Discrete Analysis Seminar

In this talk, we discuss harmonic analysis over the $\mathbb{Q}_p$. Compared to when working over $\mathbb{R}$, tools such as the uncertainty principle and wavepacket decomposition are not just useful heuristics, but rigorously true. Additionally, decoupling estimates over $\mathbb{Q}_p$ are still strong enough to imply exponential sum applications which have been key applications of real decoupling theorems. This observation along with an optimization of the Guth-Maldague-Wang argument allowed the speaker with Shaoming Guo and Po-Lam Yung to show that the discrete restriction constant for the parabola is $\lesssim_{\varepsilon} (\log N)^{2 + \varepsilon}$. No familiarity of the $p$-adic analysis will be assumed.