martedi' 08-03-2005 (15:00) - Sala dei seminari
Paolo Bellingeri (Pisa) :
Surface braids II: finite type invariants
Argomento: Geometria
Starting with the classic braid group $B_n$, Birman and Baez have
introduced the monoid $SB_n$ of singular braids where, in addition to the
usual positive and negative crossings $\sg_i$ and $\sg_i^{-1}$ of the
strands at position $i$ and $i+1$, one allows a singular crossing denoted
$\tau_i$ where the two strands intersect. In the same way one can
introduce the singular braid monoid on $\Sigma$, $SB_n(\Sigma)$, as an
extension of the surface braid group $B_n (\Sigma)$. This monoid has been
introduced by Gonz'alez-Meneses and Paris in order to define finite type
(Goussarov- Vassiliev) invariants for surface braids. They constructed a
universal finite type invariant for surface braids with integer
coefficients. This result cannot be improved. We will show that there does
not exist a universal finite type invariant for surface braids, which is
also fonctorial, i.e. there does not exist a Kontsevich integral for
surface braids. This result does not depend on the choice of the
coefficient ring and it extends naturally to tangles on handlebodies.
Finally we will discuss about the definition of finite type invariants for
braid groups and their generalisations.
Giulia Curciarello
Segreteria Didattica
tel: 050-2213219
e-mail curciare(a)dm.unipi.it
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venerdi' 11-03-2005 (16:00) - Sala Riunioni
Joan Bagaria (ICREA Barcelona) :
Natural Axioms of Set Theory and the Continuum.
As is well-known, Cantor's continuum problem, namely, what is the
cardinality of the set of real numbers, is independent of the usual ZFC
axioms of Set Theory. K. Goedel suggested that new natural axioms
should be found that would settle the problem and hinted at
large-cardinal
axioms as such. However, shortly after the invention of forcing,
it was shown that the problem remains independent even if one adds to
the standard ZFC system of Set Theory the usual axioms of
large-cardinals, like the existence of measurable cardinals, or even
supercompact cardinals, provided, of course, that these axioms are
consistent. While numerous axioms have been proposed that settle the
problem--although not always in the same way--from the Axiom of
Constructibility to strong combinatorial axioms like
the Proper Forcing Axiom or Martin's Maximum, none of them so far
has been recognized as a natural axiom and been accepted as an
appropriate solution to the continuum problem. In this talk we will
discuss some heuristic principles, which might be regarded as
Meta-Axioms of Set Theory, that provide a criterion for
assessing the naturalness of the set-theoretic axioms. Under this
criterion we then evaluate several kinds of axioms, with a special
emphasis on a class of recently introduced set-theoretic
principles, known as axioms of generic absoluteness, for which we can
reasonably argue that they constitute very natural axioms of Set Theory
and which settle Cantor's continuum problem.
Giulia Curciarello
Segreteria Didattica
tel: 050-2213219
e-mail curciare(a)dm.unipi.it
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