Schedule
The schedule is subject to change. Check back for changes and updates
Date | Description | Due | References used |
---|---|---|---|
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 |
References
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 R^{n+1} moving by mean curvature flow. Undergraduate honors thesis, Stanford University, 2011. (Based on [8].) Link 📎.