Integrable systems & mathematical physics
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)
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)
Szabo, G., Wu, J., Zacharias, J. (2019) Rokhlin dimension for actions of residually finite groups Ergodic Theory and Dynamical Systems, 39, pp. 2248-2304. (doi: 10.1017/etds.2017.113)
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)
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)
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)
Hawkins, A., Zacharias, J. (2017) Spectral metric spaces on extensions of C*-algebras Communications in Mathematical Physics, 350, pp. 475-506. (doi: 10.1007/s00220-016-2820-7)
Korff, C. (2017) Dimers, crystals and quantum Kostka numbers Seminaire Lotharingien de Combinatoire, 78B.40, pp. 12pp.
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)
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
Bhowmick, J., Voigt, C., Zacharias, J. (2015) Compact quantum metric spaces from quantum groups of rapid decay Journal of Noncommutative Geometry, 9, pp. 1175-1200. (doi: 10.4171/JNCG/220)
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
David, L., Strachan, I. A.B. (2014) Symmetries of F-manifolds with eventual identities and special families of connections Annali della Scuola Normale Superiore de Pisa: Classe di Scienze, 13, pp. 641-674. (doi: 10.2422/2036-2145.201205_010)
Strachan, I. B., Szablikowski, B. M. (2014) Novikov algebras and a classification of multicomponent Camassa-Holm equations Studies in Applied Mathematics, 133, pp. 84-117. (doi: 10.1111/sapm.12040)
Korff, C. (2014) Quantum cohomology via vicious and osculating walkers Letters in Mathematical Physics, 104, pp. 771-810. (doi: 10.1007/s11005-014-0685-2)
Strachan, I. A.B. (2012) Simple elliptic singularities: a note on their G-function Asian Journal of Mathematics, 16, pp. 409-426. (doi: 10.4310/AJM.2012.v16.n3.a3)
Athorne, C. (2012) On the equivariant algebraic Jacobian for curves of genus two Journal of Geometry and Physics, 62, pp. 724-730. (doi: 10.1016/j.geomphys.2011.12.016)
England, M., Athorne, C. (2012) Building Abelian functions with Baker-Hirota operators Symmetry, Integrability and Geometry: Methods and Applications, 8, pp. 1-36. (doi: 10.3842/SIGMA.2012.037)
Feigin, M. (2012) Generalized Calogero–Moser systems from rational Cherednik algebras Selecta Mathematica - New Series, 18, pp. 253-281. (doi: 10.1007/s00029-011-0074-y)
Feigin, M., Shramov, C. (2012) On unitary submodules in the polynomial representations of rational cherednik algebras International Mathematics Research Notices, 15, pp. 3375-3414. (doi: 10.1093/imrn/rnr140)
Athorne, C. (2011) A generalization of Baker's quadratic formulae for hyperelliptic ℘-functions Physics Letters A, 375, pp. 2689-2693. (doi: 10.1016/j.physleta.2011.05.056)
David, L., Strachan, I. A.B. (2011) Dubrovin's duality for F-manifolds with eventual identities Advances in Mathematics, 226, pp. 4031-4060. (doi: 10.1016/j.aim.2010.11.006)
Korff, C. (2011) The su(n) WZNW fusion ring as integrable model: a new algorithm to compute fusion coefficients RIMS Kokyuroku Bessatsu, B28, pp. 121-153.
Korff, C. (2010) Noncommutative Schur polynomials and the crystal limit of the Uq sl(2)-vertex model Journal of Physics A: Mathematical and Theoretical, 43, pp. 434021. (doi: 10.1088/1751-8113/43/43/434021)
Korff, C., Stroppel, C. (2010) The sl(n)k-WZNW fusion ring: a combinatorial construction and a realisation as quotient of quantum cohomology Advances in Mathematics, 225, pp. 200-268. (doi: 10.1016/j.aim.2010.02.021)
Skalski, A., Zacharias, J. (2010) Approximation properties and entropy estimates for crossed products by actions of amenable discrete quantum groups Journal of the London Mathematical Society, 82, pp. 184-202. (doi: 10.1112/jlms/jdq023)
Morrison, E.K., Strachan, I. (2010) Modular frobenius manifolds and their invariant flows International Mathematics Research Notices, 2011, pp. 3957-3982. (doi: 10.1093/imrn/rnq236)
Skalski, A., Zacharias, J. (2010) On approximation properties of Pimsner algebras and crossed products by Hilbert bimodules Rocky Mountain Journal of Mathematics, 40, pp. 609-625.
