An emerging way to protect privacy is to replace true data by synthetic data. Medical records of artificial patients, for example, could retain meaningful statistical information while preserving privacy of the true patients. But what is synthetic data, and what is privacy? How do we define these concepts mathematically? Is it possible to make synthetic data that is both useful and private? I will tie these questions to a simple-looking problem in probability theory: how much information about a random vector X is lost when we take conditional expectation of X with respect to some sigma-algebra? This talk is based on a series of papers joint with March Boedihardjo and Thomas Strohmer, mainly this one: https://arxiv.org/abs/2107.05824

In this talk we will discuss properties of integers selected according to the Riemann zeta distribution. We will emphasize two aspects of this distribution. The first is its faithful similarity to properties of an integer chosen according to the uniform distribution on a finite interval. The second aspect will be the appearance of Poisson behavior under this distribution. The Riemann zeta function is given for $\mbox{Re} z>1$ by 

$$\zeta(z)=\sum_{n=1}^\infty \frac{1}{n^z}.$$

An alternative description is given by

$$\zeta(z)=\Pi_{p\in\mathcal{P}}\lt(1-\frac{1}{p^z}\rt)^{-1},$$

where $\mathcal{P}$ denotes the set of primes.

In our discussions we will replace the complex $z$ by a real number $s>1.$ We will denote by $X_s$ a random variable with the distribution

P(X_s=n)=\frac{1}{\zeta(s)n^s},\, n=1,2,3,\cdots.

The statistical properties of $X_s$ is the focus of the talk.

The talk is based on joint work with Adrien Peltzer.

Perhaps a very basic question one asks in learning theory is as follows: we are given a  function f on the hypercube {-1,1}^n, and we are allowed to query samples (X, f(X)) where X is uniformly distributed on {-1,1}^n. After getting these samples (X_1, f(X_1)), ..., (X_N, f(X_N)) we would like to construct a function h which approximates f up to an error epsilon (say in L^2). Of course h is a random function as it involves i.i.d. random variables X_1, ... , X_N in its construction. Therefore, we want to construct such h which can only fail to approximate f with probability at most delta. So given parameters epsilon, delta  in (0,1) the goal is to minimize the number of random queries N. I will show that around log(n) random queries are sufficient to learn bounded "low-complexity" functions. Based on joint work with Alexandros Eskenazis. 

Classical matrix concentration inequalities are sharp up to a logarithmic factor. This logarithmic factor is necessary in the commutative case but unnecessary in many classical noncommutative cases. We will present some matrix concentration results that are sharp in many cases, where we overcome this logarithmic factor by using an easily computable quantity that captures noncommutativity. Joint work with Afonso Bandeira and Ramon van Handel. Paper: https://arxiv.org/abs/2108.06312

A nodal domain of a function is a connected component of the set where this function has a constant sign. Nodal domains of eignefunctions of the Laplacian are classical objects in geometry of manifolds. Their number of nodal domains corresponding to the k-th smallest eigenvalue is bounded by k and it typically increases as k gets larger. About 10 years ago, Dekel, Lee, and Linial proved that with high probability, the number of nodal domains of Erdos-Renyi graphs remains bounded as the size of the graph and the eigenvalues increase.  This runs contrary to the intuition we draw from the world of manifolds. We will survey some recent results on the structure of nodal domains of such graphs. Based in part on the joint work with Han Huang.