Welcome to my homepage!

I am a member of the Geometry research group here at UT Austin. I also interact with our research groups in Partial Differential Equations and Topology.

UT hosted the 40^th Texas Geometry and Topology Conference on October 10-12, 2008. Click here for details.

Research publications and preprints

Minimally invasive surgery for Ricci flow singularities. Coauthors: Sigurd Angenent and M. Cristina Caputo. Submitted.

Cross curvature flow on a negatively curved solid torus. Coauthors: Jason Deblois and Andrea Young. Submitted.

Convergence and stability of locally R^N-invariant solutions of Ricci flow. J. Geom. Anal. 19 (2009), no. 4, 817-846.

Estimating the trace-free Ricci tensor in Ricci flow. Proc. Amer. Math. Soc. 137 (2009), no. 9, 3099-3103.

Asymptotic stability of the cross curvature flow at a hyperbolic metric. Coauthor: Andrea Young. Proc. Amer. Math. Soc. 137 (2009), no. 2, 699-709.

Local monotonicity and mean value formulas for evolving Riemannian manifolds. Coauthors: Klaus Ecker, Lei Ni, and Peter Topping. J. Reine Angew. Math. (Crelle) 616 (2008) 89-130.

Precise asymptotics of the Ricci flow neckpinch. Coauthor: Sigurd Angenent. Comm. Anal. Geom. 15 (2007), no. 4, 773-844.

Linear stability of homogeneous Ricci solitons. Coauthors: Christine Guenther and James Isenberg. Int. Math. Res. Not. (2006), Article ID 96253, 30 pp.

Positivity of Ricci curvature under the Kaehler-Ricci flow. Commun. Contemp. Math. 8 (2006), no. 1, 123-133.

An example of neckpinching for Ricci flow on Sⁿ⁺¹. Coauthor: Sigurd Angenent. Math. Res. Lett. 11 (2004), no. 4, 493-518.

Rotationally symmetric shrinking and expanding gradient Kaehler-Ricci solitons. Coauthors: Mikhail Feldman and Tom Ilmanen. J. Differential Geom. 65 (2003), no. 2, 169-209.

A lower bound for the diameter of solutions to the Ricci flow with nonzero H¹(Mⁿ;R). Coauthor: Tom Ilmanen. Math. Res. Lett. 10 (2003), no. 2, 161-168.

Hamilton's injectivity radius estimate for sequences with almost nonnegative curvature operators. Coauthors: Bennett Chow and Peng Lu. Comm. Anal. Geom. 10 (2002), no. 5, 1151-1180.

Stability of the Ricci flow at Ricci-flat metrics. Coauthors: Christine Guenther and James Isenberg. Comm. Anal. Geom. 10 (2002), no. 4, 741-777.

New Li-Yau-Hamilton inequalities for the Ricci flow via the space-time approach. Coauthor: Bennett Chow. J. Differential Geom. 60 (2002), no. 1, 1-51.

Quasi-convergence of model geometries under the Ricci flow. Coauthor: Kevin McLeod. Comm. Anal. Geom. 9 (2001), no. 4, 879-919.

Quasi-convergence of the Ricci flow. Comm. Anal. Geom. 8 (2000), no. 2, 375-391.

Books, surveys, and expository articles

The Ricci Flow: Techniques and Applications, Part III: Geometric-Analytic Aspects. Coauthors: Bennett Chow, Sun-Chin Chu, David Glickenstein, Christine Guenther, Jim Isenberg, Tom Ivey, Peng Lu, Feng Luo, and Lei Ni. Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI. To appear.

The Ricci Flow: Techniques and Applications, Part II: Analytic Aspects. Coauthors: Bennett Chow, Sun-Chin Chu, David Glickenstein, Christine Guenther, Jim Isenberg, Tom Ivey, Peng Lu, Feng Luo, and Lei Ni. Mathematical Surveys and Monographs, Vol. 144. American Mathematical Society, Providence, RI, 2008.

