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oai:www.ideals.illinois.edu:2142/26182 2023-07-10 col_2142_5131 col_2142_16340 com_2142_5130 com_2142_16339 com_2142_8903

Dunfield, Nathan M. Leininger, Christopher J. Dunfield, Nathan M. Ahlgren, Scott Alexander, Stephanie B. Sinick, Jonah D. 2011-08-25T22:17:39Z 2011-08-25T22:17:39Z 2011-08-25T22:17:39Z 2011-08 In the first part of this thesis we generalize a theorem of Kiming and Olsson concerning the existence of Ramanujan-type congruences for a class of eta quotients. Specifically, we consider a class of generating functions analogous to the generating function of the partition function and establish a bound on the primes ℓ for which their coefficients c(n) obey congruences of the form c(ℓn + a) ≡ 0 (mod ℓ). We use this last result to answer a question of H.C. Chan. In the second part of this thesis [S2] we explore a natural analog of D. Calegari’s result that there are no hyperbolic once-punctured torus bundles over S^1 with trace field having a real place. We prove a contrasting theorem showing the existence of several infinite families of pairs (−χ, p) such that there exist hyperbolic surface bundles over S^1 with trace field of having a real place and with fiber having p punctures and Euler characteristic χ. This supports our conjecture that with finitely many known exceptions there exist such examples for each pair ( −χ, p). Item withdrawn by Mark Zulauf (zulauf@illinois.edu) on 2011-06-13T13:27:49Z Item was in collections: University of Illinois Theses & Dissertations (ID: 1) No. of bitstreams: 2 thesisfiles.zip: 70476 bytes, checksum: 1684ad96bdb2b270d88ae74905555a46 (MD5) Sinick_Jonah.pdf: 290025 bytes, checksum: 68804b242ca5b80cab18998b7e80335e (MD5) Made available in DSpace on 2011-08-25T22:17:39Z (GMT). No. of bitstreams: 4 Sinick_Jonah.pdf: 305014 bytes, checksum: c0cb500692d85e5808da5e73b64ec7c6 (MD5) license.txt: 4062 bytes, checksum: e518a69de718f2080e33f9f9218f47f2 (MD5) thesisfiles.zip: 70476 bytes, checksum: 1684ad96bdb2b270d88ae74905555a46 (MD5) 0_Sinick_Jonah.pdf: 305014 bytes, checksum: c0cb500692d85e5808da5e73b64ec7c6 (MD5) http://hdl.handle.net/2142/26182 en Copyright 2011 Jonah Sinick Number theory Modular forms Hyperbolic geometry Low dimensional topology Problems in number theory and hyperbolic geometry Mathematics 1257 Mathematics 0439 University of Illinois at Urbana-Champaign Dissertation Ph.D. PHD:Mathematics -UIUC 10KS0439PHD