Integrable systems & mathematical physics
Our research covers classical and quantum integrable systems as well as related topics in algebra and geometry. Broadly speaking, integrable systems often involve nonlinear equations but despite this allow for exact solutions.
Staff
Publications
2024
Perletti, S., Strachan, I. A.B. (2024) Regular Fmanifolds with eventual identities. Journal of Physics A: Mathematical and Theoretical,
Feigin, M., Kaminski, L., Strachan, I. A.B. (2024) Legendre transforms for type An and Bn Vsystems. Journal of Physics A: Mathematical and Theoretical,
Feigin, M., Wang, Z., Yoshinaga, M. (2024) Integral expressions for derivations of multiarrangements. International Journal of Mathematics,
Abe, T., Enomoto, N., Feigin, M., Yoshinaga, M. (2024) Free reflection multiarrangements and quasiinvariants. International Mathematics Research Notices,
Alkadhem, M., Feigin, M. (2024) Commutativity equations and their trigonometric solutions. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 480, (doi: 10.1098/rspa.2024.0168)
Castellan, S. (2024) Automorphism groups of deformations and quantizations of Kleinian singularities. Transformation Groups,
Feigin, M., Valeri, D., Wright, J. (2024) Flat coordinates of algebraic Frobenius manifolds in small dimensions. Journal of Geometry and Physics, 200, (doi: 10.1016/j.geomphys.2024.105151)
De Matteis, G., Giglio, F., Moro, A. (2024) Complete integrability and equilibrium thermodynamics of biaxial nematic systems with discrete orientational degrees of freedom. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 480, (doi: 10.1098/rspa.2023.0701)
2023
Giglio, F., Landolfi, G., Martina, L. (2023) On solutions to a novel nonevolutionary integrable 1+1 PDE. Journal of Physics A: Mathematical and Theoretical, 56, (doi: 10.1088/17518121/ad04a5)
2022
Feigin, M., Vrabec, M. (2022) Bispectrality of AG2 Calogero–Moser–Sutherland system. Mathematical Physics, Analysis and Geometry, 25, (doi: 10.1007/s11040022094407)
Gilson, C. R. (2022) A visual approach to the ultradiscrete KdV equation with noninteger site values. Proceedings of the Royal Society of London Series A: Mathematical, Physical and Engineering Sciences, 478, (doi: 10.1098/rspa.2022.0150)
Athorne, C. (2022) Gauß–Manin from scratch: theme, variations and fantasia. Proceedings of the Royal Society of London Series A: Mathematical, Physical and Engineering Sciences, 478, (doi: 10.1098/rspa.2022.0217)
Feigin, M., Hakobyan, T. (2022) Algebra of Dunkl Laplace–Runge–Lenz vector. Letters in Mathematical Physics, 112, (doi: 10.1007/s11005022015510)
2021
Giglio, F., Landolfi, G., Martina, L., Moro, A. (2021) Symmetries and criticality of generalised van der Waals models. Journal of Physics A: Mathematical and Theoretical, 54, (doi: 10.1088/17518121/ac2009)
Feigin, M. V., Hallnäs, M. A., Veselov, A. P. (2021) Quasiinvariant Hermite polynomials and LassalleNekrasov correspondence. Communications in Mathematical Physics, 386, pp. 107141. (doi: 10.1007/s00220021040368)
Stedman, R., Strachan, I. A.B. (2021) Extended ⋁systems and trigonometric solutions to the WDVV equations. Journal of Mathematical Physics, 62, (doi: 10.1063/5.0024108)
Korff, C. (2021) Cylindric Hecke characters and GromovWitten invariants via the asymmetric sixvertex model. Communications in Mathematical Physics, 381, pp. 591640. (doi: 10.1007/s0022002003906x)
Strachan, I. A.B., Bridgeland, T. (2021) Complex hyperkähler structures defined by Donaldson–Thomas invariants. Letters in Mathematical Physics, 111, (doi: 10.1007/s1100502101388z)
2020
Alkadhem, M., Feigin, M. (2020) Trigonometric Vsystems and solutions of WDVV equations. Journal of Physics A: Mathematical and Theoretical, 54, (doi: 10.1088/17518121/abccf8)
