**
Speaker: Jingrui Cheng,
**
The State University of New York at Stony Brook

**
Title:
**

**
Abstract:
**

**
Speaker: Mat Langford
**
University of Tennessee at Knoxville

**
Title:
**

**
Abstract:
**

**
Speaker: Yannick Sire, Johns Hopkins University
**

**
Title: New result on constant Q-curvature metrics
**

**
Abstract:
**
In the recent years, several major results have been obtained in the problem of finding a constant Q-curvature metric in a given conformal class in dimensions bigger than 5. This talk will cover new results concerning existence and multiplicity of such metrics. I will first present a rather general geometric approach to prove existence and multiplicity of regular metrics, giving several explicit examples. Then I will move to the case of singular metrics, i.e. complete metrics with constant Q-curvature outside of a closed set. This requires to develop several tools to handle 4th order equations (but applicable actually to higher order ones). I will also provide some explicit examples of such metrics and investigate their multiplicity. I will state open problems as well.

**
Speaker: Bruce Kleiner, Courant Institute
**

**
Title: Ricci flow and contractibility of spaces of metrics
**

**
Abstract:
**
In the lecture I will discuss recent joint work with Richard Bamler, which uses Ricci flow through singularities to construct deformations of spaces of metrics on 3-manifolds. We show that the space of metrics with positive scalar on any 3-manifold is either contractible or empty; this extends earlier work by Fernando Marques, which proved path-connectedness. We also show that for any spherical space form M, the space of metrics with constant sectional curvature is contractible. This completes the proof of the Generalized Smale Conjecture, and gives a new proof of the original Smale Conjecture for S^3.

**
Speaker: Jiaping Wang, University of Minnesota
**

**
Title: Green's function estimates and applications
**

**
Abstract:
** We plan to discuss an L^1 estimate for the Green's function on
manifolds
with positive bottom spectrum. As an application, we derive a curvature
decay estimate
for a class of steady Ricci solitons. This is a joint work with Ovidiu
Munteanu and Chiung-Jue Sung.

**
Speaker: Fang Wang, Shanghai Jiao Tong University
**

**
Title: Obata type rigidity theorems on manifold with boundary and their
applications
**

**
Abstract:
**
In this talk, I will introduce some rigidity theorems for the
(generalized)
Obata equation on manifolds with boundary with different kinds of
boundary conditions.
Then I will also give two main applications. One application is in the
rigidity theorems
of Poincare-Einstein manifolds; and the other is in the first eigenvalue
problems on
manifolds with boundary. This is joint work with Mijia Lai and Xuezhang
Chen.

**
Speaker: Paul Yang, Princeton University
**

**
Title: CR invariant area functionals and the singular CR Yamabe
solutions on the Heisenberg of $H^{1}$
**

**
**

**
Speaker: Philip Isett, MIT
**

**
Title: A Proof of Onsager's Conjecture for the Incompressible Euler Equations
**

**
Abstract:
**
In an effort to explain how anomalous dissipation of energy occurs in hydrodynamic turbulence, Onsager conjectured in 1949 that weak solutions to the incompressible Euler equations may fail to exhibit conservation of energy if their spatial regularity is
below 1/3-Holder. I will discuss a proof of this conjecture that shows that there are nonzero, (1/3-\epsilon)-Holder Euler flows in 3D that have compact support in time. The construction is based on a method known as "convex integration," which has its origins in the work of Nash on isometric embeddings with low codimension and low regularity. A version of this method was first developed for the incompressible Euler equations by De Lellis and
Szekelyhidi to build Holder-continuous Euler flows
that fail to conserve energy, and was later improved by Isett and by
Buckmaster-De Lellis-Szekelyhidi to obtain further partial results towards Onsager's conjecture. The proof of the full conjecture combines convex integration
using the "Mikado flows"
introduced by Daneri-Szekelyhidi with a
new "gluing approximation"
technique. The latter technique exploits a special structure in the linearization of the incompressible Euler equations.

**
Speaker:
Matthew J. Gursky, University of Notre Dame
**

**
Title: Some existence and non-existence results for Poincare-Einstein metrics
**

**
Abstract:
**
I will begin with a brief overview of the existence question for conformally compact Einstein manifolds with prescribed conformal infinity. After stating the seminal result of Graham-Lee, I will discuss a non-existence result (joint with Qing Han) for certain conformal classes on the 7-dimensional sphere. I will also mention some ongoing work (with Gabor Szekelyhidi) on a version of "local existence" of Poincare-Einstein metrics.

