Characterizing Boolean functions with small total influence is one of the most fundamental questions in analysis of Boolean functions. 

The seminal results of Kahn-Kalai-Linial and of Friedgut address this question for total influence $K = o(\log n)$, and show that 

a function with total influence $K$ (essentially) depends on $2^{O(K)}$ variables. 

The Fourier-Entropy Conjecture of Friedgut and Kalai is an outstanding conjecture that strengthens these results, and remains

meaningful for $K \geq \log n$. Informally, the conjecture states that the Fourier transform of a function with total influence $K$, 

is concentrated on at most $2^{O(K)}$ distinct characters. 

In this talk, we will discuss recent progress towards this conjecture. We show that functions with total influence $K$ are concentrated 

on at most $2^{O(K\log K)}$ distinct Fourier coefficients. We also mention some applications to learning theory and sharp thresholds.

Based on a joint work with Esty Kelman, Guy Kindler, Noam Lifshitz and Muli Safra.

Consider families of spherical means where the radii are restricted to a given subset of a compact interval. One is interested in the L^p improving estimates for the associated maximal operators and related objects. Results depend on several notions of fractal dimension of the dilation set, or subsets of it. There are some unexpected statements on the shape of the possible type sets. Joint works with J. Roos, and with T. Anderson, K. Hughes and J. Roos.

Log-concave sequences appear naturally in a variety of
fields. For example in convex geometry the Alexandrov-Fenchel
inequalities demonstrate the intrinsic volumes of a convex body to be
log-concave, while in combinatorics the resolution of the Mason
conjecture shows that the number of independent sets of size n in a
matroid form a log-concave sequence as well.   By Lyapunov's
inequality we refer to  the log-convexity of the (p-th power) of the
L^p norm of a function with respect to an arbitrary measure, an
immediate consequence of Holder's inequality.   In the continuous
setting measure spaces satisfying concavity conditions are known to
satisfy a sort of concavity reversal of both Lyapunov's inequality,
due to Borell, while the Prekopa-Leindler inequality gives a reversal
of Holder.  These inequalities are foundational in convex geometry,
give Renyi entropy comparisons in information theory, the Gaussian
log-Sobolev inequality, and more generally the HWI inequality in
optimal transport among other applications.  An analogous theory has
been developing in the discrete setting.  In this talk we establish a
reversal of Lyapunov's inequality for monotone log-concave sequences,
settling a conjecture of Havrilla-Tkocz and Melbourne-Tkocz. A
strengthened version of the same conjecture is disproved through
counterexamples.