Gilson, C. R., Hamanaka, M., Huang, S.C., Nimmo, J. J.C. (2020) Soliton solutions of noncommutative antiselfdual YangMills equations. Journal of Physics A: Mathematical and Theoretical, 53, (doi: 10.1088/17518121/aba72e)
Gorbounov, V.G., Korff, C., Stroppel, C. (2020) YangBaxter algebras, convolution algebras, and Grassmannians. Russian Mathematical Surveys, 75, (doi: 10.1070/RM9959)
Athorne, C. (2020) Equivariance and algebraic relations for curves. Journal of Geometry and Physics, 155, (doi: 10.1016/j.geomphys.2020.103748)
Alkadhem, M., Antoniou, G., Feigin, M. (2020) Solutions of BCn type of WDVV equations. American Mathematical Society
Korff, C., Palazzo, D. (2020) Cylindric symmetric functions and positivity. Algebraic Combinatorics, 3, pp. 191247. (doi: 10.5802/alco.90)
2019
Nimmo, J. J.C., Gilson, C. R., Willox, R. (2019) Darboux dressing and undressing for the ultradiscrete KdV equation. Journal of Physics A: Mathematical and Theoretical, 52, (doi: 10.1088/17518121/ab45cf)
Athorne, C. (2019) Equivariance in the Theory of Higher Genus ℘Functions. (doi: 10.1063/1.5125069)
Dubrovin, B., Strachan, I. A.B., Zhang, Y., Zuo, D. (2019) Extended affine Weyl groups of BCDtype: their Frobenius manifolds and their LandauGinzburg superpotentials. Advances in Mathematics, 351, pp. 897946. (doi: 10.1016/j.aim.2019.05.030)
Feigin, M., Vrabec, M. (2019) Intertwining operator for AG2 CalogeroMoserSutherland system. Journal of Mathematical Physics, 60, (doi: 10.1063/1.5090274)
Athorne, C., Yilmaz, H. (2019) Twisted laplace maps. Journal of Physics A: Mathematical and Theoretical, 52, (doi: 10.1088/17518121/ab1926)
Antoniou, G., Feigin, M. (2019) Supersymmetric Vsystems. Journal of High Energy Physics, 2019, (doi: 10.1007/JHEP02(2019)115)
Strachan, I. A.B. (2019) A construction of multidimensional DubrovinNovikov brackets. Journal of Nonlinear Mathematical Physics, 26, pp. 202213. (doi: 10.1080/14029251.2019.1591716)
2018
De Matteis, G., Giglio, F., Moro, A. (2018) Exact equations of state for nematics. Annals of Physics, 396, pp. 386396. (doi: 10.1016/j.aop.2018.07.016)
Barnett, S. M., Ferenczi, G., Gilson, C. R., Speirits, F. C. (2018) Statistics of photonsubtracted and photonadded states. Physical Review A: Atomic, Molecular and Optical Physics, 98, (doi: 10.1103/PhysRevA.98.013809)
Feigin, M. V., Veselov, A. P. (2018) ∨systems, holonomy lie algebras, and logarithmic vector fields. International Mathematics Research Notices, 2018, pp. 20702098. (doi: 10.1093/imrn/rnw289)
Korff, C., Palazzo, D. (2018) Cylindric reverse plane partitions and 2D TQFT. Séminaire Lotharingien de Combinatoire, 80B,
Athorne, C. (2018) Laplace maps and constraints for a class of third order partial differential operators. Journal of Physics A: Mathematical and Theoretical, 51, (doi: 10.1088/17518121/aaa475)
Athorne, C. (2018) Fifty years of mathematical physics: selected works of Ludwig Fadeev. Contemporary Physics, 59, pp. 421. (doi: 10.1080/00107514.2018.1538165)
Athorne, C. (2018) Fractals in probability and analysis. Contemporary Physics, 59, pp. 429. (doi: 10.1080/00107514.2018.1559228)
2017
Gorbounov, V., Korff, C. (2017) Quantum integrability and generalised quantum Schubert calculus. Advances in Mathematics, 313, pp. 282356. (doi: 10.1016/j.aim.2017.03.030)
Strachan, I. A.B., Zuo, D. (2017) Frobenius manifolds and Frobenius algebravalued integrable systems. Letters in Mathematical Physics, 107, pp. 9971026. (doi: 10.1007/s110050170939x)
Strachan, I. A.B., Stedman, R. (2017) Generalized Legendre transformations and symmetries of the WDVV equations. Journal of Physics A: Mathematical and Theoretical, 50, (doi: 10.1088/17518121/aa58b2)
Korff, C. (2017) Dimers, crystals and quantum Kostka numbers. Seminaire Lotharingien de Combinatoire, 78B.40, pp. 12pp.
