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Nevins, Thomas Pascaleff, James Dodd, Christopher Bergvelt, Maarten Penciak, Matej 2019-11-26T20:49:26Z 2019-11-26T20:49:26Z 2021-11-27T10:15:20Z 2019-07-10 2019-08 Fix a Weierstrass cubic curve \(E\), and an element $\sigma$ in the Jacobian variety $\Jac E$ corresponding to the line bundle \(\mathcal L_\sigma \). We introduce a space \(\mathsf{RS}_{\sigma, n}(E, V)\) of pure 1-dimensional sheaves living in \(S_\sigma = \PP(\OO \oplus \mathcal L_\sigma)\) together with framing data at the \(0\) and \(\infty\) sections \(E_0, E_\infty \subset S_\sigma \). For a particular choice of \(V\), we show that the space \(\mathsf{RS}_{\sigma, n}(E, V)\) is isomorphic to a completed phase space for the spin Ruijsenaars-Schneider system, with the Hamiltonian vector fields given by tweaking flows on sheaves at their restrictions to \(E_0\) and \(E_\infty\). We compare this description of the RS system to the description of the Calogero-Moser system in \cite{MR2377220}, and show that the two systems can be assembled into a universal system by introducing a \(\sigma \rightarrow 0\) limit to the CM phase space. We also shed some light on the effect of Ruijsenaars' duality between trigonometric CM and rational RS spectral curves coming from the two descriptions in terms of supports of spectral sheaves. Submission published under a 24 month embargo labeled 'U of I Access', the embargo will last until 2021-08-01 The student, Matej Penciak, accepted the attached license on 2019-07-10 at 12:25. The student, Matej Penciak, submitted this Dissertation for approval on 2019-07-10 at 12:39. This Dissertation was approved for publication on 2019-07-10 at 17:11. DSpace SAF Submission Ingestion Package generated from Vireo submission #14245 on 2019-11-26 at 13:05:18 Made available in DSpace on 2019-11-26T20:49:26Z (GMT). No. of bitstreams: 11 PENCIAK-DISSERTATION-2019.pdf: 543979 bytes, checksum: d777d3404a90c5b5e3417857692ff0ff (MD5) RSthesis.tex: 4315 bytes, checksum: c70b9599b0363bc1bc98b671953fc639 (MD5) abs.tex: 1141 bytes, checksum: 9c6367bd659b9cc5a4742abee07eec68 (MD5) ack.tex: 5308 bytes, checksum: ef56b9ebd69599d9509d2d918e7a65d3 (MD5) background.tex: 28090 bytes, checksum: 4b6a92200e2d1381b69cfb1f82e9aa86 (MD5) ded.tex: 25 bytes, checksum: e4e6c72e4e31aab44df936f35050efff (MD5) intro.tex: 23212 bytes, checksum: c56782fe9546c001ecd1efb258b6e907 (MD5) main.tex: 88463 bytes, checksum: 2e4238c79d9b735c80ef0c2e1be44aeb (MD5) thesisrefs.bib: 25400 bytes, checksum: a69ce664e5b54048809382e20126af35 (MD5) uiucecethesis09.cls: 22162 bytes, checksum: f7b508a2389c889fd082d4b2526767e7 (MD5) LICENSE.txt: 4210 bytes, checksum: 8894067df1a3ec3490d62d11989f6380 (MD5) Previous issue date: 2019-07-10 Embargo set by: Seth Robbins for item 112953 Lift date: 2021-11-26T20:49:41Z Reason: Author requested U of Illinois access only (OA after 2yrs) in Vireo ETD system U of I Only Restriction Lifted for Item 112953 on 2021-11-27T10:15:20Z. application/pdf http://hdl.handle.net/2142/105808 en Copyright 2019 Matej Penciak Ruijsenaars-Schneider, spectral curves, integrable systems, algebraic geometry A spectral description of the spin Ruijsenaars-Schneider system text text Mathematics Mathematics University of Illinois at Urbana-Champaign Dissertation Ph.D.