Du 17 au 19 septembre 2007
IRMA
Ces trois journées rendront compte des progrès récents sur la Conjecture du Volume, au carrefour de la géométrie et de la topologie quantique.
Organisateurs : Stéphane Baseilhac (IJF, Grenoble), Francesco Costantino et Gwénaël Massuyeau (IRMA, Strasbourg).
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Lundi 17 septembre 2007
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09:30 - 10:30
F. Bonahon, Los Angeles
Using quantum Teichmüller theory to represent the braid group.
Quantum Teichmuller theory is very closely related to Kashaev's original presentation of the Kashaev invariant. We will explain how to use quantum Teichmuller theory to construct projective representations of the braid groups of the sphere. We will conclude with some speculations on the Volume Conjecture in this context. -
11:00 - 12:00
S. Garoufalidis, Atlanta
Resurgence in Quantum Topology : what is it, and why should we care ?
We will formulate a general resurgence conjecture in Quantum Topology that covers the cases of manifolds with/without torus boundary and implies the Volume and Witten Conjectures to all orders, with exponentially small terms included, in an exact form. We will also discuss a proof of this resurgence conjecture for single sums of quantum factorials. -
14:00 - 15:00
B. Patureau-Mirand, Vannes
Multivariable link invariants.
We will present some multivariable link invariants related to Lie superalgebras and quantum groups at root of unity. They can be thought of as generalizations of the multivariable Alexander polynomial and Kashaev's invariants. -
15:30 - 16:30
F. Gueritaud, Paris
The Laurent phenomenon and total positivity.
An inductively defined family of rational functions in several variables can surprisingly turn out to be a family of Laurent series. Fomin and Zelevinsky gave an algebraic explanation of this ``Laurent phenomenon'', which we will outline in the lecture. Examples include rational functions which describe chart maps in Teichmuller space, and their deformations. Moreover, the Laurent series are often totally positive, although this has been shown only in a few special cases. -
Mardi 18 septembre 2007
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09:30 - 10:30
R. Kashaev, Genève
On rings associated with knots
I will talk about some ring valued invariants of knots and explain how to calculate them in few concrete examples. -
11:00 - 12:00
S. Garoufalidis, Atlanta
Concrete asymptotics of classical and quantum 6j-symbols.
The classical 6j-symbols depend on 6 parameters that can be arranged as the edges of a metric tetrahedron. We will discuss in detail and concretely the case of the regular euclidean tetrahedron with edge-lenghts 1, and we will prove the existence of a concrete asymptotic expansion with leading terms two complex numbers in the unit circle. Our formula corrects some numerical errors in J. Robert's theorem and 30+years old physics literature. Our proof (which generalizes without change to the quantum 6j-symbols, regular or not) uses elementary algebraic ideas of WZ theory, together with a recently solved resurgence conjecture for single sums of hypergeometric functions. This is a joint work with Roland van der Veen, conceived during an unexpected bus-ride to the Billund airport at the end of the June 2007 Aarhus conference. -
14:00 - 15:00
J. Dubois, Barcelona
Reidemeister torsion and Chern-Simons invariant appear in the asymptotic expansion of the colored Jones polynomial for torus knots.
Let < K >_N be the Kashaev invariant (a specialization of the colored Jones polynomial). For torus knots, the Volume Conjecture is well-known and trivially true in the sense that log(|< K >_N|)/N converges to 0 as N goes to infinity. In fact, < K >_N is equivalent to N^{3/2} for large N. -
15:30 - 16:30
F. Costantino, Strasbourg
On a shadow-approach to the Volume Conjecture.
I will propose and discuss a version of the Generalized Volume Conjecture based on shadow-state sums expressing the colored Jones polynomials of links. I will prove this version of the conjecture for all the universal hyperbolic links starting from the Murakami-Yano-Ushijima formula for the volume of a truncated hyperbolic tetrahedron. -
Mercredi 19 septembre 2007
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09:30 - 10:30
R. Benedetti, Pisa
About quantum hyperbolic partition functions
We will discuss some structural features and a few experimental evidences (about the asymptotic behaviour) of the QHFT partition functions built (in joint works with S. Baseilhac) on the matrix dilogarithms. -
11:00 - 12:00
J. Andersen, Aarhus
The mapping class groups do not have Kazhdan's Property (T)
We prove that the mapping class group of a closed oriented surface of genus at least two does not have Kazhdan's property (T). We use Reshetikhin-Turaev quantum representations of the mapping class groups to construct a Hilbert space representation of the mapping class group. By a theorem of J. Roberts this representation does not have fixed vectors. We construct an almost fixed vector for this representation by applying the theory of coherent states and the theory of Toeplitz operators to the construction of the representation via geometric quantization of moduli spaces. -
14:00 - 15:00
R. Van der Veen, Amsterdam
The Volume Conjecture for augmented knotted trivalent graphs.
We propose to extend the Volume Conjecture to knotted trivalent graphs (KTGs) and show that it holds for all augmented KTGs. By an augmented KTG we mean a KTG to which a number of unknotted, mutually unlinked components have been added in a specific way. -
15:30 - 16:30
S. Baseilhac, Grenoble
Tropical views on cusped manifolds
The 3-manifold invariants of cusped manifolds arising from quantum hyperbolic geometry are defined on a sequence of amoebas of complex curves. We will discuss its degeneration as the level goes to infinity.