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

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,

2023

Giglio, F., Landolfi, G., Martina, L. (2023) On solutions to a novel non-evolutionary integrable 1+1 PDE. Journal of Physics A: Mathematical and Theoretical, 56, (doi: 10.1088/1751-8121/ad04a5)

2022

Feigin, M., Vrabec, M. (2022) Bispectrality of AG2 Calogero–Moser–Sutherland system. Mathematical Physics, Analysis and Geometry, 25, (doi: 10.1007/s11040-022-09440-7)

Gilson, C. R. (2022) A visual approach to the ultradiscrete KdV equation with non-integer 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/s11005-022-01551-0)

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: 10.1088/1751-8121/ac2009)

Feigin, M. V., Hallnäs, M. A., Veselov, A. P. (2021) Quasi-invariant Hermite polynomials and Lassalle-Nekrasov correspondence. Communications in Mathematical Physics, 386, pp. 107-141. (doi: 10.1007/s00220-021-04036-8)

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 Gromov-Witten invariants via the asymmetric six-vertex model. Communications in Mathematical Physics, 381, pp. 591-640. (doi: 10.1007/s00220-020-03906-x)

Strachan, I. A.B., Bridgeland, T. (2021) Complex hyperkähler structures defined by Donaldson–Thomas invariants. Letters in Mathematical Physics, 111, (doi: 10.1007/s11005-021-01388-z)

2020

Alkadhem, M., Feigin, M. (2020) Trigonometric V-systems and solutions of WDVV equations. Journal of Physics A: Mathematical and Theoretical, 54, (doi: 10.1088/1751-8121/abccf8)

Gilson, C. R., Hamanaka, M., Huang, S.-C., Nimmo, J. J.C. (2020) Soliton solutions of noncommutative anti-self-dual Yang-Mills equations. Journal of Physics A: Mathematical and Theoretical, 53, (doi: 10.1088/1751-8121/aba72e)

Gorbounov, V.G., Korff, C., Stroppel, C. (2020) Yang-Baxter 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. 191-247. (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: 10.1088/1751-8121/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 BCD-type: their Frobenius manifolds and their Landau-Ginzburg superpotentials. Advances in Mathematics, 351, pp. 897-946. (doi: 10.1016/j.aim.2019.05.030)

Feigin, M., Vrabec, M. (2019) Intertwining operator for AG2 Calogero-Moser-Sutherland 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: 10.1088/1751-8121/ab1926)

Antoniou, G., Feigin, M. (2019) Supersymmetric V-systems. Journal of High Energy Physics, 2019, (doi: 10.1007/JHEP02(2019)115)

Strachan, I. A.B. (2019) A construction of multidimensional Dubrovin-Novikov brackets. Journal of Nonlinear Mathematical Physics, 26, pp. 202-213. (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. 386-396. (doi: 10.1016/j.aop.2018.07.016)

Barnett, S. M., Ferenczi, G., Gilson, C. R., Speirits, F. C. (2018) Statistics of photon-subtracted and photon-added 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. 2070-2098. (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: 10.1088/1751-8121/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. 282-356. (doi: 10.1016/j.aim.2017.03.030)

Strachan, I. A.B., Zuo, D. (2017) Frobenius manifolds and Frobenius algebra-valued integrable systems. Letters in Mathematical Physics, 107, pp. 997-1026. (doi: 10.1007/s11005-017-0939-x)

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: 10.1088/1751-8121/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. 293-300. (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. 163-182. (doi: 10.1007/s00220-016-2657-0)

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: 10.1088/2040-8978/18/6/064004)

Athorne, C. (2016) Fundamental principles of classical mechanics: a geometrical perspective. Contemporary Physics, 57, pp. 238-241. (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: 10.1088/1751-8113/49/13/135201)

Korff, C. (2016) From quantum Bäcklund transforms to topological quantum field theory. Journal of Physics A: Mathematical and Theoretical, 49, (doi: 10.1088/1751-8113/49/10/104001)

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. 1159-1176. (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: 10.1088/2040-8978/17/12/125610)

Strachan, I. A.B., Zuo, D. (2015) Integrability of the Frobenius algebra-valued Kadomtsev-Petviashvili 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: 10.1088/1751-8113/48/29/295201)

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. Wiley-VCH, 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.

Quantum spin-chains and exactly solvable lattice models (PhD)

Supervisors: Christian Korff

Relevant research groups: Algebra, Integrable Systems and Mathematical Physics

Quantum spin-chains and 2-dimensional statistical lattice models, such as the Heisenberg spin-chain and the six and eight-vertex 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 Temperley-Lieb algebra, the Virasoro algebra and Kac-Moody 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 Gromov-Witten invariants (enumerative geometry) and fusion coefficients in conformal field theory (mathematical physics).

Integrable quantum field theory and Y-systems (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 so-called Y-systems which appear in cluster algebras introduced by Fomin and Zelevinsky and the proof of dilogarithm identities in number theory.

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 quasi-invariant polynomials, study of differential operators on quasi-invariants related to non-Coxeter arrangements. Relations with quantum integrable systems of Calogero-Moser type may be explored as well. Some other possible topics may include study of quasi-invariants for non-Coxeter arrangements in relation to theory of free arrangements of hyperplanes.

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 Donaldson-Thomas invariants in terms of deformation of hyperKahler geometry and quantum-Riemann-Hilbert problems.

Almost-duality 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 almost-dual Frobenius manifold structure. The class of examples include Coxeter and Extended-Affine-Weyl 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.

Ultra-discrete Integrable Systems (PhD)

Supervisor: Claire Gilson

Relevant research groups: Integrable Systems and Mathematical Physics

An ultra-discrete 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 ultra-discretization, this is a limiting process which takes you from a continuous integrable system to an ultra-discrete system via a discrete system.  This project investigates the world of ultra-discrete 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.

Non-commutative 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 non-commutative integrable systems that are known,  these have solutions in terms of quasi-determinants (Gelfand and Retakh, vol25, p91, Functional Analysis and its Applications).  This project aims to investigate the properties of known and new non-commutative systems.

Seminars

Regular seminars relevant to the group are held as part of the Applied Mathematics seminar series. 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 cross-roads 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.