Date: October 22 – October 27, 2023
Venue: Institute for Advanced Study in Mathematics (IASM), Hangzhou, China
Organizers: Frank Nijhoff (University of Leeds), Linyu Peng (Keio University), Yang Shi (Flinders University), Da-jun Zhang (Shanghai University)
Sponsors: BIRS and IASM
Workshop webpage at BIRS; IASM-BIRS Workshops in 2023
Objectives
The workshop deals with a fast-developing novel field of research that was initiated a decade ago with the pioneering paper by Lobb/Nijhoff of 2009, and that has seen quite an impressive development since with various groups making major contributions. Since there have never been specific workshops dedicated to this particular area yet, the time is now ripe for such a specific workshop to be held. Thus, the objectives of this workshop are as follows:
1. to bring together the main experts in this field in order to consolidate the results obtained so far, and to identify the outstanding problems and challenges and formulate a strategy for moving the subject forward;
2. to bring together experts from various cognate areas (such as differential geometry, theory of invariants, special function theory and from quantum physics and loop quantum gravity) whose research exhibits common ground with the subject of the workshop; to bring them up to speed with the developments and discuss potential applications.
The following areas will be covered during the workshop:
1. foundations of the theory: the basic principles and the derivation of multiform Euler-Lagrange equations;
2. emergence of (higher) variational symmetries, conservation laws and (multiform) Noether theorem;
3. symmetry reductions (e.g. to integrable maps and Painleve hierarchies) and connections to differential and difference invariant theory;
4. classification problem of Lagrangian multiforms and pluri-Lagrangian systems - multicomponent and matrix systems, higher dimensions;
5. discrete and continuous differential geometric descriptions and the variational bicomplex;
6. the quantum formulation of the theory and Feynman path integrals;
7. connections with topological field theory and fundamental theories of physics (e.g., loop quantum gravity).
Confirmed participants
Vincent Caudrelier | University of Leeds | UK |
Xiangke Chang | Chinese Academy of Sciences | China |
Yong Chen | East China Normal University | China |
Jipeng Cheng
| China University of Mining and Technology | China |
Masashi Hamanaka | Nagoya University | Japan |
Yuji Kodama | Shandong University of Science and Technology, & Ohio State University | China/USA |
Thanadon Kongkoom | Naresuan University | Thailand |
Chuanzhong Li | Shangdong University of Science and Technology | China |
Shangshuai Li | Shanghai University | China |
Shihao Li | Sichuan University | China |
Xing Li | Jiangsu Normal University | China |
Jin Liu | Shanghai University | China |
Xingyu Liu | Shanghai University | China |
Yaqing Liu | Beijing Information Science and Technology University | China |
Frank Nijhoff | University of Leeds | UK |
Yusuke Ono | Keio University | Japan |
Linyu Peng | Keio University | Japan |
Changzheng Qu | Ningbo University | China |
Nicolai Reshethikin | Tsinghua University | China |
Jacob Richardson | University of Leeds | UK |
Leilei Shi | Shanghai University | China |
