Giles Auchmuty Professor of Mathematics - University of Houston Office Location: PGH 696 Telephone: (713) 743-3475 Fax: (713) 743-3505 E-mail: auchmuty@uh.edu My physical, and mail, addresses are University of Houston Department of Mathematics PGH Rm 696 4800 Calhoun Ave, Houston, TX 77204-30

Teaching:

         In Spring  2012,   I will teach  Math 6327 Partial Differential Equations.   For more information about the  courses and a listing of courses taught in recent years please  select the  Teaching  link above.

Research Interests:

             My research centers on the mathematical analysis of problems that arise in physical or engineering applications. Currently my primary interest  is in the solution of div-curl systems  which arise in both electromagnetic field theory and in fluid mechanics. Also various non-standard eigenvalue problems and questions of how to  describe the dependence of the solutions of partial differential equations on non-trivial boundary data. The mathematical basis of these theory centers on issues about Sobolev spaces and associated trace spaces. Recently I developed a spectral theory of trace spaces and have obtained new trace inequalities. Also I have published papers on traces for mixed boundary problems and for interface and immersed boundary problems.  This analysis  involves the theory of Steklov and other eigenproblems and Reproducing Kernel Hilbert spaces.

          Another continuing research interest is in the theory of potentials for various classes of 3-dimensional vector fields when various types of boundary conditions are imposed. Recently I have found extensions of the Sobolev imbedding theorems for 3-d fields where integrability conditions are imposed on the Laplacian of a potential - but not on the function itself. This is the physical  situation for gravitational and electrostatic potentials. Together with Qi Han, we are studying the represntation of functions and fields in exterior regions in space.

        This research is currently supported by NSF award DMS 1108754 to study "Steklov spectra and Div-curl analysis".   Maxwell's equations are the most important examples of div-curl systems. See the link to Research for more information. 

      The work on these issues is theoretical mathematics, involving functional analysis and variational principles. It  does not involve  computational studies  and I do not have positions available for  programmers.

Research Students:    

        A list of  former Ph.D. Students who have graduated. Currently I am advising Qi Han, Manki Cho and Liu Puchen on their doctoral research.   I have also supervised many M.S. theses and some undergraduate honors students and other research  projects.  I will be happy to talk to students who are interested in possible research projects and know some mathematical analysis.

 Editorial:

        This section associated with SIAM Review welcomes unsolved problems in applied and applicable mathematics. Contributors should send proposed problems to Cecil C Rousseau. If you can solve posted problems please let the proposer and SIAM know about your solutions.

Houston Journal of Mathematics.
       
        The Houston Journal of Mathematics publishes original research results in many areas of mathematics. See their web-page Houston J. of Mathematics for details of the submission process.