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Arithmetic Aspects of Noncommutative Geometry

Title: Arithmetic Aspects of Noncommutative Geometry: Motives of Noncommutative Tori and Phase Transitions on GL(n) and Shimura Varieties Systems.
Name(s): Shen, Yunyi, author
Marcolli, Matilde, professor co-directing dissertation
Aluffi, Paolo, 1960-, professor co-directing dissertation
Chicken, Eric, 1963-, university representative
Bowers, Philip L., 1956-, committee member
Petersen, Kathleen L., committee member
Florida State University, degree granting institution
College of Arts and Sciences, degree granting college
Department of Mathematics, degree granting department
Type of Resource: text
Genre: Text
Doctoral Thesis
Issuance: monographic
Date Issued: 2017
Publisher: Florida State University
Place of Publication: Tallahassee, Florida
Physical Form: computer
online resource
Extent: 1 online resource (69 pages)
Language(s): English
Abstract/Description: In this dissertation, we study three important cases in noncommutative geometry. We first observe the standard noncommutative object, noncommutative torus, in noncommutative motives. We work with the category of holomorphic bundles on a noncommutative torus, which is known to be equivalent to the heart of a nonstandard t-structure on coherent sheaves of an elliptic curve. We then introduce a notion of (weak) t-structure in dg categories. By lifting the nonstandard t-structure to the t-structure that we defined, we find a way of seeing a noncommutative torus in noncommutative motives. By applying the t-structure to a noncommutative torus and describing the cyclic homology of the category of holomorphic bundle on the noncommutative torus, we finally show that the periodic cyclic homology functor induces a decomposition of the motivic Galois group of the Tannakian category generated by the associated auxiliary elliptic curve. In the second case, we generalize the results of Laca, Larsen, and Neshveyev on the GL2-Connes-Marcolli system to the GLn-Connes-Marcolli systems. We introduce and define the GLn-Connes-Marcolli systems and discuss the existence and uniqueness questions of the KMS equilibrium states. Using the ergodicity argument and Hecke pair calculation, we classify the KMS states at different inverse temperatures β. Specifically, we show that in the range of n − 1 < β ≤ n, there exists only one KMS state. We prove that there are no KMS states when β < n − 1 and β ̸= 0, 1, . . . , n − 1,, while we actually construct KMS states for integer values of β in 1 ≤ β ≤ n − 1. For β > n, we characterize the extremal KMS states. In the third case, we push the previous results to more abstract settings. We mainly study the connected Shimura dynamical systems. We give the definition of the essential and superficial KMS states. We further develop a set of arithmetic tools to generalize the results in the previous case. We then prove the uniqueness of the essential KMS states and show the existence of the essential KMS stats for high inverse temperatures.
Identifier: FSU_SUMMER2017_Shen_fsu_0071E_13982 (IID)
Submitted Note: A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy.
Degree Awarded: Summer Semester 2017.
Date of Defense: July 3, 2017.
Keywords: CM Systems, Connected Shimura Varieties, Motives, Noncommutative Goemetry, Noncommutative tori
Bibliography Note: Includes bibliographical references.
Advisory Committee: Matilde Marcolli, Professor Co-Directing Dissertation; Paolo Aluffi, Professor Co-Directing Dissertation; Eric Chicken, University Representative; Philip Bowers, Committee Member; Kathleen Petersen, Committee Member.
Subject(s): Mathematics
Persistent Link to This Record:
Host Institution: FSU

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Shen, Y. (2017). Arithmetic Aspects of Noncommutative Geometry: Motives of Noncommutative Tori and Phase Transitions on GL(n) and Shimura Varieties Systems. Retrieved from