Institute of Mathematics

RESEARCH

NUMBER THEORY AND ALGEBRA

• Study the arithmetic properties of global function fields. This includes exploring the algebraic-geometric struc-tures defined over these function fields, and examining properties of the individual object as well as properties of the moduli spaces.
• Interpret the analytic proofs of classical theorems on plane projective geometry using algebraically defined resides. Study Riemann-Roch theorem via residual complexes.
• Linnik-Selberg Problem about sums of Kloosterman Sums. We have computed its second moments. It remains for us to obtain the higher moments.

Mathematical Experiment

COMPUTATIONAL MATHEMATICS AND DYNAMIC SYSTEMS

• Continue our work on the phenomena of point bifur-cations of periodic points for families of continuous maps.
• High quality scientific computing based on reliable numerical schemes. The interested area is dynamical systems related to fluid and biology.
• Grid generation makes up an essential part of a numerical method. The structured grid generation, despite its popularity, is inadequate when dealing with certain types of computation. Unstructured grids have several advantages over structured grids, such as the flexibility for complex geometries, grid adaptation, relative ease and speed-up use. Therefore, the development and use of unstructured grid technique will be attemped. Also, due to the difficulty of the vectorization of unstruc-tured grid, the possibility of parallelization of the technique will be investigated.
• The first quantization of classical mechanics can be interpreted as a high frequency approximation of the Schrodinger equation. Following the same idea, we will analyze the problem of quantization of the Maxwell- Dirac equation. We hope the analysis will be helpful to understanding the origin of the divergence difficulty of the present second quantizationprocess and, possibly, to solving the difficulty.

DISCRETE MATHEMATICS

• Continue the study of the equitable coloring problem of graphs. We will focus on what happens when multiple colors are assigned to vertices. We will study the relationship between the multiple equitable chromatic number and the maximum degree of a graph.
• Employing the character sums to study permutation polynomials over finite fields.
• Study polynomial properties of graphs.
• Study number theory, algebra, and combinatorial structure using combinatorial method, and then compute their approximation behavior using complex variable method.
• Study applications of the combinatorics in any area.

THEORY OF SEVERAL COMPLEX VARIABLES AND GEOMETRY

• Analyze the tangent space of the moduli space of certain pseudohermitian hyperbolic Seifert manifolds of dimension 3 to retrieve the dimension formula and the complex structure of the universal Picard variety Pic.
• Study the topology of diffeomorphism group of those Seifert 3-manifolds via understanding the topology.
• Give a differential-geometric approach to the compac-tification of Pic, which has been done previously by algebraic methods.
• Study complex, algebraic geometries related to mirror symmetry and the Yang-Baxter equation in mathema-tical physics.
• Investigate the tangential Cauchy-Riemann equations on CR manifolds with codimension greater than one.

ANALYSIS AND DIFFERENTIAL EQUATIONS

• Fractional calculus and its applications to the theory of univalent functions and the summation of infinite series.
• Investigate one dimensional or high dimensional in-verse scattering transform and the associated evolution equations.
• Study various types of wavelet bases. We intend to apply the wavelet bases to singular integral operators and compression of images.
• Study of harmonic maps from C to hyperbolic disk, and also of spectrum problem of Laplacian on complete noncompact manifolds.
• Derive further estimates of geodesic curvature and gradient of surfaces of constant mean curvature via Weierstrass representation.
• Use concentration-compactness principle to obtain exi-stence results and analyze the structure of surfaces of constant mean curvature defined unbounded domains.
• Study the motion of liquid in thin films and the formation of singularities.

Workshop

PROBABILITY THEORY

• Continue our study on using Markov diffusion processes to approximate a distribution in Euclidean space. We will use first order perturbations and study their convergence rates. We also consider the problem in 2-dim torus to see if the convergence rate can be improved without limitation.
• Study game theory. The game theory has been well developed since 1944 after the work of Von Neumann and Morgenstein. We know that the theory also has many applications in social sciences. We will focus our study on topics related to probability theory such as stochastic game, repeated game, etc.
• Study problems related to the theory of diffusions on graphs.
• Consider the particle systems on infinite lattice and study how the external force determines the property of the system.
• Consider Markov chains indexed by trees. We will study the relation between the growth of the tree and the properties of the Markov chain such as transience, recurrence, etc., especially when there is a phase tran-sition.

PERSONNEL AND FACILITIES

PERSONNEL

Currently, the Institute has 15 research fellows, 10 associate research fellows and 6 assistant research fellows.

Libary

LIBRARY

A good library is as important to a mathematics research institute as a laboratory to an experimental scientific institute. Ever since the foundation of the institute, great effort has been made to build a complete, if not perfect, library. Our library now has the following valuable collections:

• Periodicals: There are more than 1,000 titles and except the discontinued ones, they are almost complete. Among them, twenty seven date back to the 19th century. We also update our lists as soon as possible.
• Books: There are more than 33,500 volumes (mostly in English).
• Videos: 66 reels.
• CD-ROM Discs:
  1. Math\Sci Database
  2. Compact Math Database

The Stock Room

The library implemented the INNOPAC library automation system. In order to share our resources, our service is not only available to our research staff, but also to all people related to mathematics. We also provide inter-library loans.

