LSC Masterclass 2024: Symmetries and computation
Symmetries and computation
The course begins with an introduction to the linear representation theory of finite groups with a short emphasis on the symmetric group and then continues towards
computational Fourier Analysis on finite and compact groups.
Representation theory allows understanding groups with means of linear algebra but also provides a set of tools leading to more efficient computations.
Fourier analysis is deeply rooted in the theory of group representations, whereas the classical theory is based on Locally Compact Abelian groups (LCA). This perspective clarifies many aspects of computational Fourier theory and signal processing, and it paves the road for useful generalizations to non-commutative groups.
Lecturers: Hans Munthe-Kaas and Cordian Riener
Participation: The course is open to all students from Norwegian institutions and selected participants through collaboration with CIMPA. Please enregister here: Registration
In case you are not from a Norwegian institution but want to participate, please feel free to reach out.
Setup of the course
The course has weekly online lectures and two in-person meetings (one at UiT and one at UiB). The lectures are aimed to give a solid understanding of the basic concepts and the in-person meetings will seek to consolidate this understanding. Additionally, the participants will be asked to work in groups of 3-4 on more advanced topics and present these to the other participants during the second gathering. There will be an oral exam at the end with a grade system A-F.
Contents of the course:
1) Representation Theory (finite groups)
- Representations
- Characters
- Irreducible representations
- Schur’s Lemma and consequences
- Group Algera and regular representation
2) Representation Theory of S_n
- Young Diagrams
- Young Modules
- Specht Modules
- Young’s Theorem
- Applications
3) LCA, mostly focusing on the classical cases Z, Z/nZ, R, R/TR.
- The dual space, Pontryagin duality
- The Fourier basis for L^2(G)
- Sampling and interpolation, band-limited functions and reconstructions
- Fast Fourier Transform
- Applications
- Generalised Chebyshev polynomials (short discussion)
4) Fourier analysis on finite groups
- Peter Weyl theorem
- Fast transforms
- Exploring symmetry in numerical linear algebra
- Applications in medical image registration
5) A short discussion of compact Lie groups and unimodular groups
- Peter Weyl theorem (probably without proof)
- Applications
Literature:
We will discuss parts of the following:
• Serre, Jean Pierre, „Linear „Representation Theory of finite group“ Springer
• Fulton, William and Harris, Joe. "Representation Theory". Graduate Texts in Mathematics. New York, NY: Springer New York
• Sagan, Bruce The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions (Graduate Texts in Mathematics, Vol. 203).
• Munthe-Kaas, Hans Z., Morten Nome, and Brett N. Ryland. "Through the kaleidoscope: symmetries, groups and Chebyshev-approximations from a computational point of view." Foundations of computational mathematics, Budapest 403 (2011): 188-229.
• Åhlander, Krister, and Hans Munthe-Kaas. "Applications of the generalized Fourier transform in numerical linear algebra." BIT Numerical Mathematics 45 (2005): 819-850.
• Rudin, Walter. Fourier analysis on groups. Courier Dover Publications, 2017.
• Dym, Harry, and McKean HP. "Fourier series and integrals." (1972).
• Tao, Terence 254A, Notes 3: Haar measure and the Peter-Weyl theorem, Online: https://terrytao.wordpress.com/2011/09/27/254a-notes-3-haar-measure-
and-the-peter-weyl-theorem/
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