On the regularity of varifold mean curvature flow, Yoshi Tonegawa.
In his wellknown book published in 1978, Brakke initiated the theory of
mean curvature flow (MCF) using the notion of varifold and studied the
existence and regularity properties. While there have been many advances
in the understanding on his version of MCF, the full proof of his
regularity theorem remained out of reach even for specialists of MCF.
Recently we gave a new and complete proof of Brakke's regularity theorem
which also comes with a natural generalization. The generalization fits
well with its stationary counterpart, Allard's regularity theorem, and
is useful to prove the partial regularity of Brakke's MCF in general
Riemannian manifolds. Starting from the definition, I will explain the
outline of the proof of the regularity theorem.

Structure of measures in Lipschitz differentiability spaces, David Bate.
This talk will present results showing the equivalence of two
very different ways of generalising Rademacher's theorem to metric
measure spaces. The first was introduced by Cheeger and is based upon
differentiation with respect to another, fixed, chart function. The
second approach is new for this generality and originates in some ideas
of Alberti. It is based upon forming partial derivatives along a very
rich structure of Lipschitz curves, analogous to the differentiability
theory of Euclidean spaces. By examining this structure further, we
naturally arrive to several descriptions of Lipschitz differentiability
spaces.

Convergence of harmonic maps, Zahra Sinaei.
In this talk I will present a compactness theorem for a sequence of
harmonic maps which are defined on a converging sequence of Riemannian
manifolds. The sequence of manifolds will be considered in the space of
compact ndimensional Riemannian manifolds with bounded sectional
curvature and bounded diameter, equipped with measured GromovHausdorff
topology.

Vector fields on
metric measure spaces, and 1rectifiable structure, Andrea Schioppa.
In this talk we describe a correspondence between two
different structures associated to a metric measure space (X,mu):
Weaver derivations and Alberti representations. The module of Weaver
derivations is an algebraic structure which describes, roughly speaking,
the measurable vector fields on (X,mu). An Alberti representation of
the measure mu is a generalized Lebesgue decomposition of mu in terms
of 1rectifiable measures. As an application of this correspondence we
obtain a characterization of the differentiability spaces in the sense
of Cheeger which is, roughly speaking, a quantitative version of a
recent characterization due to Bate.

Regularity of isometries of subRiemannian manifolds, Enrico Le Donne.
We consider Lie groups equipped with distances for which every pair of
points can be join with an arc with length equal to the distance of
the two points. These distances are generalizations of Riemannian
distances. They are completely described as subFinsler structures, by
the work of Gleason, Montgomery, Zippin, and Berestowski. We are
interested in studying the isometries of such metric spaces. As for
the Riemannian case, we show that a (global) isometry is uniquely
determined by the blownup map at a point. The blownup map is an
isometry between the tangent metric spaces, which in this case are
particular groups called Carnot groups. Generalizing a result of U.
Hamenstädt, we also show that an isometry between open sets of Carnot
groups are affine maps. A key point in the argument is in showing
smoothness of such isometries.
The work is in collaboration with L. Capogna and A. Ottazzi..

Gluing constructions for constant mean curvature (hyper)surfaces, Christine Breiner.
Constant mean curvature (CMC) surfaces are critical points to the area
functional subject to an enclosed volume constraint. Classic examples
include the round sphere, the cylinder, and a family of rotationally
symmetric solutions discovered by Delaunay. More than 150 years later,
Kapouleas determined a generalized gluing construction that produced
infinitely many new examples of CMC surfaces. Building on and refining this
work, we produce infinitely many new embedded CMC surfaces and
hypersurfaces. In this talk I will outline the main steps of the gluing
construction and explain some of the difficulties involved in solving such a
problem. This work is joint with N. Kapouleas.

Ricci flow through singularities, Bruce Kleiner.
It has been a longstanding problem in geometric analysis to find a good definition of generalized solutions to the Ricci flow equation that would formalize the heuristic idea of flowing through singularities. I will discuss a notion in the 3d case that has good analytical properties, enabling one to prove existence and compactness of solutions, as well as a number of structural results. It may also be used to partly address a question of Perelman concerning the convergence of Ricci flow with surgery to a canonical flow through singularities.
This is joint work with John Lott.

Embedded minimal tori in S^3, Simon Brendle.
In 1970, Blaine Lawson constructed an infinite family of embedded
minimal surfaces in S^3 which have higher genus. He also proved that there
are many immersed minimal surfaces in S^3 of genus 1. We show that there is
only one embedded minimal surface of genus 1 up to ambient isometries. The
proof involves an application of the maximum principle to a function that
depends on pairs of points.

The hypo elliptic Laplacian in real and complex geometry, JeanMichel Bismut.
The hypoelliptic Laplacian is supposed to be a deformation of a standard
elliptic Laplacian, that acts on the total space of the tangent bundle of a
Riemann manifold, and interpolates between the elliptic Laplacian and the
generator of the geodesic flow. Its construction involves a deformation of
the underlying geometric and analytic structures. The hypoelliptic Laplacian
is neither elliptic nor selfadjoint in the classical sense, but it is
selfadjoint with respect to a Hermitian form of signature
$(\infty,\infty)$. On locally symmetric spaces, the hypoelliptic deformation
preserves the spectrum of the elliptic Laplacian.
I will explain its construction in the case of the circle, its applications
to Selberg's trace formula, and also to the proof of a
RiemannRochGrothendieck theorem in BottChern cohomology.

Homotopical effects of kdilation, Larry Guth.
The kdilation of a map measures how much it stretches
kdimensional areas. If Dil_k f < L, then it means that for any
kdimensional submanifold S in the domain, Vol_k (f(S)) is at most L
Vol_k(S). We discuss how the kdilation restricts the homotopy type of a
map. Our main theorem concerns maps between unit spheres, from S^{m} to
S^{m1}. If k > (m+1)/2, then there are homotopically nontrivial maps S^m
to S^{m1} with arbitrarily small kdilation. If k is at most (m+1)/2,
there every homotopically nontrivial map from S^m to S^{m1} has kdilation
at least c(m) > 0.
In this talk, I want to focus on the case k at most (m+1)/2. The
nontrivial homotopy type of a map S^m to S^{m1} is detected by a certain
Steenrod square. The main issue is how to connect Steenrod squares with
quantitative estimates about kdimensional volumes. This involves a mix of
topology and geometry  on the geometrical side the tools are related to
isoperimetric inequalities/geometric measure theory. (We don't assume
familiarity with Steenrod squares.).
