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Kedem, Rinat Di Francesco, Philippe Bergvelt, Maarten Yong, Alexander Vichitkunakorn, Panupong 2017-08-10T19:14:47Z 2017-08-10T19:14:47Z 2017-04-13 2017-05 This dissertation presents connections between cluster algebras and discrete integrable systems, especially T-systems and their specializations/generalizations. We give connections between the T-system or the octahedron relation, and the pentagram map and its various generalizations. A solution to the T-system with quasi-periodic boundary conditions gives rise to a solution to a higher pentagram map. In order to obtain all the solutions of higher pentagram map, we define T-systems with principal coefficients from cluster algebra aspect. Combinatorial solutions of the T-systems with principal coefficients with respect to any valid initial condition are shown to be partition functions of perfect matchings, non-intersecting paths and networks. This also provides a solution to other systems with various choices of coefficients on T-systems including Speyer's octahedron recurrence (Speyer 2007), generalized lambda-determinants (Di Francesco 2013) and (higher) pentagram maps (Schwartz 1992, Ovsienko et al. 2010, Glick 2011, Gekhtman et al. 2016). We study a discrete dynamic on weighted bipartite graphs on a torus, analogous to dimer integrable systems of Goncharov and Kenyon 2013. We show that all Hamiltonians, partition functions of all weighted perfect matchings with a common homology class, are invariant under a move on the weighted graph. This move coincides with a cluster mutation, analog to Y-seed mutation in dimer integrable systems. Q-systems are reductions of T-systems by forgetting one of the parameters. We construct graphs for Q-systems of type A and B and show that the Hamiltonians are conserved quantities of the systems. The conserved quantities can be written as partition functions of hard particles on a certain graph. For type A, they Poisson commute under a nondegenerate Poisson bracket. Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2017-08-10 without embargo terms The student, Panupong Vichitkunakorn, accepted the attached license on 2017-04-07 at 08:14. The student, Panupong Vichitkunakorn, submitted this Dissertation for approval on 2017-04-07 at 08:33. This Dissertation was approved for publication on 2017-04-13 at 12:17. DSpace SAF Submission Ingestion Package generated from Vireo submission #10667 on 2017-08-10 at 13:38:56 Made available in DSpace on 2017-08-10T19:14:47Z (GMT). No. of bitstreams: 3 VICHITKUNAKORN-DISSERTATION-2017.pdf: 1272043 bytes, checksum: db015bf04320f0542ec691aacbbfcafd (MD5) LICENSE.txt: 4220 bytes, checksum: 559f20a3f4d163a1d070eb45739a89f0 (MD5) PROQUEST_LICENSE.txt: 4566 bytes, checksum: b147c72a907b33fcb4c67aad8eaece67 (MD5) Previous issue date: 2017-04-13 application/pdf http://hdl.handle.net/2142/97314 en Copyright 2017 Panupong Vichitkunakorn Cluster algebras Discrete integrable systems Cluster algebras and discrete integrable systems text text Mathematics Mathematics University of Illinois at Urbana-Champaign Dissertation Ph.D.