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 A\u0398 with n unitary generators U1, . . . , Un which satisfy UkUj = e2\u03c0i\u03b8j,kUjUk and Uj* = Uj-1, where \u0398 \u2208 Mn(\u211d) is skew-symmetric with upper triangular entries that are irrational and linearly independent over \u211a. 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.
", "doi": "10.7907/C4QF-GF45", "publication_date": "2019", "thesis_type": "phd", "thesis_year": "2019" }, { "id": "thesis:9799", "collection": "thesis", "collection_id": "9799", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05272016-153537481", "primary_object_url": { "basename": "NiThesis-2016.pdf", "content": "final", "filesize": 749282, "license": "other", "mime_type": "application/pdf", "url": "/9799/1/NiThesis-2016.pdf", "version": "v2.0.0" }, "type": "thesis", "title": "Rota-Baxter Algebras, Renormalization on Kausz Compactifications and Replicating of Binary Operads", "author": [ { "family_name": "Ni", "given_name": "Xiang", "clpid": "Ni-Xiang" } ], "thesis_advisor": [ { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "thesis_committee": [ { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Ni", "given_name": "Yi", "orcid": "0000-0002-5287-4258", "clpid": "Ni-Yi" }, { "family_name": "Flach", "given_name": "Matthias", "orcid": "0000-0002-4523-9467", "clpid": "Flach-M" }, { "family_name": "Fathizadeh", "given_name": "Farzad", "orcid": "0000-0002-7863-4009", "clpid": "Fathizadeh-F" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "This thesis is divided into two parts:
\r\n\r\nIn the first part, we consider Rota-Baxter algebras of meromorphic forms with poles along a (singular) hypersurface in a smooth projective variety and the associated Birkhoff factorization for algebra homomorphisms from a commutative Hopf algebra. In the case of a normal crossings divisor, the Rota-Baxter structure simplifies considerably and the factorization becomes a simple pole subtraction. We apply this formalism to the unrenormalized momentum space Feynman amplitudes, viewed as (divergent) integrals in the complement of the determinant hypersurface. We lift the integral to the Kausz compactification of the general linear group, whose boundary divisor is normal crossings. We show that the Kausz compactification is a Tate motive and the boundary divisor is a mixed Tate configuration. The regularization of the integrals that we obtain differs from the usual renormalization of physical Feynman amplitudes, and in particular it gives mixed Tate periods in cases that have non-mixed Tate contributions in the usual form. This part is based on joint work with Matilde Marcolli (see (80)).
\r\n\r\nIn the second part, we consider the notions of the replicators, including the duplicator and triplicator, of a binary operad. We show that taking replicators is in Koszul dual to taking successors in (9) for binary quadratic operads and is equivalent to taking the white product with certain operads such as Perm. We also relate the replicators to the actions of average operators. After the completion of this work (in 2012; see (85)), we realized that the closely related notions di-Var-algebra and tri-Var-algebra have been introduced independently in (48) (in 2011; see also (63; 64)) by Kolesnikov and his coauthors. In fact their notions also apply to not necessarily binary operads (64). In this regard, the second part of this thesis provides an alternative and more detailed treatment of these notations for binary operads. This part is based on joint work with Chengming Bai, Li Guo, and Jun Pei (see (85)).
", "doi": "10.7907/Z9SB43QZ", "publication_date": "2016", "thesis_type": "phd", "thesis_year": "2016" }, { "id": "thesis:8875", "collection": "thesis", "collection_id": "8875", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05212015-124038753", "primary_object_url": { "basename": "Kasatkin_Victor_2015_thesis.pdf", "content": "final", "filesize": 1050740, "license": "cc_by", "mime_type": "application/pdf", "url": "/8875/1/Kasatkin_Victor_2015_thesis.pdf", "version": "v3.0.0" }, "type": "thesis", "title": "Some Constructions, Related to Noncommutative Tori; Fredholm Modules and the Beilinson\u2013Bloch Regulator", "author": [ { "family_name": "Kasatkin", "given_name": "Victor", "clpid": "Kasatkin-Victor" } ], "thesis_advisor": [ { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "thesis_committee": [ { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Fathizadeh", "given_name": "Farzad", "orcid": "0000-0002-7863-4009", "clpid": "Fathizadeh-F" }, { "family_name": "Mantovan", "given_name": "Elena", "clpid": "Mantovan-E" }, { "family_name": "Rains", "given_name": "Eric M.", "clpid": "Rains-E-M" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "A noncommutative 2-torus is one of the main toy models of noncommutative geometry, and a noncommutative n-torus is a straightforward generalization of it. In 1980, Pimsner and Voiculescu in [17] described a 6-term exact sequence, which allows for the computation of the K-theory of noncommutative tori. It follows that both even and odd K-groups of n-dimensional noncommutative tori are free abelian groups on 2n-1 generators. In 1981, the Powers-Rieffel projector was described [19], which, together with the class of identity, generates the even K-theory of noncommutative 2-tori. In 1984, Elliott [10] computed trace and Chern character on these K-groups. According to Rieffel [20], the odd K-theory of a noncommutative n-torus coincides with the group of connected components of the elements of the algebra. In particular, generators of K-theory can be chosen to be invertible elements of the algebra. In Chapter 1, we derive an explicit formula for the First nontrivial generator of the odd K-theory of noncommutative tori. This gives the full set of generators for the odd K-theory of noncommutative 3-tori and 4-tori.
\r\n\r\nIn Chapter 2, we apply the graded-commutative framework of differential geometry to the polynomial subalgebra of the noncommutative torus algebra. We use the framework of differential geometry described in [27], [14], [25], [26]. In order to apply this framework to noncommutative torus, the notion of the graded-commutative algebra has to be generalized: the \"signs\" should be allowed to take values in U(1), rather than just {-1,1}. Such generalization is well-known (see, e.g., [8] in the context of linear algebra). We reformulate relevant results of [27], [14], [25], [26] using this extended notion of sign. We show how this framework can be used to construct differential operators, differential forms, and jet spaces on noncommutative tori. Then, we compare the constructed differential forms to the ones, obtained from the spectral triple of the noncommutative torus. Sections 2.1-2.3 recall the basic notions from [27], [14], [25], [26], with the required change of the notion of \"sign\". In Section 2.4, we apply these notions to the polynomial subalgebra of the noncommutative torus algebra. This polynomial subalgebra is similar to a free graded-commutative algebra. We show that, when restricted to the polynomial subalgebra, Connes construction of differential forms gives the same answer as the one obtained from the graded-commutative differential geometry. One may try to extend these notions to the smooth noncommutative torus algebra, but this was not done in this work.
\r\n\r\nA reconstruction of the Beilinson-Bloch regulator (for curves) via Fredholm modules was given by Eugene Ha in [12]. However, the proof in [12] contains a critical gap; in Chapter 3, we close this gap. More specifically, we do this by obtaining some technical results, and by proving Property 4 of Section 3.7 (see Theorem 3.9.4), which implies that such reformulation is, indeed, possible. The main motivation for this reformulation is the longer-term goal of finding possible analogs of the second K-group (in the context of algebraic geometry and K-theory of rings) and of the regulators for noncommutative spaces. This work should be seen as a necessary preliminary step for that purpose.
\r\n\r\nFor the convenience of the reader, we also give a short description of the results from [12], as well as some background material on central extensions and Connes-Karoubi character.
", "doi": "10.7907/Z91R6NGH", "publication_date": "2015", "thesis_type": "phd", "thesis_year": "2015" } ]