## Distinguished Lecture Series 2023: Joseph Silverman

09 Feb 2023, by Events in10 – 12 May 2023 (In person. 16.00 – 17.00 LG.02, Fry Building)

Organised in collaboration with the School of Mathematics, University of Bristol, UK

Joseph Silverman is a world-leading mathematician working in the general area of Number Theory, and more specifically in Diophantine and arithmetic geometry, elliptic curves and cryptography. Joseph is the recipient of several awards and he is a Fellow of the American Mathematical Society.

Joseph Silverman, Brown University, USA

**Lecture 1 followed by wine reception **(Colloquium Style)

**Arithmetic Geometry: Integral and Rational Points on Elliptic Curves**

The problem of solving polynomial equations using integers or rational numbers dates back to antiquity, but the modern theory using a combination of arithmetic and geometry may plausibly be dated to the work of Mordell and Siegel starting in the 1920s. For the past century, elliptic curves have been the proving ground for arithmetic geometry, and their study has spawned deep theorems and intriguing conjectures that continue to inform the field in its second century. In this talk I will explain what an elliptic curve is, describe some of the classical theorems of the 20th century, and discuss some more recent results and conjectures that are currently driving the subject. As is typical in much of number theory, few prerequisites are required to understand and appreciate this beautiful subject, although as is also typical, the proofs (which I will omit) tend to be quite challenging.

**Lecture 2**

**Arithmetic Dynamics: Rational Points in Orbits**

Discrete dynamical systems is the study of iteration of functions. Number theory is the study of properties of integers. Combine the two and you land in the brave new world of arithmetic dynamics, wherein we study number theoretic properties of orbits of integers and rational numbers under iteration of polynomials and rational functions. Building on analogies from arithmetic geometry, the last few decades have seen great progress and great challenges in arithmetic dynamics. In this talk, I will describe some of the motivating problems and some of the recent progress in the field. No number theoretic or dynamical prerequisites will be assumed.

**Lecture 3**

**Arithmetic Complexity: Rational Points on Varieties and in Orbits**

How complicated is a number? A rough way to measure the complexity of a positive integer N is by the number of bits it takes to describe N, so roughly log_2(N). What happens to the complexity of a number when we (repeatedly) apply a function? Given a multi-variable polynomial equation, how do the complexities of the solutions grow? Are there more refined ways to measure complexity so that these questions have nicer answers? In this talk I will discuss some of the ways in which complexity of numbers is used to formulate and study problems in arithmetic geometry and arithmetic dynamics, including results both classical and modern. No number theoretic or dynamical prerequisites will be assumed.