Transformation Groups in Geometry and their Applications 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.

(in Polish) Dyscyplina naukowa, do której odnoszą się efekty uczenia się

mathematics

(in Polish) E-Learning

(in Polish) E-Learning (pełny kurs) z podziałem na grupy

(in Polish) Grupa przedmiotów ogólnouczenianych

(in Polish) nie dotyczy

Course coordinators

Term 2026/27_L:

Dorota Dąbrowska
Omid Makhmali

Term 2025/26_L:

Omid Makhmali

(in Polish) Opis nakładu pracy studenta w ECTS

Term 2025/26_L:

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

Term 2026/27_L:

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, Total: 50 hours 1.4 ECTS – face-to-face instruction 0.6 ECTS – independent study TUTORIALS class attendance 30 hours, consultations 5 hours, preparation for classes 20 hours, preparation for tests/exams 20 hours, Total: 75 hours 1.4 ECTS – face-to-face instruction 1.6 ECTS – independent study

Subject level

intermediate

Learning outcome code/codes

(in Polish) wykład: MA2_W01, MA2_W03, MA2_W13 ćwiczenia: MA2_U01, MA2_U03, MA2_U04, MA2_U23, MA2_K06 OPIS SYMBOLI - Absolwent: wykład MA2_W01 - posiada pogłębioną wiedzę z zakresu podstawowych działów matematyki, MA2_W03 - zna najważniejsze twierdzenia i hipotezy z głównych działów matematyki, MA2_W13 - zna język angielski na poziomie wystarczającym do studiowania literatury fachowej i uczestniczenia w dyskusjach prowadzonych w tym języku (poziom B2+ lub wyższy). ćwiczenia MA2_U01 - posiada umiejętności konstruowania rozumowań matematycznych: dowodzenia twierdzeń, jak i obalania hipotez poprzez konstrukcje i dobór kontrprzykładów, MA2_U03 - posiada umiejętność sprawdzania poprawności wnioskowań w budowaniu dowodów formalnych, MA2_U04 - w zagadnieniach matematycznych dostrzega struktury formalne związane z podstawowymi działami matematyki i rozumie znaczenie ich własności, MA2_U23 - potrafi korzystać z literatury fachowej dla wybranej dziedziny matematyki zarówno w języku polskim jak i w wybranych językach obcych, MA2_K06 - potrafi samodzielnie wyszukiwać informacje w literaturze, także w językach obcych.

Type of subject

obligatory

Preliminary Requirements

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

Learning outcomes

LECTURE

Student:

W1 - knows and understands fundamental and advanced concepts and theorems concerning matrix Lie groups and Lie algebras, including standard examples, the exponential map, one-parameter subgroups, and the relationship between Lie groups and their Lie algebras (MA2_W04, MA2_W05).

W2 - knows and understands the connections between transformation group theory and algebra, geometry, and topology, in particular the geometric significance of group actions, orbits, stabilizers, and homogeneous spaces (MA2_W07).

TUTORIALS

Student:

U1 - is able to apply algebraic and linear-algebraic methods to computations involving matrix Lie groups and Lie algebras, in particular to the analysis of subgroups, Lie algebras, the exponential map, and standard examples (MA2_U10).

U2 - is able to recognize and use geometric and topological structures arising from group actions, in particular orbits, stabilizers, and homogeneous spaces, in concrete examples (MA2_U08, MA2_U13).

U3 - is able to apply methods of Lie group and transformation group theory to solve problems in geometry and algebra and to present the resulting mathematical reasoning correctly (MA2_U13).

U4 - is able to construct proofs of selected results concerning Lie groups, Lie algebras, and group actions, using tools from linear algebra, algebra, geometry, and topology when necessary (MA2_U14).

Assessment criteria

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.

Additional information

Additional information (registration calendar, class conductors, localization and schedules of classes), might be available in the USOSweb system:

Description of WM-MA-S2-E4-TG in USOSweb