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 4800 Calhoun Ave, Houston, TX 77204-3008

Teaching:

         In Spring  2013,   I am teaching  Math 6367  Optimization Theory and Variational Methods.    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 elliptic boundary value problems and issues that arise in classical field theories - especially electromagnetic and gravitational fields.

          Recent papers have proved new results about harmonic functions and fields, boundary values (traces) of Sobolev functions, the existence and properties of scalar and vector potentials and the well-posedness of  div-curl systems  that arise in either electromagnetic field theory or in fluid mechanics.

       Many of these results have involved either elliptic spectral theory, especially the theory of Steklov eigenproblems, or the theory of Reproducing Kernel Hilbert Spaces. The results have often depended on proving new inequalities and developing special representation results. Recently I proved new 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. With a recent graduate student Qi Han, we are studying the representation of functions and fields in exterior regions in space and developing well-posedness results for  elliptic boundary value problems on exterior regions.  

          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.

        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:    

 Here is a listing of  Ph.D. graduatets whose theses I supervised.  Currently   Manki Cho, Mauricio Rivas  and Liu Puchen are working on their doctoral research with me.   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.