Yang Shi | Flinders University | Australia |
Ying Shi | Zhejiang University of Science and Technology | China |
Pengyu Sun | Shanghai University | China |
Yingying Sun | University of Shanghai for Science and Technology | China |
Kongkoom Thanadon | Naresuan University | Thailand |
Kelei Tian | Hefei University of Technology | China |
Shigeru Tomita | Keio University | Japan |
Mats Vermeeren | Loughborough University | UK |
Chengfa Wu | Shenzhen University | China |
Yuancheng Xie | Peking University | China |
Xiaoxue Xu | Zhengzhou University | China |
Lingling Xue | Ningbo University | China |
Bo Yang | Ningbo University | China |
Di Yang | University of Science and Technology in China | China |
Sikarin Yoo-Kong | Naresuan University | Thailand |
Cheng Zhang | Shanghai University | China |
Da-jun Zhang | Shanghai University | China |
Danda Zhang | Ningbo University | China |
Zejun Zhou | University of Science and Technology of China | China |
Dafeng Zuo | University of Science and Technology of China | China |
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Online participants (confirmed) |
Huai-Liang Chang | Hong Kong University of Science and Technology | Hong Kong |
Bianca Dittrich | Perimeter Institute Canada | Canada |
Adam Doliwa | University of Warmia and Mazury | Poland |
Anton Dzhamay | The University of Northern Colorado | USA |
Joao Faria Martins | University of Leeds | UK |
Evgeny Ferapontov | Loughborough University | UK |
Wei Fu | East China Normal University | China |
Felix Gunther | Technische Universität Berlin | Germany |
Jarmo Hietarinta | University of Turku | Finland |
Andrew Kels | University of New South Wales | Australia |
Irina Kogan | North Carolina State University | USA |
Atsuo Kuniba | University of Tokyo | Japan |
Nobutaka Nakazono | Tokyo University of Agriculture and Technology | Japan |
Qingping Liu | China University of Mining and Technology (Beijing) | China |
Peter Olver | University of Minnesota -- Twin Cities | USA |
Reinout Quispel | La Trobe University | Australia |
Wolfgang Schief | University of New South Wales | Australia |
Anup Anand Singh | University of Leeds | UK |
Yuri Suris | Technical University of Berlin | Germany |
Benoit Vicedo | University of York | UK |
Pavlos Xenitidis | Liverpool Hope University | UK |
Jing Ping Wang | University of Kent | UK |
Songlin Zhao | Zhejiang University of Technology | China |
Programme
https://workshops.birs.ca/events/23w5043/schedule
Related papers
First papers:
[1] S. Lobb, F.W. Nijhoff, Lagrangian multiforms and multidimensional consistency, J. Phys. A: Math Theor., 42 (2009) 454013
[2] S. Lobb, F.W. Nijhoff, R. Quispel, Lagrangian multiform structure for the lattice KP system, J. Phys. A: Math Theor., 42 (2009) 472002
[3] S. Lobb, F.W. Nijhoff, Lagrangian multiform structure for the lattice Gel'fand-Dikii hierarchy, J. Phys. A: Math. Theor., 43 (2010) 072003
[4] A. Bobenko, Yu.B. Suris, On the Lagrangian structure of integrable quad-equations, Lett. Math. Phys., 92 (2010) 17-31
[5] P. Xenitidis, F.W. Nijhoff, S. Lobb, On the Lagrangian formulation of multidimensionally consistent systems, Proc. Roy. Soc., A467 (2011) 3295-3317
[6] S. Yoo-Kong, S. Lobb, F.W. Nijhoff, Discrete-time Calogero-Moser system and Lagrangian 1-form structure, J. Phys. A: Math. Theor., 44 (2011) 365203
