Please use this identifier to cite or link to this item: http://arks.princeton.edu/ark:/88435/dsp01rf55zb86g
 Title: Families and dichotomies in the circle method Authors: Wang, Victor Young Advisors: Sarnak, Peter C Contributors: Mathematics Department Keywords: circle methodcubic formdichotomiesL-functionsRandom Matrix Theoryrational points Subjects: Mathematics Issue Date: 2022 Publisher: Princeton, NJ : Princeton University Abstract: In the eighties, Hooley applied the Grand Riemann Hypothesis, and what practically amounts to the general Langlands reciprocity (automorphy) conjecture, in a fresh new way, over certain families of cubic threefolds.This eventually led to conditional near-optimal bounds for the number $N$ of integral solutions to $x_1^3+\dots+x_6^3 = 0$ in expanding boxes. Elsewhere, building on Hooley's work, we have given new applications of large-sieve hypotheses, the Square-free Sieve Conjecture, and predictions of Random Matrix Theory type, over the same geometric families---for instance, conditional optimal asymptotics for $N$ in a large class of regions, with applications to sums of three cubes.The underlying harmonic analysis---which in rough form goes back to Kloosterman---picks up equally significant contributions from the classical major and minor arcs in the circle method. Here, we mainly provide extended summaries, commentary, and other complementary material, leaving complete traditional accounts to papers available elsewhere.Two central themes of this thesis are families (of arithmetic or analytic objects) and dichotomies (between structure and randomness). We especially consider (mainly in relation to the aforementioned cubic questions) families of regions, weights, point counts, oscillatory integrals, exponential sums, Hasse--Weil $L$-functions, and quadratic equations; and dichotomies for point counts over finite and infinite fields. URI: http://arks.princeton.edu/ark:/88435/dsp01rf55zb86g Alternate format: The Mudd Manuscript Library retains one bound copy of each dissertation. Search for these copies in the library's main catalog: catalog.princeton.edu Type of Material: Academic dissertations (Ph.D.) Language: en Appears in Collections: Mathematics