Abstract: Stein's Fourier restriction conjecture concerns the L^p behavior of functions whose Fourier transform is supported on a smooth hypersurface. In this talk, I will survey this conjecture, and discuss a Kakeya-type incidence inequality that would imply the full restriction conjecture. The talk is based on joint work with Hong Wang.

Abstract: The study of curved model of Triangular Hilbert Transform, the maximal variant, and the related Roth type problem has direct connection to a form of smoothing inequality. I’ll survey the related history, present a proof sketch for the smoothing inequality along parabola, and commented on how O-minimal structure serves as a nice environment to perform the analysis, leading to our general result. This talk is based on joint work with Fred Yu-Hsiang Lin.

Abstract: Given a probability space (X,mu), a square integrable function f on such space and a (unilateral or bilateral) shift operator T, we prove under suitable assumptions that the ergodic means N^{-1}sum_{n=0}^{N-1} T^n f converge pointwise almost everywhere to zero with a speed of convergence which, up to a small logarithmic transgression, is essentially of the order of N^{-1/2}. We also provide a few applications of our results. From a joint work with N. Chalmoukis, L. Colzani and A. Monguzzi.

Abstract: For $c\in(1,2)$ we consider the following operators

\[

\mathcal{C}_{c}f(x) \coloneqq \sup_{\lambda \in [-1/2,1/2)}\bigg| \sum_{n \neq 0}f(x-n) \frac{e^{2\pi i\lambda \lfloor |n|^{c} \rfloor}}{n}\bigg|\text{,}\quad    \mathcal{C}^{\mathsf{sgn}}_{c}f(x) \coloneqq \sup_{\lambda \in [-1/2,1/2)}\bigg| \sum_{n \neq 0}f(x-n) \frac{e^{2\pi i\lambda \mathsf{sign(n)} \lfloor |n|^{c} \rfloor}}{n}\bigg|,

 \]

and prove that both extend boundedly on $\ell^p(\mathbb{Z})$, $p\in(1,\infty)$. The second main result is establishing almost everywhere pointwise convergence for the following ergodic averages

\[

A_Nf(x)\coloneqq\frac{1}{N}\sum_{n=1}^Nf(T^nS^{\lfloor n^c\rfloor}x)\,

\]

where $T,S\colon X\to X$ are commuting measure-preserving transformations on a $\sigma$-finite measure space $(X,\mu)$, and $f\in L_{\mu}^p(X)$, $p\in(1,\infty)$. The point of departure for both proofs is the study of exponential sums with phases $\xi_2 \lfloor |n^c|\rfloor+ \xi_1n$ through the use of a simple variant of the circle method.

This talk is based on joint work with Leonidas Daskalakis.

Abstract: Suppose that n is 0 or 4 mod 6. We show that there are infinitely many primes of the form p^2 + nq^2 with both p and q prime, and obtain an asymptotic for their number. In particular, when n = 4 we verify the `Gaussian primes conjecture' of Friedlander and Iwaniec.

Joint w. Ben Green (Oxford).

Abstract: Given a $p$-random $A \subseteq \mathbb{Z}_n$, the random Cayley graph $\Gamma_p$ is defined to have vertex set $\mathbb{Z}_n$ and an edge between two distinct vertices $x, y \in \mathbb{Z}_n$ if $x + y \in A$. For $p=1/2$ Green and Morris showed that the independence number of $\Gamma_{1/2}$ is asymptotically equal to $\alpha(G(n, 1/2))$. In this talk I'll show that the independence number of $\Gamma_p$ matches that of $G(n,p)$ for $p \geq (\log n)^{-1/80}$.

This is joint work with Gabriel Dahia and João Pedro Marciano.

Abstract: The talk will be based on a recent joint work with Bo'az Klartag in which we solve positively Bourgain's slicing conjecture, from the early 80s. Namely we prove that any convex set of volume 1 in dimension n admits a hyperplane section whose (n-1)-dimensional volume is bounded below by a universal constant. Our proof combines a recent breakthrough by Qingyang Guan concerning Eldan's stochastic localization technique with more classical tools from convex geometry such as the notion of Milman ellipsoid. In the talk I'll try to review briefly the history of the problem and to lay down the main steps of our proof.

Abstract:  Hindman's finite sums theorem from 1970's states that whenever the natural numbers are partitioned into finitely many cells, one of those cells must contain an infinite set, together with the sum of any finitely many elements of the set. While not every set with positive density contains such a configuration, at the time Erdos asked what (if any) is the density version of this theorem. In the last few years, in joint work with Kra, Richter and Robertson, we have used ergodic theory methods to investigate this question. In the talk I will briefly survey our current knowledge, and present a very recent result showing that every positive integer k, every set of positive density contains, up to a shift, an infinite set together with the sums of any i of its elements for all i up to k.