**
Speaker: Yanyan Li, Rutgers University
**

**
Title:
**
Blow up analysis of solutions of conformally invariant fully nonlinear elliptic equations

**
Abstract:
**
We establish blow-up profiles for any blowing-up sequence of solutions of genera
l conformally invariant fully nonlinear elliptic equations on Euclidean domains.
We prove that (i) the distance between blow-up points is bounded from below by
a universal positive number, (ii) the solutions are very close to a single stand
ard bubble in a universal positive distance around each blow-up point, and (iii)
the heights of these bubbles are comparable by a universal factor. As an applic
ation of this result, we establish a quantitative Liouville theorem.
This is a joint work with Luc Nguyen.

**
Speaker: Yi Wang, Johns Hopkins University
**

**
Title:
**
A fully nonlinear Sobolev trace inequality

**
Abstract:
**
The $k$-Hessian operator $sigma_k$ is the $k$-th elementary symmetric function of the eigenvalues of the Hessian. It is known that the $k$-Hessian equation $sigma_k(D^2 u)=f$ with Dirichlet boundary condition $u=0$ is variational; indeed, this problem can be studied by means of the $k$-Hessian energy $int -u sigma_k(D^2 u)$. We construct a natural boundary functional which, when added to the $k$-Hessian energy, yields as its critical points solutions of $k$-Hessian equations with general non-vanishing boundary data. As a consequence, we prove a sharp Sobolev trace inequality for $k$-admissible functions $u$ which estimates the $k$-Hessian energy in terms of the boundary values of $u$.
This is joint work with Jeffrey Case.

**
Speaker: Changfeng Gui, University of Connecticut
**

**
Title:
**
Moser-Trudinger type inequalities, mean field equations and Onsager vortices

**
Abstract:
**
In this talk, I will present a recent work confirming the best constant of a Moser-Trudinger type inequality conjectured by A. Chang and P. Yang. The proof is based on
a new and powerful lower bound of total mass for mean field equations. Other applications of the lower bound include the classification of certain Onsager vortices
on the sphere, the radially symmetry of solutions to Gaussian curvature equation on the plane, classification of solutions for mean field equations on tori and the sphere, etc.
The resolution of several interesting problems in these areas will be presented. The work is jointly done with Amir Moradifam from UC Riverside.

**
Speaker: Jacob Bernstein, Johns Hopkins University
**

**
Title:
**
Hypersurfaces of low entropy

**
Abstract:
**
The entropy is a natural geometric quantity introduced by Colding and Minicozzi and may be thought of as a rough measure of the complexity of a hypersurface of Euclidean space. It is closely related to the mean curvature flow. On the one hand, the entropy controls what types of singularities the flow develops. On the other, the flow may be used to study the entropy. In this talk I will survey some recent results with Lu Wang that show that hypersurfaces of low entropy can't be too complicated.

**
Speaker: Andrea Malchiodi, Scuola Normale Superiore
**

**
Title:
**
Embedded Willmore tori in three-manifolds with small area constraint

**
Abstract:
**
While there are lots of contributions on Willmore surfaces in the three-dimensional
Euclidean space, the literature on curved manifolds is still relatively limited. One of the main
aspects of the Willmore problem is the loss of compactness under conformal transformations.
We construct embedded Willmore tori in manifolds with a small area constraint by analyzing
how the Willmore energy under the action of the Mobius group is affected by the curvature of
the ambient manifold. The loss of compactness is then taken care of using minimization arguments
or Morse theory.

**
Speaker: Daniela De Silva, Columbia University
**

**
Title:
**
The two membranes problem

**
Abstract:
**
We will
consider the two membranes obstacle problem for two different operators,
possibly non-local. In the case when the two operators have different
orders, we discuss how to obtain $C^{1,\gamma}$ regularity of the
solutions. In particular, for two fractional Laplacians of different
orders, one obtains optimal regularity and a characterization of the
boundary of the coincidence set. This is a joint work with L.
Caffarellii and O. Savin.

