Course detail

Modern Mathematical Methods in Informatics

FIT-MIDAcad. year: 2019/2020

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). 

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.

Recommended optional programme components

Not applicable.

Recommended or required reading

G. Grätzer, Lattice Theory, Birkhäuser, 2003
K.Denecke and S.L.Wismath, Universal Algebra and Applications in Theoretical Computer Science, Chapman & Hall, 2002
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
S. Roman, Lattices and Ordered Sets, Springer, 2008.

Planned learning activities and teaching methods

Not applicable.

Assesment methods and criteria linked to learning outcomes

Tests during the semester

Language of instruction

Czech, English

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.  

Classification of course in study plans

  • Programme VTI-DR-4 Doctoral

    branch DVI4 , any year of study, winter semester, 0 credits, optional

  • Programme VTI-DR-4 Doctoral

    branch DVI4 , any year of study, winter semester, 0 credits, optional

  • Programme VTI-DR-4 Doctoral

    branch DVI4 , any year of study, winter semester, 0 credits, optional

  • Programme VTI-DR-4 Doctoral

    branch DVI4 , any year of study, winter semester, 0 credits, optional

Type of course unit

 

Lecture

26 hours, optionally

Teacher / Lecturer

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  

eLearning