
MiniSymposium at the 6th European Congress of Mathematics, Krakow, Poland
Discrete Structures in Algebra, Geometry, Topology and Computer Science
Date: Tuesday, July 3rd, 2012,
Time: 15:4517:35
Place: Small Hall
Organizers:
Eva Maria Feichtner (University of Bremen, Germany)
Dmitry FeichtnerKozlov (University of Bremen, Germany)
Speakers:
Martin Raussen (Aalborg University, Denmark)
Louis Rowen (BarIlan University, Israel)
Mario Salvetti (University of Pisa, Italy)
Schedule:
15:4516:15  Mario Salvetti (University of Pisa, Italy) 
Discrete methods for the topological study of some Configuration Spaces 
We apply some methods from "discrete topology" to the study of some
"configuration Spaces." In particular, we are interested in (co)homological computations for
onfiguration spaces associated with Artin and Coxeter groups.


16:2516:55  Martin Raussen (Aalborg University, Denmark) 
Directed Algebraic Topology and Applications 
Directed Algebraic Topology deals with topological spaces in which only a
certain system of (directed) paths is allowed. This occurs naturally when
the time flow is crucial, e.g. in relativity theory or for concurrent
computations. The topological analysis gets far more complicated than
in the classical situation; for example, the fundamental group has to be
replaced by the fundamental category of a directed space. Nevertheless,
it is possible to apply techniques and invariants from classical algebraic
topology after modification. For example, it has been shown how to model
the space of executions (directed paths) in Higher Dimensional Automata
(a particular model for concurrency) by a simplicial complex; this allows
to calculate (homological) invariants with important interpretations. 

17:0517:35  Louis Rowen (BarIlan University, Israel) 
Structures in tropical algebra 
The rapidly developing topic called "tropical mathematics", has
been based on two main approaches. Primarily, tropical curves have
been defined as domains of nondifferentiability of polynomials
over the maxplus algebra, and also tropical mathematics has been
viewed in terms of valuation theory applied to curves over Puiseux
series. Unfortunately, semirings such as the maxplus algebra
possess a limited algebraic structure theory, and also do not
reflect these valuationtheoretic properties, thereby forcing
researchers to turn to combinatoric arguments.
The object of this talk is to present an algebraic structure (studied
jointly with Z. Izhakian and M. Knebusch) more compatible with
algebraic structure theory and valuation theory than the maxplus algebra.
We present a "layered" structure, "sorted" by a semiring which
permits varying ghost layers, and indicate how it permits a direct
algebraic description of tropical varieties, and show how its
coordinate semiring reflects the geometric properties. We also discuss
factorization of polynomials, linear algebra, properties of the resultant,
discriminant, and multiple roots of polynomials. 
