Lectures/Seminars

Four stages of invariant theory


Department: Matematický Ústav MFF UK
Partner: Univerzita Karlova v Praze

Four stages of invariant theory

by
Roger Howe

These talks will survey progress on classical invariant theory since Weyl’s book, with an emphasis on more recent developments.

They will begin with a brief review of Weyl’s results, including the notion of classical action, and Weyl’s First Fundamental Theorem for such actions. This will be followed by the introduction of the Weyl algebra, and the notion of dual pair of Lie subalgebras of the symplectic Lie algebra. These ideas permit description of the full isotopic decomposition for classical actions, not just the invariants, and also give rise to decomposition of polynomial rings into invariants and harmonics relative to a given classical action. These phenomena suggest the concept of stable range: when relevant parameters are restricted appropriately, various phenomena of interest become more tractable. For example, in the stable range, the full polynomial ring can be expressed as the tensor product of the invariants and the harmonics. This is referred to as separation of variables. Seeing classical actions as part of a dual pair structure enables consideration of the relationship between different actions, and leads to the study of branching rules and to reciprocity laws. These are formulated in terms of certain algebras, the branching algebras. Finer investigation of these issues is aided by ideas from commutative algebra, especially term orders, SAGBI theory and Hibi rings. These allow detailed description of the rings of invariants and of harmonics, and provide a natural generalization of Hodge’s standard monomial theory. They also enable detailed descriptions of branching algebras, including a representation-theoretic approach to the Littlewood-Richardson Rule.

Finally, the extent to which separation of variable fails outside the stable range will be considered. This is the subject of current research.




Lecture I (Due to a technical glitch, the video is frozen after eighth minute. Slides and sound are OK.)
Lecture IIa
Lecture IIb
Lecture in Trest
Lecture IIIa
Lecture IIIb
Lecture IIIc
Lecture IVa
Lecture IVb

Seminar on knot invariants


Website: http://www.karlin.mff.cuni.cz/~stovicek/..
Person: Mgr. Martin Doubek Ph.D.
Department: Matematický Ústav MFF UK
Partner: Univerzita Karlova v Praze

Time and place: Friday 10:40-12:10 in the seminar room of the Mathematical Institute

Description: The aim of the seminar is to understand various invariants of knots, links and tangles. Starting with an elementary aproach and polynomial invariants (Jones polynomial and variants), we should step by step learn more about quantum groups and R-matrices, Khovanov homology and categorification, and hopefully also about connections to statistical physics, cobordisms and Witten's work.

Snaeks, knots and pairs of pants


Person: Dany Majard Ph.D.
Partner: Masarykova Univerzita

Central European seminar


Website: http://eci.math.muni.cz/ECIces.html
Person: Prof. RNDr. Jan Slovák DrSc.
Department: Department of Mathematics and Statistics, Masaryk University
Partner: Masarykova Univerzita

Seminar in theoretical physics and astrophysics


Website: http://www.physics.muni.cz/?q=node/6
Department: Ústav teoretické fyziky a astrofyziky, Masarykova Univerzita
Partner: Masarykova Univerzita

Seminář z algebry


Website: http://www.math.muni.cz/veda-a-vyzkum/po..
Person: prof. RNDr. Jiří Rosický DrSc.
Partner: Masarykova Univerzita

Seminář z diferenciální geometrie


Website: http://www.math.muni.cz/veda-a-vyzkum/po..
Person: Prof. RNDr. Jan Slovák DrSc.
Partner: Masarykova Univerzita

Harmonic analysis


Website: http://muuk.karlin.mff.cuni.cz/cs/node/1..
Person: prof. RNDr. Vladimír Souček DrSc.
Department: Matematický Ústav MFF UK
Partner: Univerzita Karlova v Praze

Friday 10:40-12:10 at seminar room of Mathematical Institute.

Differential geometry


Website: http://muuk.karlin.mff.cuni.cz/cs/node/1..
Person: prof. RNDr. Vladimír Souček DrSc.
Department: Matematický Ústav MFF UK
Partner: Univerzita Karlova v Praze

Friday 9:00-10:30 at seminar room of Mathematical Institute.

Algebra seminar


Website: http://www.karlin.mff.cuni.cz/katedry/ka..
Person: prof. RNDr. Jan Trlifaj CSc. DSc.
Department: Katedra Algebry MFF UK
Partner: Univerzita Karlova v Praze

Time and place: Monday 3.30pm in the seminar room of the Department of Algebra.

Description: The seminar serves as a platform for presentation of recent research of the visitors to the ECI and to the Department of Algebra, as well as members of the Department and their students.