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Pentagrams, inscribed polygons, and Prym varieties
1. | Department of Mathematics, University of Toronto, Canada |
References:
[1] |
R. Felipe and G. Marí Beffa, The pentagram map on Grassmannians, to appear in Ann. Inst. Fourier, arXiv:1507.04765, (2015). |
[2] |
V. V. Fock and A. Marshakov, Loop groups, clusters, dimers and integrable systems, arXiv:1401.1606, (2014). |
[3] |
D. Fuchs and S. Tabachnikov, Self-dual polygons and self-dual curves, Funct. Anal. Other Math., 2 (2009), 203-220.
doi: 10.1007/s11853-008-0020-5. |
[4] |
M. Gekhtman, M. Shapiro, S. Tabachnikov and A. Vainshtein, Integrable cluster dynamics of directed networks and pentagram maps, with an appendix by A. Izosimov, Adv. Math., 300 (2016), 390-450.
doi: 10.1016/j.aim.2016.03.023. |
[5] |
M. Glick, The pentagram map and Y-patterns, Adv. Math., 227 (2011), 1019-1045.
doi: 10.1016/j.aim.2011.02.018. |
[6] |
M. Glick, The Devron property, J. Geom. Phys., 87 (2015), 161-189.
doi: 10.1016/j.geomphys.2014.07.029. |
[7] |
M. Glick and P. Pylyavskyy, Y-meshes and generalized pentagram maps, Proc. London Math. Soc. (3), 112 (2016), 753-797.
doi: 10.1112/plms/pdw007. |
[8] |
R. Kedem and P. Vichitkunakorn, T-systems and the pentagram map, J. Geom. Phys., 87 (2015), 233-247.
doi: 10.1016/j.geomphys.2014.07.003. |
[9] |
B. Khesin and F. Soloviev, Integrability of higher pentagram maps, Math. Ann., 357 (2013), 1005-1047.
doi: 10.1007/s00208-013-0922-5. |
[10] |
B. Khesin and F. Soloviev, The geometry of dented pentagram maps, J. Eur. Math. Soc., 18 (2016), 147-179.
doi: 10.4171/JEMS/586. |
[11] |
G. Marí Beffa, On integrable generalizations of the pentagram map, Int. Math. Res. Not., (2015), 3669-3693.
doi: 10.1093/imrn/rnu044. |
[12] |
V. Ovsienko, R. Schwartz and S. Tabachnikov, The pentagram map: A discrete integrable system, Comm. Math. Phys., 299 (2010), 409-446.
doi: 10.1007/s00220-010-1075-y. |
[13] |
V. Ovsienko, R. Schwartz and S. Tabachnikov, Liouville-Arnold integrability of the pentagram map on closed polygons, Duke Math. J., 162 (2013), 2149-2196.
doi: 10.1215/00127094-2348219. |
[14] |
R. Schwartz, The pentagram map, Exp. Math., 1 (1992), 71-81. |
[15] |
R. Schwartz, The pentagram map is recurrent, Exp. Math., 10 (2001), 519-528.
doi: 10.1080/10586458.2001.10504671. |
[16] |
R. Schwartz, The Poncelet grid, Adv. Geom., 7 (2007), 157-175.
doi: 10.1515/ADVGEOM.2007.010. |
[17] |
R. Schwartz, Discrete monodromy, pentagrams, and the method of condensation, J. Fixed Point Theory Appl., 3 (2008), 379-409.
doi: 10.1007/s11784-008-0079-0. |
[18] |
R. Schwartz, The pentagram integrals for Poncelet families, J. Geom. Phys., 87 (2015), 432-449.
doi: 10.1016/j.geomphys.2014.07.024. |
[19] |
R. Schwartz and S. Tabachnikov, Elementary surprises in projective geometry, Math. Intelligencer, 32 (2010), 31-34.
doi: 10.1007/s00283-010-9137-8. |
[20] |
R. Schwartz and S. Tabachnikov, The pentagram integrals on inscribed polygons, Electron. J. Combin., 18 (2011), Paper 171, 19pp. |
