A design is a finite set of points in a space on which every "simple" functions averages to its global mean. Illustrative examples of simple functions are low-degree polynomials on the Euclidean sphere or on the Hamming cube.

We prove lower bounds on designs in spaces with a large group ... more >>>

A distance estimator is a code together with a randomized algorithm. The algorithm approximates the distance of any word from the code by making a small number of queries to the word. One such example is the Reed-Muller code equipped with an appropriate algorithm. It has polynomial length, polylogarithmic alphabet ... more >>>

For two polynomials $f \in \mathbb{F}[x_1, x_2, \ldots, x_n, y]$ and $p \in \mathbb{F}[x_1, x_2, \ldots, x_n]$, we say that $p$ is a root of $f$, if $f(x_1, x_2, \ldots, x_n, p) \equiv 0$. We study the relation between the arithmetic circuit sizes of $f$ and $p$ for general circuits ... more >>>