# Seminars

## Lezioni LEONARDESCHE

- Terence TAO (University of California, Los Angeles), June 22, 2018, 2.30pm, room U4-08. (Video of the lecture)

Titolo: *VAPORIZING AND FREEZING THE RIEMANN ZETA FUNCTION*

Abstract: In 1950, de Bruijn studied the effect of evolving the Riemann zeta function (or more precisely, a closely related function known as the Riemann xi function) by the (backwards) heat equation. His analysis, together with later work by Newman, showed that there existed a finite constant Lambda, at most 1/2 in value, such that the Riemann hypothesis for this evolved function was true at times greater than or equal to Lambda, and false below that threshold. Thus the Riemann hypothesis for the zeta function is equivalent to Lambda being non-positive. Recently, in joint work with Brad Rodgers, I was able to establish the complementary estimate that Lambda is non-negative, confirming a conjecture of Newman; thus, the Riemann hypothesis for zeta, if true, is only “barely so”. The proof relies on an analysis of the dynamics of zeroes of entire functions under heat flow; it turns out that as one evolves forward in time, the zeroes “freeze” into approximate arithmetic progressions, while if one evolves backwards, the zeroes “vaporize” to leave the critical line. In followup work in an online collaborative “Polymath” project, the upper bound on Lambda has also been improved. We describe these results and their proofs in this talk.

ABOUT THE SPEAKER: Terence Tao is nothing short of a mathematical superstar whose breadth, depth and innovative spirit is rapidly changing the mathematical landscape. Gifted with a prodigious talent which propelled him to a gold medal at the International Mathematical Olympiad at age 13 and a Full Professorship at UCLA at the age of 24, Tao has combined this talent with his stated vision that “mathematics is a unified subject and I am particularly happy when I get the opportunity to work on a project that involves several fields at once”. Tao’s spectacular problem solving skills in a have earned him a list of prizes and awards too long to mention, but it is telling that the citation for his 2006 Fields Medal reads “for his contributions to partial differential equations, combinatorics, harmonic analysis and additive number theory” which shows the unprecedented range of his activity that continues to expand. Among his most well-known achievements, Tao together with various collaborators has established: existence of arithmetic sequences of prime numbers of arbitrary length, the first entropy estimates in analytic number theory, wide ranging results for nonlinear Schrödinger equations, L^p harmonic analysis and critical Sobolev regularity for wave maps, the solution of the Kakeya problem in geometric measure theory and the Horn conjecture on eigenvalue inequalities for Hermitian matrices. Tao is also a pioneer in mathematical exposition and interaction in the internet age with an unprecedented success in mathematical blogs which have stimulated countless textbooks and collaborations worldwide. About him, fellow Fields medalist Timothy Gowers speculated that Terence Tao may be the modern day David Hilbert who comes to know all of mathematics. Tao is currently holds the James and Carol Collins Chair in Mathematics at UCLA.