We prove that the rank gradient vanishes for mapping class groups, Aut(Fn) for all n, Out(Fn), n > 2 and any Artin group whose underlying graph is connected. We compute the rank gradient and verify that it is equal to the first L2-Betti number for some classes of Coxeter groups.
Many geophysical flows over topography can be modeled by two-dimensional
depth-averaged fluid dynamics equations. The shallow water equations
are the simplest example of this type, and are often sufficiently
accurate for simulating tsunamis and other large-scale flows such
as storm surge. These hyperbolic partial differential equations
can be modeled using high-resolution finite volume methods. However,
several features of these flows lead to new algorithmic challenges,
e.g. the need for well-balanced methods to capture small perturbations
to the ocean at rest, the desire to model inundation and flooding,
and that vastly differing spatial scales that must often be modeled,
making adaptive mesh refinement essential. I will discuss some of
the algorithms implemented in the open source software GeoClaw that
is aimed at solving real-world geophysical flow problems over
topography. I'll also show results of some recent studies of the
11 March 2011 Tohoku Tsunami and discuss the use of tsunami modeling
in probabilistic hazard assessment.
A map betweem metric spaces is a bilipschitz homeomorphism if it
is Lipschitz and has a Lipschitz inverse; a map is a bilipschitz embedding
if it is a bilipschitz homeomorphism onto its image. Given metric spaces
X and Y, one may ask if there is a bilipschitz embedding X--->Y, and if
so, one may try to find an embedding with minimal distortion, or at least
estimate the best bilipschitz constant. Such bilipschitz embedding
problems arise in various areas of mathematics, including geometric group
theory, Banach space geometry, and geometric analysis; in the last 10
years they have also attracted a lot of attention in theoretical computer
science.
The lecture will be a survey bilipschitz embedding in Banach spaces from
the viewpoint of geometric analysis.
I’ll report on my recent work (with co-authors Holt and Ciobanu) on Artin
groups of large type, that is groups with presentations of the form
G = hx1, . . . , xn | xixjxi · · · = xjxixj · · · , 8i 3. (In fact, our results still hold when some, but not all
possible, relations with mij = 2 are allowed.)
Recently, Holt and I characterised the geodesic words in these groups, and
described an effective method to reduce any word to geodesic form. That
proves the groups shortlex automatic and gives an effective (at worst quadratic)
solution to the word problem. Using this characterisation of geodesics, Holt,
Ciobanu and I can derive the rapid decay property for most large type
groups, and hence deduce for most of these that the Baum-Connes conjec-
ture holds; this has various consequence, in particular that the Kadison-
Kaplansky conjecture holds for these groups, i.e. that the group ring CG
contains no non-trivial idempotents.
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In recent years there has been extensive interest in word maps on groups, and various results were obtained, with emphasis on simple groups. We shall focus on some new results on word maps for more general families of finite and infinite groups.