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3:00pm to 3:50pm - - Analysis John, N. Treuer - (University of Texas A&M) The Levi Core and Compactness in the $\overline{\partial}$-Neumann problem The $\overline{\partial}$-Neumann problem is one of the fundamental PDEs in several complex variables. The solution operator, called the $\overline{\partial}$-Neumann operator, $N$ can give information on the inhomogeneous Cauchy-Riemann equation as well as information on the boundary regularity of biholomorphic mappings. One important question is when is $N$ compactBy the work of Catlin and Sibony, it is known that compactness holds if a potential theoretic condition Property ($P$) holds for the boundary of a smooth, pseudoconvex domain. In this talk, we consider a recent construction due to Dall’Ara and Mongodi called the Levi core and show that compactness holds if Property ($P$) holds for the support of the Levi core. |
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2:00pm to 3:00pm - 510R Rowland Hall - Combinatorics and Probability Haonan Zhang - (UCI) Learning quantum observables of low degrees from a logarithmic number of random queries A fundamental problem from computational learning theory is to well-reconstruct an unknown function on the discrete hypercubes. One classical result of this problem for the random query model is the low-degree algorithm of Linial, Mansour and Nisan in 1993. This was improved exponentially by Eskenazis and Ivanisvili in 2022 using a family of polynomial inequalities going back to Littlewood in 1930. Recently, quantum analogs of such polynomial inequalities were conjectured by Rouzé, Wirth and Zhang (2022). This conjecture was resolved by Huang, Chen and Preskill (2022) without knowing it when studying learning problems of quantum dynamics. In this talk, I will discuss another proof of this conjecture that is simpler and gives better estimates. As an application, it allows us to recover the low-degree algorithm of Eskenazis and Ivanisvili in the quantum setting. This is based on arXiv:2210.14468, joint work with Alexander Volberg (MSU). |
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10:00am to 11:00am - https://uci.zoom.us/j/95304920920 - Number Theory Xiannan Li - (Kansas State University) Quadratic twists of modular L-functions The behavior of quadratic twists of modular L-functions is at the critical point is related both to coefficients of half integer weight modular forms and data on elliptic curves. Here we describe a proof of an asymptotic for the second moment of this family of L-functions, previously available conditionally on the Generalized Riemann Hypothesis by the work of Soundararajan and Young. Our proof depends on deriving an optimal large sieve type bound. |