Transformation groups in geometry and their applications- zaj. w j. angielskim WM-MA-S2-E4-TG

The notion of transformation groups, in particular matrix groups, are essential in many fields of science, engineering and industry. They have a wide range of applications, extending from representation theory and geometric structures, to image processing, computer vision, and robotics.

This course introduces transformation groups in geometry starting with the framework of matrix Lie groups. In this course, we start with real and complex matrix groups and their standard examples, and develop the exponential map, one‑parameter subgroups, and the associated Lie algebras, and then interpret matrix groups as smooth manifolds (Lie groups). We study smooth group actions and use orbit methods to construct and analyze homogeneous spaces (including projective spaces and Grassmannians), with geometric applications and structural results. Time permitting, we continue to compact connected Lie groups via maximal tori and the Weyl group, leading toward the classification viewpoint through root systems and Dynkin diagrams.

The format of the lectures and tutorials will be in a classroom with a blackboard and there will not be any mandatory programming involved.

Dyscyplina naukowa, do której odnoszą się efekty uczenia się

matematyka

E-Learning

E-Learning (pełny kurs) z podziałem na grupy

Grupa przedmiotów ogólnouczenianych

nie dotyczy

Opis nakładu pracy studenta w ECTS

For the lecture: class attendance 30 hours, participation in tests/exams outside of class 3 hours, consultations 2 hours, preparation for classes 5 hours, preparation for tests/exams 10 hours, All together 50 h, equivalent to 2 ECTS credits For the laboratory: class attendance 30 hours, consultations 5 hours, preparation for classes 20 hours, preparation for tests/exams 20 hours, All together 75 h, equivalent to 3 ECTS credits

Poziom przedmiotu

średnio-zaawansowany

Symbol/Symbole kierunkowe efektów uczenia się

MA2_W04, MA2_W05, MA2_W06, MA2_W07, MA2_W16, MA2_U02, MA2_U08, MA2_U10, MA2_U13, MA2_U14, MA2_U15, MA2_U23, MA2_U24 MA2_K01, MA2_K02, MA2_K07

Wymagania wstępne

Linear Algebra, basic abstract algebra, multivariable calculus It would be helpful to know some elementary ODEs and point-set topology.

Koordynatorzy przedmiotu

Omid Makhmali

Efekty kształcenia

W1 – Understands matrix Lie groups and Lie algebras, including core definitions/theorems and standard examples, and the role of these structures in geometry. (MA2_W04, MA2_W05, MA2_W16)

W2 – Understands how transformation groups connect Lie theory with other areas (geometry/topology/algebra) and can formulate selected themes that naturally lead to open or research-level questions and applications. (MA2_W06, MA2_W07)

U1 – Can carry out computations and solve problems using algebraic tools (e.g., Lie algebra calculations, subgroup/representation examples, linear-algebraic methods) and justify steps rigorously. (MA2_U10, MA2_U13, MA2_U14)

U2 – Can recognize and use geometric/topological structures arising from group actions (orbits, stabilizers, homogeneous spaces) in concrete examples and basic applications. (MA2_U08, MA2_U13)

U3 – Can present correct mathematical reasoning about transformation groups clearly in writing and orally (definitions, statements, proofs, examples). (MA2_U02)

U4 – Can use specialist literature (incl. English sources) to extend the course material, identify directions of interest, and apply the techniques of the chosen specialization to representative applications of transformation groups. (MA2_U15, MA2_U23, MA2_U24)

K1 – Is prepared to recognize gaps in knowledge, pursue independent learning, and ask precise questions that complete missing parts of an argument. (MA2_K01, MA2_K02)

K2 – Is prepared to formulate and defend informed opinions about mathematical issues related to transformation groups, including topics with a research flavor. (MA2_K07)

Kryteria oceniania

For all learning outcomes, the following assessment criteria are adopted for all forms of verification:

grade 5: fully achieved (no obvious shortcomings),

grade 4.5: achieved almost fully and criteria for awarding a higher grade are not met,

grade 4: largely achieved and the criteria for a higher grade are not met,

grade 3.5: largely achieved - with a clear majority of positives - and the criteria for granting a higher grade are not met,

grade 3: achieved for most of the cases covered by the verification and criteria for a higher grade are not met,

grade 2: not achieved for most of the cases covered by the verification.

Więcej informacji

Dodatkowe informacje (np. o kalendarzu rejestracji, prowadzących zajęcia, lokalizacji i terminach zajęć) mogą być dostępne w serwisie USOSweb:

Strona przedmiotu WM-MA-S2-E4-TG w USOSweb