Schedule, Notes/Handouts & Videos

Over the past few decades uniformly rectifiability emerged as a natural geometric condition, necessary and sufficient for classical estimates in harmonic analysis, boundedness of singular integrals in L^2, and, in the presence of some background topological assumptions, for suitable scale invariant estimates on harmonic functions closely related to the solvability of the Dirichlet problem. In the first lecture will discuss the state of the art in the case of co-dimension one. The second one will concentrate on the new analogue of harmonic measure, recent results, and many remaining mysteries for sets of higher co-dimension

Inequalities with full affine invariance are rare. Fundamental examples include the inequalities of Brunn-Minkowski, Young, Riesz-Sobolev, and Hausdorff-Young. For each of these, a sharp form with an optimal constant is known, including a characterization of all extremizing functions, or sets. This course will discuss refinements of some of these sharp inequalities. These refinements quantify the uniqueness of extremizers. The first lecture will be a general introduction, reviewing several inequalities, stating refinements, and introducing associated ideas. The second lecture will outline a proof of a sharpened Riesz-Sobolev inequality. In contrast to earlier work of the speaker which focused on the exploitation of ideas from (finite) additive combinatorics, this proof is rooted firmly in the continuum. It emphasizes the role of the affine group as a symmetry group of the set of cosets of one-parameter subgroups of the Euclidean group.

In the last few years, Jean Bourgain and Ciprian Demeter have proven a variety of striking ``decoupling'' theorems in Fourier analysis. I think this is an important development in Fourier analysis. As a corollary, they were able to give very sharp estimates for the L^p norms of various trigonometric sums. These sums appear in PDE when one studies the Schrodinger equation on a torus, and they appear in analytic number theory in connection with the circle method. In this first lecture, I will explain what decoupling theorems say, look at some examples, and discuss applications. I will try to describe why I think the theorems are important, and to say something about what makes the problems difficult. In the next two lectures, we will discuss how to prove decoupling theorems. We will focus on the simplest decoupling theorem: decoupling for the parabola in the plane.

The goal of the lectures will be to provide an overview of the proof of the Threshold Conjecture for the hyperbolic Yang-Mills equations.

They involve nonlinear covariant wave equations, Lie algebras, gauge questions, microlocal analysis in the Lie algebra setting, bilinear and multilinear estimates,

renormalization and parametrices, induction on energy, etc (or as much of this as I have time to cover). 

Inequalities with full affine invariance are rare. Fundamental examples include the inequalities of Brunn-Minkowski, Young, Riesz-Sobolev, and Hausdorff-Young. For each of these, a sharp form with an optimal constant is known, including a characterization of all extremizing functions, or sets. This course will discuss refinements of some of these sharp inequalities. These refinements quantify the uniqueness of extremizers. The first lecture will be a general introduction, reviewing several inequalities, stating refinements, and introducing associated ideas. The second lecture will outline a proof of a sharpened Riesz-Sobolev inequality. In contrast to earlier work of the speaker which focused on the exploitation of ideas from (finite) additive combinatorics, this proof is rooted firmly in the continuum. It emphasizes the role of the affine group as a symmetry group of the set of cosets of one-parameter subgroups of the Euclidean group

The goal of the lectures will be to provide an overview of the proof of the Threshold Conjecture for the hyperbolic Yang-Mills equations.

They involve nonlinear covariant wave equations, Lie algebras, gauge questions, microlocal analysis in the Lie algebra setting, bilinear and multilinear estimates,

renormalization and parametrices, induction on energy, etc (or as much of this as I have time to cover). 

We show that the classical results about rotating a line segment in arbitrarily small area, and the existence of a Besicovitch and a Nikodym set hold if we replace the line segment by an arbitrary rectifiable set. This is a joint work with Alan Chang

The Euler-Lagrange equation of the functional \[ \int [|\nabla v|^2 + F(v)] \] is $2\Delta u = F'(u)$. At large scale, solutions to these equations for suitable functions $F$ resemble variational solutions associated with one or the other of two natural, scale-invariant, so-called singular limits, namely, the Alt-Caffarelli energy functional for free boundaries or the area functional for minimal surfaces. In these lectures, we will describe the relationship between the functionals and characterize the property of stability for each by finding the second variation. We will then pursue the analogy between minimal surfaces and free boundaries, already a powerful device in work in 1980 of Alt and Caffarelli. James Simons's regularity theory for stable minimal surfaces will lead us to a regularity theorem for stable free boundaries in dimensions 3 and 4. The analogy will guide our investigation into what can be said about minimizers in high dimensions and higher critical points in dimensions 2 and 3