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 -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)
|
|
|