Uniqueness of weak solutions to Ricci flow, and Perelman's convergence conjecture, Bruce Kleiner.
In his proof of Thurston's geometrization conjecture, Perelman proved the existence of a Ricci flow with surgery starting from any given compact smooth Riemannian 3manifold. In the same papers, he conjectured that when the surgery parameters are sent to zero, the flow with surgery converges to a limiting "flow through singularities", yielding a canonical generalized Ricci flow. The lecture will briefly cover some background on uniqueness questions for weak solutions to geometric evolution equations (Ricci flow, mean curvature flow and harmonic map heat flow), and then discuss recent joint work of Richard Bamler and myself, giving a proof of Perelman's convergence conjecture.

Poincare inequalities via quantitative connectivity, and differentiability in metric measure spaces, Sylvester ErikssonBique.
The talk will start by discussing some background of Poincare
inequalities on metric measure spaces. We will then introduce a new
condition that is equivalent to admitting a Poincare inequality. This
metric condition is flexible enough to have a number of applications, and
we immediately obtain new classes of spaces admitting Poincare
inequalities. We also observe a connection to classical analysis on
Euclidean space through OrliczPoincare inequalities and Muckenhoupt
weights.
In the second half of the talk we introduce an asymptotic version of our
condition, and explain how it guarantees a new type of rectifiability
result in terms of spaces admitting Poincare inequalities. This result is
applied to resolve, in a particular context, a conjecture of Cheeger and
Kleiner on so called differentiability spaces. The talk should be
understandable without particular knowledge of the applications.

Differentiability and rectifiability on metric planes, Guy David.
Since the work of Cheeger, many nonsmooth metric measure spaces are now known to support a differentiable structure for Lipschitz functions. The talk will discuss this structure on metric measure spaces with quantitative topological control: specifically, spaces whose blowups are topological planes. We show that any differentiable structure on such a space is at most 2dimensional, and furthermore that if it is 2dimensional the space is 2rectifiable. This is partial progress on a question of Kleiner and Schioppa, and is joint work with Bruce Kleiner.

Fibrations, subsurface projections and veering triangulations, Yair Minsky. When a hyperbolic 3manifold fibers over the circle, its geometric features can be read from the fine structure of its monodromy map, specifically the "subsurface projections" of the stable and unstable foliations to the arc complexes of subsurfaces of the fiber. While this correspondence is useful when the topological type of the fiber is fixed, it is not wellunderstood in general. A good laboratory for studying this is a single 3manifold that fibers in infinitely many different ways, as organized by Thurston's norm on homology. In this setting there are canonical triangulations due to Agol, which can be studied very explicitly via a construction of Gueritaud. We explore how the subsurface projections of monodromies for all the fibers can be seen in the structure of this triangulation, and how this leads to a nice combinatorial picture with estimates that do not depend on complexity of the fibers. Joint work with Sam Taylor.

A new verticalversushorizontal isoperimetric inequality on the Heisenberg group, with applications to metric geometry and approximation algorithms, Assaf Naor. In this talk we will show that for every measurable subset of the
Heisenberg group of dimension at least 5, an appropriately defined notion of
its ``vertical perimeter" is at most a constant multiple of its horizontal
(Heisenberg) perimeter. We will explain how this new isoperimetrictype
inequality solves open questions in analysis (an endpoint estimate for a
certain singular integral on W^{1,1}), metric geometry (sharp
nonembeddability into L_1) and approximation algorithms (asymptotic
evaluation of the performance of the GoemansLinial algorithm for the
Sparsest Cut problem). Joint work with Robert Young.

The structure of currents in R^n and in metric spaces, Andrea Schioppa.
I will talk about recent progress in understanding the structure of normal currents in Euclidean spaces and of AmbrosioKirchheim metric currents in general metric spaces. In particular, I will focus on the solution of the following question of Giovanni Alberti: Can a kdimensional normal current be represented as an integral of krectifiable currents?