[7] J. Atkinson, S.B. Lobb, F.W. Nijhoff, An integrable multicomponent quad equation and its Lagrangian formalism, Theor. Math. Phys., 173 (2012) 1644-1653
[8] S. Yoo-Kong, F.W. Nijhoff, Discrete-time Ruijsenaars-Schneider system and Lagrangian 1-form structure, arXiv:1112.4576 (Dec. 2011)
[9] Yu.B. Suris, Variational formulation of commuting Hamiltonian flows: multi-time Lagrangian 1-forms, J. Geom. Mech., 2013, 5(3): 365-379, arXiv: 1212.3314 (Dec. 2012)
[10] R. Boll, M. Petrera, Yu. B. Suris, Multi-time Lagrangian 1-forms for families of Bäcklund transformations. Toda-type systems, arXiv:1302.7144 (Feb. 2013)
[11] R. Boll, M. Petrera, Yu.B. Suris, Multi-time Lagrangian 1-forms for families of Bäcklund transformations. Relativistic Toda-type systems, J. Phys A: Math. Theor., 48 (2015) 085203
[12] R. Boll, M. Petrera Yu.B. Suris, What is integrability of discrete variational systems? Proc, Roy Soc. A470 (2014) 20130550
[13] S. Lobb, F.W. Nijhoff, A variational principle for discrete integrable systems, SIGMA, 14 (2018) 041, arXiv: 1312.1440
EL equations and continuum limits:
[14] Yu.B. Suris, Variational fomrtulation of commuting Hamiltonian flows: multi-time Lagrangian 1-forms, J. Geom. Mech, 5 (2013) 365--379
[15] M. Vermeeren, Continuum limits of pluri-Lagrangian, J. Int. Syst. 4 (2019) 1-34
[16] M. Vermeeren, A variational perspective on continuum limits of ABS and lattice GD eqations, SIGMA, 15 (2019) 044
The pluri-Lagrangian formulation:
[17] A. Bobenko, Yu.B. Suris, Discrete pluriharmonic functions as solutions of linear pluri-Lagrangian systems, Comm. Mah. Phys., 336 (2015) 199-215
[18] Yu.B. Suris, Variational symmetries and pluri-Lagrangian systems, in: Dynamical systems, Number Theory and Appls. (Springer Verlag), 2016) 255-266
[19] Yu.B. Suris, M. Vermeeren, On the Lagrangian structure of integrable hierarchies, in: Advances in Discrete Differential Geometry (Springer Verlag, 2016) 347-378
Some recent developments:
Lagrangian multiform formulation of the Lax pair:
[20] D. Sleigh, F.W. Nijhoff, V. Caudrelier, A variational approach to Lax representations, J. Geom. Phys., 142 (2019) 66--79
Role of variational symmetries in multiform context:
[21] Yu.B. Suris, Variational symmetries and pluri-Lagrangian systems, in "Dynamical Systems, Number Theory and Applications, (Springer Verlag, 2016) 255-266
[22] D. Sleigh, F.W. Nijhoff, V. Caudrelier, Variational symmetries and Lagrangian multiforms, Lett. Math. Phys., 110 (2020) 805--826
[23] M. Petrera, M. Vermeeren, Variational symmetries and pluri-Lagrangian strucures for integrable hierarchies of PDEs, Eur. J. Math., 7 (2021) 741-765
Lagrangian 3-form structure for the KP hierarchy:
[24] D. Sleigh, F.W. Nijhoff, V. Caudrelier, Lagrangian multiforms for the Kadomtsev-Petviashvili (KP) and the Gel'fand-Dickey hierarchy, Int. Math. Res. Not., 2021 (2021) 1-41
[25] S. Lobb, F.W. Nijhoff, R. Quispel, Lagrangian multiform structure for the lattice KP system, J. Phys. A: Math. Theor., 42 (2009) 472002
[26] S. Lobb, F.W. Nijhoff, Lagrangian multiform structure for the lattice Gel'fand-Dikii hierarchy, J. Phys. A: Math. Theor., 43 (2010) 072003
[27] F.W. Nijhoff, Lagrangan 3-form structure for the Darboux systems and the KP hierarchy, Lett. Math. Phys., (2023) arxiv: 2206.14338
