Centre for Geometry and Applications

A digital image of a shape created by Artie Prendergast-Smith using surfer.

Geometry lies at the heart of modern science. It unifies diverse branches of mathematics such as algebra, analysis and number theory, and connects to applications in theoretical physics, computer science, statistics and more.

Loughborough University has a strong tradition of research in geometry and related areas. The Centre for Geometry and Applications builds on this through an extensive programme of activities including international conferences, world-leading academic visitors and new links with industrial partners utilising applications of geometry.

Research interests

  • Higher dimensional birational geometry
  • Positivity
  • Fano varieties
  • K3 surfaces and Calabi-Yau threefolds
  • Toric geometry
  • Secant varieties
  • Projective geometry
  • Lie sphere geometry
  • Web geometry
  • Generalised conformal geometry
  • Submanifold geometry
  • Special holonomy
  • Hamiltonian and non-holonomic mechanics
  • Lagrangian fibrations
  • Soliton equations
  • Symmetry methods
  • Dispersionless integrability
  • Hamiltonian formalism
  • Algebraic reduction method
  • Frobenius manifolds
  • Discrete and quantum integrable systems
  • Special functions
  • Algebro-geometric approach to integrability

Associate members

Former members

  • Dr Hamid Abban
  • Professor Sara Lombardo
  • Dr Elisa Postinghel

Current postdocs

Completed

  • Dr Matteo Casati
  • Dr Fabien Clery
  • Dr Drew Duffield
  • Dr Maxime Fairon
  • Dr Enrico Fatighenti
  • Dr Liana Heuberger

Current research students

Completed

  • Tim Grange (2022): Birational geometry and positivity of blow-ups of products of projective spaces (supervisors E. Postinghel and A. Prendergast-Smith)
  • Tiago Guerreiro (2022): Explicit birational geometry of Fano 3-folds of high index (supervisor H. Abban)
  • Ronan Kerr (2022): On spherical type singularities in integrable systems (supervisors A. Bolsinov and A. Veselov)
  • Laura Mallinson (2022): Classification of 3-fold flips in codimension 2 (supervisor H. Abban)
  • Luis Jose Santana-Sanchez (2022): On blow-ups of projective spaces at points on a rational normal curve (supervisor E. Postinghel)
  • Guangxiu Zhao (2021): Integrability of two-component systems of partial differential equations (supervisor V. Novikov)
  • Jinrong Bao (2020): Some generalised models in rigid body dynamics (supervisors A. Bolsinov and A. Neishtadt)
  • Erik Paemurru (2020): Birational models of terminal sextic double solids (supervisor H. Abban)
  • Yiru Ye (2020): Classical and Quantum Integrability in Dimensions 2 and 3 (supervisor A. Bolsinov and A. Veselov)
  • Norbert Pintye (2019): Positivity of Cycle Classes on Linear Blowing-Ups (supervisor A. Prendergast-Smith)
  • Katie Spalding (2018): Growth and integrability in multivalued dynamics (supervisor A. Veselov)
  • Jonathan Moss (2016): Linear degeneracy in multidimensions (supervisor E. Ferapontov)
  • Andrea Savoldi (2016): On Poisson structures of hydrodynamic type and their deformations (supervisors E. Ferapontov and  P. Lorenzoni)
  • William Haese-Hill  (2015): Spectral properties of integrable Schroedinger operators with singular potentials (supervisors M. Hallnas and A. Veselov)
  • Ilia Roustemoglou (2015): Classification of integrable differential/difference equations in 2+1 dimensions (supervisors E. Ferapontov and V. Novikov)
  • Veronika Schreiber (2014): Special vector configurations in geometry and integrable systems (supervisor A. Veselov)
  • Dragomir Tsonev (2013): Realisation of holonomy algebras on pseudo-Riemannian manifolds by means of Manakov operators (supervisor A. Bolsinov)
  • Graham Kemp (2013): Algebra and geometry of Dirac magnetic monopole (supervisor A. Veselov)
  • Adrian Hemery (2012): The spectral properties and singularities of monodromy-free Schrodinger operators  (supervisor A Veselov);
  • Nikola Stoilov (2012): Hamiltonian systems of hydrodynamic type in 2+1 dimensions (supervisor E. Ferapontov)
  • Pumei Zhang (2012): Algebraic aspects of compatible Poisson structures (supervisor  A. Bolsinov)
  • Anton Izosimov (2012): Singularities of bihamiltonian systems and the multidimensional rigid body (supervisor A. Bolsinov)
  • Pavel Burovskiy (2009): On the integrability of three-dimensional second order quasilinear PDEs (supervisors E. Ferapontov and A. Veselov)
  • Lenos Hadjikos (2009): Integrable equations of the dispersionless Hirota type (supervisors E. Ferapontov and K. Khusnutdinova)
  • David Marshall (2008): Differential-geometric approach to the integrability of hydrodynamic chains (supervisors E. Ferapontov and K. Khusnutdinova)

