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. / A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Summer Semester 2017. / July 3, 2017. / CM Systems, Connected Shimura Varieties, Motives, Noncommutative Goemetry, Noncommutative tori / Includes bibliographical references. / Matilde Marcolli, Professor Co-Directing Dissertation; Paolo Aluffi, Professor Co-Directing Dissertation; Eric Chicken, University Representative; Philip Bowers, Committee Member; Kathleen Petersen, Committee Member.
| Identifer | oai:union.ndltd.org:fsu.edu/oai:fsu.digital.flvc.org:fsu_552130 |
| Contributors | Shen, Yunyi (authoraut), 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 departmentdgg) |
| Publisher | Florida State University |
| Source Sets | Florida State University |
| Language | English, English |
| Detected Language | English |
| Type | Text, text, doctoral thesis |
| Format | 1 online resource (69 pages), computer, application/pdf |
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