SCIENTIFIC PROGRAMS AND ACTIVITIES

January 21, 2018

THE FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES

July-December 2014
Thematic Program on
Variational Problems in Physics, Economics and Geometry

November 6-7
Minischool on Variational Problems in Geometry

Organizing Committee
  Almut Burchard (Toronto)
Panagiota Daskalopoulos (New York)
Young-Heon Kim (Vancouver)
Ludovic Rifford (Nice)
Neil Trudinger (Canberra)

PROVISIONAL SPEAKERS, TITLE AND ABSTRACT INFO FOR MINISCHOOL

Thursday, November 6 (Stewart Library)

8:00-8:45

Morning coffee and on site registration

8:45-9;00

Welcoming Remarks

9:00-10:30

Ludovic Rifford (University of Nice, Sophia Antipolis) (SLIDES)
Geometric control and Applications 1

10:30-11:00

Coffee break

11:00-12:00

Gerhard Huisken (University of Tuebingen)
Mean curvature flow with surgery 1

12:00-2:00

Lunch

2:00-3:00

Almut Burchard (University of Toronto)
Geometric stability problems for non-local functionals 1

3:00-3:30

Tea break
3:30-5:00
Young-Heon Kim (University of British Columbia)
Monge-Ampere type equations 1
Friday, November 7 (Stewart Library)
9:00-10:30

Almut Burchard (University of Toronto)
Geometric stability problems for non-local functionals 2

10:30-11:00

Coffee break
11:00-12:00
Ludovic Rifford (University of Nice, Sophia Antipolis)(SLIDES)
Geometric control and Applications 2
12:00-2:00
Lunch

2:00-3:00

Young-Heon Kim (University of British Columbia)
Monge-Ampere type equations 2

3:00-3:30

Tea break
3:30-5:30

Gerhard Huisken (University of Tuebingen)
Mean curvature flow with surgery 2

 

Almut Burchard (University of Toronto)
Geometric stability problems for non-local functionals

Geometric stability results where a "deficit" (the deviation of a functional from its optimal value) controls some measure of "asymmetry" (the distance from the family of optimizers) have been established for many classical inequalities, starting with Bonnesen's quantitative improvements of the isoperimetric inequality in the plane from the 1920s. Most results in that direction since the 1990s are for "local" functionals that involve gradients. Less is known for "non-local" functionals that involve convolutions, even though stability of those has important applications in Mathematical Physics. I will describe some open problems and very recent advances in the field.
References:
1) F. Maggi, Some methods for studying stability in isoperimetric type problems, Bull. Amer. Math. Soc. 45:367-408 (2008).
2) E. A. Carlen, Lecture notes from the Ischia Summer School, (2011).
3) M. Christ, Near equality in the Riesz-Sobolev inequality. Preprint arXiv:1309.5856 (2013).
4) A. Figalli and D. Jerison,Quantitative stability for sumsets in R^n.JEMS, to appear (2014).
5) A. Burchard and G. R. Chambers, Geometric stability of the Coulomb energy. Preprint arXiv:1407.1918 (2014).

Gerhard Huisken (University of Tuebingen)
Mean curvature flow with surgery

We describe joint work with Simon Brendle on mean curvature flow with surgery for embedded meanconvex surfaces in 3-manifolds.
This builds on older work with C. Sinestrari in the higher dimensional case plus several new estimates: for embedded surfaces, gradient estimates, and a new pseudolocality principle, and includes both analytical and geometric aspects.

Ludovic Rifford (University of Nice, Sophia Antipolis)
Geometric control and Applications

This course is an introduction to geometric control theory with an emphasis on applications. Geometric control refers to the study of control systems, that is dynamical systems with parameters, on finite-dimensional manifolds. Given a point, each open loop control gives rise to a trajectory starting from that point. Under what assumptions can we insure that this point can be joined to any other point through a trajectory of the control system ? This is a controllability problem. How to do it in an optimal way with respect to some cost ? This is an optimal control problem. We will focus on the study on controllability and optimality properties of control systems and their applications in analysis and dynamics.

Young-Heon Kim (University of British Columbia)
Monge-Ampere type equations

Optimal transportation theory studies phenomena where mass distributions are matched in an efficient way, with respect to a given transportation cost. In the most standard case, optimal transport maps are given by the gradients of convex functions that solve the Monge-Ampere equation. We explain some of the most basic concepts and techniques for regularity theory of the Monge-Ampere equation. This minicourse is intended for people who do not have background either in optimal transport or in fully nonlinear PDEs.

 




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