Spring 2019 Analysis Student Seminar

      This semester, we will learn about regularity of minimal surfaces in higher codimension. We will start with an introduction of Geometric Measure Theory, more specifically the Federer-Fleming theory of integral currents. Then, we will cover the classical theory of regularity in codimension 1. After understanding what are the obstructions to generalize this theory to higher condimensions, we will move onto regularity of minimal surfaces in codimension bigger than 1. There are no prerequisites besides basic knowledge of functional analysis, measure theory, and differential geometry.

      References

      • "The regularity theory of minimal surfaces in higher codimension" (a survey) by Camillo De Lellis. Link here.
      • "Almgren's center manifold in a simple setting" (PCMI Lectures) by Camillo De Lellis. Link here. This set of notes come with some exercise sheets: one, two, three, and four.
      • "Almgren's Q-Valued Functions Revisited", Proceedings of the ICM, 2010, by Camillo De Lellis. Link here.
      • Many other papers, survey articles, lecture notes on related topics by De Lellis can be found here.
      • Ton of books, old and new, including Leon Simon's book on GMT.

      Schedule

      All talks will be given in 5-127 on Wednesdays at 4:00, and we'll try to end around 5:15. The following is the tentative schedule, along with the references we will follow that day.


      Date

      Speaker

      Topic

      Reading

      Feb. 6

      Silvia Ghinassi

      Federer-Fleming theory of integral currents Part I

      De Lellis Survey Section 2

      Feb. 13

      Jack Burkart

      Federer-Fleming theory of integral currents Part II

      De Lellis Survey Section 2

      Feb. 20

      Jacob Mazor

      First considerations in the regularity theory

      De Lellis Survey Section 3

      Feb. 27

      Silvia Ghinassi

      First considerations in the regularity theory - again

      De Lellis Survey Section 3

      Mar. 6

      Jordan Rainone

      The monotonicity formula and its consequences

      De Lellis Survey Section 3

      Mar. 13

      Jae Ho Cho

      The regularity theory in codimension 1

      De Lellis Survey Section 4

      Mar. 20

      Enjoy your Spring break!

      Mar. 27

      No seminar (Silvia is traveling)

      Apr. 3

      No seminar (Silvia is sick)

      Apr. 10

      No seminar (Silvia forgot)

      Apr. 17

      Silvia Ghinassi

      Federer's theorem and the failure of ε-regularity in codimension >1

      De Lellis Survey Section 4&5

      Apr. 24

      Jack Burkart

      Almgren's stratification

      De Lellis Survey Section 6

      May 1

      Paul Sweeney

      Multiple valued functions minimizing the Dirichlet energy

      De Lellis Survey Section 7

      May 8

      No seminar (Silvia's defense)