| Bemerkung |
Course: Integral transformations in Mathematical-Physics
Marco Valerio d’Agostino
April 2024
The primary objective of this course is to provide an overview of mathematical-physics problems that
can be explored through integral transforms. The central focus will be on analyzing the classification of
phases of matter in topological insulators. These physical systems exhibit a high symmetry structure,
specifically a crystalline structure, and the Schrödinger operator describing electron propagation can
be diagonalized through the Bloch-Floquet transform—an extension of the Fourier transform.
What is particularly intriguing is the emergence of geometrical objects from this approach, enabling
the detection of topological invariants within the system. These topological obstructions, such as the
potential hindrance to the existence of exponentially localized Wannier functions (the eigenfunctions
of the Schrödinger operator representing molecular orbitals), are obtained through the computation of
the Chern numbers of a suitable vector bundle.
Here is a list of the topics covered in the course:
• Classical Fourier transform and dispersion analysis.
• Generalization of the Fourier transform to topological groups: Harmonic analysis approach.
• Bloch-Floquet transform. The Bloch-Floquet transform will be discussed from various perspec-
tives.
• Spectral analysis and unbounded self-adjoint operators. Analysis of the Schrödinger operator
and the Elasticity operator.
• Symmetries in Physics.
• Crystallographic groups and time-reversal symmetry.
• Schrödinger operator and topological insulators.
• Topological phases of matter: K-theory, characteristic classes, and C∗−algebras.
• Discussion about open problems and new research directions in the field.
Here the list of references for the course:
References
[1] M. A. Aguilar, S. Gitler, C. Prieto, and M. Aguilar. Algebraic topology from a homotopical viewpoint. Vol. 2002.
Springer, 2002.
[2] H. Brézis. Functional analysis, Sobolev spaces and partial differential equations. Vol. 2. 3. Springer, 2011.
[3] K. S. Brown. Cohomology of groups. Vol. 87. Springer Science & Business Media, 2012.
[4] A. Deitmar and S. Echterhoff. Principles of harmonic analysis. Springer, 2014.
[5] G. B. Folland. A course in abstract harmonic analysis. Vol. 29. CRC press, 2016.
[6] D. Monaco, G. Panati, A. Pisante, and S. Teufel. “Optimal decay of Wannier functions in Chern and quantum
Hall insulators”. Communications in Mathematical Physics 359.1 (2018), 61–100.
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[7] G. Panati. “Triviality of Bloch and Bloch–Dirac bundles”. Annales Henri Poincaré. Vol. 8. 5. Springer. 2007,
995–1011.
[8] M. Reed and B. Simon. I: Functional analysis. Vol. 1. Academic press, 1981.
[9] M. Reed and B. Simon. II: Fourier Analysis, Self-Adjointness. Vol. 2. Elsevier, 1975.
[10] M. Reed and B. Simon. IV: Analysis of Operators. Vol. 4. Elsevier, 1978.
[11] K. Schmüdgen. Unbounded self-adjoint operators on Hilbert space. Vol. 265. Springer Science & Business Media,
2012. |
