We establish a quantitative result on norm convergence of triple ergodic averages with respect to three general commuting transformations by proving an r-variation estimate, r > 4, in the norm. We approach the problem via real harmonic analysis, using the recently developed techniques for bounding singular Brascamp-Lieb forms. It is not known whether such norm-variation estimates hold for all r>=2 as in the analogous cases for one or two commuting transformations, or whether such estimates hold for any r<infinity for more than three commuting transformations. This is joint work with Christoph Thiele and Lenka Slavikova.

Using the method of Rudin-Shapiro polynomials we prove the bi-analytic version of the Mitiagin - DeLeeuw - Mirkhil non-inequality for complex partial differential operators with constant coefficients on bi-disc

We present a Johnson-Lindenstrauss-type dimension reduction algorithm with additive error for incompressible subsets of $\ell_p$. The proof relies on a derandomized version of Maurey’s empirical method and a combinatorial idea of Ball.

Bohnenblust--Hille (BH) inequalities are an extension of Littlewood's 4/3 inequality and have found many applications to harmonic analysis. A variant of BH inequalities for Boolean cubes has been proven with constants that are dimension-free and subexponential in degree. Such inequalities have found great applications in learning low-degree Boolean functions. Motivated by learning quantum observables, a quantum analog of BH inequality for Boolean cubes was recently conjectured and resolved unaware of the conjecture. In this talk, we give a simpler proof with better constants. As applications, we study learning problems for quantum observables of low degrees. Joint work with Alexander Volberg.

Given a polynomial $P$ of constant degree in $d$ variables and consider the oscillatory integral $$I_P = \int_{[0,1]^d} e(P(\xi)) \mathrm{d}\xi.$$ Assuming $d$ is also fixed, what is a good upper bound of $|I_P|$? In this talk, I will introduce a ``stationary set'' method that gives an upper bound with simple geometric meaning. The proof of this bound mainly relies on the theory of o-minimal structures. As an application of our bound, we obtain the sharp convergence exponent in the two dimensional Tarry's problem for every degree via additional analysis on stationary sets. Consequently, we also prove the sharp $L^{\infty} \to L^p$ Fourier extension estimates for every two dimensional Parsell-Vinogradov surface whenever the endpoint of the exponent $p$ is even. This is joint work with Saugata Basu, Shaoming Guo and Pavel Zorin-Kranich.