Research publications and preprints

Books, surveys, and expository articles

The

Mathematics Department, RLM 8.100

2515 Speedway, Stop C1200

§ Office: RLM 9.152

§ Email: danknopf {at} math {dot} utexas {dot} edu

§ Phone: 512.471.8131

§ Fax: 512.471.9038

§ Instructor office hours: 1:30-3:30 Thursdays and by appointment

§ Graduate Adviser office hours: 2:00-4:00 Wednesdays in RLM 8.146

§ 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.

§ Ricci flow neckpinches without rotational symmetry. Coauthors: James Isenberg and Natasa Sesum.

§ Universality in mean curvature flow neckpinches. Coauthor: Zhou Gang.

§ Neckpinch dynamics for asymmetric surfaces evolving by mean curvature flow. Coauthors: Zhou Gang and Israel Michael Sigal.

§
Degenerate neckpinches in Ricci flow. Coauthors: Sigurd Angenent and James
Isenberg. **J. Reine
Angew. Math. (Crelle)**
To appear.

§
Minimally invasive
surgery for Ricci flow singularities. Coauthors: Sigurd
Angenent and M. Cristina Caputo. **J. Reine Angew.
Math. (Crelle)** **672** (2012) 39-87.

§
Formal matched asymptotics for degenerate Ricci flow neckpinches.
Coauthors: Sigurd Angenent
and James Isenberg. **Nonlinearity** **24** (2011), 2265-2280.

§
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 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 S^{n+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 H^{1}(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.

§
Neckpinching for asymmetric surfaces moving by mean
curvature. *Nonlinear Evolution
Problems. *Mathematisches Forschungsinstitut
Oberwolfach Report No. 26/2012. (DOI:
10.4171/OWR/2012/26)

§
__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 *Mathematical
Surveys and Monographs, *Vol. 135. American Mathematical Society,

§
An introduction to
the Ricci flow neckpinch. Geometric Evolution
Equations. Edited by Shu-Cheng Chang, Bennett Chow,
Sun-Chin *Contemporary
Mathematics. *Vol. 367, 141-148. American Mathematical Society,

§
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,

§ M310P - Plan II Mathematics: Through the Lens of Mathematics (Fall 2014)

§ M427K - Advanced Calculus for Applications I – Math Honors (Spring 2014)

§ M427K – Advanced Calculus for Applications I (Fall 2013)

§ M427K
– Advanced Calculus for Applications I – *Math Honors* (Spring 2013)

§ TC310 – Plan II 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 – Plan II 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)

§ 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)

§ My wife, Stephanie Cawthon, is also a faculty
member at UT.

**SAGE** (__S__ymposia
on __A__nalysis of __G__eometric __E__volution) was the name of a
series of annual workshops hosted by UT-Austin and designed to integrate
research, graduate education, and undergraduate outreach. SAGE was 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.

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, Edinburg, 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.