**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.