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: TBA.

Abstract: TBA.

Abstract:  TBA.