Connection with classical r-matrices:
[28] V. Caudrelier, M. Stoppato, B. Vicedo, Classical Yang-Baxter equation, Lagrangian multiforms and ultralocal integrable hierarchies, arXiv:2201.08286
[29] V. Caudrelier, M. Stoppato, Multiform description of the AKNS hierarchy and classical r-matrix, J. Phys. A: Math. Theor., 54 (2022) 235204, arXiv:2010.07163
[30] V. Caudrelier, M, Stoppato, Hamiltonian multiform of an integrable hierarchy, J. Math. Phys., 61 (2020) 123506, arxiv:2004.01164
Lagrangian structure of non-commuting flows:
[31] V. Caudrelier, F. Nijhoff, D. Sleigh, M. Vermeeren, Lagrangian multiforms on Lie groups and non-commuting flows, J. Geom. Phys., 187 (2023) 104807, arxiv:2204.09663
Towards quantum multiform theory:
[32] S.D. King, F.W. Nijhoff, Quantum variational principle and quantum multiform theory, Nucl. Phys. B947 (2019) 114686, arxiv:1702.08709 [math-ph]
[33] T. Kongkoom, S. Yoo-Kong, Quantum integrability: Lagrangian 1-form case, Nucl. Phys. B987 (2023) 116101, arxiv:2203.16914 [math-ph]
[34] C. Puttarprom, W. Piensuk, S. Yoo-Kong, Integrable Hamiltonian hierarchies and Lagangian 1-forms, arxiv:1904.00582
[35] A. Sridhar, Yu.B. Suris, Commutativity in Lagrangian and Hamiltonian Mechanics, J. Geom. Phys., 137 (2019) 154-161, arxiv: 1801.06076 [math-ph]
Connection with Loop Quantum Gravity:
[36] B. Bahr, B. Dittrich, Breaking and restoring of diffeomorphism symmetry in discrete gravity, arxiv:0909.5688 [gr-qc]
[37] B. Bahr, B. Dittrich, Improved and perfect actions in discrete gravity, Phys. Rev. D, 80 (2009) 124030, arxiv: 0907.4323 [gr-qc]
[38] C. Rovelli, On the structure of a background independent quantum theory: Hamilton function, transition amplitudes, classical limit and continuous limit, arxiv:1108.0832 [gr-qc]
[39] S. Ariwahjoedi, J.S. Kosasih, C. Rovelli, F.P. Zen, Degrees of freedom in discrete geometry, arxiv: 1607.07963 [gr-qc]
[40] C. Rovelli, Discretizing parametrized systems: the magic of Ditt–invariance, Phys. Rev. D 106 (2022) 104062, arxiv:1107.2310 [hep-lat]
Connections with Chern-Simons theory and topological field theory:
[41] L. Martina, Kur. Myrzakulov, R. Myrzakulov, G. Soliani, Deformation of surfaces, integrable systems and Chern-Simons theory, J. Math. Phys., 42 (2001) 1397-1417, arXiv:nlin/0006039
[42] K.J. Costello, Integrable lattice models from four-dimensional field theories, Proc. Symp. Pure Math., 88 (2014) 3-23. arxiv:1308.0370
[43] K. Costello, E. Witten, M. Yamazaki, Gauge theory and integrability, I, ICCM Not. 6 (2018) 46–119, arxiv:1709.09993 [hep-th]
[44] K. Costello, E. Witten, M. Yamazaki, Gauge theory and integrability, II, ICCM Not. 6 (2018) 120–149, arxiv:1802.01579 [hep-th]
[45] K. Costello, M. Yamazaki, Gauge theory and integrability, III, arXiv:1908.02289 [hep-th]
[46] F. Delduc, S. Lacroix, M. Magro, B. Vicedo, A unifying 2d action for integrable σ-models from 4d Chern-Simons theory, Lett. Math. Phys., 110 (2020) 1645–1687
[47] F. Nijhoff, Integrable hierarchies, Lagrangian structures and noncommuting flows, published in Eds. M.J. Ablowitz, B. Fuchssteiner and M. Kruskal, Topics in Soliton Theory and Exactly Solvable Nonlinear Equations, pp. 150–181, Signapore, World Scientific Publ. Co., 1987.