[21] |
F. Soloviev, Integrability of the pentagram map, Duke Math. J., 162 (2013), 2815-2853.
doi: 10.1215/00127094-2382228. |
show all references
References:
[1] |
R. Felipe and G. Marí Beffa, The pentagram map on Grassmannians, to appear in Ann. Inst. Fourier, arXiv:1507.04765, (2015). |
[2] |
V. V. Fock and A. Marshakov, Loop groups, clusters, dimers and integrable systems, arXiv:1401.1606, (2014). |
[3] |
D. Fuchs and S. Tabachnikov, Self-dual polygons and self-dual curves, Funct. Anal. Other Math., 2 (2009), 203-220.
doi: 10.1007/s11853-008-0020-5. |
[4] |
M. Gekhtman, M. Shapiro, S. Tabachnikov and A. Vainshtein, Integrable cluster dynamics of directed networks and pentagram maps, with an appendix by A. Izosimov, Adv. Math., 300 (2016), 390-450.
doi: 10.1016/j.aim.2016.03.023. |
[5] |
M. Glick, The pentagram map and Y-patterns, Adv. Math., 227 (2011), 1019-1045.
doi: 10.1016/j.aim.2011.02.018. |
[6] |
M. Glick, The Devron property, J. Geom. Phys., 87 (2015), 161-189.
doi: 10.1016/j.geomphys.2014.07.029. |
[7] |
M. Glick and P. Pylyavskyy, Y-meshes and generalized pentagram maps, Proc. London Math. Soc. (3), 112 (2016), 753-797.
doi: 10.1112/plms/pdw007. |
[8] |
R. Kedem and P. Vichitkunakorn, T-systems and the pentagram map, J. Geom. Phys., 87 (2015), 233-247.
doi: 10.1016/j.geomphys.2014.07.003. |
[9] |
B. Khesin and F. Soloviev, Integrability of higher pentagram maps, Math. Ann., 357 (2013), 1005-1047.
doi: 10.1007/s00208-013-0922-5. |
[10] |
B. Khesin and F. Soloviev, The geometry of dented pentagram maps, J. Eur. Math. Soc., 18 (2016), 147-179.
doi: 10.4171/JEMS/586. |
[11] |
G. Marí Beffa, On integrable generalizations of the pentagram map, Int. Math. Res. Not., (2015), 3669-3693.
doi: 10.1093/imrn/rnu044. |
[12] |
V. Ovsienko, R. Schwartz and S. Tabachnikov, The pentagram map: A discrete integrable system, Comm. Math. Phys., 299 (2010), 409-446.
doi: 10.1007/s00220-010-1075-y. |
[13] |
V. Ovsienko, R. Schwartz and S. Tabachnikov, Liouville-Arnold integrability of the pentagram map on closed polygons, Duke Math. J., 162 (2013), 2149-2196.
doi: 10.1215/00127094-2348219. |
[14] |
R. Schwartz, The pentagram map, Exp. Math., 1 (1992), 71-81. |
[15] |
R. Schwartz, The pentagram map is recurrent, Exp. Math., 10 (2001), 519-528.
doi: 10.1080/10586458.2001.10504671. |
[16] |
R. Schwartz, The Poncelet grid, Adv. Geom., 7 (2007), 157-175.
doi: 10.1515/ADVGEOM.2007.010. |
[17] |
R. Schwartz, Discrete monodromy, pentagrams, and the method of condensation, J. Fixed Point Theory Appl., 3 (2008), 379-409.
doi: 10.1007/s11784-008-0079-0. |
[18] |
R. Schwartz, The pentagram integrals for Poncelet families, J. Geom. Phys., 87 (2015), 432-449.
doi: 10.1016/j.geomphys.2014.07.024. |
[19] |
R. Schwartz and S. Tabachnikov, Elementary surprises in projective geometry, Math. Intelligencer, 32 (2010), 31-34.
doi: 10.1007/s00283-010-9137-8. |
[20] |
R. Schwartz and S. Tabachnikov, The pentagram integrals on inscribed polygons, Electron. J. Combin., 18 (2011), Paper 171, 19pp. |
[21] |
F. Soloviev, Integrability of the pentagram map, Duke Math. J., 162 (2013), 2815-2853.
doi: 10.1215/00127094-2382228. |
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