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mazur

Einstein Lecture by Mazur

The 2019 Einstein Lecture will take place at the university’s Kennedy Theatre in March. It is a public lecture; help us spread the news across campus, to local higher education institutions, to local high schools, and to other interested people.

The American Mathematical Society Presents
The 2019 AMS Einstein Public Lecture in Mathematics

BARRY MAZUR
Harvard University

ON THE ARITHMETIC OF CURVES

SATURDAY, MARCH 23 AT 5:30 P.M.

Kennedy Theatre, University of Hawai’i at Manoa

Reception to follow.

Barry Mazur is Gerhard Gade University Professor at Harvard University. Winner of many awards and prizes, including the National Medal of Science, Mazur has done outstanding work in many areas of mathematics—his work provided a foundation for the proof of Fermat’s Last Theorem—and is known for being able to communicate those results to non-mathematicians and to relish doing so. His former dean, Jeremy Knowles, said:

“Barry is not only a brilliant mathematician, but a wonderful teacher who engages biologists, physicists, economists, and others and seduces them into an understanding of the beauty and use of mathematics.”

In this public lecture, Mazur will give a survey of current approaches, results, and conjectures in the vibrant subject of algebraic curves.

The Einstein Lecture is part of the 2019 AMS Spring Central and Western Joint Sectional Meeting (March 22–24) at the University of Hawaii at Manoa.

Event details

Sectional details

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1994 result of Smith and Csordas inspires solution to Newman’s Conjecture

Newly (re-)elected department chair Wayne Smith with Varga and our professor emeritus George Csordas in a 1994 paper gave results on the de Bruijn-Newman constant $\Lambda$, where the statement $$\Lambda\le 0$$ is equivalent to the Riemann Hypothesis (some of Riemann’s notes on this shown above). Now in 2018 Terence Tao and Rodgers have shown $$\Lambda\ge 0.$$ Their primary inspiration, judging from the abstract, was Smith et al.’s work:
Wayne Smith in 2012

For each $t \in {\bf R}$, define the entire function $$ H_t(x) := \int_0^\infty e^{tu^2} \Phi(u) \cos(xu)\ du$$ where $\Phi$ is the super-exponentially decaying function $$ \Phi(u) := \sum_{n=1}^\infty (2\pi^2 n^4 e^{9u} – 3\pi n^2 e^{5u} ) \exp(-\pi n^2 e^{4u} ).$$ Newman showed that there exists a finite constant $\Lambda$ (the de Bruijn-Newman constant) such that the zeroes of $H_t$ are all real precisely when $t \geq \Lambda$. The Riemann hypothesis is the equivalent to the assertion $\Lambda \leq 0$, and Newman conjectured the complementary bound $\Lambda \geq 0$.
In this paper we establish Newman’s conjecture. The argument proceeds by assuming for contradiction that $\Lambda < 0$, and then analyzing the dynamics of zeroes of $H_t$ (building on the work of Csordas, Smith, and Varga) to obtain increasingly strong control on the zeroes of $H_t$ in the range $\Lambda< t \leq 0$, until one establishes that the zeroes of $H_0$ are in local equilibrium, in the sense that locally behave (on average) as if they were equally spaced in an arithmetic progression, with gaps staying close to the global average gap size. But this latter claim is inconsistent with the known results about the local distribution of zeroes of the Riemann zeta function, such as the pair correlation estimates of Montgomery.

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Conformally removable sets

Assistant Professor Malik Younsi has been awarded an NSF grant for 2017-2020 to work on removable sets and questions in geometric function theory.

The research project involves the study of the geometric properties of conformally removable sets, including problems related to rigidity of circle domains, conformal welding and fingerprints of lemniscates. It also includes questions dealing with shapes of Julia sets and the subadditivity of analytic capacity