Modern Mathematical Methods in Informatics

• 26 hrs lectures

• 100 pts final exam

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.  
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.

• Discrete Mathematics (IDM)
• Mathematical Structures in Computer Science (MAT)

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

• G. Grätzer, Lattice Theory, Birkhäuser, 2003
• P.T. Johnstone, Stone Spaces, Cambridge University Press, 1982
• N.M. Martin and S. Pollard, Closure Spaces and Logic, Kluwer, 1996
• S. Roman, Lattices and Ordered Sets, Springer, 2008.
• V.K.Garg, Introduction to Lattice Theory with Computer Science Applications, Wiley, 2015
• T. Y. Kong, Digital topology; in L. S. Davis (ed.), Foundations of Image Understanding, pp. 73-93. Kluwer, 2001

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  

Tests during the semester
The subject is evaluated according to the result of the final exam, the minimum for passing the exam is 50/100 points.