A point process is a random scatter of points in space; natural examples are zeros of random polynomials, eigenvalues of random matrices, and the Poisson point process. Given a point process M in the plane, the Voronoi tessellation allocates a polygon (of different area) to each point of M. How can we modify this to a "fair" allocation, which allocates the same area to each point? ## UNIVERSITY OF PENNSYLVANIA

## Department of Mathematics

Fall 2004 -Hans Rademacher Lectures in Mathematics## Departments of Statistics and Mathematics, University of California, Berkeley

Yuval Peres

will deliver four lectures on

## POINT PROCESSES, THE STABLE MARRIAGE ALGORITHM, AND GAUSSIAN POWER SERIES

Point Processes and Tessellations

Monday....October 18, 2004....4:30pm

For a translation-invariant point process M, we show that there is a unique "fair" allocation that is "stable" in the sense of the Gale-Shapley stable marriage problem. The Gale-Shapley algorithm, which is also used to assign residents to hospitals, plays a crucial role. However, the geometry of these fair allocations is mysterious: see the picture Fair Allocations

Tuesday....October 19, 2004....4:30pm

Discrete and continuous point processes where the joint intensities are determinants arise in Combinatorics (Random spanning trees) and Physics (Fermions, eigenvalues of Random matrices). For these processes the number of points in a region can be represented as a sum of independent, zero-one valued variables, one for each eigenvalue of the relevant operator. This provides an elementary, yet powerful, instance of a quantum-classical correspondence. Determinantal Processes

Wednesday....October 20, 2004....4:30pm

Zeros of power series with complex Gaussian coefficients have recently been investigated intensively. In the case of independent coefficients with equal variance, we show that the zeros form an isometry-invariant determinantal process in the hyperbolic plane. A partition identity of Euler, and a permanent-determinant identity of Borchardt (1855) are crucial in this proof. The repulsion between zeros "smoothes" the corresponding stable allocation. This repulsion can be illuminated via a dynamic version where the coefficients perform Brownian motion (We'll see a movie). Zeros of the I.I.D. Gaussian Power Series

Thursday....October 21, 2004....4:00pm

Previous Rademacher Lecturers

Lectures on Monday, Tuesday and Wednesday will be held in room A-6 of the David Rittenhouse Laboratory,

Thursday's lecture will be held in A-5 of the David Rittenhouse Laboratory

S.E. corner of 33rd and Walnut Streets, Philadelphia, PA.

Tea: 4E17 David Rittenhouse Laboratory, at 4PM on Monday, Tuesday, and Wednesday, and at 3:30pm on Thursday.For further information, please call the Department of Mathematics at the University of Pennsylvania - 215-898-8627.