VIGRE (2014/5) - Minimal Surfaces, Conformal Maps and friends.

Group leader: Dr. Saar Hersonsky

Time/Place: R, 2:15-3:30, Boyd 326.

Introduction: Minimal surfaces can be tracked back to Euler and Lagrange and the beginning of the calculus of variations - a mathematical discipline that may be described as a general theory of extreme values. The name of this discipline does not reflect the type of problems it is connected with but with rather a specific technique that is widely used in it - the technique of variation. This will discuss in length, via severalcalssical examples, so as we derive certain necessary conditions for the existence of extreme values.

What are "variational problems"? The Dirichlet problem, Minimal surfaces, isoperimetric problems, existence of geodesics in Riemannian manifolds, and optimal control problems are just a few classical and important examples. An exciting new theme in this discipline, which will start our study, is related to the theme of recognizing conformal maps and constant curvature in two/three dimensions via discrete schemes. This is intimately related to Cannon's Conjecture, Ricci flow on surfaces, and discrete harmonic maps.

Students will be expected to choose a paper/topic of interest, discuss it with me and to give lectures (1-2) to the group later on during the semester.

 

 

Date

Speaker

 Topic
Fall 2014    
8.21.14 Dr. Saar Hersonsky Introuduction
8.28.14 Eric Burgess (UGA) Introduction to the Steiner Problem
9.4.14 Canceled  
9.11.14 Eric Burgess (UGA) Continued
9.18.14 Tom Needham (UGA) Basic notions of the geometry of surfaces
9.25.14 Tom Needham (UGA) Continued
10.2.14 Tom Needham (UGA) First example of Minimal Surfaces - The Catenoid
10.9.14 Eric Perkerson (UGA)
 The Helicoid
10.16.14 Eric Perkerson (UGA) continued.
10.30.14  Harrison Chapman (UGA) The Unified Surface Ricci Flow (after Zhang, Guo, Luo, Yau, and Gu)
11.6.14  Harrison Chapman (UGA)
 Continued.
11.13.14 Alex Newman (UGA) The minimal surface equation.
11.20.14  Alex Newman (UGA) Bernstein's problem; The Scherk surface.
     
Spring 2015    
1.8.15    
1.15.15    
1.22.15    
1.29.15    
2.5.15    
2.12.15    
2.19.15    
2.26.15    
3.5.15    
3.19.15    
3.26.15    
4.2.15    
4.9.15    
4.16.15    
4.23.15    

References:

Here are a few sources for you to look at. They are arranged according to the order of the talks I have (plan to) given so far. As we progress, expect a few more.

A Course in Minimal Surfaces, by T. Colding and W. P. Minicozzi II.

Minimal Surfaces, by U. Dierkes, S. Hildebradet and F. Sauvigny.

Dirichlet's Principle, Conformal Mapping, and Minimal Surfaces, by R. Courant.

Elements of the geometry and topology of minimal surfaces in three-dimensional space, by A.T. Fomenko and A.Tuzhilin

Squaring rectangles: the finite Riemann mapping theorem, by J. Cannon, W. Floyd, and W. Parry, Contemp. Math., 169, Amer. Math. Soc., Providence, RI, 1994.

The Theory of Negatively Curved Spaces and Groups, by J. Cannon in Ergodic Theory, Symbolic dynamics, and hyperbolic spaces, Edited by T. Bedford, M. Keane and C. Series, Oxford University Press 1991.

Boundary value problems on planar graphs and flat surfaces with integer cone singularities, I: The Dirichlet problem, by S. Hersonsky, J. Reine Angew. Math. 670 (2012) 65--92.

The unified surface Ricci flow, by M. Zhang, R. Guo, W. Zeng, F. Luo, S.T. Yau and X. Gu, Graphical Models Vol 76 Issue 5, (2014) 321--339.