Research Overview

I graduated from the University of Pennsylvania in 1996 (adviser: Herman Gluck) with a Ph.D. in Mathematics (specializing in Differential Geometry, particularly in Calibrated Geometry, M.S. in Math 1993) and another M.S.E. in Computer Science (specializing in Computer Aided Geometric Design). Since then, my research directions have been developed into the following four major areas.

  • Differential Geometry & Topology This research field is one of the main focuses of my mathematical activity, which include the development of techniques for the method of calibrations as an effective tool for identifying volume-minimizing cycles in Riemannian manifolds such as Grassmann manifolds, Calabi-Yau 4-folds and exceptional holonomy manifolds. This part of my work is presented in [Gu1], [GrG], [GrGu] [GG], [GJ], [GP], [GS], [GW], [Gu2], [Gu3], [GHe], [GHa], [GSo] and [Gu4]. This research field developed starting with my Ph.D. thesis. I have since developed a research program in differential geometry with applications in string theory.
  • Geometric Modeling and Design This part of my work is presented in [CG1], [CG2], [CG3], [CG4], [DGG], [FGM], [GGo] and [WG]. This research field developed starting with a wish to build a bridge between differential geometry and computer aided geometric design when I was graduate student at UPenn. I have since initiated a research program supported now by an NSF grant (NSF 058663, \$424,135) conducting research with CGU math faculty members and Claremont students on problems using geometric modeling techniques that include the level sets method.
  • Applications to Industrial Mathematics Applying Mathematics and Computer Science to Solve Industrial Problems: This research field started with my desire to learn how to supervise a mathematics Clinic project about finding efficient algorithms for picking up garbage bins by a truck with a robotic arm. Since then, I have supervised 6 clinic projects, and my recent summer work at Hewlett-Packard, improving color technology, is in the process of being patented.
  • Applications to Math-Biology These applications mainly are Tumor and HIV Modeling using Dynamical Analysis, Optimal Control, and Geometric Modeling with Computer Graphic Simulation. This part of my work is presented in [HG1], [DGR], [DG], [HG2], [DGF1], [DGF2], [DGF3]. This research field was developed when I began auditing Professor L.~de Pillis’ Math-Bio class. Now it is a major research field of mine. The research program (together with Prof. Pillis) on tumor modeling is supported by an NSF grant (NSF 041401, \$328,283).

Research Topics in Detail