This year's Paul Turán Memorial Lectures will be delivered by Peter Sarnak of Princeton University.

The lectures are organized by the János Bolyai Mathematical Society and will take place in the Main Lecture Hall of the Rényi Institute.

**Program of the lectures:**

**Lecture 1 (Tuesday 26 November, 2pm):**Sums of squares and Hilbert's 11th problem**Lecture 2 (Wednesday 27 November, 2pm):**The spectra of modular surfaces**Lecture 3 (Thursday 28 November, 2pm):**The spectra of modular surfaces continued.

**Abstract:** Hilbert's 11th problem asks about sums of
squares (or more generally representability by integral
quadratic forms) of integers in a number field. Recent
progress has led to its solution. In the first of these
lectures we will give an introduction to this problem
and outline the key developments that led to its solution.

In the second and third lectures we describe the spectral theory of the Laplacian on modular surfaces indicating how the latter is used in the solution of the problem of Lecture 1. We also describe other aspects and applications of this spectral theory, for example to problems of "Quantum Chaos".