**
Speaker: Gregory Seregin, The University of Oxford
**

**
Title:
**
Ancient solutions to Navier-Stokes equations

**
Abstract:
**
In the talk, I shall try to explain the relationship between the so-called ancient (backwards) solutions to the Navier-Stokes equations in the space or in a half space and the global well-posedness of initial boundary value problems for these equations. Ancient solutions itself are an interesting part of the the theory of PDE's. Among important questions to ask are classification, smoothness, existence of non-trivial solutions, etc. The latter problem is in fact a Liouville type theory for non-stationary Navier-Stokes equations. The essential part of the talk will be addressed the so-called mild bounded ancient solutions. The Conjecture is that {\it any mild bounded ancient solution is a constant}, which should be identically zero in the case of the half space. The validity of the Conjecture would rule out Type I blowups that have the same kind of singularity as possible self-similar solutions. I am going to list known cases for which the Conjecture has been proven: the Stokes system, the 2D Navier-Stokes system, axially symmetric solutions in the whole space. Very little is known in the case of the half space. Other type of ancients solutions to the Navier-Stokes equations will be mentioned as well.

**
Speaker: Fernando Marques, Princeton University
**

**
Title:
**
Multiparameter sweepouts and the existence of minimal hypersurfaces

**
Abstract:
**
It follows from the work of Almgren in the 1960s that the space
of
unoriented closed hypersurfaces, in a compact
Riemannian manifold M, endowed with the flat topology, is weakly
homotopically equivalent to the infinite dimensional real projective
space. Together with Andre Neves, we have used this nontrivial structure,
and previous work of Gromov and Guth on the associated multiparameter
sweepouts, to prove the existence of infinitely many smooth
embedded closed minimal hypersurfaces in manifolds with positive Ricci
curvature and dimension at most 7. This is motivated by a conjecture of
Yau (1982).
We will discuss this result, the higher dimensional case and current work
in progress on the problem of the Morse index.

**
Speaker: William Minicozzi, MIT
**

**
Title:
**
Uniqueness of blowups and Lojasiewicz inequalities

**
Abstract:
**
The mean curvature flow (MCF) of any closed hypersurface becomes
singular in finite time. Once one knows that singularities occur, one
naturally wonders what the singularities are like. For minimal
varieties the first answer, by Federer-Fleming in 1959, is that they
weakly resemble cones. For MCF, by the combined work of Huisken,
Ilmanen, and White, singularities weakly resemble shrinkers.
Unfortunately, the simple proofs leave open the possibility that a
minimal variety or a MCF looked at under a microscope will resemble
one blowup, but under higher magnification, it might (as far as anyone
knows) resemble a completely different blowup. Whether this ever
happens is perhaps the most fundamental question about singularities.
We will discuss the proof of this long standing open question for MCF
at all generic singularities and for mean convex MCF at all
singularities. This is joint work with Toby Colding.

**
Speaker: Luis Silvestre, University of Chicago
**

**
Title:
**
$C^{1,\alpha}$ regularity for the parabolic homogeneous
p-Laplacian equation

**
Abstract:
**
It is well known that p-harmonic functions are $C^{1,\alpha}$ regular,
for some $\alpha>0$. The classical proofs of this fact uses
variational methods. In a recent work, Peres and Sheffield construct
p-Harmonic functions from the value of a stochastic game. This
construction also leads to a parabolic versions of the problem.
However, the parabolic equation derived from the stochastic game is
not the classical parabolic p-Laplace equation, but a homogeneous
version. This equation is not in divergence form and variational
methods are inapplicable. We prove that solutions to this equation are
also $C^{1,\alpha}$ regular in space. This is joint work with Tianling Jin.

**
Speaker: Sigurd Angenent, University of Wisconsin, Madison
**

**
Title:
**
Mean Curvature Flow of Cones

**
Abstract:
**
For smooth initial hypersurfaces one has short time existence and uniqueness of solutions to Mean Curvature Flow. For general initial data Brakke showed that varifold solutions exist, but that they need not be unique if the initial data are non smooth. In this talk I will discuss the multitude of solutions to MCF that exist if the initial hypersurface is a cone that is smooth except at the origin. Some of the examples go back to older work with Chopp, Ilmanen, and Velazquez, other examples are recent.

**
Speaker: John Lott, University of California, Berkeley,
**

**
Title:
**
Geometry of the space of probability measures

**
Abstract:
**
The space of probability measures, on a compact Riemannian
manifold,
carries the Wasserstein metric coming from optimal transport.
Otto
found a remarkable
formal Riemannian metric on this infinite-dimensional space.
It is a
challenge to make
rigorous sense of the ensuing formal calculations, within the framework
of metric
geometry. I will describe what is known about geodesics, curvature,
tangent spaces
(cones) and parallel transport.

