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Research Interests

My research is in geometric analysis and its connections with mathematical physics. This rapidly developing field involves intriguing combinations of ideas and techniques from several different fields, including algebraic geometry, differential geometry, topology, and partial differential equations. My recent work uses analytic methods to study Gromov-Witten invariants.

Teaching:

Fall 09 - on sabbatical
Spring 09 - Math 133 (Sections 21-24) Calculus II

Earlier Courses

My Ph.D. Students

1994 Liviu Nicolaescu (Associate Professor, Notre Dame University).
1996 Eleny-Nicoleta Ionel (Professor, Stanford University).
2001 Junho Lee (Assistant Professor, Central Florida University).
2005 Jens Von Bergmann (Visiting Researcher, University of Calgary).

Honors

Phi Beta Kappa Teaching Award, Harvard University, 1982
Invited Hour Speaker, Regional A.M.S. Meeting,Worchester, MA, 1989
Frame Teaching Award, MSU Mathematics Department, 1998
Member, Institute for Advanced Study, Princeton, NJ, 2001-2002.

Selected Publications and Preprints

• An obstruction bundle relating Gromov-Witten invariants of curves and Kahler Surfaces (with Junho Lee). Browse
• Geodesics and Approximate Heat Kernels.
• A Structure Theorem for the Gromov-Witten Invariants of Kahler Surfaces, J. Diff. Geometry, 77 (2007), 483-513 (with Junho Lee).
• Symplectic Gluing and Family Gromov-Witten Invariants, in Geometry and topology of manifolds, 147-172, Fields Inst. Commun., 47, AMS, 2005 (with Junho Lee).
• The Symplectic Sum Formula for Gromov-Witten Invariants, Annals of Math., 159 (2004), 935-1025 (with E.-N. Ionel).
• Relative Gromov-Witten Invariants, Annals of Math., 157 (2003), 1-52 (with E.-N. Ionel).
• What is a Bubble Tree?, Notices of the AMS 50 (2003), 666-667.
• Gromov Invariants and Symplectic Maps, Math. Annalen. 314 (1999), 127-158 (with E.-N. Ionel).
• Gromov-Witten Invariants of Symplectic Sums, Math. Research Letters 5 (1998), 563-576 (with E.-N. Ionel).
• The Gromov Invariants of Ruan-Tian and Taubes, Math. Research Letters 4 (1997), 521-532 (with E.-N. Ionel).
• Bubble Tree Convergence for Harmonic Maps, J. Diff. Geometry 44 (1996), 595-633.
• Pseudo-Holomorphic Maps and Bubble Trees, Jour. of Geometric Analysis, 3 (1993), 63-98 (with J. Wolfson).
• Non-Minimal Yang-Mills Fields and Dynamics, Invent. Math. 107 (1992), 397-420.
• The Yamabe Problem, Bulletin of the AMS 17 (1987), 37-91 (with J. M. Lee).
• On Witten's Proof of the Positive Mass Theorem, Commun. Math. Phys. 84 (1982), 223-238 (with C. H. Taubes).
• Gauge Theories on Four-Dimensional Riemannian Manifolds, Commun. Math. Phys. 85 (1982), 563-602.

Mathematics Education Publications

Elementary Mathematics for Teachers (with S. Baldridge). A textbook for a ``Mathematics for Elementary School Teachers'' course taught in a mathematics department, designed to be used in conjunction with the "Primary Mathematics" elementary school texts from Singapore. Sefton-Ash Publishing, 2004. Available online.

Elementary Geometry for Teachers (with S. Baldridge). A sequel to the above textbook for the second semester course on mathematics for elementary and middle school teachers, focusing on measurement and geometry and including probability and statistics. It also is used in conjunction with the Singapore "Primary Mathematics" texts. Sefton-Ash Publishing, 2008. Available online.

Instructor resources for the above two textbooks are freely available at this site.

A Study of Core-Plus Students Attending Michigan State University (with R. Hill). A study involving over 3000 Michigan students found that students arriving at Michigan State University from four high schools which began using the Core-Plus Mathematics program placed into, and enrolled in, increasingly lower level courses as the implementation progressed. The existence of a downward trend is statistically statistically very robust (p<.0001). The grades these students earned in their mathematics courses were also below average (p<.01). American Math. Monthly, 113 (2006), 905-921.

The State of State Math Standards 2005 (by David Klein, Bastiaan J. Braams, Thomas H. Parker, William Quirk, Wilfried Schmid, W. Stephen Wilson, Chester E. Finn, Jr., Justin Torres, Lawrence Braden, Ralph A. Raimi). Evaluations of each state's K-12 Mathematics Standards, written by mathematicians. Published by the Thomas B. Fordham Foundation, 2005.

Links

Geometry / Topology at MSU

Geometric Analysis Seminar

Symplectic Geometry Seminar