Workshop on Hyperbolic volumes
December 9 (Tue.) -- December 11 (Thu.), 2003
at Third conference room, Second floor, S55 building
Waseda University (Ohkubo campus)
Workshop top <-- Abstracts --> Program
Yunhi Cho (University of Seoul);
"Extended hyperbolic space and complex volume of simplex",
We study the extended hyperblic space which contains hyperbolic space as a subspace and is the analytic continuation of the hyperbolic space. From the new space, we can induce an another kind of nice geometry and we can see that various hyperbolic formulas are satisfied on the space. The extended hyperbolic space gives the volumes of simplicies as complex values. So we need to explain the naturality of the space and the complex valued phenomena.
Kazuhiro Hikami (University of Tokyo);
"Quantum invariant and modular form",
I will speak about a relationship between quantum invariants for torus knot/link and modular forms.
Ruth Kellerhals (University of Fribourg);
1st talk: "Non-euclidean volume: Introduction and survey",
I speak about the differential formula, polylogarithms, even- versus odd-dimensional cases, spherical verus hyperbolic, Schlaefli's conjecture, etc
2nd talk: "Hyperbolic volume: Higher dimensional computations and open problems",
I speak about the results and problems in higher dimensions, the function theoretical problems involved, etc.
Gregory Leibon (Dartmouth College);
Lecture 1: "Scissors congruence: An introduction",
Lecture 1: "Scissors congruence: The birth of hyperbolic volume",
Lecture 3: "Scissors congruence: Volume symmetries",
Two polyhedra P and Q are called scissors congruent if we can chop P up into a finite number of pieces, then glue the pieces back together in order to form Q. Hilbert's third problem was to find a pair Euclidean polyhedra which had the same volume, but failed to be scissors congruent. Dehn quickly (1901) found such an example, and, in the process, discovered what is known today as the Dehn invariant. The question then became: if two polyhedra share the same volume and Dehn invariant, then are they scissors congruent? This question was answered in the affirmative by Sydler (1965). The corresponding question in hyperbolic space is one of the major open problems of low dimensional geometry. In this talk, we introduce the notion of scissors congruence, and present a bit about what is known concerning this fundamental conjecture.
The Milnor conjecture concerns the number theoretic properties of the Lobachevski function, a function closely related to hyperbolic volume. Due to this close relationship, the analytic identities showing up in this conjecture, as well as other analytic identities related to hyperbolic volume, often have interesting scissors congruence interpretations. Identifying these relationships can sometimes have startling geometric consequence, like Suslin's proof that if nP is scissors congruent to nQ, then P is scissors congruent to Q (in hyperbolic space, 1991). In this talk, we explore these scissors congruence interpretations of volume identities. In particular, we discuss some new identities discovered by J. Murakami and M. Yano, and provide a scissors congruence interpretation of these identities, due to Doyle and Leibon.
Koji Ohnuki (Waseda University);
"Link invariant as critical values of the dilogarithm functions",
In my talk I introduce a link invariant by using the dilogarithm function and show that this invariant for some knots gives the volume and the Chern-Simons invariant of the complement of the knot in 3-sphere.
Jun Murakami (Waseda University);
"Volume of hyperbolic tetrahedron and quantum 6j-symbols",
I explain a formula for the volumes of hyperbolic and spherical tetrahedra obtained from the quantum 6j-symbols. I also explain the background of this formula, that is the hyperbolic volume conjecture for the Turaev-Viro invariant of 3-manifolds.
Naoko Tamura (Tokyo Metroporitan University);
"On an ideal triangulation and the A-polynomial of a knot",
We explain the Yokota's method to construct an ideal triangulation of a knot complement and how to compute the A-polynomials by using it.
Akira Ushijima (University of Warwick);
"Proofs of volume formulae with respect to the dihedral angles and the edge lengths",
Following Murakami's talk, I will explain in this talk how to prove two volume formulae for hyperbolic tetrahedra, one is related to the dihedral angles and the other to the edge lengths, are really formulae. Both proofs are done similarly and the key tools are two known facts: one is the so-called Schlafli's differential formula, and the other is the relationship between dihedral angles and edge lengths of hyperbolic simplices.
Yoshiyuki Yokota (Tokyo Metroporitan University);
"On the Jones polynomial and the Neumann-Zagier function of hyperbolic knots",
I shall explain how the Neumann-Zagier function on the deformation space of a hyperbolic knot complement is derived from the colored Jones polynomial of the knot.