(in Polish) Kwantowe niezmienniki węzłów
General data
Course ID: | 1000-1M20KNW |
Erasmus code / ISCED: |
11.1
|
Course title: | (unknown) |
Name in Polish: | Kwantowe niezmienniki węzłów |
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): |
6.00
|
Language: | English |
Main fields of studies for MISMaP: | biology |
Type of course: | elective monographs |
Prerequisites (description): | The prerequisites for the lecture are minimal, as basic algebra and topology courses. All concepts will remain explained during the lecture if necessary. |
Mode: | Remote learning |
Short description: |
The course aims to introduce into the theory of quantized and categorified invariants of knots. After explaining the classical Alexander, Conway, Jones and HOMFLY-PT polynomials, we will quantize them with the use of representations of quantum groups and the Reshetikhin- Turaev functor and categorify to the Khovanov-Lee homology. All technicalities will be explained on the spot when needed. The course is based on selected fragments of the literature listed below. The suggestions for further reading will be provided. |
Full description: |
1. Reidemeister theorem 2. Tricolorability, Knot Group 3. Kauffman bracket, Jones polynomial 4. Braid group, Burau representation 5. Temperley-Lieb algebra 6. Jones polynomial through braid representations 7. Khovanov homology, categorification of the Jones polynomial 8. Frobenius algebras and Topological Quantum Field Theories 9. Tangles and Hopf algebras, graphical calculus 10. Quasitriangular, modular, and ribbon Hopf algebras 11. Quantum groups and representations of tangles 12. Coloring ribbon graphs by representations and modular categories 13. Reshetikhin-Turaev invariants of ribbon graphs from quantum groups 14. Knots and 3-dimensional manifolds 15. Physics, chemistry and biology of knots |
Bibliography: |
M. F. Atiyah, The geometry and physics of knots, Cambridge University Press, Cambridge, 1990. D. Bar-Natan. On Khovanov’s categorification of the Jones polynomial. Algebr. Geom. Topol., 2:337–370, 2002. P. Etingof, O. Schiffmann: Lectures on Quantum Groups. International Press (2002) V. Jones, A polynomial invariant for knots via von Neumann algebras, Bull. Amer. Math. Soc. (N.S.) 12 (1985) 103--111. L. Kauffman, Knots and physics, World Scientific Publishing, 3rd edition, 1993. T. Ohtsuki: Quantum Invariants. World Scientific (2001) V. V. Prasolov and A. B. Sossinsky, Knots, links, braids and 3-Manifolds. Translations of Mathematical Monographs 154, Amer. Math. Soc., Providence, RI, 1997. N. Reshetikhin and V. Turaev, Ribbon graphs and their invariants derived from quantum groups, Comm. Math. Phys. 127 (1990) 1--26. |
Learning outcomes: |
1. Knowledge of the basic concepts of knot theory, including topological, algebraic, and categorial ones. 2. Understanding the basic results of knot theory connecting their invariants with constructions in representation theory, algebra, and noncommutative geometry. In particular, understanding the equivalence of Jones polynomial definitions, derived from seemingly independent points of view. 3. Knowledge of the relationship of knot theory with other scientific disciplines, such as mathematical physics, chemistry, and biology of proteins and DNA, etc. 4. Readiness of the listener for independent reading of contemporary scientific literature in the field. |
Assessment methods and assessment criteria: |
Active participation in classes. |
Classes in period "Summer semester 2023/24" (in progress)
Time span: | 2024-02-19 - 2024-06-16 |
Navigate to timetable
MO TU W TH FR WYK
CW
|
Type of class: |
Classes, 30 hours
Lecture, 30 hours
|
|
Coordinators: | Tomasz Maszczyk | |
Group instructors: | Tomasz Maszczyk | |
Students list: | (inaccessible to you) | |
Examination: | Examination | |
Main fields of studies for MISMaP: | biology |
|
Short description: |
The course aims to introduce into the theory of quantized and categorified invariants of knots. After explaining the classical Alexander, Conway, Jones and HOMFLY-PT polynomials, we will quantize them with the use of representations of quantum groups and the Reshetikhin- Turaev functor and categorify to the Khovanov-Lee homology. All technicalities will be explained on the spot when needed. The course is based on selected fragments of the literature listed below. The suggestions for further reading will be provided. |
|
Full description: |
1. Reidemeister theorem 2. Tricolorability, Knot Group 3. Kauffman bracket, Jones polynomial 4. Braid group, Burau representation 5. Temperley-Lieb algebra 6. Jones polynomial through braid representations 7. Khovanov homology, categorification of the Jones polynomial 8. Frobenius algebras and Topological Quantum Field Theories 9. Tangles and Hopf algebras, graphical calculus 10. Quasitriangular, modular, and ribbon Hopf algebras 11. Quantum groups and representations of tangles 12. Coloring ribbon graphs by representations and modular categories 13. Reshetikhin-Turaev invariants of ribbon graphs from quantum groups 14. Knots and 3-dimensional manifolds 15. Physics, chemistry and biology of knots |
|
Bibliography: |
M. F. Atiyah, The geometry and physics of knots, Cambridge University Press, Cambridge, 1990. D. Bar-Natan. On Khovanov’s categorification of the Jones polynomial. Algebr. Geom. Topol., 2:337–370, 2002. P. Etingof, O. Schiffmann: Lectures on Quantum Groups. International Press (2002) V. Jones, A polynomial invariant for knots via von Neumann algebras, Bull. Amer. Math. Soc. (N.S.) 12 (1985) 103--111. L. Kauffman, Knots and physics, World Scientific Publishing, 3rd edition, 1993. |
Classes in period "Summer semester 2024/25" (future)
Time span: | 2025-02-17 - 2025-06-08 |
Navigate to timetable
MO TU W TH FR |
Type of class: |
Classes, 30 hours
Lecture, 30 hours
|
|
Coordinators: | Tomasz Maszczyk | |
Group instructors: | Tomasz Maszczyk | |
Students list: | (inaccessible to you) | |
Examination: | Examination |
Copyright by University of Warsaw.