Course detail

Modern Mathematical Methods in Informatics

FIT-MIDAcad. year: 2017/2018

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

Czech

Number of ECTS credits

0

Mode of study

Not applicable.

Guarantor

prof. RNDr. Josef Šlapal, CSc.

Department

Dept. of Algebra and Discrete Mathematics (ÚM OAAG)

Learning outcomes of the course unit

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.

Prerequisites

Basic knowledge of set theory, mathematical logic and general algebra.

Co-requisites

Not applicable.

Planned learning activities and teaching methods

Not applicable.

Assesment methods and criteria linked to learning outcomes

Study evaluation is based on marks obtained for specified items. Minimimum number of marks to pass is 50.

Course curriculum

Syllabus of lectures:
  1. Naive and axiomatic (Zermelo-Fraenkel) set theories, finite and countable sets.
  2. Cardinal arithmetic, continuum hypothesis and axiom of choice.
  3. Partially and well-ordered sets, isotone maps, ordinals.
  4. Varieties of universal algebras, Birkhoff theorem.
  5. Lattices and lattice homomorphisms
  6. Adjunctions of ordered sets, fix-point theorems and their applications
  7. Partially ordered sets with suprema of directed sets (DCPO) and their applications in informatics
  8. Scott information systems and domains, category of domains
  9. Closure operators, their basic properties and applications (in logic)
  10. Basics og topology: topological spaces and continuous maps, separation axioms
  11. Connectedness and compactness in topological spaces
  12. Special topologies in informatics: Scott and Lawson topologies
  13. Basics of digital topology, Khalimsky topology  

Work placements

Not applicable.

Aims

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.  

Specification of controlled education, way of implementation and compensation for absences

Tests during the semester

Recommended optional programme components

Not applicable.

Prerequisites and corequisites

Basic 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.     

Recommended reading

  • 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.

 

 

Classification of course in study plans

  • Programme VTI-DR-4 Doctoral

    branch DVI4 , any year of study, summer semester, elective

Type of course unit

 

Lecture

26 hours, optionally

Teacher / Lecturer

prof. RNDr. Josef Šlapal, CSc.

Syllabus

  1. Naive and axiomatic (Zermelo-Fraenkel) set theories, finite and countable sets.
  2. Cardinal arithmetic, continuum hypothesis and axiom of choice.
  3. Partially and well-ordered sets, isotone maps, ordinals.
  4. Varieties of universal algebras, Birkhoff theorem.
  5. Lattices and lattice homomorphisms
  6. Adjunctions of ordered sets, fix-point theorems and their applications
  7. Partially ordered sets with suprema of directed sets (DCPO) and their applications in informatics
  8. Scott information systems and domains, category of domains
  9. Closure operators, their basic properties and applications (in logic)
  10. Basics og topology: topological spaces and continuous maps, separation axioms
  11. Connectedness and compactness in topological spaces
  12. Special topologies in informatics: Scott and Lawson topologies
  13. Basics of digital topology, Khalimsky topology