Course details
Modern mathematical methos in informatics
MID Acad. year 2009/2010 Summer semester
Categories and diagrams, special objects and morphisms, limits (products and pullbacks), functors, natural transformations and adjunctions, exponentiation in categories, applications (deduction systems and typed lambda-calculus). Partially ordered sets, infima, suprema, DCPO, domains. Fix point theorems and applications. Closure spaces and topological spaces, applications in informatics (Scott, Lawson and Khalimsky topologies).
Guarantor
Language of instruction
Completion
Time span
- 26 hrs lectures
Department
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.
The graduates will be able to use modrn and efficient mathematical methods in their scientific work.
Learning objectives
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, category theory, algebra and topology will be discussed.
Recommended prerequisites
Prerequisite knowledge and skills
Basic knowledge of set theory, mathematical logic and general algebra.
Study literature
- 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.
Fundamental literature
- 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
- S. Roman, Lattices and Ordered Sets, Springer, 2008.
Syllabus of lectures
- Definitions, basic properties and examples of categories
- Special objects and morphisma, diagrams and limits (products and pullbacks)
- Functors, natural transformations and adjunctions
- Applications of categories - deduction systems and typed lambda-calculus
- Partially ordered sets, suprema and infima, isotone maps
- 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
Progress assessment
Study evaluation is based on marks obtained for specified items. Minimimum number of marks to pass is 50.
Controlled instruction
Tests during the semester
Course inclusion in study plans