國家理論科學研究中心(南區)數學組

Optimization and Numerical Methods in Mathematics

COORDINATOR:

Soon Yi Wu (NCKU)

Optimization, Functional Analysis, Linear Programming

soonyi@mail.ncku.edu.tw

Tel: +886-6-2757575 Ext. 65133

 

Topics:

Optimization is the science of rational decision-making based on quantitative analysis. It has the broadest application spreading out in many areas, such as military operation, environmental protection, transportation management, inventory control, human resources governance, medicare resource allocation, etc., many of which are important issues faced by our society today. We will focus on some major topics in optimization as follows:

  1. Infinite and semi-infinite programming
    Topics include nonlinear semi-infinite programming, generalized semi-infinite programming, optimal control problems, and continuous linear programming. We will study the potential algorithms for these problems. Since the continuous transportation problem, and continuous two person game problems are part of linear programming on measure spaces or Lp spaces, we will study more about the theory of linear programming on measure spaces or Lp spaces. Moreover, duality theory for all above related problems is also part of our theme.
  2. Generalized variational inequality and duality
    We will study the property of solutions and gap function for generalized variational inequality problems. Moreover, the generalized complementary problems and the conditions for the existence of solutions for generalized variational inequality problems are part of our theme.

Besides the above two themes, nonlinear analysis and integer programming will also be our themes.

  1. We will study the following three types of problems in electronmagnetics:
    The hyperbolic PDEs in Maxwell equations and the Euler equations;
    The Navier-Stokes equations with a mixed-hyperbolic-parabolic system;
    The Schrodinger wave equations.

 

 

Functional Analysis, Operator Theory, and Algebraic Methods in Analysis

COORDINATOR:

Ngai-Ching Wong (NSYSU)

Functional Analysis, Operator Algebras,Operator Theory

wong@math.nsysu.edu.tw

Tel: +886-7-5252000 Ext. 3818

 

Topics:

In 2007, we will continue our emphases on the study of Jordan structures in analysis. Jordan algebras were first introduced by Jordan, von Neumann and Wigner around 1934 for an axiomatic study of quantum mechanics. Since then, Jordan structures have found many important applications, besides physics, in diverse areas of mathematics, notably in Lie algebras, group theory, geometry, complex and functional analysis. Among them, Zelmanov's solution of the restricted Burnside problem awarded him a Fields Medal in 1994.

We are mainly interested in the possible applications of the Jordan structures to complex variables and geometry in the infinite dimension setting. For instance, in the theory of homogeneous Banach manifolds, Kaup and Upmeier have successfully made use of the tools from Jordan triples, Jordan pairs and Jordan algebras to generalize the famous Cartan's classification for bounded symmetric domains to infinite dimension. It might be interesting to note that in the finite dimensional case, both Lie and Jordan approaches give rise to the same theory, while only the Jordan tools are applicable in the infinite dimensional case. We will explore into the area of the bounded symmetric domains in Banach spaces, Cartan factors, and JB*-triples. A possibly theme might be on the theory of differential operators on bounded symmetric domains and Shimura varieties.

In April 2006, there held a very stimulating international conference in “Jordan Structures in Geometry and Analysis” in NSYSU and NCKU. Kaup, Upmeier, Edwards, Chu and many other key figures in Jordan theory presented their new results in this event. To follow the most recent advance in this subject, we will place our foci in 2006 again at:

 

  1. Banach manifolds and Jordan structure theory.
  2. Differential operators on subharmonic functions in bounded symmetric domains.
  3. General theory of operator algebras and operator spaces.
  4. General theory of linear preservers, composition operators, Jordan homomorphisms and Lie homomorphisms on function modules and operator algebras, as well as, general algebras and rings.
  5. Hankel and Toeplitz operators and their spectral theory.
  6. Degree theory, optimization, and variational problems in infinite dimension.

 

Participants

Soon Yi Wu (National Cheng Kung University)

Jen Chin Yao (National Sun Yat-sen University)

Ruey Lin Sheu (National Cheng Kung University)

Chun Hao Teng (National Cheng Kung University)

Chern Shuh Wang (National Cheng Kung University)

Ngai-Ching Wong (National Sun Yat-sen University)

Mark C. Ho (National Sun Yat-sen University)

Tsai-Lien Wong (National Sun Yat-sen University)

Jhy-Shyang Jeang (ROC Military University)

Yuen Fong (National Cheng Kung University)

Mue-Ming Wong (Meiho Institute of Technology)

Hong Kun Xu (National Sun Yat-sen University)