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Dan Knopf

DanAtUTProfessor

The University of Texas at Austin

Contact information

Curriculum vitae

Research interests

Research publications and preprints

Books, surveys, and expository articles

Current and recent courses

Current students

Interesting links

SAGE

Fun stuff

 

Contact information

The University of Texas at Austin

Mathematics Department, RLM 8.100

2515 Speedway, Stop C1200

Austin, TX 78712-1202

 

  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

Curriculum vitae

CV

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

  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 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(M;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. 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, 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

  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)

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) 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.

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, 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.

 

This page last updated on Sunday, July 13, 2014.
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.