UNIVERSITY OF PENNSYLVANIA
Department of Mathematics
Fall 2004 - Hans Rademacher Lectures in
Mathematics
Yuval Peres
Departments of Statistics and Mathematics, University of California, Berkeley
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
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?
Fair Allocations
Tuesday....October 19, 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
Determinantal Processes
Wednesday....October 20, 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.
Zeros of the I.I.D. Gaussian Power Series
Thursday....October 21, 2004....4:00pm
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).
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.
Previous Rademacher Lecturers