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: 3:00-5:00 Tuesdays and by
appointment
Graduate
Adviser office hours: 3:00-5:00 Wednesdays 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
Degenerate neckpinches in Ricci flow. Coauthors: Sigurd Angenent and James
Isenberg.
Neckpinch
dynamics for asymmetric surfaces evolving by mean curvature flow.
Coauthors: Zhou Gang and Israel Michael Sigal.
Formal matched asymptotics for degenerate Ricci flow neckpinches.
Coauthors: Sigurd Angenent
and James Isenberg. Nonlinearity 24 (2011), 2265-2280.
Minimally invasive
surgery for Ricci flow singularities. Coauthors: Sigurd
Angenent and M. Cristina Caputo. J. Reine Angew.
Math. (Crelle). In press.
Cross
curvature flow on a negatively curved solid torus. Coauthors: Jason Deblois and Andrea Young. Algebr. Geom. Topol. 10 (2010), 343-372.
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
Neckpinching for asymmetric surfaces moving by mean
curvature. Nonlinear Evolution
Problems. Oberwolfach Reports. To appear.
The
Ricci Flow: Techniques and Applications, Part IV: Long Time Solutions and
Related Topics. 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. To appear.
The Ricci Flow:
Techniques and Applications, Part III: Geometric-Analytic 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. 163. American Mathematical Society,
Providence, RI, 2010.
The Ricci Flow:
Techniques and Applications, Part II: Analytic 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. 144. American Mathematical Society,
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
M427K
– Advanced Calculus for Applications I – Math
Honors (Spring 2013)
TC310
– Modes of Reasoning: Optimal Geometry in Nature, Art, and Mathematics
(Fall 2012)
M427K
– Advanced Calculus for Applications I – Math
Honors (Spring 2012)
M427K –
Advanced Calculus for Applications I (Fall 2011)
M408C
– Differential and Integral Calculus (Spring 2011)
M392C –
Riemannian Geometry (Fall 2010)
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)
Current and former students
Haotian
Wu (PhD May 2013, Oregon postdoc 2013–16)
Davi
Maximo (PhD May 2013, Stanford postdoc 2013–16)
Michael Bradford Williams (PhD
May 2011, UCLA postdoc 2011–14)
Interesting links
My
wife, Stephanie
Cawthon, is also a faculty member at UT.
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 took place September 1-3, 2010. Its topics included: Optimal Transport and Riemannian Geometry,
in particular, Bakry-Emery curvature, Ma-Trudinger-Wang
curvature, and applications to Ricci flow. Click here to learn
more.
The
fifth (and final) workshop took place January 11-12, 2012. Its topics were: Mean curvature flow and its stationary
solutions. 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, Buffalo, China, Cologne, Corinth, Dublin, Earth, Egypt, Holland, Iraan, Italy, London, Memphis,
Miami, Moscow, Nevada, Newark, Palestine, Paris, Pasadena, Princeton, Rhome, San
Diego, Scotland,
and Turkey.
Here
is an example of how not
to teach math.
Here
are some tongue-in-cheek applications
of graduate mathematics.
And
here is a resource in
case you feel a post-modernist urge to deconstruct LaTeX.
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.