Course: Selected topics from geometry

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Course title Selected topics from geometry
Course code KMA/XVPGE
Organizational form of instruction no contact
Level of course Doctoral
Year of study not specified
Semester Winter and summer
Number of ECTS credits 30
Language of instruction Czech
Status of course Compulsory-optional
Form of instruction Face-to-face
Work placements This is not an internship
Recommended optional programme components None
Lecturer(s)
  • Pech Pavel, prof. RNDr. CSc.
Course content
Afinne varieties, Hilbert's basis theorem, Ideal, Grőbner bases of ideals, Ideal - variety correspondence, Solving polynomial equations, resultant, Buchberger's algorithm, Elimination of variables, Algebraic surfaces, Implicit and parametric equation of a surface, computer methods, The use of surfaces in practice, examples of cubic surfaces and surfaces of a higher order, Surfaces as loci of points.

Learning activities and teaching methods
Monologic (reading, lecture, briefing), Dialogic (discussion, interview, brainstorming), Monitoring
Learning outcomes
Afinne varieties, Hilbert's basis theorem, Ideal, Grőbner bases of ideals, Ideal - variety correspondence, Solving polynomial equations, resultant, Buchberger's algorithm, Elimination of variables, Algebraic surfaces, Implicit and parametric equation of a surface, computer methods, The use of surfaces in practice, examples of cubic surfaces and surfaces of a higher order, Surfaces as loci of points.
Knowledge of basic algorithms of computer algebra and their application.
Prerequisites
Knowledge of geometry in the extent of a univerity basic course of geometry.

Assessment methods and criteria
Oral examination, Written examination

Understanding of basic notions of the theory, knowledge of definitions, theorems and basic proofs. Ability to solve typical problems of a given theory. Written part of the exam is particularly oriented to verification whether a student can solve individual problems which come from a given theory. The aim of the oral part of the exam is to verify student's understanding of the theory and his/her ability to apply it.
Recommended literature
  • Cox, D., Little, J., O'Shea, D. Ideals, Varieties and Algorithms. Springer Verlag, 1979..
  • Cox, D., Little, J., O'Shea, D. Using Algebraic Geometry. Birkhäuser, 2005..


Study plans that include the course
Faculty Study plan (Version) Category of Branch/Specialization Recommended year of study Recommended semester
Faculty: Faculty of Education Study plan (Version): Theory of Mathematics Education (2) Category: Pedagogy, teacher training and social care - Recommended year of study:-, Recommended semester: -
Faculty: Faculty of Education Study plan (Version): Theory of Mathematics Education (2) Category: Pedagogy, teacher training and social care - Recommended year of study:-, Recommended semester: -