Modern Mathematical Methods in Informatics
MID Acad. year 2010/2011 Summer semester
Naive and axiomatic (Zermelo-Fraenkel) set theories, finite and countable sets, cardinal arithmetic, continuum hypothesis and axiom of choice. Partially and well-ordered sets and ordinals. Varieties of universal algebras, Birkhoff theorem. Lattices and lattice homomorphisms. Adjunctions, fixed-point theorems and their applications. Partially ordered sets with suprema of directed sets, (DCPO), Scott domains. Closure spaces and topological spaces, applications in informatics (Scott, Lawson and Khalimsky topologies).
Language of instruction
Subject specific learning outcomes and competences
Students will learn about modern mathematical methods used in informatics and will be able to use the methods in their scientific specializations.
Generic learning outcomes and competences
The graduates will be able to use modrn and efficient mathematical methods in their scientific work.
The aim of the subject is to acquaint students with modern mathematical methods used in informatics. In particular, methods based on the theory of ordered sets and lattices, algebra and topology will be discussed.
Prerequisite kwnowledge and skills
Basic knowledge of set theory, mathematical logic and general algebra.
- G. Grätzer, Lattice Theory, Birkhäuser, 2003
- K.Denecke and S.L.Wismath, Universal Algebra and Applications in Theoretical Computer Science, Chapman & Hall, 2002
- S. Roman, Lattices and Ordered Sets, Springer, 2008
- J.L. Kelley, general Topology, Van Nostrand, 1955.
- G. Grätzer, Universal Algebra, Springer, 2008
- B.A. Davey, H.A. Pristley, Introduction to Lattices ad Order, Cambridge University Press, 1990
- P.T. Johnstone, Stone Spaces, Cambridge University Press, 1982
- S. Willard, General Topology, Dover Publications, Inc., 1970
- N.M. Martin and S. Pollard, Closure Spaces and Logic, Kluwer, 1996
- T. Y. Kong, Digital topology; in L. S. Davis (ed.), Foundations of Image Understanding, pp. 73-93. Kluwer, 2001
Syllabus of lectures
- Naive and axiomatic (Zermelo-Fraenkel) set theories, finite and countable sets.
- Cardinal arithmetic, continuum hypothesis and axiom of choice.
- Partially and well-ordered sets, isotone maps, ordinals.
- Varieties of universal algebras, Birkhoff theorem.
- Lattices and lattice homomorphisms
- Adjunctions of ordered sets, fix-point theorems and their applications
- Partially ordered sets with suprema of directed sets (DCPO) and their applications in informatics
- Scott information systems and domains, category of domains
- Closure operators, their basic properties and applications (in logic)
- Basics og topology: topological spaces and continuous maps, separation axioms
- Connectedness and compactness in topological spaces
- Special topologies in informatics: Scott and Lawson topologies
- Basics of digital topology, Khalimsky topology
Tests during the semester
Course inclusion in study plans