Welcome to my homepage!
Dan Knopf
Assistant Professor*
University of Texas at Austin
* Promotion to Associate Professor with tenure effective September 1, 2008.
Research
publications and preprints
Books,
surveys, and expository articles
Contact information
Department of Mathematics
1 University Station, C1200
Office: RLM 9.152
Email: danknopf {at} math {dot} utexas {dot} edu
Phone: 512.471.8131
Fax: 512.471.9038
Office hours: Tuesdays 2:00..3:00, Fridays
3:00..4:00, and by appointment
Curriculum vitae
Research interests
Geometric analysis
Differential geometry
Geometric partial differential equations
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.
Research publications and preprints
Convergence and stability
of locally RN-invariant solutions of Ricci flow. Submitted.
Estimating the
trace-free Ricci tensor in Ricci flow. Submitted.
Asymptotic stability of the
cross curvature flow at a hyperbolic metric. Coauthor: Andrea Young. Proc. Amer. Math. Soc. To appear.
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). To appear.
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, doi:10.1155/IMRN/2006/96253.
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
The
Ricci Flow: Techniques and Applications, Part II: Analytic Aspects. Coauthors:
Bennett Chow, Sun-Chin
The
Ricci Flow: Techniques and Applications, Part I: Geometric Aspects.
Coauthors: Bennett Chow, Sun-Chin
An introduction to
the Ricci flow neckpinch. Geometric Evolution Equations. Edited by
Shu-Cheng Chang, Bennett Chow, Sun-Chin
The
Ricci flow: An Introduction. Coauthor: Bennett Chow. Mathematical Surveys and Monographs, Vol. 110. American
Mathematical Society,
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,
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,
Currently teaching
M382D –
Differential Topology (Spring 2008)
M427K – Advanced Calculus for Applications I
(Fall 2008) Syllabus coming soon.
M427K – Advanced Calculus for Applications I
(Spring 2009)
M365G – Curves and Surfaces (Spring 2009)
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.
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.
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.