Welcome to my homepage!
Dan Knopf
Professor, Department of Mathematics
The University of Texas at Austin
Research publications and preprints
Books, surveys, and expository articles
Contact information
- Office location: PMA 9.152
- Email address: danknopf {at} math {dot} utexas {dot} edu
- Phone number: 512.471.8131
- Fax number: 512.471.9038
- Office hours: By appointment.
- Mailing address:
Mathematics Department, PMA 8.100
2515 Speedway, Stop C1200
Austin, TX 78712-1202
Curriculum vitae
Research interests
- Geometric analysis
- Differential geometry
- Geometric partial differential equations
I am a member of the Geometry research group at UT-Austin. I also interact with our research groups in Partial Differential Equations and Topology.
Research publications and preprints
Asymptotic behavior of unstable perturbations of the Fubini-Study metric in Ricci flow. Coauthors: David Garfinkle, James Isenberg, and Haotian Wu. Submitted (arxiv:2403.06427v1)
A numerical stability analysis of mean curvature flow of noncompact hypersurfaces with Type-II curvature blowup: II. Coauthors: David Garfinkle, James Isenberg, and Haotian Wu. Exp. Math. (2023) (DOI: 10.1080/10586458.2023.2201958)
Ricci solitons, conical singularities, and nonuniqueness. Coauthor: Sigurd Angenent. Geom. Funct. Anal. (GAFA) 32 (2022), no. 3, 411-489. (DOI:10.1007/s00039-022-00601-y)
Singularity formation of complete Ricci flow solutions. Coauthors: Timothy Carson, James Isenberg, and Natasa Sesum. Adv. Math. 403 (2022) 108326.
A numerical stability analysis of mean curvature flow of noncompact hypersurfaces with Type-II curvature blowup. Coauthors: David Garfinkle, James Isenberg, and Haotian Wu. Nonlinearity 34 (2021), no. 9, 6539-6560. (DOI:10.1088/1361-6544/ac15a9)
Non-Kaehler Ricci flow singularities modeled on Kaehler-Ricci solitons. Coauthors: James Isenberg and Natasa Sesum. Pure Appl. Math. Q. 15 (2019), no. 2, 749-784.
Dynamic instability of CPN under Ricci flow. Coauthor: Natasa Sesum. J. Geom. Anal. 29 (2019), no. 1, 902-916. (DOI: 10.1007/s12220-018-0022-6)
Sphere Bundles with 1/2-pinched Fiberwise Metrics. Coauthors: Thomas Farrell, Zhou Gang, and Pedro Ontaneda. Trans. Amer. Math. Soc. 369 (2017), no. 9, 6613-6630.
Ricci flow neckpinches without rotational symmetry. Coauthors: James Isenberg and Natasa Sesum. Comm. Partial Differential Equations 41 (2016), no. 12, 1860-1894.
Universality in mean curvature flow neckpinches. Coauthor: Zhou Gang. Duke Math. J. 164 (2015), no. 12, 2341-2406.
Neckpinch dynamics for asymmetric surfaces evolving by mean curvature flow. Coauthors: Zhou Gang and Israel Michael Sigal. Mem. Amer. Math. Soc. 253 (2018), no. 1210, 1-78.
Degenerate neckpinches in Ricci flow. Coauthors: Sigurd Angenent and James Isenberg. J. Reine Angew. Math. (Crelle) 709 (2015), 81-117.
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 096253.
Positivity of Ricci curvature under the Kaehler-Ricci flow. Commun. Contemp. Math. 8 (2006), no. 1, 123-133.
Corrigendum to: An example of neckpinching for Ricci flow on Sn+1.
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, Vol. 206. American Mathematical Society, Providence, RI, 2015.
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.
Recent courses
M310P – Plan II Mathematics: Through the Lens of Mathematics (Fall 2019)
M310P – Plan II Mathematics: Through the Lens of Mathematics (Fall 2018)
M310P – Plan II Mathematics: Through the Lens of Mathematics (Fall 2017)
M310P – Plan II Mathematics: Through the Lens of Mathematics (Fall 2016)
M427J – Differential Equations with Linear Algebra – Math Honors (Spring 2016)
M310P – Plan II Mathematics: Through the Lens of Mathematics (Fall 2015)
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
Max Stolarski (PhD, May 2019)
Tim Carson (PhD, May 2018)
Haotian Wu (PhD, May 2013)
Davi Maximo (PhD, May 2013)
Michael Bradford Williams (PhD, May 2011)
Bradley Anderson (MA, May 2008)
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, Beverly Hills, Buffalo, Canadian, China, Cologne, Corinth, Dublin, Earth, Edinburg, Egypt, Holland, Iraan, Italy, London, Memphis, Miami, Moscow, Nevada, Newark, Palestine, Paris, Pasadena, Princeton, Rhome, San Diego, Santa Fe, Scotland, and Turkey.
Here is an example of how not to teach math.
And here is a resource in case you feel a post-modernist urge to deconstruct LaTeX.
The Klein Bottle Company is my favorite source for non-orientable 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.