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Introduction to Hochschild cohomology and deformation theory (MSTR Elective 2) - NMAG499

Title:	Výběrová přednáška z MSTR 2
Guaranteed by:	Department of Algebra (32-KA)
Faculty:	Faculty of Mathematics and Physics
Actual:	from 2024
Semester:	winter
E-Credits:	3
Hours per week, examination:	winter s.:2/0, Ex [HT]
Capacity:	unlimited
Min. number of students:	unlimited
4EU+:	no
Virtual mobility / capacity:	no
State of the course:	taught
Language:	English, Czech
Teaching methods:	full-time
Additional information:	https://sites.google.com/view/sebastianopper
Note:	you can enroll for the course repeatedly

Guarantor:	Sebastian Opper, Ph.D.
Teacher(s):	Sebastian Opper, Ph.D.
Class:	M Mgr. MSTR M Mgr. MSTR > Volitelné
Classification:	Mathematics > Algebra

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Annotation -

Non-repeated universal elective course. In winter 2024/25: Introduction to Hochschild cohomology and deformation theory

Last update: Žemlička Jan, doc. Mgr. et Mgr., Ph.D. (06.06.2024)

Course completion requirements

Oral exam.

Last update: Jeřábek Emil, Mgr. et Mgr., Dr., Ph.D. (27.12.2023)

Literature -

Thomas F. Fox, An introduction to algebraic deformation theory, Journal of Pure and Applied Algebra, Volume 84, Issue 1, 1993, Pages 17-41, ISSN 0022-4049, https://doi.org/10.1016/0022-4049(93)90160-U.

Manetti, Marco, Lie methods in deformation theory, Springer Monographs in Mathematics, Springer, Singapore 2022, https://doi.org/10.1007/978-981-19-1185-9

Last update: Žemlička Jan, doc. Mgr. et Mgr., Ph.D. (06.06.2024)

Requirements to the exam -

The exam will be oral.

Last update: Žemlička Jan, doc. Mgr. et Mgr., Ph.D. (30.09.2024)

Syllabus -

This course offers an introductory account on Hochschild cohomology and its importance for the study of deformations of algebraic structures. Although the course primarily focuses on deformations of algebras, we also explain the role of Lie algebras in deformation problems and illustrate the surprising significance of this relationship to other areas of mathematics including geometry and topology.

Summary:

(1) Introduction to Hochschild cohomology of algebras (Hochschild complex, interpretation as derived functors, examples).

(2) Structures on the Hochschild complex (differential, composition and cup product,

Lie bracket).

(3) Introduction to deformation theory (deformations of algebras and affine varieties,

examples, deformation problems as functors).

(4) Deformations via dg Lie algebras (Maurer-Cartan elements, gauge equivalence

and gauge group).

Last update: Žemlička Jan, doc. Mgr. et Mgr., Ph.D. (06.06.2024)

Entry requirements -

Some familiarity with the following concepts will be preferable, although central aspects

will be reviewed during the course as they become relevant.

(1) Category theory: categories, functors, natural transformations, Yoneda embedding.

(2) Homological algebra: complexes, homology, derived functors.

(3) Representation Theory: algebras and their modules.

In addition, a basic familiarity with algebraic geometry, e.g. affine varieties and their

relation to algebras, would be helpful to motivate certain concepts.

Last update: Žemlička Jan, doc. Mgr. et Mgr., Ph.D. (06.06.2024)