Research

My research focuses on geometric analysis and its applications to mathematical physics.

Gromov-Witten Invariants

Relating VFCs on thin compactifications (with E.-N. Ionel), Math. Annalen 375 (2019), 845–893. 📘 DOI
Thin compactifications and relative fundamental classes (with E.-N. Ionel), J. Symplectic Geometry 17(3), 2019. 📘 Journal 🧪 arXiv
The Gopakumar–Vafa formula for symplectic manifolds (with Eleny Ionel), Annals of Mathematics 187 (2018), 1–64. 📘 Journal
A natural Gromov–Witten virtual fundamental class (with E.-N. Ionel). 🧪 arXiv
Spin Hurwitz numbers and the Gromov–Witten invariants of Kähler Surfaces (with Junho Lee), Communications in Analysis & Geometry 21 (2013). 📘 DOI
An obstruction bundle relating Gromov–Witten invariants of curves and Kähler Surfaces (with Junho Lee), American Journal of Mathematics 134 (2012), 453–506. 📘 Journal 🧪 arXiv
A Structure Theorem for the Gromov–Witten Invariants of Kähler Surfaces (with Junho Lee), Journal of Differential Geometry 77 (2007), 483–513. 📘 Journal
Symplectic Gluing and Family Gromov–Witten Invariants (with Junho Lee), Fields Inst. Commun. 47 (2005). 📘 Journal
The Symplectic Sum Formula for Gromov–Witten Invariants (with E.-N. Ionel), Annals of Mathematics 159 (2004), 935–1025. 📘 Journal Corrigendum (with E.-N. Ionel). 🧪 arXiv
Relative Gromov–Witten Invariants (with E.-N. Ionel), Annals of Mathematics 157 (2003), 1–52. 📘 Journal
Gromov Invariants and Symplectic Maps (with E.-N. Ionel), Mathematische Annalen 314 (1999), 127–158. 📘 Journal
Gromov–Witten Invariants of Symplectic Sums (with E.-N. Ionel), Math. Res. Letter 5 (1998), 563–576. 📘 Journal
The Gromov Invariants of Ruan–Tian and Taubes (with E.-N. Ionel), Math. Res. Letters 4 (1997), 521–532. 📘 Journal

Harmonic and Pseudo-holomorphic maps

Bubble Tree Convergence for Harmonic Maps, J. of Differential Geometry 44 (1996), 595–633. 📘 Journal
What is a Bubble Tree?, Notices of the AMS 50 (2003), 666–667. 📘 Journal
Pseudo‑Holomorphic Maps and Bubble Trees (with J. Wolfson), J. of Geometric Analysis 3 (1993), 63–98. 📘 Journal
Conformal Fields and Stability, Math. Zeitschrift. , 185 (1984) 305-319. 📘 Journal

Yang-Mills and Gauge Theory

Gauge Theory on Four‑Dimensional Riemannian Manifolds, Comm. Math. Physics 85 (1982), 563–602. 📘 Journal
Non‑Minimal Yang–Mills Fields and Dynamics, Inventiones Math. 107 (1992), 397–420. 📘 Journal
A Morse Theory for Equivariant Yang–Mills, Duke Math Journal 66 (1992), 337–356. 📘 Journal 📘 Erratum
Equivariant Sobolev Theorems and Yang-Mills-Higgs Fields, in Global Analysis in Modern Geometry, Publish or Perish, Houston, (1993), 33-72. 📘 Book Chapter
Sharp Decay Estimates for Yang-Mills Fields (with D. Groisser), Comm. in Analysis and Geometry, 5, (1997), 439-474. 📘 Journal
Semiclassical Yang-Mills Theory I: Instantons (with D. Groisser), Commun. Math. Phys., 135 (1990) 101-140. 📘 Journal
The Geometry of the Yang-Mills Moduli Space for Definite Manifolds (with D. Groisser), J. of Differential Geometry 29 (1989), 499–544. 📘 Journal
The Riemannian Geometry of the Yang-Mills Moduli Space (with D. Groisser), Commun. Math. Phys. , 112 (1987) 663-689. 📘 Journal

The Yamabe Problem, General Relativity and other subjects

The Yamabe Problem (with J. M. Lee), Bulletin of the AMS 17 (1987), 37–91. 📘 Journal
On Witten's Proof of the Positive Mass Theorem (with C. H. Taubes), Comm Math Physics 84 (1982), 223–238. 📘 Journal
Gauge Choice in Witten’s Energy Expression, Comm Math Physics 100 (1985) 471-480 📘 Journal
Invariants of Conformal Laplacians (with S. Rosenberg), Jour. Diff. Geometry, , 25 (1987) 199-222. 📘 Journal
Geodesics and Approximate Heat Kernels. 📘

Mathematics Education

Elementary Mathematics for Teachers (with S. Baldridge), Sefton‑Ash Publishing, 2004. 📘 Publisher
Elementary Geometry for Teachers (with S. Baldridge), Sefton‑Ash Publishing, 2008. 📘 Publisher
A Study of Core‑Plus Students Attending Michigan State University (with R. Hill), American Math. Monthly 113 (2006), 905–921. 📘 Journal