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<DIV><FONT face=Arial size=2><FONT face="Times New Roman" size=3>Dylan Thurston
(Harvard) sara' ospite del Dipartimento durante il mese<BR>di giugno<BR>e terra'
una serie di seminari. Questi si svolgeranno il mercoledi' e/o<BR>il giovedi'
alle ore<BR>15:00 in sala dei seminari (di volta in volta sara' precisato
quando).<BR>Il primo seminario e'<BR>previsto per giovedi' 5/6.<BR><BR>Here is a
title for the series and a tentative plan for the talks.<BR><BR>SHADOW SURFACES
AND SPINES OF 3- AND
4-MANIFOLDS<BR>------------------------------------------------<BR>Lecture 1:
The algebra of knotted trivalent graphs<BR>Lecture 2: Spines, shadows, and
calculi for them<BR>Lecture 3: Notions of complexity for 3- and
4-manifolds<BR>Lecture 5: Remarks on Witten's asymptotics and hyperbolic
volume<BR>conjectures<BR>Lecture 4: Associators and foundations of finite-type
invariants<BR>Lecture 6: Curves in surfaces and surfaces in
3-manifolds<BR><BR>Abstarcts for the first two lectures:<BR><BR>Lecture 1: The
algebra of knotted trivalent
graphs<BR>--------------------------------------------------<BR>Knotted
trivalent graphs (KTGs) form a rich algebra with a few simple<BR>operations:
connected sum, unzip, and bubbling. With these<BR>operations, KTGs are
generated by two simple graphs, the unknotted<BR>tetrahedron and two Mbius
strips. Almost all previously known<BR>representations of knots, including
knot diagrams and non-associative<BR>tangles, can be turned into KTG
presentations in a natural way.<BR><BR>Often two sequences of KTG operations
produce the same output on all<BR>inputs. These relations can be subtle:
for instance, there is a<BR>planar algebra of KTGs with a distinguished
cycle. Studying these<BR>relations naturally leads us to Turaev's
\emph{shadow surfaces}, a<BR>combinatorial representation of 3-manifolds based
on simple 2-spines<BR>of 4-manifolds. In particular, for every KTG
presentation of a knot<BR>we construct a shadow surface for the knot complement,
and KTG<BR>presentations which yield the same shadow surface produce the
same<BR>knot.<BR><BR>Lecture 2: Spines, shadows, and calculi for
them<BR>------------------------------------------------<BR>We contrast spines
for 3-manifolds (which are related to, but more<BR>general than, duals to
triangulations of 3-manifolds) and shadow<BR>surfaces for 3- and 4-manifolds,
studying in particular the natural<BR>spines and shadows related to a knot
diagram. We find calculi that<BR>give sequences of moves that relate any
two spines of the same<BR>3-manifold, any two shadow surfaces for the same
3-manifold, but not<BR>for any two shadow surfaces for the same 4-manifold
(except modulo<BR>some equivalences). The failure in the last case is
closely related<BR>to the Andrews-Curtis
conjecture.</FONT><BR></FONT></DIV></BODY></HTML>