The Ricci Flow: Techniques and Applications, Part I: Geometric Aspects. Coauthors: Bennett Chow, Sun-Chin Chu, David Glickenstein, Christine Guenther, James Isenberg, Tom Ivey, Peng Lu, Feng Luo, and Lei Ni. Mathematical Surveys and Monographs, Vol. 135. American Mathematical Society, Providence, RI, 2007.

An introduction to the Ricci flow neckpinch. Geometric Evolution Equations. Edited by Shu-Cheng Chang, Bennett Chow, Sun-Chin Chu, and Chang-Shou Lin. Contemporary Mathematics. Vol. 367, 141-148. American Mathematical Society, Providence, RI. 2005.

The Ricci flow: An Introduction. Coauthor: Bennett Chow. Mathematical Surveys and Monographs, Vol. 110. American Mathematical Society, Providence, RI, 2004.

Singularity models for the Ricci flow: an introductory survey. Variational Problems in Riemannian Geometry: Bubbles, Scans and Geometric Flows. Edited by Paul Baird, Ahmad El Soufi, Ali Fardoun, and Rachid Regbaoui. Progress in Nonlinear Differential Equations and Their Applications, Vol. 59, 67-80. Birkhaeuser, Basel, 2004.

An injectivity radius estimate for sequences of solutions to the Ricci flow having almost nonnegative curvature operators. Coauthors: Bennett Chow and Peng Lu. Proceedings of ICCM 2001. Edited by Chang-Shou Lin, Lo Yang, and Shing-Tung Yau. New Studies in Advanced Mathematics, Vol. 4, 249-256. International Press, Somerville, MA, 2004.

Current and recent courses

TC310 – Modes of Reasoning: Optimal Geometry in Nature, Art, and Mathematics (Spring 2010) [Syllabus coming soon.]

M408C – Differential and Integral Calculus (Fall 2009)

M427K – Advanced Calculus for Applications I - Engineering Honors (Spring 2009)

M365G – Curves and Surfaces (Spring 2009)

M427K – Advanced Calculus for Applications I (Fall 2008)

M382D – Differential Topology (Spring 2008)

M392C- Riemannian Geometry (Fall 2007)

SAGE

SAGE (Symposia on Analysis of Geometric Evolution) is the name of a series of annual workshops at UT-Austin designed to integrate research, graduate education, and undergraduate outreach. SAGE is supported by the National Science Foundation (NSF Career grant DMS-0545984).

The first workshop took place May 7-11, 2007. Its topics included: Kaehler-Ricci solitons, Kaehler-Ricci flow, and Fano manifolds. Click here to learn more.

The second workshop took place May 5-8, 2008. Its topics included: Asymptotics and singularity formation of geometric evolution equations. Click here to learn more.

The third workshop took place May 21-24, 2009. Its topics included: Homogeneous solitons and large-time Ricci flow behavior. Click here to learn more.

Fun stuff

Never before in the course of human history have there been as many opportunities to waste time as we enjoy today - all thanks to the Internet.

Here are some place you can visit, all without leaving Texas: Athens, Atlanta, Egypt, Holland, Italy, London, Moscow, Newark, Paris, Pasadena, Princeton, and Rhome.

Here is an example of how not to teach math.

And here are some tongue-in-cheek applications of graduate mathematics.

Stephen Colbert ponders the Poincare Conjecture.

The Klein Bottle Company is my favorite source for nonorientable surfaces.

The Continental Drift Cam provides up-to-the-minute updates on plate tectonics.

The Daily Texan informs the UT community.

The Texas Travesty entertains us. (Warning: this is a highly irreverent humor publication.)

Our friends in the natural sciences have graciously provided many opportunities to be frivolous: we can enjoy biological puns, sing physics songs, or study chemistry gone awry.

When you are done wasting time, you may conserve valuable electrons by shutting down the Internet.

This page last updated on Thursday, October 1, 2009.

Author supported in part by the NSF. Any opinions, findings, conclusions, or recommendations on this page are those of the author and do not necessarily reflect the views of the National Science Foundation.