Abstract: Fully nonlinear elliptic PDEs include the Monge-Ampere equation from optimal transport and the PDEs for constructing minimal surfaces of high codimension.  Such PDEs can be solved in the weak, viscosity sense, so the question is whether such solutions are smooth, or what kinds of singularities are possible.  In the past, these questions were solved for each equation using very different approaches.  In this talk, we indicate a unified approach to these questions and equations, based on the idea of a doubling inequality.

We discuss a new upper bound for the Heilbronn triangle problem, showing that for sufficiently large n, in every configuration of n points chosen inside a unit square, there exists a triangle of area less than n^{-8/7-1/2000}.

This is joint work with Alex Cohen and Dmitrii Zakharov.

The number of real roots for random algebraic polynomials is a topic with a long history and contributions of many authors. In this talk, I will discuss a brief history of the topic and some recent developments.

Given a hypersurface S\subset \mathbb{R}^{2d}, we study the bilinear averaging operator that averages a pair of functions over S, as well as more general bilinear multipliers of limited decay and various maximal analogs. Of particular interest are bilinear maximal operators associated to a fractal dilation set E\subset [1,2]; in this case, the boundedness region of the maximal operator is associated to the geometry of the hypersurface and various notions of the dimension of the dilation set. In particular, we determine Sobolev smoothing estimates at the exponent L^{2}\times L^{2}\rightarrow L^2 using Fourier-analytic methods, which allow us to deduce additional L^{p} improving bounds for the operators and sparse bounds and their weighted corollaries for the associated multi-scale maximal functions. We also extend the method to study analogues of these questions for the triangle averaging operator and biparameter averaging operators. In addition, some necessary conditions for boundedness of these operators are obtained.

Fix a real number 0 < s <= 1. A set E in the plane is a s-Furstenberg set if there exists a line in every direction that intersects E in a set with Hausdorff dimension s. For example, a planar Kakeya set is a special case of a 1-Furstenberg set, and indeed we know that 1-Furstenberg sets have Hausdorff dimension 2. However, obtaining a sharp lower bound for the Hausdorff dimension of s-Furstenberg sets for any 0 < s < 1 has been a challenging open problem for half a century. In this talk, I will illustrate the rich connections between the Furstenberg sets conjecture and other important topics in geometric measure theory and harmonic analysis, and show how exploring these connections can fully resolve the Furstenberg conjecture. Joint works with Yuqiu Fu and Hong Wang.

Carleson's theorem on the convergence of Fourier series is

equivalent to the weak-$L^2$-boundedness of the maximally modulated

Hilbert transform, and adaptions of the proof show more generally

weak-$L^2$-boundedness of maximally modulated Calderón-Zygmund operators.

This talk is about the open problem of whether this result can be extended

to singular Radon transforms, such as the Hilbert transform along the

parabola $H_P$. I will discuss the main ingredients used in the proof of

Carleson's theorem, and to what extent they can be adapted for $H_P$. A

corollary are improved quantitative estimates for maximal modulations of

operators approximating $H_P$.

The Assouad dimension is a notion of dimension which captures the worst-case scaling of a set at all locations and all scales. However, in many situations the Assouad dimension measures scaling in a way which is too coarse, and quantifying the precise resolution at which larger-than-average scaling occurs has been important in applications. In this talk, I will give an introduction and overview of recent work on variations of the Assouad dimension. I will also touch on some recent applications in the literature including: large deviations of branching processes, smoothness of iterated function system attractors, quasi-conformal distortion of sets, and $L^p$-improving properties of maximal operators with restricted dilation sets.

A domain Ω ⊂ C^n is said to be pseudoconvex if − log(−δ(z)) is plurisub-harmonic in Ω, where δ is a signed distance function of Ω. The study of global regularity of  ̄∂-Neumann problem on bounded pseudoconvex domains is dated back to the 1960s. However, a complete understanding of the regularity is still absent. On the other hand, the Diederich–Fornæss index was introduced in 1977 originally for seeking bounded plurisubharmonic functions. Through decades, enormous evidence has indicated a relationship between global regularity of the  ̄∂-Neumann problem and the Diederich–Fornæss in-dex. Indeed, it has been a long-lasting open question whether the trivial Diederich–Fornæss index implies global regularity. In this talk, we will intro-duce the backgrounds and motivations. The main theorem of the talk proved recently by Emil Straube and me answers this open question for (0, n − 1) forms.