[48] V. Caudrelier, M. Stoppato, B. Vicedo, On the Zakharov-Mkhailov action: 4d Chern-Simons origin and covariant Poisson algebra of the Lax connection, arXiv:2012.04431 [hep-th]
Connection with quantum geometry:
[49] V.V. Bazhanov, A.P. Kels, S.M. Sergeev, Quasi-classical expansion of the star-triangle relation and integrable systems on quad-graphs, J. Phys. A: Math. Theor., 49 (2016) 464001, arXiv:1602.07076 [math-ph]
[50] A.P. Kels, M. Yamazaki, Lens elliptic gamma function solution of the Yang-Baxter equation at roots of unity, J. Stat. Mech., (2018) 023108, arXiv:1709.07148 [math-ph]
[51] A. Bobenko, F. Gunther, On discrete integrable equations with convex variational principles, Lett. Math. Phys., 102 (2012) 181-202, arXiv:1111.6273 [nlin.SI]
[52] V.V. Bazhanov, S.M. Sergeev, A master solution of the quantum Yang-Baxter equation and classical discrete integrable equations, Adv. Theor. Math, Phys., 16 (2012) 65-95, arxiv: 1006.0651 [math-ph]
[53] V.V. Bazhanov, V.V. Mangazeev, S.M. Sergeev, Quantum geometry of 3-dimensional lattices and tetrahedron equation, J. Stat. Mech., (2008) P07004, arxiv: 0911.3693 [math-ph]
References on variational bicomplex:
For Differential Equations:
[54] I.M. Anderson, Introduction to the variational bicomplex, in Mathematical Aspects of Classical Field Theory, Contemp. Math. 132 (1992) 51–73
[55] I. M. Anderson, The Variational Bicomplex, (book manuscript), Utah State University, 1989.
[56] L.A. Dickey, Soliton Equations and Hamiltonian Systems, 2nd edition, World Scientific, 2003
[57] I. Kogan, P.J. Olver, Invariant Euler-Lagrange equations and the invariant variational Bicomplex, Acta Appl. Math.,76 (2003) 137-193
[58] I.S. Krasil'shchik, A.M. Vinogradov (eds), Symmetries and Conservation Laws for Differential Equations of Mathematical Physics, Providence, RI: AMS Publications, Monograph, 1999.
[59] P.J. Olver, Applications of Lie Groups to Differential Equations, (2nd edn), New York: Springer-Verlag, 1993.
[60] T. Tsujishita, On variation bicomplexes associated to differential equations, Osaka J. Math. 19 (1982) 311–363.
[61] W.M. Tulczyjew, The Lagrange complex, Bulletin la S. M. F. 105 (1977) 419-431.
[62] W.M. Tulczyjew, The Euler-Lagrange resolution, in Lecture Notes in Mathematics 836, Differential Geometrical Methods in Mathematical Physics (Eds. A. Dold, B. Eckmann), pp 22–48 (Springer-Verlag, New York 1980).
[63] A.M. Vinogradov, A spectral sequence associated with a non-linear differential equation, and the algebro-geometric foundations of Lagrangian field theory with constraints, Sov. Math. Dokl. 19 (1978) 144–148.
[64] A.M. Vinogradov, The b-spectral sequence, Lagrangian formalism, and conservation laws. I: The linear theory, J. Math. Anal. Appl., 100 (1984) 1-40.
[65] A.M. Vinogradov, The b-spectral sequence, Lagrangian formalism, and conservation laws. II: The nonlinear theory, J. Math. Anal. Appl., 100 (1984) 41-129.
[66] R. Vitolo, Finite order variational bicomplexes, Mathematical Proceedings of the Cambridge Philosophical Society 125 (1999) 321-333, arXiv:math-ph/0001009
For discrete equations:
[67] P.E. Hydon, E.L. Mansfield, A variational complex for difference equations, Found. Comput. Math., 4 (2004) 187-217.
[68] B.A. Kupershmidt, Discrete Lax Equations and Differential-Difference Calculus, Paris: Asterisque 123, SMF, 1985.
[69] E.L. Mansfield, P.E. Hydon, Diffeence forms, Found. Comput. Math., 8 (2008) 427-467.
[70] L. Peng, From Differential to Difference: The Variational Bicomplex and Invariant Noether's Theorems, Ph.D. Thesis, University of Surrey, 2013.
[71] L. Peng, P.E. Hydon, The difference variational bicomplex and multisymplectic systems, arXiv:2307.13935 [math-ph].