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Higher pentagram maps, weighted directed networks, and cluster dynamics
1. | Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, United States |
2. | Department of Mathematics, Michigan State University, East Lansing, MI 48823, United States |
3. | Department of Mathematics, Penn State University, University Park, PA 16802 |
4. | Department of Mathematics and Department of Computer Science, University of Haifa, Haifa, Mount Carmel 31905, Israel |
References:
[1] |
A. Bobenko and T. Hoffmann, Hexagonal circle patterns and integrable systems: Patterns with constant angles, Duke Math. J., 116 (2003), 525-566.
doi: 10.1215/S0012-7094-03-11635-X. |
[2] |
A. Bobenko and Yu. Suris, "Discrete Differential Geometry. Integrable Structure," Amer. Math. Soc., Providence, RI, 2008. |
[3] |
H. Busemann and P. Kelly, "Projective geometry and projective metrics," Academic Press, New York, 1953. |
[4] |
B. Dubrovin, I. Krichever and S. Novikov, Integrable systems. I, in "Dynamical Systems," IV, Encyclopaedia Math. Sci., 4, Springer, Berlin, (2001), 177-332. |
[5] |
L. Faddeev and L. Takhtajan, "Hamiltonian Methods in the Theory of Solitons," Springer-Verlag, Berlin, 1987. |
[6] |
S. Fomin and A. Zelevinsky, Cluster algebras. IV. Coefficients, Compos. Math., 143 (2007), 112-164.
doi: 10.1112/S0010437X06002521. |
[7] |
M. Gekhtman, M. Shapiro and A. Vainshtein, Poisson geometry of directed networks in a disk, Selecta Math., 15 (2009), 61-103.
doi: 10.1007/s00029-009-0523-z. |
[8] |
M. Gekhtman, M. Shapiro and A. Vainshtein, "Cluster Algebras and Poisson Geometry," Amer. Math. Soc., Providence, RI, 2010. |
[9] |
M. Gekhtman, M. Shapiro and A. Vainshtein, Poisson geometry of directed networks in an annulus, J. Europ. Math. Soc., preprint, arXiv:0901.0020. |
[10] |
M. Gekhtman, M. Shapiro and A. Vainshtein, Generalized Bäcklund-Darboux transformations for Coxeter-Toda flows from a cluster algebra perspective, Acta Math., 206 (2011), 245-310.
doi: 10.1007/s11511-011-0063-1. |
[11] |
M. Glick, The pentagram map and $Y$-patterns, Adv. Math., 227 (2011), 1019-1045.
doi: 10.1016/j.aim.2011.02.018. |
[12] | |
[13] |
A. Goncharov and R. Kenyon, Dimers and cluster integrable systems, preprint, arXiv:1107.5588. |
[14] |
B. Khesin and F. Soloviev, Integrability of a space pentagram map, in preparation. |
[15] |
B. Konopelchenko and W. Schief, Menelaus' theorem, Clifford configurations and inversive geometry of the Schwarzian KP hierarchy, J. Phys. A, 35 (2002), 6125-6144.
doi: 10.1088/0305-4470/35/29/313. |
[16] |
G. Mari Beffa, On generalizations of the pentagram map: Discretizations of AGD flows, preprint, arXiv:1103.5047. |
[17] |
S. Morier-Genoud, V. Ovsienko and S. Tabachnikov, 2-frieze patterns and the cluster structure of the space of polygons, Ann. Inst. Fourier, preprint, arXiv:1008.3359. |
[18] |
F. Nijhoff and H. Capel, The discrete Korteweg-de Vries equation, Acta Appl. Math., 39 (1995), 133-158.
doi: 10.1007/BF00994631. |
[19] |
M. Olshanetsky, A. Perelomov, A. Reyman and M. Semenov-Tian-Shansky, Integrable systems. II, in "Dynamical systems," VII, Encycl. Math. Sci., 16, Springer, Berlin, (1994), 83-259. |
[20] |
V. Ovsienko, R. Schwartz and S. Tabachnikov, Quasiperiodic motion for the Pentagram map, Electron. Res. Announc. Math. Sci., 16 (2009), 1-8. |
[21] |
V. Ovsienko, R. Schwartz and S. Tabachnikov, The Pentagram map: A discrete integrable system, Commun. Math. Phys., 299 (2010), 409-446.
doi: 10.1007/s00220-010-1075-y. |
[22] |
V. Ovsienko, R. Schwartz and S. Tabachnikov, Liouville-Arnold integrability of the pentagram map on closed polygons, preprint, arXiv:1107.3633. |
[23] |
A. Postnikov, Total positivity, Grassmannians, and networks, preprint, arXiv:math.CO/0609764. |
[24] |
O. Schramm, Circle patterns with the combinatorics of the square grid, Duke Math. J., 86 (1997), 347-389.
doi: 10.1215/S0012-7094-97-08611-7. |
[25] |
R. Schwartz, The pentagram map, Experiment. Math., 1 (1992), 71-81. |
[26] |
R. Schwartz, The pentagram map is recurrent, Experiment. Math., 10 (2001), 519-528. |
[27] |
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. |
[28] |
R. Schwartz and S. Tabachnikov, Elementary surprises in projective geometry, Math. Intelligencer, 32 (2010), 31-34.
doi: 10.1007/s00283-010-9137-8. |
[29] |
R. Schwartz and S. Tabachnikov, The pentagram integrals on inscribed polygons, Electron. J. Comb., 18 (2011), 171. |
[30] |
F. Soloviev, Integrability of the pentagram map, preprint, arXiv:1106.3950. |
[31] |
Yu. Suris, On some integrable systems related to the Toda lattice, J. Phys. A, 30 (1997), 2235-2249.