Skalski, A., Zacharias, J. (2010) Poisson transform for higher-rank graph algebras and its applications Journal of Operator Theory, 63, pp. 425-454.
Strachan, I.A.B. (2010) Weyl groups and elliptic solutions of the WDVV equations Advances in Mathematics, 224, pp. 1801-1838. (doi: 10.1016/j.aim.2010.01.013)
Feigin, M., Korff, C., Strachan, I. (2009) Workshop island 3: algebraic aspects of integrability. Introduction to an additional volume of selected papers arising from the conference on algebraic aspects of integrable systems, Island 3, Islay 2007 Glasgow Mathematical Journal, 51, pp. 1-3. (doi: 10.1017/S0017089508004734)
Gilson, C.R., Hamanaka, M., Nimmo, J.J.C. (2009) Bäcklund transformations for noncommutative anti-self-dual Yang-Mills equations Glasgow Mathematical Journal, 51, pp. 83-93. (doi: 10.1017/S0017089508004801)
Korff, C., Stroppel, C. (2009) A combinatorial description of the sl(n) fusion ring Oberwolfach Reports, 6, pp. 846-850. (doi: 10.4171/OWR/2009/15)
Athorne, C. (2009) Applications of Transvectants Springer-Verlag
Gilson, C. R., Hamanaka, M., Nimmo, J. J. C. (2009) Backlund transformations and the Atiyah-Ward ansatz for non-commutative anti-self-dual Yang-Mills equations Proceedings of the Royal Society of London Series A: Mathematical, Physical and Engineering Sciences, 465, pp. 2613-2632. (doi: 10.1098/rspa.2008.0515)
Winter, W., Zacharias, J. (2009) Completely positive maps of order zero Munster Journal of Mathematics, 2, pp. 311-324.
Strachan, I.A.B. (2009) Differential and Functional Identities for the Elliptic Trilogarithm Symmetry, Integrability and Geometry: Methods and Applications, 5, pp. 031. (doi: 10.3842/SIGMA.2009.031)
Gilson, C.R., Macfarlane, S.R. (2009) Dromion solutions of noncommutative Davey–Stewartson equations Journal of Physics A: Mathematical and Theoretical, 42, pp. 235202. (doi: 10.1088/1751-8113/42/23/235202)
Feigin, M. (2009) Trigonometric solutions of WDVV equations and generalized Calogero-Moser-Sutherland systems Symmetry, Integrability and Geometry: Methods and Applications, 5, pp. 088. (doi: 10.3842/SIGMA.2009.088)
Athorne, C. (2008) Identities for hyperelliptic ℘-functions of genus one, two and three in covariant form Journal of Physics A: Mathematical and Theoretical, 41, pp. 5202. (doi: 10.1088/1751-8113/41/41/415202)
Korff, C. (2008) PT symmetry of the non-Hermitian XX spin-chain: non-local bulk interaction from complex boundary fields Journal of Physics A: Mathematical and Theoretical, 41, pp. 295206. (doi: 10.1088/1751-8113/41/29/295206)
Korff, C. (2008) Turning the quantum group invariant XXZ spin-chain Hermitian: a conjecture on the invariant product Journal of Physics A: Mathematical and Theoretical, 41, pp. 194013. (doi: 10.1088/1751-8113/41/19/194013)
Ferguson, J., Strachan, I.A.B. (2008) Logarithmic deformations of the rational superpotential/Landau-Ginzburg construction of solutions of the WDVV equations Communications in Mathematical Physics, 280, pp. 1-25. (doi: 10.1007/s00220-008-0464-y)
Gilson, C.R., Nimmo, J.J.C., Sooman, C.M. (2008) On a direct approach to quasideterminant solutions of a noncommutative modified KP equation Journal of Physics A: Mathematical and Theoretical, 41, pp. 085202. (doi: 10.1088/1751-8113/41/8/085202)
Skalski, A., Zacharias, J. (2008) Entropy of shifts on higher-rank graph C*-algebras Houston Journal of Mathematics, 34, pp. 269-282.
Skalski, A., Zacharias, J. (2008) Noncommutative topological entropy of endomorphisms of Cuntz algebras Letters in Mathematical Physics, 86, pp. 115-134. (doi: 10.1007/s11005-008-0279-y)
Feigin, M., Veselov, A.P. (2008) On the geometry of V-systems American Mathematical Society
Skalski, A., Zacharias, J. (2008) Wold decomposition for representations of product systems of C* corrrespondences International Journal of Mathematics, 19, pp. 455-479. (doi: 10.1142/S0129167X08004765)
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
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
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
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
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
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.