**
Speaker: Panagiota Daskalopoulos, Columbia University
**

**
Title:
**
Ancient solutions to geometric flows

**
Abstract:
**
We will discuss ancient or eternal solutions to geometric parabolic
partial differential equations.
These are special solutions that appear as blow up limits near a
singularity. They often
represent models of singularities.
We will address the classification of ancient solutions to geometric
flows such as the
Mean Curvature flow, the Ricci flow and the Yamabe flow, as well as
methods of constructing
new ancient solutions from the gluing of two or more solitons.
We will also include future research directions.

**
Speaker: Lan-Hsuan Huang, University of Connecticut
**

**
Title:
**
Geometry of asymptotically flat graphical hypersurfaces in
Euclidean space

**
Abstract:
**
We consider a special class of asymptotically flat manifolds of
nonnegative scalar curvature that can be isometrically embedded in
Euclidean space as graphical hypersurfaces. In this setting, the scalar
curvature equation becomes a fully nonlinear equation with a divergence
structure, and we prove that the graph must be weakly mean convex. The
arguments use some intriguing relation between the scalar curvature and
mean curvature of the graph and the mean curvature of its level sets.
Those
observations enable one to give a direct proof of the positive mass
theorem
in this setting in all dimensions, as well as the stability statement that
if the ADM masses of a sequence of such graphs approach zero, then the
sequence converges to a flat plane in both Federer-Flemings flat topology
and Sormani-Wenger's intrinsic flat topology.

**
Speaker: Fanghua Lin, Courant Institute
**

**
Title:
**
Large N asymptotics of Optimal partitions of Dirichlet eigenvalues

**
Abstract:
**
In this talk, we will discuss the following problem: Given a bounded domain . i
n R^n, and a positive energy N, one divides . into N subdomains, .j,j=1,2,...,N.
We consider the so-called optimal partitions that give the least possible value
for the sum of the first Dirichelet eigenvalues on these sumdomains among all a
dmissible partitions of $\Omega$.

**
Speaker: Bruce Kleiner, Courant Institute
**

**
Title:
**
Ricci flow through singularities

**
Abstract:
**
It has been a long-standing 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 3-d 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.

**
Speaker: Peter Constantin, Princeton University
**

**
Title:
**
Long time behavior of forced 2D SQG equations

**
Abstract:
**
We prove the absence of anomalous dissipation of energy for the forced
critical surface quasi-geostrophic equation (SQG) in {\mathbb {R}}^2 and
the existence of a compact finite dimensional golbal attractor in {\mathbb
T}^2. The absence of anomalous dissipation can be proved for rather rough
forces, and employs methods that are suitable for situations when uniform
bounds for the dissipation are not available. For the finite
dimensionality
of the attractor in the space-periodic case, the global regularity of the
forced critical SQG equation needs to be revisited, with a new and final
proof. We show that the system looses infinite dimensional information, by
obtaining strong long time bounds that are independent of initial data.
This is joint work with A. Tarfulea and V. Vicol.

**
Speaker: Mihalis Dafermos, Princeton University
**

**
Title:
**
The linear stability of the Schwarzschild solution under gravitational perturbations in general relativity

**
Abstract:
**
I will discuss joint work with G. Holzegel and I. Rodnianski showing the linear stability of the
celebrated Schwarzschild black hole solution in general relativity.

**
Speaker: Gaoyong Zhang, Polytechnic Institute of New York University
**

**
Title:
**
The logarithmic Minkowski problem

**
Abstract:
**
The logarithmic Minkowski problem asks for necessary
and sufficient conditions
in order that a nonnegative finite Borel measure in (n-1)-dimensional projective
space be the cone-volume measure of the unit ball of an n-dimensional Banach spa
ce. The solution to this problem is presented. Its relation to conjectured
geometric inequalities that are stronger
than the classical Brunn-Minkowski inequality
will be explained.

