Welcome to my homepage!
Dan
Knopf
Associate Professor
University of Texas
at Austin
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
Instructor
office hours: 10:00-12:00 Mondays, and by appointment
Graduate
Adviser office hours: 3:00-5:00 Mondays in RLM 8.146
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
Minimally invasive
surgery for Ricci flow singularities. Coauthors: Sigurd Angenent and M.
Cristina Caputo. J. Reine Angew. Math.
(Crelle). To appear.
Cross
curvature flow on a negatively curved solid torus. Coauthors: Jason Deblois
and Andrea Young. Algebr. Geom. Topol.
To appear.
Convergence and stability
of locally RN-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
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,
Current
and recent courses
TC310 – Modes
of Reasoning: Optimal Geometry in Nature, Art, and Mathematics (Spring
2010)
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.
The
fourth workshop will take place in early autumn, 2010. Details coming soon.
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.