2016
Giglio, F., Landolfi, G., Moro, A. (2016) Integrable extended van der Waals model. Physica D: Nonlinear Phenomena, 333, pp. 293300. (doi: 10.1016/j.physd.2016.02.010)
Feigin, M. (2016) The Hilbert series of Λr,s(m). Appendix in Etingof, P., Rains, E. On Cohen–Macaulayness of algebras generated by generalized power sums. Communications in Mathematical Physics, 347, pp. 163182. (doi: 10.1007/s0022001626570)
Barnett, S. M., Allen, L., Cameron, R. P., Gilson, C. R., Padgett, M. J., Speirits, F. C., Yao, A. M. (2016) On the natures of the spin and orbital parts of optical angular momentum. Journal of Optics, 18, (doi: 10.1088/20408978/18/6/064004)
Athorne, C. (2016) Fundamental principles of classical mechanics: a geometrical perspective. Contemporary Physics, 57, pp. 238241. (doi: 10.1080/00107514.2015.1048299)
Athorne, C., Yilmaz, H. (2016) Invariants of hyperbolic partial differential operators. Journal of Physics A: Mathematical and Theoretical, 49, (doi: 10.1088/17518113/49/13/135201)
Korff, C. (2016) From quantum Bäcklund transforms to topological quantum field theory. Journal of Physics A: Mathematical and Theoretical, 49, (doi: 10.1088/17518113/49/10/104001)
Adler, V.E., Berest, Y. Y., Buchstaber, V.M., Grinevich, P.G., Dubrovin, B.A., Krichever, I.M., Novikov, S.P., Sergeev, A.N., Feigin, M.V., Felder, G., Ferapontov, E.V., Chalykh, O.A., Etingof, P.I. (2016) Alexander Petrovich Veselov (on his 60th birthday) Russian Mathematical Surveys, 71, pp. 11591176. (doi: 10.1070/RM9734)
2015
Cameron, R. P., Speirits, F. C., Gilson, C. R., Allen, L., Barnett, S. M. (2015) The azimuthal component of Poynting's vector and the angular momentum of light. Journal of Optics, 17, (doi: 10.1088/20408978/17/12/125610)
Strachan, I. A.B., Zuo, D. (2015) Integrability of the Frobenius algebravalued KadomtsevPetviashvili hierarchy. Journal of Mathematical Physics, 56, (doi: 10.1063/1.4935936)
Feigin, M., Hakobyan, T. (2015) On Dunkl angular momenta algebra. Journal of High Energy Physics, 2015, (doi: 10.1007/JHEP11(2015)107)
Gilson, C.R., Nimmo, J.J.C., Nagai, A. (2015) A direct approach to the ultradiscrete KdV equation with negative. Journal of Physics A: Mathematical and General, 48, (doi: 10.1088/17518113/48/29/295201)
Strachan, I., Brendle, T., Wilson, A. (2015) Online assessment and feedback: how to square the circle.
Feigin, M. (2015) Propagation of sound waves. Mathematical Etudes. Institute of Mathematics of the Russian Academy of Sciences
Athorne, C. (2015) Special functions. WileyVCH, Verlag GmbH & Co. KGaA
Postgraduate research students
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Integrable Systems and Mathematical Physics  Example Research Projects
Information about postgraduate research opportunities and how to apply can be found on the Postgraduate Research Study page. Below is a selection of projects that could be undertaken with our group.
There is also the possibility of applying to The EPSRC Centre for Doctoral Training in Algebra, Geometry, and Quantum Fields (AGQ CDT) is a collaborative effort by University of Glasgow, University of Edinburgh, and HeriotWatt University. The programme is at the forefront of mathematical innovation, leveraging the power of symmetry, geometry, and quantum physics to shape the technologies of tomorrow.
Cherednik Algebras and related topics (PhD)
Supervisors: Misha Feigin
Relevant research groups: Algebra, Integrable Systems and Mathematical Physics
The project is aimed at clarifying certain questions related to Cherednik algebras. These questions may include study of homomorphisms between rational Cherednik algebras for particular Coxeter groups and special multiplicity parameters, defining and studying of new partial spherical Cherednik algebras and their representations related to quasiinvariant polynomials, study of differential operators on quasiinvariants related to nonCoxeter arrangements. Relations with quantum integrable systems of CalogeroMoser type may be explored as well. Some other possible topics may include study of quasiinvariants for nonCoxeter arrangements in relation to theory of free arrangements of hyperplanes.
Ultradiscrete Integrable Systems (PhD)
Supervisor: Claire Gilson
Relevant research groups: Integrable Systems and Mathematical Physics
An ultradiscrete equation is an equation where not only are the independent variables x and t discrete but the dependent variable are also discrete and often take on the values 0 or 1. There are many interesting results known about these systems. They are usually obtained by the process of ultradiscretization, this is a limiting process which takes you from a continuous integrable system to an ultradiscrete system via a discrete system. This project investigates the world of ultradiscrete systems and soliton cellular automata both periodic and non periodic. Looking in particular at how one can build different systems and extract the conserved quantities of these systems.
