Scissors congruences

Spring School

April 30 - May 4, 2009
This spring school is part of the study program of the Graduiertenkolleg "Homotopy and Cohomology" at the universities of Bonn, Bochum and Düsseldorf. It takes place at the "Universitätskolleg" Bommerholz (TU-Dortmund).

Scissors congruences

Two polytopes in euclidean n-space are called scissors congruent if they can be subdivided into finitely many pieces such that each piece in the first polytope is congruent to exactly one piece in the second polytope.

Elementary geometric considerations show that polytopes in the plane are scissors congruent if and only if they have the same area. Hilbert's 3rd problem was the question whether volume determines the scissors congruence class also in 3-space. The answer was given by Max Dehn almost immediately: In 1900, he described an invariant with values in R ⊗Z R/Z which shows that the answer is no. Only 1965 J. P. Sydler proved that volume and Dehn invariant together determine the scissors congruence class in 3-space. Higher dimensional analogues are still unsolved. There are variants for spherical and hyperbolic geometry, which are open even in dimension 3.

From a modern point of view, these classical questions are closely related to the computation of the homology of Lie groups considered as discrete groups. Furthermore there are interesting connections to deep questions about the algebraic K-theory of the complex numbers.

The basic reference for the school is the monograph
[D] J.L. Dupont, Scissors congruences, group homology and characteristic classes, World Scientific.

Further references occuring below are

[M] Milnor, On the homology of Lie groups made discrete. Commentarii Mathematici Helvetici, Vol. 58, No. 1, 72--85, 1983

[S] Suslin, A.A., Algebraic K-theory of fields. Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), 222--244, Amer. Math. Soc., Providence, RI, 1987

Program

The meeting begins with dinner on April 30th. The program starts early the next morning and ends on May 4th late afternoon.

Time
 Title
 Speaker
 References
Friday
 May 1st
9:30-10:30
 (1) Introduction
 J. Dupont
11:00-12:00
 (2) A homological interpretation of scissors congruence groups
 B. Dimitrova
 [D] Ch. 2
15:30-16:30
 (3) Flag complexes
 K. Hutschenreuter
 [D] Ch. 3, ignore the spherical case
17:00-18:00
 (4) Translational scissors congruences
 F. Lenhardt
 [D] Ch. 4
Saturday
 May 2nd
9:30-10:30
 (5) Euclidean scissors congruences
 M. Ullmann
 [D] Ch. 5
11:00-12:00
 (6) Hyperbolic scissors congruences I
 J. Matz
 [D] Ch. 8
15:30-16:30
 (7) Hyperbolic scissors congruences II
 V. Ozornova
 [D] Ch. 8
17:00-18:00
 (8) Survey and summary
 J. Dupont
Sunday
 May 3rd
9:30-10:30
 (9) The group H2 (SO(3), R³) vanishes
 A. Beckers
 [D] Ch. 6
11:00-12:00
 (10) Some homological stability results
 I. Patchkoria
 [D] Ch. 9, in particular Proof of Theorem 6.1, euclidean case, only report the hyperbolic case
15:30-16:30
 (11) Homology of Lie groups made discrete
 J. Moellers
 [M]
17:00-18:00
 (12) Chern-Simons classes and volume
 A. Angel
 [D] Ch. 10
Monday
 May 4th
9:00-10:00
 (13) Polytopes with algebraic vertices
T. Kuessner
 [D] Ch. 10
10:00-10:45
 (14) A Dilogarithm formula for the Cheeger-Chern-Simons class
 J. Dupont

11:00-12:00
 (15) Hyperbolic 3-manifolds, the extended Bloch group and algebraic K-theory
 C. Zickert
14:00-15:00
 (16) Suslin's stable results about the Friedlander-Milnor conjecture
 H. Reich
 [S]

Registration

If you are interested in participation please send an E-Mail containing the following information: Registration deadline has expired.

Last Updated: 2009/05/08 - F. Deniz