Ma 193b Winter 2012: Topics in Geometry and Physics
Brief Course Description
This class will cover recent approaches to geometric modeling for high energy physics, including noncommutative geometry models for particle physics and algebro geometric models for quantum field theory. Methods based on noncommutative and algebraic geometry in statistical mechanics will also be covered, with both physical and mathematical applications.
Notes of classes
- Jan 5: C* algebras, abelian C* algebras and locally compact
Hausdorff spaces, Gelfand transform
- Jan 10: aperiodic tilings, Penrose tilings, parameter space, bad quotients
and convolution algebras, the AF-algebra of Penrose tilings, K-theory of C*
algebras, K0-group of the algebra of Penrose tilings
- Jan 12: the algebraic geometry of Penrose tilings (after S.Paul Smith):
NC spaces as algebras of functions or as categories of sheaves, an NC
algebraic variety of Penrose tiling, recovering the algebra, equivalence
of categories of modules, the Grothendieck group of an abelian category;
matching the K0 calculations.
- Jan 17: The classical Bloch theory of electrons in solid,
discretized Schroedinger problem, Bloch variety, Fermi curves,
spectral density as period; magnetic fields and the noncommutative
- Jan 19: Magnetic translations and the noncommutative torus,
the range of the trace on K-theory and gap labeling.
- Jan 24: Orbifold geometries, twisted group C* algebras, K-theory and index theory, orbifold Euler characteristics, conductance cocycle and Kubo formula, fractional Quantum Hall Effect.
- Jan 26: Index theory: the heat kernel proof of the Atiyah-Singer index theorem.
- Feb 7: Hochschild and cyclic (co)homology and the index pairing.
- Feb 9: Fredholm modules and spectral triples.
- Feb 14: Properties and examples of spectral triples: real structures and KO-dimension, the case of noncommutative tori, commutative and almost commutative geometries.
- Feb 16: Particle physics and gravity models from spectral triples, the spectral action functional and its asymptotic expansion, zeta functions, residues, heat kernel expansion.
- Feb 21: Finite geometries, moduli spaces of Dirac operators, Yukawa parameters, product geometry, fermions and bosons, quantum numbers; Spectral action and the bosonic terms in the Standard Model coupled to gravity.
- Feb 23: Slow-roll inflation cosmology: Friedmann equation and Hubble constant, scaling factor, matter/energy/vacuum dominated cases; Spectral action and the Poisson summation formula, slow-roll potential from scalar fluctuations of the Dirac operator, sphere case, homogenous isotropic cosmic topologies.
- Feb 28: Quantum Statistical mechanics, KMS states, Cuntz and Toeplitz algebras, Bost-Connes system.
List of possible topics to choose from
- Noncommutative geometry in condensed matter physics (Quantum
Hall effect, quasi-crystals)
- Algebro-geometric structures in Perturbative Quantum Field Theory.
- Quantum Statistical Mechanics: operator algebras and KMS states,
examples of QSM systems and classification of KMS states, number
- Classical statistical mechanics: Potts models and the problem of
phase transitions, a motivic approach.
- Spectral triples and Noncommutative Riemannian Geometry: examples,
and applications to particle physics, cosmology, quantum gravity.
- Noncommutative motives and string theory.
- ... more topics to be added later (upon request from course participants)
- J.Varilly, "An introduction to noncommutative geometry", EMS 2006.
- M.Marcolli, "Feynman Motives", World Scientific 2010.
- A.Connes, M.Marcolli, "Noncommmutative Geometry, Quantum Fields and
Motives", AMS 2008.
Additional reading suggestions (papers) will be added as the course
- S.Paul Smith, "The space of Penrose tilings and the non-commutative curve with homogeneous coordinate ring k x,y mod (y^2)", arXiv:1104.3811
- Bellissard, Jean Noncommutative geometry and quantum Hall effect.
Proceedings of the International Congress of Mathematicians, Vol. 1, 2
(Zuerich, 1994), 1238--1246, Birkhaeuser, 1995.
- Bellissard, J.; van Elst, A.; Schulz-Baldes, H. The noncommutative
geometry of the quantum Hall effect. Topology and physics. J. Math. Phys.
35 (1994), no. 10, 5373-5451.
- M.Marcolli, V.Mathai, Towards the fractional quantum Hall effect: a noncommutative geometry perspective, arXiv:cond-mat/0502356
- D.Gieseker, H.Knoerrer, E.Trubowitz,
An overview of the geometry of algebraic Fermi curves. in
"Algebraic geometry: Sundance 1988", 19-46, Contemp. Math., 116, Amer. Math. Soc., Providence, RI, 1991.
A seminar for the class will meet Thursdays at 5 pm
- January 12: Branimir Cacic, "Morita equivalences and a category of noncommutative spaces"
- January 19: Kevin Teh, "The Einstein-Yang-Mills system"
- January 26: Gjergji Zaimi, "Deletion-contraction relations and Orlik-Solomon algebras"
- February 2: Victor Kasatkin, "C*-algebras of singular integral operators on a real line with symbols having discontinuities in coordinates and momenta"
- February 9: Xiang Ni, "Manin products and Rota Baxter operators"
- February 16: (cancelled)
- February 23: Eric Samperton, "Noncommutative Heegard splittings"
- February 23: Bogdan Stoica, "Higher spin black holes"
- March 1: Maria Nastasescu, "Monopoles in
non-abelian Yang-Mills-Higgs system"
- March 1: Melissa Young, "Braids, Train Tracks, and Fluid Dynamics"
- March 8: Mark Greenfield
- March 8: Justin Khim
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