About

I am a fourth year Ph.D. student in the math department. I completed my B.S. and M.A. in mathematics at UCLA in 2021. My research interests are in harmonic analysis and partial differential equations. Here is my CV.

My coordinates are:

• Department of Mathematics
• University of California, Berkeley
• 1062 Evans Hall
• Berkeley, CA 94720
• Email: zhao [dot] j [at] berkeley [dot] edu

Preprints and Publications (click title to show/hide abstract)

• Local well-posedness for dispersive equations with bounded data, arXiv:2409.04706 [math.AP].

  Abstract: Given sufficiently regular data without decay assumptions at infinity, we prove local well-posedness for non-linear dispersive equations of the form \[ \partial_t u + \mathsf A(\nabla) u + \mathcal Q(|u|^2) \cdot \nabla u= \mathcal N (u, \overline u), \] where \(\mathsf A(\nabla)\) is a Fourier multiplier with purely imaginary symbol of order \(\sigma + 1\) for \(\sigma > 0\), and polynomial-type non-linearities \(\mathcal Q(|u|^2)\) and \(\mathcal N(u, \overline u)\). Our approach revisits the classical energy method by applying it within a class of local Sobolev-type spaces \(\ell^\infty_{\mathsf A(\xi)} H^s (\mathbb R^d)\) which are adapted to the dispersion relation in the sense that functions \(u\) localised to dyadic frequency \(|\xi| \approx N\) have size \[ ||u||_{\ell^\infty_{\mathsf A(\xi)} H^s} \approx N^s \sup_{{\operatorname{diam}(Q) = N^\sigma}} ||u||_{L^2_x (Q)}. \] In analogy with the classical \(H^s\)-theory, we prove \(\ell^\infty_{\mathsf A(\xi)} H^s\)-local well-posedness for \(s > \tfrac{d}2 + 1\) for the derivative non-linear equation, and \(s > \tfrac{d}2\) without the derivative non-linearity. As an application, we show that if in addition the initial data is spatially almost periodic, then the solution is also spatially almost periodic.

• Determining optimal test functions for \(2\)-level densities, with Elzbieta Boldyriew, Fangu Chen, Charles Devlin VI, Steven J. Miller. Research in Number Theory Vol. 9 (2023) arXiv:2011.10140 [math.NT] .

  Abstract: Katz and Sarnak conjectured a correspondence between the \(n\)-level density statistics of zeros from families of \(L\)-functions with eigenvalues from random matrix ensembles, and in many cases the sums of smooth test functions, whose Fourier transforms are finitely supported over scaled zeros in a family, converge to an integral of the test function against a density \(W_{n, G}\) depending on the symmetry \(G\) of the family (unitary, symplectic or orthogonal). This integral bounds the average order of vanishing at the central point of the corresponding family of \(L\)-functions.

  We can obtain better estimates on this vanishing in two ways. The first is to do more number theory, and prove results for larger \(n\) and greater support; the second is to do functional analysis and obtain better test functions to minimize the resulting integrals. We pursue the latter here when \(n=2\), minimizing \[ \frac{1}{\Phi(0, 0)} \int_{{\mathbb R}^2} W_{2,G} (x, y) \Phi(x, y) dx dy \] over test functions \(\Phi : {\mathbb R}^2 \to [0, \infty)\) with compactly supported Fourier transform. We study a restricted version of this optimization problem, imposing that our test functions take the form \(\phi(x) \psi(y)\) for some fixed admissible \(\psi(y)\) and \(\operatorname{supp}{\widehat \phi} \subseteq [-1, 1]\). Extending results from the \(1\)-level case, namely the functional analytic arguments of Iwaniec, Luo and Sarnak and the differential equations method introduced by Freeman and Miller, we explicitly solve for the optimal \(\phi\) for appropriately chosen fixed test function \(\psi\). We conclude by discussing further improvements on estimates by the method of iteration.

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