(in Polish) Algorytmy w algebrze

General data

Course ID:	1000-1M18AA
Erasmus code / ISCED:	11.1 Kod klasyfikacyjny przedmiotu składa się z trzech do pięciu cyfr, przy czym trzy pierwsze oznaczają klasyfikację dziedziny wg. Listy kodów dziedzin obowiązującej w programie Socrates/Erasmus, czwarta (dotąd na ogół 0) – ewentualne uszczegółowienie informacji o dyscyplinie, piąta – stopień zaawansowania przedmiotu ustalony na podstawie roku studiów, dla którego przedmiot jest przeznaczony. / (0541) Mathematics The ISCED (International Standard Classification of Education) code has been designed by UNESCO.
Course title:	(unknown)
Name in Polish:	Algorytmy w algebrze
Organizational unit:	Faculty of Mathematics, Informatics, and Mechanics
Course groups:	(in Polish) Przedmioty obieralne na studiach drugiego stopnia na kierunku bioinformatyka Elective courses for 2nd stage studies in Mathematics
ECTS credit allocation (and other scores):	(not available) Basic information on ECTS credits allocation principles: the annual hourly workload of the student’s work required to achieve the expected learning outcomes for a given stage is 1500-1800h, corresponding to 60 ECTS; the student’s weekly hourly workload is 45 h; 1 ECTS point corresponds to 25-30 hours of student work needed to achieve the assumed learning outcomes; weekly student workload necessary to achieve the assumed learning outcomes allows to obtain 1.5 ECTS; work required to pass the course, which has been assigned 3 ECTS, constitutes 10% of the semester student load. view allocation of credits
Language:	English
Type of course:	elective monographs
Prerequisites:	Commutative algebra 1000-135ALP
Prerequisites (description):	The student should complete the course Commutative algebra before or at the same time as Algorithms in algebra.
Short description:	The course will concern mathematical results which can be used to enable or simplify computations in algebra and using the computer to analysing examples coming from algebra and (algebraic) geometry. The main goal is to explore currently important methods in algebraic computations, especially theoretical results which make such computations possible and efficient.
Full description:	The course is designed mainly for students interested in commutative algebra and algebraic geometry, but should be also useful for these interested in algebraic topology and homological algebra. We will study how to apply computer algebra systems to access and understand certain properties of rings, ideals and varieties, how to interpret these results and, in some cases, make computations more efficient. We assume good knowledge of basic algebra (as in the Algebra I and Commutative algebra courses), especially ring theory. No experience in programming is required. Tentative course plan: 1 Groebner bases (3-4 lectures): Buchberger's algorithm, examples and applications, in particular elimination theory and computing homomorphism kernel. 2 Resolutions, syzygies, deformations (4-6 lectures), Betti numbers, many examples (including ones coming from geometry), Hilbert function and Hilbert polynomial, resolutions in families. 3 Properties of rings and ideals (2-3 lectures): normality and Serre's criterion, localisation, saturation, homogenisation. 4 Additional topics (selection): (a) Finite groups: classification, representations, character tables (GAP) (b) Invariants of finite group actions (Singular) (c) Khovanskii bases - Groebner-type bases for algebras (d) Basic toric geometry - algorithms for cones and polytopes (Macaulay, Singular, Polymake) (e) Basic toric geometry - varieties, morphisms (Magma) (f) Computing tropicalisations (g) Symmetric polynomials
Bibliography:	"Ideals, Varieties, and Algorithms", David A. Cox, John B. Little, Donal O'Shea "Computational Commutative Algebra", M. Kreuzer, L. Robbiano "A Singular Introduction to Commutative Algebra", G.-M. Greuel and G. Pfister "Computations in Algebraic Geometry with Macaulay2", D. Eisenbud, D.R. Grayson, M. Stillman, B. Sturmfels (eds) "Commutative Algebra", D. Eisenbud "Geometry of Syzygies", D. Eisenbud "Commutative Algebra", H. Matsumura "Groebner Bases and Convex Polytopes", B. Sturmfels "Combinatorial Commutative Algebra", E. Miller, B. Sturmfels manuals of Macaulay2, Singular, Magma, GAP
Learning outcomes:	Students understand most important theorem and algorithms on which computer algebra systems are based. Moreover, they know how to approach solving problems, in particular analysing examples coming from algebra and (algebraic) geometry using computer algebra systems.
Assessment methods and assessment criteria:	Oral exam.

This course is not currently offered.

Course descriptions are protected by copyright.
Copyright by University of Warsaw.