Dates
Time/Location

The course will be taught on the following dates:
25. and 26. September
16. and 17. October, 2012
6. and 7. November, 2012
27. and 28. November, 2012
18. and 19. December, 2012
Tuesdays (dates above): 13:1517:00, Univ. of Fribourg, Building Math II, Room 0.101
Wednesdays (dates above): 10:1512:00 and 13:15  17:00, Univ. of Fribourg, Physics Building, Room 2.52

Outline of the course:
In this graduatelevel class I will present the modern theory of flat chains with coefficients in an Abelian group G, with emphasis on the ambient space being an arbitrary Banach space. The theory was initiated in Euclidean spaces by W.H. Fleming and recent developments are due to R. Hardt and myself. It is relevant to solving general Plateau type problems with Dirichlet boundary data or in homology classes of some metric spaces.
I will cover the basic material emphasizing the approximation procedure of flat chains by polyhedral chains, and of Lipschitz maps by PL maps, thereby illustrating the analogy with simplicial homology. Completion with respect to the flat norm adds powerful tools from analysis. As an example, I will treat the theory of slicing of flat chains, originating in differential topology.
The main results I will prove in detail are the deformation theorem, the compactness theorem, the slicenullity theorem, the rectifiability theorem, and the isoperimetric inequality.
Literature:
T. Adams: Flat chains in Banach spaces, J. Geom. Anal. 18 (2008), no. 1, 1  28
L. Ambrosio: Metric space valued functions of bounded variation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 17 (1990), no. 3, 439  478
Th. De Pauw, R. Hardt: Rectifiable and flat G chains in a metric space, Amer. J. Math. 134 (2012), no. 1, 1  69
H. Federer: Flat chains with positive densities, Indiana Univ. Math. J. 35 (1986), no. 2, 413  424
W. H. Fleming: Flat chains over a finite coefficient group, Trans. Amer. Math. Soc. 121 (1966) 160  186
B. Kirchheim: Rectifiable metric spaces: local structure and regularity of the Hausdorff measure, Proc. Amer. Math. Soc. 121 (1994), no. 1, 113  123
C. Riedweg: Virtual flat chains and homologies in metric spaces, Ph.D. Thesis ETH Zurich, 2011
M. Snipes: Flat forms in Banach spaces, Ph.D. Thesis University of Michigan, 2009
S. Wenger: Isoperimetric inequalities of Euclidean type in metric spaces, Geom. Funct. Anal. 15 (2005), no. 2, 534  554
B. White: The deformation theorem for flat chains, Acta Math. 183 (1999), no. 2, 255  271
H. Whitney: Geometric integration theory, Princeton University Press, Princeton, N. J., 1957