Noncommutative Integrable Systems (PhD)
Supervisor: Claire Gilson
Relevant research groups: Integrable Systems and Mathematical Physics
Integrable Systems are a very special class of differential equations with exact (soliton) solutions. There are several well known integrable systems, most of which are commutative (i.e the order in which the dependent variables are written down in the equation doesn't matter). Most of these equations have multiple soliton solutions that can be written down in terms of determinants such as Wronskians or Grammians. There are many fewer noncommutative integrable systems that are known, these have solutions in terms of quasideterminants (Gelfand and Retakh, vol25, p91, Functional Analysis and its Applications). This project aims to investigate the properties of known and new noncommutative systems.
Quantum spinchains and exactly solvable lattice models (PhD)
Supervisors: Christian Korff
Relevant research groups: Algebra, Integrable Systems and Mathematical Physics
Quantum spinchains and 2dimensional statistical lattice models, such as the Heisenberg spinchain and the six and eightvertex models remain an active area of research with many surprising connections to other areas of mathematics.
Some of the algebra underlying these models deals with quantum groups and Hecke algebras, the TemperleyLieb algebra, the Virasoro algebra and KacMoody algebras. There are many unanswered questions ranging from very applied to more pure topics in representation theory and algebraic combinatorics. For example, recently these models have been applied in combinatorial representation theory to compute GromovWitten invariants (enumerative geometry) and fusion coefficients in conformal field theory (mathematical physics).
Integrable quantum field theory and Ysystems (PhD)
Supervisors: Christian Korff
Relevant research groups: Algebra, Integrable Systems and Mathematical Physics
The mathematically rigorous and exact construction of a quantum field theory remains a tantalising challenge. In 1+1 dimensions exact results can be found by computing the scattering matrices of such theories using a set of functional relations. These theories exhibit beautiful mathematical structures related to Weyl groups and Coxeter geometry.
In the thermodynamic limit (volume and particle number tend to infinity while the density is kept fixed) the set of functional relations satisfied by the scattering matrices leads to socalled Ysystems which appear in cluster algebras introduced by Fomin and Zelevinsky and the proof of dilogarithm identities in number theory.
Longrange quantum integrability (PhD)
Supervisors: Jules Lamers
Relevant research groups: Algebra, Integrable Systems and Mathematical Physics
This project deals with mathematical physics, or more precisely, representation theory with a view towards quantum physics. Spin chains are models for magnetic materials based on quantum mechanics. While research often assumes interactions between the spins of adjacent atoms only, longrange spin interactions are relevant for cuttingedge experiments, quantum information and computing, and theoretical highenergy physics.
If you like algebra or representation theory, and think you might want to help develop the powerful toolkit of quantum integrability in the longrange context, please send me an email! A background in physics is not required.
Keywords for possible topics are: Hecke algebras, Macdonald polynomials, quantum groups, elliptic generalisations, quantum spin chains, quantum manybody systems.
qDT invariants and deformations of hyperKahler geometry (PhD)
Supervisor: Ian Strachan
Relevant research groups: Geometry and Topology, Integrable Systems and Mathematical Physics
The project seeks to understand and exploit the integrable structure behind quantum DonaldsonThomas invariants in terms of deformation of hyperKahler geometry and quantumRiemannHilbert problems.
Almostduality for arbitrary genus Hurwitz spaces (PhD)
Supervisor: Ian Strachan
Relevant research groups: Geometry and Topology, Integrable Systems and Mathematical Physics
The space of rational functions (interpreted as the space of holomorphic maps from the Riemann sphere to itself) may be endowed with the structure of a Frobenius manifolds, and hence there also exists an almostdual Frobenius manifold structure. The class of examples include Coxeter and ExtendedAffineWeyl orbit group spaces. This extends to spaces of holomorphic maps between the torus and the sphere, where one can proved stronger results than just existence results. The project will seek to extend this to the explicit study of the space of holomorphic maps from an arbitrary genus Riemann surface to the Riemann sphere.
Seminars
The group holds its own regular research seminars. You can find a full list of the seminars within the school on the main seminars page, where you can subscribe to their respective calendars.
Integrable systems is a branch of mathematics which first came to prominence in the mid 1960s with the (mathematical) discovery of the soliton by Kruskal and Zabusky while studying dispersive shallow water waves. Broadly speaking the focus is on systems for which  despite their nonlinear behaviour  exact solutions can be derived. Today the field has come to embrace many different aspects of mathematical physics and is at the crossroads with other modern branches of pure and applied mathematics.
Our group is one of the largest in the UK, is part of an LMS network on quantum and classical integrability and hosts its own series of international conferences, ISLAND. Staff members have a diverse range of interests including topics in algebra and geometry. We welcome applications by prospective PhD students; please see Postgraduate opportunities or see below for examples of possible projects, and for further information see how to apply.