Schedule

The schedule is subject to change. Check back for changes and updates

Semester Timeline
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:

  1. W. Rudin - Principles of mathematical analysis. Third edition. International Series in Pure and Applied Mathematics. McGraw-Hill Book Co., New York-Auckland-Düsseldorf, 1976.
  2. M. P. do Carmo - Differential geometry of curves and surfaces. Prentice-Hall, Inc., Englewood Cliffs, N.J., 1976.
  3. L. C. Evans - Partial differential equations. Second edition. Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 2010.
  4. K.-S. Chou, X.-P. Zhu - The curve shortening problem. Chapman & Hall/CRC, Boca Raton, FL, 2001.
  5. C. Mantegazza - Lecture notes on mean curvature flow. Progress in Mathematics, 290. Birkhäuser/Springer Basel AG, Basel, 2011.
  6. R. Osserman - A survey of minimal surfaces. Dover Publications, Inc., New York, 1986.
  7. T. H. Colding, W. P. Minicozzi II - A course in minimal surfaces. Graduate Studies in Mathematics, 121. American Mathematical Society, Providence, RI, 2011.
  8. 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:

  1. B. White - Topics in mean curvature flow (Spring 2015 notes by O. Chodosh). Link 📎.
  2. B. White - Topics in minimal surfaces (Spring 2013 notes by O. Chodosh, C. Mantoulidis). Link 📎.
  3. T. Tao - 245C, Notes 4. Link 📎.
  4. R. Haslhofer - Lectures on curve shortening flow. Link 📎.
  5. L. Nirenberg - A strong maximum principle for parabolic equations. Comm. Pure Appl. Math. 6 (1953), no. 2, 167--177. Link 📎.
  6. C. Mantoulidis - Regularity of hypersurfaces in Rn+1 moving by mean curvature flow. Undergraduate honors thesis, Stanford University, 2011. (Based on [8].) Link 📎.