Hermann Weyl Lecture

Date:
Nov
07
2022

Hermann Weyl Lectures

Simultaneous small fractional parts of polynomials
2:00pm|Simonyi 101 and Remote Access

Given several real numbers $\alpha_1,...,\alpha_k$, how well can you simultaneously approximate all of them by rationals which each have the same square number as a denominator? Schmidt gave a clever iterative argument which showed that this can be...

Nov
09
2022

Hermann Weyl Lectures

The Duffin-Schaeffer Conjecture
2:00pm|Simonyi 101 and Remote Access

Given any non-negative function $\f:\mathbb{Z}\rightarrow\mathbb{R}$, it follows from basic ergodic ideas that either 100% of real numbers $\alpha$ have infinitely many rational approximations $a/q$ with $a,q$ coprime and $|\alpha-a/q|

I'll describe a recent resolution of this conjecture, which recasts the problem in combinatorial language, and then uses a general 'structure vs randomness' principle combined with an iterative argument to solve this combinatorial problem.

Nov
11
2022

Hermann Weyl Lectures

Half-isolated zeros and the Density Hypothesis
2:00pm|Simonyi 101 and Remote Access

Many important consequences of the Riemann Hypothesis would remain true even if there were some zeros off the critical line, provided these exceptions to the Riemann Hypothesis are suitably rare. We can unconditionally prove some results on the...