COMPUTING FACILITIES

The main computation facilities in the institute include IBM RISC6000/580 workstations and several Sun Sparc 20 workstations. Softwares like Mathematica and Matlab, accompanied with various toolboxes loaded on these two stations are accessible to all research members and visiting mathematicians. Each room has a LAN-WAN connected PC (80486 or higher) or Macintosh connected to the Internet.

There are four special working groups: Numerical Computation Group working with problems about fluid dynamics, Probability Group engaged with the problem of simulation, Wavelet Group ongoing research on applications of wavelets, and Symbolic Computation Group. Large symbolic computation packages, such as Maple, Macaulay, Pari-gp and SIMATH, are also available on various machines.

MAJOR RESULTS OF RESEARCH

NUMBER THEORY AND ALGEBRA

• We proved a geometric analogue of the Ankeny-Artin-Chowla conjecture for the hyperelliptic curves over finite fields. This gives non-vanishing values for the relative differential of the fundamental unit in question. When the characteristic case is odd we have an inequality relating the genus with units. This inequality also holds in characteristic 0.
• We studied the maximum number of solutions of additive equations: small zero's problem.
• Over a finite field, we studied various properties of a certain class of mappings induced from Vandermonde determinant, as well as the conditions for these mappings to be bijections. The study results in more understanding and combinatorial application of finite field theory.

COMPUTATIONAL MATHEMATICS AND DYNAMIC SYSTEMS

• We showed that, for some suitably chosen constant, the one-parameter family has a point bifurcation of periodic points of some period equal or greater than 3 and the bifurcation diagram has bubbles.
• Numerical studies of domain decomposition and explicit scheme for high Reynolds number flow.
• A two-dimensional incompressible flow solver using stream function-vorticity formulation has been established. The capability of the method and the program has been demonstrated via simulations of unsteady flow passing bluff body such as stationary circular cylinder and ellipse. Since the upwind technology was involved within the computation of vorticity, the Reynold numbers of these incompressible flow simulations were raised to 40,000 and higher. We have also been working on the calculation of flow field of a rotating disc.

DISCRETE MATHEMATICS

• A signed graph is a graph in which every edge is labeled with either a minus or a plus sign. The algebraic sum of signs of the edges incident to a vertex is the signed degree of that vertex. We obtained necessary and sufficient conditions for an integral sequence to be the signed degree sequence of a signed graph or a signed tree.
• A chip firing game is defined on graphs as follows. Each vertex contains a number of chips. A move consists of selecting a vertex with at least as many chips as its degree and sending one chip from the vertex to each of its neighbors. We obtained methods to decide whether a game on some special classes of graphs will terminate in finite steps.
• We completely factorized Dickson polynomials of the first and the second kind over finite fields.
• We computed Wiener indexes of mixed polygonal and two-dimensional hexagonal chemical graphs and studied properties of Wiener polynomials.
• On each function graph, we computed the number of independent sets so that the cardinality of each set equals the independence number ofthe graph.

THEORY OF SEVERAL COMPLEX VARIABLES AND GEOMETRY

• A differential-geometric study of the universal Picard variety: we gave a brief description of the universal Picard variety in terms of CR and pseudohermition geometries. Namely, there is a one-one correspondence between the universal Picard variety and the moduli space of certain pseudohermitian hyperbolic Seifert manifolds of dimension 3. We computed the tangent space of this moduli space.
• In the study of mirror symmetry, we dealt with elliptic curves. The mirror symmetry of an elliptic curve possesses a similar representation formula as the modular function.
• In studying the Yang-Baxter equation and the quantum solvable models, we obtain partial results about finite lattices relative to the Bethe Ansatz equation.

ANALYSIS AND DIFFERENTIAL EQUATIONS

• A new type of Lusin property was obtained with application to interpolation of sobolev spaces.
• Existence of equilibrium measures and points for two systems of functions were obtained. This has close relations to the Frobenius problem for matrices, as well as to the Von Neumann equilibriums for expanding economy.
• Univalent function theory in several complex variable and Banach spaces.
• Asymptotic behaviors of solutions of some differential (difference) equations.
• Using Darboux-Carton moving frame method to inter-pret some 1+1-dimensional soliton equations.

PROBABILITY THEORY

• We revised our work on some quadratic perturbation of Ornstein-Uhlenbeck process.
• We obtained the asymptotics for the first eigenvalue and eigenfunction of a nearly first order operator with large potential.
• We studied the diffusion processes on graphs and their large deviation properties.
• We used asymptotic analysis for ordinary differential equation to obtain the asymptotics for the exit time and exit distribution of simulated annealing process from a set.

During the period of May 1995 to April 1996, we sponsored one hundred colloquia and seminars. We published forty-six research papers, four issues of the BULLETIN of the Institute of Mathematics, Academia Sinica, four issues of MATHMEDIA. Our researchers participated in thirty-six academic activities abroad, including international conferences, workshops and visits. We also hosted twen-tynine foreign visitors.

Two international conferences took place at our institute: "Workshop on Probability and Mathematical Physics," from July 24 to 29, 1995, and "Workshop on Geometry with Emphasis on Symplectic, Contact, Complex, and CR Structures," from April 1 to 5, 1996.

RESEARCH STAFF

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