Fri, 08 May 2009
16:30
L2

Eigenvalues of large random trees

Professor Steven N. Evans
(Berkeley)
Abstract

A common question in evolutionary biology is whether evolutionary processes leave some sort of signature in the shape of the phylogenetic tree of a collection of present day species.

Similarly, computer scientists wonder if the current structure of a network that has grown over time reveals something about the dynamics of that growth.

Motivated by such questions, it is natural to seek to construct``statistics'' that somehow summarise the shape of trees and more general graphs, and to determine the behaviour of these quantities when the graphs are generated by specific mechanisms.

The eigenvalues of the adjacency and Laplacian matrices of a graph are obvious candidates for such descriptors.

I will discuss how relatively simple techniques from linear algebra and probability may be used to understand the eigenvalues of a very broad class of large random trees. These methods differ from those that have been used thusfar to study other classes of large random matrices such as those appearing in compact Lie groups, operator algebras, physics, number theory, and communications engineering.

This is joint work with Shankar Bhamidi (U. of British Columbia) and Arnab Sen (U.C. Berkeley).

 

Fri, 24 Oct 2008
14:15
Oxford-Man Institute

(JOINTLY WITH OXFORD-MAN) Equilibrium in Continuous-Time Financial Markets: Endogenously Dynamically Complete Markets

Robert Anderson
(Berkeley)
Abstract

We prove existence of equilibrium in a continuous-time securities market in which the securities are potentially dynamically complete: the number of securities is at least one more than the number of independent sources of uncertainty. We prove that dynamic completeness of the candidate equilibrium price process follows from mild exogenous assumptions on the economic primitives of the model. Our result is universal, rather than generic: dynamic completeness of the candidate equilibrium price process and existence of equilibrium follow from the way information is revealed in a Brownian filtration, and from a mild exogenous nondegeneracy condition on the terminal security dividends. The nondegeneracy condition, which requires that finding one point at which a determinant of a Jacobian matrix of dividends is nonzero, is very easy to check. We find that the equilibrium prices, consumptions, and trading strategies are well-behaved functions of the stochastic process describing the evolution of information.

We prove that equilibria of discrete approximations converge to equilibria of the continuous-time economy

Thu, 12 Jun 2008
16:00
L3

Characterizing Z in Q with a universal-existential formula

Bjorn Poonen
(Berkeley)
Abstract

Refining Julia Robinson's 1949 work on the undecidability of the first order theory of Q, we prove that Z is definable in Q by a formula with 2 universal quantifiers followed by 7 existential quantifiers. It follows that there is no algorithm for deciding, given an algebraic family of Q-morphisms, whether there exists one that is surjective on rational points.

Wed, 28 Mar 2007
11:00
L3

From Polynomial Interpolation to the Complexity of Ideals

David Eisenbud, MSRI
(Berkeley)
Abstract

 

 

One natural question in interpolation theory is: given a finite set of points

in R^n, what is the least degree of polynomials on R^n needed to induce every

function from the points to R? It turns out that this "interpolation degree" is

closely related to a fundamental measure of complexity in algebraic geometry

called Castelnuovo-Mumford regularity. I'll explain these ideas a new

application to projections of varieties.

 

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