**
Speaker: Sergiu Klainerman, Princeton University
**

**
Title:
**
On the Reality of Black Holes

**
Speaker: Natasa Sesum, Rutgers University
**

**
Title:
**
Yamabe flow, its singularity profiles and ancient solutions

**
Abstract:
**
We will discuss conformally flat complete Yamabe flow and show that in some case
s we can give the precise description of singularity profiles close to the extin
ction time of the solution. We will also talk about a construction of new compac
t ancient solutions to the Yamabe flow.
This is a joint work with Daskalopoulos, King and Manuel del Pino

**
Speaker: Jeff Viaclovsky, University of Wisconsin-Madison
**

**
Title:
**
Critical metrics on connected sums of Einstein four-manifolds

**
Abstract:
**
I will discuss a gluing procedure designed to obtain canonical metrics on connected sums of Einstein four-manifolds. The main application is an existence result, using two well-known Einstein manifolds as building blocks: the Fubini-Study metric on CP^2, and the product metric on S^2 x S^2. Using these metrics in various gluing configurations, critical metrics are found on connected sums for a specific Riemannian functional, which depends on the global geometry of the factors.
This is joint work with Matt Gursky.

**
Speaker: Camillo De Lellis, Zurich
**

**
Title:
**
Quantitative rigidity estimates

**
Abstract:
**
For many classical rigidity questions in differential geometry it is natural to
ask to which extent they are stable. I will review several recent results in the
literature. A typical example is the following: there is a constant $C$ such th
at, if $Sigma$ is a $2$-dimensional embedded closed surface in $R^3$, then $min_
lambda |A- lambda g|_{L^2} leq C |A - {rm tr A} g/2|_{L^2}$, where $A$ is the se
cond fundamental form of the surface and $g$ the Riemannian metric as a submanif
old of $R^3$.

**
Speaker: Xiaochun Rong, Rutgers University
**

**
Title:
**
Degenerations of Ricci Flat Kahler Metrics under extremal transitions and flops

**
Abstract:
**
We will discuss degeneration of Ricci-flat Kahler metrics on Calabi-Yau manifold
s under algebraic geometric surgeries: extremal transitions or flops. We will pr
ove a version of Candelas and de la Ossa's conjecture: Ricci-flat Calabi-Yau man
ifolds related via extremal transitions and flops can be connected by a path con
sisting of continuous families of Ricci-flat Calabi-Yau manifolds and a compact
metric space in the Gromov-Hausdorff topology.
This is joint work with Yuguang Zhang

**
Speaker: Jian Song, Rutgers University
**

**
Title:
**
Analytic minimal model program with Ricci flow

**
Abstract:
**
I will introduce the analytic minimal model program proposed by Tian and myself
to study formation of singularities of the Kahler-Ricci flow. We also construct
geometric and analytic surgeries of codimension one and higher codimensions equ
ivalent to birational transformations in algebraic geometry by Ricci flow.

**
Speaker: Antonio Ache, Princeton University
**

**
Title:
**
On the uniqueness of asymptotic limits of the Ricci flow

**
Abstract:
**
Given a compact Riemannian manifold we consider a solution of a normalization of the Ricci flow which exists for all time and such that both the full curvature tensor and the diameter of the manifold are uniformly bounded along the flow. It was proved by Natasa Sesum that any such solution of the normalized Ricci flow is sequentially convergent to a shrinking gradient Ricci soliton and moreover the limit is independent of the sequence if one assumes that one of the limiting solitons satisfies a certain integrability condition. We prove that this integrability condition can be removed using an idea of Sun and Wang for studying the stability of the Kaehler-Ricci flow near a Kaehler-Einstein metric. The method relies on the monotonicity of Perelman's W-functional along the Ricci flow and a Lojasiewicz-Simon inequality for the mu-functional. If time permits we will compare this result with recent Theorems on the stability of the Ricci flow.

**
Speaker: Jie Qing, University of California, Santa Cruz
**

**
Title:
**
Hypersurfaces in hyperbolic space and conformal metrics on domains in sphere

**
Abstract:
**
In this talk I will introduce a global correspondence between properly immersed
horospherically convex hyper surfaces in hyperbolic space and complete conforma
l metrics on subdomains in the boundary at infinity of hyperbolic space. I will
discuss when a horospherically convex hypersurface is proper, when its hyperboli
c Gauss map is injective, and when it is embedded. These are expected to be usef
ul to the understandings of both elliptic problems of Weingarten hypersurfaces i
n hyperbolic space and elliptic problems of complete conformal metrics on subdom
ains in sphere.

