## MINI-CURSO

## Directed combinatorial homology and noncommutative
geometry

#### Local:
Departamento de Matemática
do IST, sala 3.10

Data e horário:
9 a 12 de Fevereiro de 2004 (segunda a quinta), das 11.00 às 12.30

#### Resumo:

We will present a brief study of the homology of cubical sets, with two
main purposes. First, this combinatorial structure is viewed as representing
directed spaces, breaking the intrinsic symmetries of topological spaces.
Cubical sets have a directed homology, consisting of preordered abelian
groups where the positive cone comes from the structural cubes. But cubical
sets can also express topological facts missed by ordinary topology. This
happens, for instance, in the study of group actions or foliations, where
a topologically-trivial quotient (the orbit set or the set of leaves) can
be enriched with a natural cubical structure whose directed homology agrees
with Connes' analysis in noncommutative geometry. Thus, cubical sets can provide
a sort of 'noncommutative topology', without the metric information of C*-algebras
[G1]. This similarity can be made stricter by introducing 'normed cubical
sets' and their normed directed homology, formed of normed preordered abelian
groups. The normed cubical sets associated with irrational rotations have
thus the same classification up to isomorphism as the well-known irrational
rotation C*-algebras [G2]. Finally, we will see that an elementary part
of these results can also be obtained with a simpler structure, using D.
Scott's equilogical spaces [G3, G4].

#### Bibliografia:

[G1] M. Grandis, Directed
combinatorial homology and noncommutative tori, Math. Proc. Cambridge
Philos. Soc., to appear. [Dip. Mat. Univ. Genova, Preprint 480 (2003), 29
p.]

[G2] M. Grandis, Normed
combinatorial homology and noncommutative tori, Dip. Mat. Univ. Genova,
Preprint 484 (2003), 14 p.

[G3] M. Grandis, Equilogical
spaces, homology and noncommutative geometry, Dip. Mat. Univ. Genova,
Preprint 493 (2003), 24 p.

[G4] M. Grandis,
Inequilogical spaces, directed homology and noncommutative geometry,
Dip. Mat. Univ. Genova, Preprint 494 (2004), 22 p.

[G5] M. Grandis, Directed combinatorial homology and noncommutative geometry (course notes).

#### Apoio:
FEDER,
FCT (POCTI),
CAMGSD,
CLC