# 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. L~.de Pillis) on tumor modeling
is supported by an NSF grant (NSF
041401, \$328,283).