[CvGmt News] Avviso Serie Seminari

Giulia Curciarello curciare at dm.unipi.it
Tue Jun 3 13:35:55 CEST 2003


Dylan Thurston (Harvard) sara' ospite del Dipartimento durante il mese
di giugno
e terra' una serie di seminari. Questi si svolgeranno il mercoledi' e/o
il giovedi' alle ore
15:00 in sala dei seminari (di volta in volta sara' precisato quando).
Il primo seminario e'
previsto per giovedi' 5/6.

Here is a title for the series and a tentative plan for the talks.

SHADOW SURFACES AND SPINES OF 3- AND 4-MANIFOLDS
------------------------------------------------
Lecture 1: The algebra of knotted trivalent graphs
Lecture 2: Spines, shadows, and calculi for them
Lecture 3: Notions of complexity for 3- and 4-manifolds
Lecture 5: Remarks on Witten's asymptotics and hyperbolic volume
conjectures
Lecture 4: Associators and foundations of finite-type invariants
Lecture 6: Curves in surfaces and surfaces in 3-manifolds

Abstarcts for the first two lectures:

Lecture 1: The algebra of knotted trivalent graphs
--------------------------------------------------
Knotted trivalent graphs (KTGs) form a rich algebra with a few simple
operations: connected sum, unzip, and bubbling.  With these
operations, KTGs are generated by two simple graphs, the unknotted
tetrahedron and two Mbius strips.  Almost all previously known
representations of knots, including knot diagrams and non-associative
tangles, can be turned into KTG presentations in a natural way.

Often two sequences of KTG operations produce the same output on all
inputs.  These relations can be subtle: for instance, there is a
planar algebra of KTGs with a distinguished cycle.  Studying these
relations naturally leads us to Turaev's \emph{shadow surfaces}, a
combinatorial representation of 3-manifolds based on simple 2-spines
of 4-manifolds.  In particular, for every KTG presentation of a knot
we construct a shadow surface for the knot complement, and KTG
presentations which yield the same shadow surface produce the same
knot.

Lecture 2: Spines, shadows, and calculi for them
------------------------------------------------
We contrast spines for 3-manifolds (which are related to, but more
general than, duals to triangulations of 3-manifolds) and shadow
surfaces for 3- and 4-manifolds, studying in particular the natural
spines and shadows related to a knot diagram.  We find calculi that
give sequences of moves that relate any two spines of the same
3-manifold, any two shadow surfaces for the same 3-manifold, but not
for any two shadow surfaces for the same 4-manifold (except modulo
some equivalences).  The failure in the last case is closely related
to the Andrews-Curtis conjecture.
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