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Analysis on Vector Bundles over Noncommutative Tori


Tao, Jim (2019) Analysis on Vector Bundles over Noncommutative Tori. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/C4QF-GF45.


Noncommutative geometry is the study of noncommutative algebras, especially C*-algebras, and their geometric interpretation as topological spaces. One C*-algebra particularly important in physics is the noncommutative n-torus, the irrational rotation C*-algebra AΘ with n unitary generators U1, . . . , Un which satisfy UkUj = e2πiθj,kUjUk and Uj* = Uj-1, where Θ ∈ Mn(ℝ) is skew-symmetric with upper triangular entries that are irrational and linearly independent over ℚ. We focus on two projects: an analytically detailed derivation of the pseudodifferential calculus on noncommutative tori, and a proof of an index theorem for vector bundles over the noncommutative two torus. We use Raymond's definition of an oscillatory integral with Connes' construction of pseudodifferential operators to rederive the calculus in more detail, following the strategy of the derivations in Wong's book on pseudodifferential operators. We then define the corresponding analog of Sobolev spaces on noncommutative tori, for which we prove analogs of the Sobolev and Rellich lemmas, and extend all of these results to vector bundles over noncommutative tori. We extend Connes and Tretkoff's analog of the Gauss-Bonnet theorem for the noncommutative two torus to an analog of the McKean-Singer index theorem for vector bundles over the noncommutative two torus, proving a rearrangement lemma where a self-adjoint idempotent e appears in the denominator but does not commute with the k2 already there from the rearrangement lemma proven by Connes and Tretkoff.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Pseudodifferential operators, oscillatory integrals, Sobolev spaces, vector bundles, Serre--Swan analogy, index theory, Hadamard products, projections, Connes--Chern number
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Marcolli, Matilde
Thesis Committee:
  • Rains, Eric M. (chair)
  • Marcolli, Matilde
  • Markovic, Vladimir
  • Fathizadeh, Farzad
Defense Date:8 May 2019
Non-Caltech Author Email:jtao (AT)
Errata:Reported by author on Sept. 24, 2019. Minor, easily fixable error in Mathematica files on CaltechDATA and CaltechTHESIS. "In the first notebook (argumentoftrace.nb), I fix the rule for differentiating k^{-1} in the function read[stuff_]. In my initial submission, there was the error where I had accidentally deleted the ^{-1} exponent on one of the k's. The other notebooks are updated to reflect this fix."
Funding AgencyGrant Number
Perimeter Institute for Theoretical PhysicsUNSPECIFIED
Record Number:CaltechTHESIS:05092019-193947900
Persistent URL:
Related URLs:
URLURL TypeDescription I as posted on arXiv. I as it appeared in ISQS25 conference proceedings. II as posted on arXiv. notebooks cited in Chapter II.
Tao, Jim0000-0002-0751-9273
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:11505
Deposited By: Jim Tao
Deposited On:14 May 2019 21:44
Last Modified:28 Oct 2021 19:00

Thesis Files

PDF (Full Thesis) - Final Version
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[img] Archive (ZIP) (supplementary electronic material (update 2020-01-22)) - Supplemental Material
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[img] Mathematica Notebook (NB) (case_k=1.nb) - Supplemental Material
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[img] Mathematica Notebook (NB) (argumentoftrace.nb) - Supplemental Material
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[img] Mathematica Notebook (NB) (argumentoftrace_difference.nb) - Supplemental Material
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[img] Mathematica Notebook (NB) (argumentoftrace_firstoperator.nb) - Supplemental Material
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[img] Mathematica Notebook (NB) (argumentoftrace_secondoperator.nb) - Supplemental Material
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