**
Speaker: Alessio Figalli, University of Texas, Austin
**

**
Title:
**
Regularity Results For Optimal Transport Maps

**
Abstract:
**
Knowing whether optimal maps are smooth or not is an important step towards a qualitative understanding of them. In the 90's Caffarelli developed a regularity theory on R^n for the quadratic cost, which was then extended by Ma-Trudinger-Wang and Loeper to general cost functions which satisfy a suitable structural condition. Unfortunately, this condition is very restrictive, and when considered on Riemannian manifolds with the cost given by the squared distance, it is satisfied only in very particular cases. Hence the need to develop a partial regularity theory: is it true that optimal maps are always smooth outside a "small" singular set? The aim of this talk is to first review the "classical" regularity theory for optimal maps, and then describe some recent results about their partial regularity.

**
Speaker: Andre Neves, Imperial College
**

**
Title:
**
Min-max theory and the Willmore Conjecture

**
Abstract:
**
In 1965, T. J. Willmore conjectured that the integral of the square of the mean
curvature of any torus immersed in Euclidean three-space is at least 2 pi^2. I w
ill talk about my recent joint work with Fernando Marques in which we prove this
conjecture using the min-max theory of minimal surfaces.

**
Speaker: Paul Yang, Princeton University
**

**
Title:
**
Compactness of conformally compact Einstein metrics

**
Abstract:
**

**
Speaker: Ovidiu Savin, Columbia University
**

**
Title:
**
The thin one-phase problem

**
Abstract:
**
We discuss regularity properties of solutions and their free boundaries for minimizers of the thin Bernoulli problem. We show that Lipschitz free boundaries are classical and we obtain a bound on the Hausdorff dimension of the singular set of the free boundary of minimizers. This is a joint work with D. De Silva.

**
Speaker: Haim Brezis, Rutgers University
**

**
Title:
**
Sobolev maps with values into the circle

**
Abstract:
**
Real-valued Sobolev functions are well-understood and play an immense role. By c
ontrast, the theory of Sobolev maps with values into the unit circle is not yet
sufficiently developed. Such maps occur in a number of physical problems. The re
ason one is interested in Sobolev maps, rather than smooth maps is to allow maps
with point singularities, such as x/|x| in 2-d, or line singularities in 3-d wh
ich appear in physical problems. It turns out that these classes of maps have a
rich structure. Geometrical and topological effects are already conspicuous, eve
n in this very simple framework. On the other hand, the fact that the target spa
ce is the circle (as opposed to higher-dimensional manifolds) offers the option
to study their lifting and raises some tough questions in Analysis.

**
Speaker: Gang Tian, Princeton University
**

**
Title:
**
Bounding scalar curvature along Kahler-Ricci flow

**
Abstract:
**

**
Speaker: Nassif Ghoussoub , University of British Columbia
**

**
Title:
**
A self-dual polar decomposition for vector fields

**
Abstract:
**
I shall explain how any non-degenerate vector field on a bounded domain of $R^n
$ is monotone modulo a measure preserving involution $S$ (i.e., $S2=Identity$).
This is to be compared to Brenier's polar decomposition which yields that any su
ch vector field is the gradient of a convex function (i.e., cyclically monotone)
modulo a measure preserving transformation. Connections to mass transport --whi
ch is at the heart of Brenier's decomposition-- is elucidated. This is joint wor
k with A. Momeni.

**
Speaker: Aaron Naber, MIT
**

**
Title:
**
Quantitative Stratification and regularity for Einstein manifolds, harmonic maps and minimal surfaces

**
Abstract:
**
In this talk we discuss new techniques for taking ineffective local, e.g. tangent cone, understanding and deriving from this effective estimates on regularity. Our primary applications are to Einstein manifolds, harmonic maps between Riemannian manifolds, and minimal surfaces. For Einstein manifolds the results include, for all p<2, 'apriori' L^p estimates on the curvature |Rm| and the much stronger curvature scale r_{|Rm|}(x)=max{r>0:sup_{B_r(x)}|Rm|leq r^{-2}}. If we assume additionally that the curvature lies in some L^q we are able to prove that r^{-1}_{|Rm|} lies in weak L^2q. For minimizing harmonic maps f we prove W^{1,p}cap W^{2,p/2} estimates for p<3 for f and the stronger likewise defined regularity scale. These are the first gradient estimates for p>2 and the first L^p estimates on the hessian for any p. The estimates are sharp. For minimizing hypersurfaces we prove L^p estimates for p<7 for the second fundamental form and its regularity scale. The proofs include a new quantitative dimension reduction, that in the process stengthens hausdorff estimates on singular sets to minkowski estimates.
This is joint work with Jeff Cheeger.