure; Hopf-Galois theory is a branch of Hopf algebraic representation theory, and many algebras at home and abroad are engaged in research, which is a very active research field; Galois theory of finite field has a good application in modern coding theory; Galois theory on the field has a good application in discussing the radical solution of the equation, and there is still research in this field at present.
Research results:
(1), galois theorem of projective group rings, Mathematical Yearbook17a: 6 (1996) 737-744;
② On the generalization of noncommutative Hopf-Galois, Journal of Sun Yat-sen University, Natural Science Edition, Volume 39, No.6, 2000;
(3), H- separable rings and their Hopf-Galois extensions, Mathematical Yearbook19b: 3 (1998) 311-320;
6. Complex analysis
Research content: mainly study Teichmuller space and related disciplines, including quasi-* * shape mapping, Klein group, Riemannian surface, three-dimensional manifold, hyperbolic geometry, harmonic mapping and so on.
Research achievements: Some research achievements have been made in Teichmuller space and related fields.
7. Harmonic analysis
Research contents: The main research directions are nonsmooth kernel singular integral operator theory and its application, function space related to differential operators, and functional calculus of operators.
Preliminary knowledge: The mathematical foundation mainly includes calculus, linear algebra, ordinary differential equations, partial differential equations, complex variable functions, real analysis, functional analysis and so on.
Research results: A series of important progress has been made in the function space related to differential operators, such as BMO space, Hardy space and singular integral operator theory with non-smooth kernel. The main thesis is as follows.
1, duality of Hardy and BMO spaces related to thermonuclear bound operators, J. Amer. mathmatics Socialist 18 (2005), 943-973.
2. New BMO-type function space, John nirenberg inequality, interpolation and application, general pure applied mathematics. 58 (2005), 1375- 1420.
3. Littlewood-Paley function related to second-order elliptic operator, mathematics. Z. 246 (2004), pp. 655-666.
8, partial differential equation function theory method
Research contents: The boundary value problems of singular integral operators and equations and analytic functions and their practical applications are studied.
Preliminary knowledge: The mathematical basis mainly includes calculus, linear algebra, ordinary differential equations, partial differential equations, complex variable functions, real analysis and measure theory, functional analysis, etc.
Application fields: mechanical problems, mathematical physics (nonlinear equation, Painleve equation, random matrix).
Research results: singular integral operator and its application in elastic problems. The asymptotic analysis of integral mainly includes Stokes phenomenon, uniform asymptotic, Riemann-Hilbert method and related problems in application analysis, especially in mathematical physics.
9. Asymptotic analysis
Research contents: The Stokes phenomenon of integral, uniform asymptotic expansion of integral and orthogonal polynomial system, Riemann-Hilbert analysis, Painleve function and the application of asymptotic analysis method in mathematical physics are studied.
Preliminary knowledge: The mathematical basis mainly includes calculus, linear algebra, ordinary differential equations, partial differential equations, complex variable functions, real analysis and measure theory, functional analysis, etc.
Application fields: mechanical problems, mathematical physics (nonlinear equation, Painleve equation, random matrix).
10, partial differential equation
Research content: theory and application of partial differential equations and related topics. At present, we mainly study the free boundary problem and nonlinear evolution equation of tumor growth. In the next few years, we will mainly study the oscillation integral and Fourier integral operator theory in Fourier analysis and the well-posedness and global existence theory of solutions of various nonlinear evolution equations related to it.
Preparatory knowledge: partial differential equations, ordinary differential equations, functional analysis, harmonic analysis, etc.
Application fields: physics, mechanics, chemistry, biology, etc.
Research results: Look up mathscinet and enter "Cui, Shang Bin" in the column of "Author", and you can find almost all the research work.
1 1, algebra and its application
Research contents: Hopf algebra and quantum group, and related Lie algebra and Kac-Moody algebra, commutative or noncommutative ring theory and module theory, homology algebra and algebraic representation theory.
Preliminary knowledge: abstract algebra. (Knowledge of geometry and physical background is preferred)
Applications: theoretical physics and noncommutative algebraic geometry, coding, cryptography and computing.
Research achievements: Quantum commutative algebra and its duality, China Science, 1997. Twisted product and quantum couple of Hopf algebra, Science Bulletin, 1999.
12, number theory and its application
Research content: Diophantine approximation and Diophantine equation: mainly study the effective algebraic approximation of algebraic numbers and the solutions of some Diophantine equations, and use Diophantine equations to study the class number of quadratic fields. At the same time, the irrationality and transcendence of the sequence are also studied. Difference set theory: the nonexistence of some difference sets is mainly studied by algebraic number theory. Theoretical basis of cryptography: Some problems in cryptography are mainly studied by using the theory of finite field and cyclotomic field.
Preparatory knowledge: number theory, algebra, complex analysis. You need a good foundation in number theory and algebra, or in number theory and complex analysis.
Application: Good programming ability, computing ability, and good number theory foundation.
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