(HUJI)

I will explain a very intriguing connection between low
dimensional
topology, knot invariants, and quantum computation: It turns out that in
some well defined sense, quantum computation is *equivalent* to certain
approximations of the Jones polynomial. In Computer Science language: the
approximation of the Jones polynomial at certain points is BQP complete.
This means that:
*non-unitary* representations of the Temperley Lieb algebras, and provide
algorithms for many more problems, including an additive approximation
of the partition function of the q-state Potts model.
The talk is based on several joint works with Arad, Eban, Jones and
Landau, following works of Kitaev, Freedman, Larsen and Wang. No prior
knowledge of quantum computation, topology, or physics, will be assumed.

- There is an efficient quantum algorithm to approximate the Jones polynomial of any given link, at certain roots of unity;
- The approximation of the Jones polynomial is the hardest problem a quantum computer can possibly solve;
- Shor's factoring algorithm can be presented as the approximation of the Jones polynomial of a certain link.

(Princeton)

(IHES)

In nonequilibrium statistical mechanics close to equilibrium, the linear
response of a statistical mechanics equilibrium state under a perturbation
of the original dynamics is given by the Green-Kubo formula. Away from
equilibrium we face the mathematical problem of obtaining a linear response
theory for a general dynamical system. For such a general time evolution
the observed steady states are believed to be described by so-called
SRB measures, and linear response theory should describe the change of
SRB measures under perturbations of the dynamics. We review the theory
of SRB measures, what is known mathematically of their dependence on
parameters (in particular, the differentiability question in relation
with bifurcations for systems that are not uniformly hyperbolic) and
the possible existence of a linear response formula. The lecture should
be accessible to a general scientifically educated audience.

(TAU)

How far off the edge of the table can we reach by stacking n
identical, homogeneous, frictionless blocks of length 1? A classical
solution achieves an overhang of (1/2)H_n, where H_n=1/1+1/2+..+1/n
is the n-th harmonic number. This solution is widely believed to be
optimal. We show, however, that it is, in fact, exponentially far
from optimality by constructing simple n-block stacks that achieve
an overhang of cn^{1/3}, for some constant c>0. We also show, under
reasonable assumptions, that larger overhangs are impossible.

Joint work with Mike Paterson, Yuval Peres, Mikkel Thorup and Peter Winkler.

Joint work with Mike Paterson, Yuval Peres, Mikkel Thorup and Peter Winkler.