Georgia Institute of Technology College of Sciences

We improve on some recent results of Sagiv and Steinerberger that quantify the following uncertainty principle: for a function f with mean zero, then either the size of the zero set of the function or the cost of transporting the mass of the positive part of f to its negative part must be big. We also provide a sharp upper estimate of the transport cost of the positive part of an eigenfunction of the Laplacian.

The weak-type (1,1) estimate for Calderón-Zygmund operators is fundamental in harmonic analysis. We investigate weak-type inequalities for Calderón-Zygmund singular integral operators using the Calderón-Zygmund decomposition and ideas inspired by Nazarov, Treil, and Volberg. We discuss applications of these techniques in the Euclidean setting, in weighted settings, for multilinear operators, for operators with weakened smoothness assumptions, and in studying the dimensional dependence of the Riesz transforms.

Artificial neural networks have gained widespread adoption as a powerful tool for various machine learning tasks in recent years. Training a neural network to approximate a target function involves solving an inherently non-convex problem. In practice, this is done using stochastic gradient descent with random initialization.

Abstract: Form methods are most efficient to prove generation theorems for semigroups but also for proving selfadjointness. So far those theorems are based on a coercivity notion which allows the use of the Lax-Milgram Lemma. Here we consider weaker "essential" versions of coerciveness which already suffice to obtain the generator of a semigroup S or a selfadjoint operator. We also show that one of these properties, namely essentially positive coerciveness implies a very special asymptotic behaviour of S, namely asymptotic compactness; i.e.

  We are interested in arithmetic progressions in positive measure subsets of [0,1]^d. After a counterexample by Bourgain, it seemed as if nothing could be said about the longest interval formed by sizes of their gaps. However, Cook, Magyar, and Pramanik gave a positive result for 3-term progressions if their gaps are measured in the l^p-norm for p other than 1, 2, and infinity, and the dimension d is large enough. We establish an appropriate generalization of their result to longer progressions.

Let $K$ be a n dimensional convex body with of volume $1$. and barycenter of $K$ is the origin.  It is known that $|K \cap -K|>2^{-n}$.  Via thin shell estimate by Lee-Vempala (earlier versions were done by Guedon-Milman, Fleury, Klartag), we improve the bound by a sub-exponential factor.  Furthermore, we can improve  the Hadwiger’s Conjecture in the non-symmetric case by a sub-exponential factor.  This is a joint work with Boaz A. Slomka, Tomasz Tkocz, and Beatrice-Helen Vritsiou.