Innovations and NEW Ideas in my Scientific Work

This page is under construction and will be corrected and extended from time to time.

Abbreviations used:

ACCAT     Applied and Computational Category Theory
ANN       Artificial Neural Network
CAT       Category Theory
Cat       Category
CA         Computer Algebra
LogFib     Logical Fiberings
MAS     Multiagent System(s)
NCG       Noncommutative Geometry

Innovative Work and NEW Ideas

Field of Work:     Noncommutative Geometry, NCG
(introduced by Johnanes André in the 1970s as natural generalization of
Classical Affine Geometry).
I moved to this new field when I started to work on my doctoral thesis (in 1979).
NEW Idea and NEW Model:
I introduced a new model for noncommutative geometric spaces, completely new was the introduction of the Parallel Map   <,> : X x X --> R   which represents the entire structure of a space X. Geometric axioms and configurations can be expressed by corresponding equations in that new ``<,>-Calculus''.
This leads to a huge variety of examples of spaces and opens the possibility of computer applications, e.g. for computational treatment of problems concerning automated verification of geometric configurations and axioms.
Work on systematic construction of finite geometric spaces with prescribed properties has been done with computer support.
Solving OLDER PROBLEMS:
Motivated by the powerful concept and theory of Fiber Bundles, I introduced Fibered Geometric Spaces and Group Fiberings and this helped me to solve some older open questions and problems which were discussed then in noncommutative geometry.
A brief discussion of these topics can be found in the article "A Category of Geometric Spaces", provided as .pdf file below.

A basic geometric configuration and axiom in NCG deals with parallel shifting of triangles (Tamaschke condition), I interpreted this as shifting of a geometric 2-simplex. Analogously, the noncommutative version of the famous Axiom of Desargues suggested to me to consider shifting of a geometric 3-simplex (tetraeder). This observation motivated me to introduce generalizations in terms of shiftig of a q-simplex. In Algebraic Topology, q-simplices are elementary buildung blocks in topological spaces, they play a basic role in Homology and Cohomology Theory.
Thus, I introduced
NEW Geometric Axioms:     q-Simplex Conditions and corresponding Axioms.
The new q-simplex conditions (axioms) were the basis of a proof of the
"Hauptsatz of NCG" (formulated by J.André):
Find a pure geometric criterion for a geometric space to be a group space.
J.André gave a proof for finite spaces.
For more details cf.   J.Pfalzgraf: A note on simplices as geometric configurations.
Archiv der Mathematik, Vol. 49, 134-140 (1987).
A short exposition on the Simplicial Structure of Geometric Spaces and on
Geometric Fiberings can be found in   A Category of Geometric Spaces.




NEW Theory:     Logical Fiberings (LogFib)
Original Idea:
Replace the typical fiber in a Fiber Bundle, e.g. Vector Bundle, by a logical space, originally a 2-valued logic. Thus, LogFib is a fibering, a fiber bundle of local 2-valued logics the global space of the bundle leads to a many-valued logic.
I was highly inspired by the powerful mathematical theory of Fiber Bundles.




ACCAT     Applied and Computational Category Theory
In 1990 I established a NEW working group with that title when I started to work as a faculty at RISC-Linz, University of Linz. I gave two university courses, these were a starting point.
ACCAT reflects my own interest in CAT and computational aspects and all kinds of interesting and challenging applications of CAT modeling. Of course, ACCAT is an important topic of work in my working group here in Salzburg.
More information: ACCAT.

Jointly with my colleague Hartmut Ehrig (TU Berlin) I began to establish an own international ACCAT Workshop series. In the year 2006 we started with our 1st ACCAT Workshop organized as Satellite Event at the big European conferences ETAPS: European Joint Conferences on Theory and Practice of Software.
Further ACCAT Workshops followed at ETAPS-2007, ETAPS-2008, ETAPS-2009 - for more information cf. 5th ACCAT Workshop at ETAPS-2010.




NEW Modeling Approach:   ANN   Structure Modeling of Neural Networks


Introduction of the category GeoNET   -   Geometric Networks,   based on the category of Noncommutative Geometric spaces (NCG).
A Learning step can be expressed by a corresponding morphism in that Cat.   This new modeling approach opens the view to a NEW Theory of LEARNING, very generally.
This mathematical ANN Structure Model was deployed in neural network simulation and the corresponding network simulator was applied in an industrial project on optical quality control.
An unexpected experience was the observation that this abstract model led to considerable reduction of the project costs - (learning became "cheaper"..... CAT helped to save money).     Cf.   ACCAT and Theoretical Neurobiology,
ACCAT and Neural Networks.   This contribution received a Best Paper Award.




MAS   Multiagent Systems
MAS research is in rapid development. Almost everywhere one can find systems of cooperating units (agents). I consider MAS theory as a universal modeling paradigm for a huge variety of systems in the world.
Consequently, I introduced a CAT based model (definition) for MAS and the notion of a Base Diagram of a MAS representing the complete communication structure, itself being a category.   Base Diagram of a MAS.
IDEA:   Use Algebraic Graph Transformation (H.Ehrig and his group, TU Berlin). My original idea was to apply such techniques for the manipulation of a base diagram D with the aim to find an "optimal" structure of D representing the general communication structure of the underlying MAS in an optimal way.
  Thomas Soboll started to work in this direction, he proved interesting results and made Java implementations. He constructed and implemented Robot Scenario Demonstrators with which he can illustrate simultaneously the transformation steps of corresponding base diagrams.

NEW Logical Model:
Using logical fiberings (LogFib), I introduced a logical modeling approach for MAS in terms of distributed logics assigned to a MAS.
IDEA:   Every agent obtains a local logic (logical fiber) and the collection of all these fibers forms the global logical fiber bundle, i.e. the global logical state space of the MAS,   cf.   CAT and LogFib modeling in MAS.




Work in ROBOTICS:
Mathematical modeling methods. Symbolic methods using CA. Connectionist approaches using ANN and network simulations. Hybrid methods for applications to Inverse Kinematics problems.   For first steps cf.   AISC-2008 Conference.
Based on LogFib and transjunctions (nonclassical logical operations), I introduced the notion of a "fibered logical controller" for cooperating robots scenarios.
More information on work in robotics can be found in ROBOTICS.