The schedule is subject to change. Check back for changes and updates
|Feb 6||Introduction, I||Background survey 📎||2/§1.1-1.3, 4|
|Feb 8||Introduction, II||Assigment 1 📎||2/§2.1-2.4, §3.1-3.2, 4, 5, 7|
|Feb 13||Arzelà-Ascoli, Hölder spaces (Patrick, Yonah)||1/§7, 11/§1|
|Feb 15||Existence of CSF solutions (June, Kundan)||Assigment 2 📎||4/§1.1-1.2|
|Feb 20||No class (institute policy)|
|Feb 22||Parabolic maximum principles (Bernardo, Jose)||Assigment 3 📎||3/§7.1.4, 4/§1.2, 13|
|Feb 27||Uniqueness, embeddedness, convexity in CSF (Andrew, Julius)||4/§1.1, 1.3, 3.2|
|Mar 1||Controlling all derivatives in CSF by the curvature (June)||Assignment 4 📎||8/Prop. 3.22|
|Mar 6||Gage-Hamilton, I: convex curves, maximal existence (Bernardo, Patrick)||Project proposal 📎||4/§3.1, 3.2|
|Mar 8||Gage-Hamilton, II: convex curves shrink to a point (Yonah, Jose)||Assignment 5 📎||4/§3.2, 3.3|
|Mar 13||No class (institute snow day)|
|Mar 15||Gage-Hamilton, III: renormalized flows, I (Kundan, Julius)||Assignment 6 📎||4/§3.3|
|Mar 20||Gage-Hamilton, IV: renormalized flows, II (Andrew, June)||4/§3.3|
|Mar 22||Submanifolds: introduction, Gauss map, mean curvature (Bernardo, Jose)||Assignment 7 📎||2/§2.2-2.4, 3.2|
|Mar 27||No class (holiday)|
|Mar 29||No class (holiday)|
|Apr 3||Submanifolds: integration, first variation (Yonah, Julius)||14/§2.2, 1/§10|
|Apr 5||Submanifolds: differentiation, maximum principles, MCF (Andrew, Patrick)||Assignment 8 📎||14/§2.2, 2.3|
|Apr 10||MCF: Huisken's monotonicity formula, Gauss density (Julius, Kundan)||14/§3.1, 3.4, 3.5, 3.7|
|Apr 12||MCF: Type I/II singularities, parabolic rescaling, shrinkers (Bernardo, Jose)||Project draft 1||14/§4.1, 2.4, 5/§2.4|
|Apr 17||No class (holiday)|
|Apr 19||Writing workshop|
|Apr 24||Minimal surfaces: examples, maximum principle implications (Kundan, Yonah)||7/§1.2, 1.3.1|
|Apr 26||Minimal surfaces: monotonicity, isothermal coordinates (June, Patrick)||7/§1.3.2, 6/§3, 4, 8|
|May 1||Conclusion, I: minimal/conformal/harmonic maps, Weierstrass-Enneper representation||Project draft 2|
|May 3||Conclusion, II: second variation formula, stability, index, open questions|
|May 8||Project Presentations: Kundan, Julius|
|May 10||Project Presentations: Bernardo, Jose|
|May 15||Project Presentations: June, Patrick|
|May 17||Student Presentations: Andrew, Yonah||Project in final form|
The following texts are on reserve at the library:
- W. Rudin - Principles of mathematical analysis. Third edition. International Series in Pure and Applied Mathematics. McGraw-Hill Book Co., New York-Auckland-Düsseldorf, 1976.
- M. P. do Carmo - Differential geometry of curves and surfaces. Prentice-Hall, Inc., Englewood Cliffs, N.J., 1976.
- L. C. Evans - Partial differential equations. Second edition. Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 2010.
- K.-S. Chou, X.-P. Zhu - The curve shortening problem. Chapman & Hall/CRC, Boca Raton, FL, 2001.
- C. Mantegazza - Lecture notes on mean curvature flow. Progress in Mathematics, 290. Birkhäuser/Springer Basel AG, Basel, 2011.
- R. Osserman - A survey of minimal surfaces. Dover Publications, Inc., New York, 1986.
- T. H. Colding, W. P. Minicozzi II - A course in minimal surfaces. Graduate Studies in Mathematics, 121. American Mathematical Society, Providence, RI, 2011.
- K. Ecker - Regularity theory for mean curvature flow. Progress in Nonlinear Differential Equations and their Applications, 57. Birkhäuser Boston, Inc., Boston, MA, 2004.
Some additional references:
- B. White - Topics in mean curvature flow (Spring 2015 notes by O. Chodosh). Link 📎.
- B. White - Topics in minimal surfaces (Spring 2013 notes by O. Chodosh, C. Mantoulidis). Link 📎.
- T. Tao - 245C, Notes 4. Link 📎.
- R. Haslhofer - Lectures on curve shortening flow. Link 📎.
- L. Nirenberg - A strong maximum principle for parabolic equations. Comm. Pure Appl. Math. 6 (1953), no. 2, 167--177. Link 📎.
- C. Mantoulidis - Regularity of hypersurfaces in Rn+1 moving by mean curvature flow. Undergraduate honors thesis, Stanford University, 2011. (Based on .) Link 📎.