doi: 10.1088/0305-4470/30/6/041. |
show all references
References:
[1] |
A. Bobenko and T. Hoffmann, Hexagonal circle patterns and integrable systems: Patterns with constant angles, Duke Math. J., 116 (2003), 525-566.
doi: 10.1215/S0012-7094-03-11635-X. |
[2] |
A. Bobenko and Yu. Suris, "Discrete Differential Geometry. Integrable Structure," Amer. Math. Soc., Providence, RI, 2008. |
[3] |
H. Busemann and P. Kelly, "Projective geometry and projective metrics," Academic Press, New York, 1953. |
[4] |
B. Dubrovin, I. Krichever and S. Novikov, Integrable systems. I, in "Dynamical Systems," IV, Encyclopaedia Math. Sci., 4, Springer, Berlin, (2001), 177-332. |
[5] |
L. Faddeev and L. Takhtajan, "Hamiltonian Methods in the Theory of Solitons," Springer-Verlag, Berlin, 1987. |
[6] |
S. Fomin and A. Zelevinsky, Cluster algebras. IV. Coefficients, Compos. Math., 143 (2007), 112-164.
doi: 10.1112/S0010437X06002521. |
[7] |
M. Gekhtman, M. Shapiro and A. Vainshtein, Poisson geometry of directed networks in a disk, Selecta Math., 15 (2009), 61-103.
doi: 10.1007/s00029-009-0523-z. |
[8] |
M. Gekhtman, M. Shapiro and A. Vainshtein, "Cluster Algebras and Poisson Geometry," Amer. Math. Soc., Providence, RI, 2010. |
[9] |
M. Gekhtman, M. Shapiro and A. Vainshtein, Poisson geometry of directed networks in an annulus, J. Europ. Math. Soc., preprint, arXiv:0901.0020. |
[10] |
M. Gekhtman, M. Shapiro and A. Vainshtein, Generalized Bäcklund-Darboux transformations for Coxeter-Toda flows from a cluster algebra perspective, Acta Math., 206 (2011), 245-310.
doi: 10.1007/s11511-011-0063-1. |
[11] |
M. Glick, The pentagram map and $Y$-patterns, Adv. Math., 227 (2011), 1019-1045.
doi: 10.1016/j.aim.2011.02.018. |
[12] | |
[13] |
A. Goncharov and R. Kenyon, Dimers and cluster integrable systems, preprint, arXiv:1107.5588. |
[14] |
B. Khesin and F. Soloviev, Integrability of a space pentagram map, in preparation. |
[15] |
B. Konopelchenko and W. Schief, Menelaus' theorem, Clifford configurations and inversive geometry of the Schwarzian KP hierarchy, J. Phys. A, 35 (2002), 6125-6144.
doi: 10.1088/0305-4470/35/29/313. |
[16] |
G. Mari Beffa, On generalizations of the pentagram map: Discretizations of AGD flows, preprint, arXiv:1103.5047. |
[17] |
S. Morier-Genoud, V. Ovsienko and S. Tabachnikov, 2-frieze patterns and the cluster structure of the space of polygons, Ann. Inst. Fourier, preprint, arXiv:1008.3359. |
[18] |
F. Nijhoff and H. Capel, The discrete Korteweg-de Vries equation, Acta Appl. Math., 39 (1995), 133-158.
doi: 10.1007/BF00994631. |
[19] |
M. Olshanetsky, A. Perelomov, A. Reyman and M. Semenov-Tian-Shansky, Integrable systems. II, in "Dynamical systems," VII, Encycl. Math. Sci., 16, Springer, Berlin, (1994), 83-259. |
[20] |
V. Ovsienko, R. Schwartz and S. Tabachnikov, Quasiperiodic motion for the Pentagram map, Electron. Res. Announc. Math. Sci., 16 (2009), 1-8. |
[21] |
V. Ovsienko, R. Schwartz and S. Tabachnikov, The Pentagram map: A discrete integrable system, Commun. Math. Phys., 299 (2010), 409-446.
doi: 10.1007/s00220-010-1075-y. |
[22] |
V. Ovsienko, R. Schwartz and S. Tabachnikov, Liouville-Arnold integrability of the pentagram map on closed polygons, preprint, arXiv:1107.3633. |
[23] |
A. Postnikov, Total positivity, Grassmannians, and networks, preprint, arXiv:math.CO/0609764. |
[24] |
O. Schramm, Circle patterns with the combinatorics of the square grid, Duke Math. J., 86 (1997), 347-389.
doi: 10.1215/S0012-7094-97-08611-7. |
[25] |
R. Schwartz, The pentagram map, Experiment. Math., 1 (1992), 71-81. |
[26] |
R. Schwartz, The pentagram map is recurrent, Experiment. Math., 10 (2001), 519-528. |
[27] |
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. |
[28] |
R. Schwartz and S. Tabachnikov, Elementary surprises in projective geometry, Math. Intelligencer, 32 (2010), 31-34.
doi: 10.1007/s00283-010-9137-8. |
[29] |
R. Schwartz and S. Tabachnikov, The pentagram integrals on inscribed polygons, Electron. J. Comb., 18 (2011), 171. |
[30] |
F. Soloviev, Integrability of the pentagram map, preprint, arXiv:1106.3950. |
[31] |
Yu. Suris, On some integrable systems related to the Toda lattice, J. Phys. A, 30 (1997), 2235-2249.
doi: 10.1088/0305-4470/30/6/041. |
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