The talk will be about certain isoperimetric inequalities on
the Hamming cube near and at the critical exponent 1/2 and closely
related L^1 Poincare inequalities. The proofs involve some
Bellman-type functions and computer-assisted methods. This is joint
work with Polona Durcik and Paata Ivanisvili.

Let $p$ be any polynomial of degree $2$ on $n$-dimensional discrete hypercubes. We prove dimension-free upper bounds for the absolute sum of all level-$k$ Fourier coefficients of Boolean functions $f(x)=(-1)^{p(x)}$. This is a joint work with L. Becker, J. Slote and A. Volberg.

Many problems of error analysis in quantum information processing can be formulated as deviation inequalities of random matrices. In this talk, I will talk about how complex interpolations of various Lp spaces can be an effective tool in establishing error estimates in information tasks such as quantum soft covering, privacy amplification, convex splitting and quantum decoupling.  This talk is based on joint works with Hao-Chung Cheng, Yu-Chen Shen, Frédéric Dupuis and Mario Berta. 

On the vector space of matrices equipped with the p-Schatten norm, consider the unit ball normalized to have Lebesque volume 1. Let $ W$ be the random matrix uniformly distributed on this set. We compute sharp upper and lower bounds for the moments of marginals of the random matrix $W$. As an application, we characterize subgaussian and supergaussian directions, estimate the volume of sections of these balls, and provide precise tail estimates for the singular values of the matrix $W$.  Based on a joint work with Kavita Ramanan. 

Let us think of a convex body in R^n (convex, compact set, with non-empty interior) as an opaque object, and let us place point light sources around it, wherever we want, to illuminate its entire surface. What is the minimum number of light sources that we need? The Hadwiger-Boltyanski illumination conjecture from 1960 states that we need at most as many light sources as for the n-dimensional hypercube, and more generally, as for n-dimensional parallelotopes. For the latter their illumination number is exactly 2^n, and they are conjectured to be the only equality cases.

The conjecture is still open in dimension 3 and above, and has only been fully settled for certain classes of convex bodies (e.g. zonoids, bodies of constant width, etc.). Moreover, there are some rare examples for which a basic, folklore argument could quickly lead to the upper bound 2^n, while at the same time understanding the equality cases has remained elusive for decades. One such example would be convex bodies very close to the cube, which was settled by Livshyts and Tikhomirov in 2017.

In this talk I will discuss another such instance, which comes from the class of 1-unconditional convex bodies, and which also 'forces' us to settle the conjecture for a few more cases of 1-unconditional bodies. This is based on joint work with Wen Rui Sun, and our arguments are primarily combinatorial.