Selected recent publications

2022

  • Bolsinov, Alexey V.; Konyaev, Andrey Yu.; Matveev, Vladimir S. Nijenhuis geometry. Adv. Math. 394 (2022), Paper No. 108001, 52 pp.
  • Berjawi, S.; Ferapontov, E. V.; Kruglikov, B. S.; Novikov, V. S. Second-order PDEs in 3D with Einstein-Weyl conformal structure. Ann. Henri Poincaré 23 (2022), no. 7, 2579–2609.
  • Ferapontov, E. V.; Pavlov, M. V. Kinetic equation for soliton gas: integrable reductions. J. Nonlinear Sci. 32 (2022), no. 2, Paper No. 26, 22 pp.
  • Gormley, B.; Ferapontov, E. V.; Novikov, V. S.; Pavlov, M. V. Integrable systems of the intermediate long wave type in 2+1 dimensions. Phys. D 435 (2022), Paper No. 133310, 9 pp.
  • Mikhailov, Alexander V.; Novikov, Vladimir S.; Wang, Jing Ping Perturbative symmetry approach for differential-difference equations. Comm. Math. Phys. 393 (2022), no. 2, 1063–1104.
  • Buchstaber, V. M.; Veselov, A. P.; Gaifullin, A. A. Classification of involutive commutative two-valued groups. (Russian) Uspekhi Mat. Nauk 77 (2022), no. 4(466), 91–172.

2021

  • H. Abban,. M. Fedorchuk, I. Krylov, Stability of fibrations over one-dimensional bases, Duke Math Journal (to appear), 34 pages.
  • A. Bolsinov, A. Veselov, Y. Ye, Chaos and integrability in SL(2,ℝ)-geometry. (Russian) Uspekhi Mat. Nauk 76 (2021), no. 4(460), 3–36.
  • A. Bolsinov, A. Konyaev, V. Matveev, Applications of Nijenhuis geometry II: maximal pencils of multi-Hamiltonian structures of hydrodynamic type.Nonlinearity 34 (2021), no. 8, 5136–5162.
  • B. Gormley, E. Ferapontov, V. Novikov, On a class of integrable Hamiltonian equations in 2+1 dimensions. Proc. A. 477 (2021), no. 2249, Paper No. 20210047, 17 pp.
  • E. Ferapontov, M. Pavlov, L. Xue, Second-order integrable Lagrangians and WDVV equations. Lett. Math. Phys. 111 (2021), no. 2, Paper No. 58, 33 pp.
  • E. Ferapontov, B. Kruglikov, Dispersionless integrable hierarchies and GL(2,ℝ) geometry. Math. Proc. Cambridge Philos. Soc. 170 (2021), no. 1, 129–154.
  • N. Pintye, A. Prendergast-Smith, Effective cycles on some linear blowups of projective spaces. Nagoya Math. J. 243 (2021), 243–262.
  • V. Alexeev, A. Thompson, ADE surfaces and their moduli. J. Algebraic Geom. 30 (2021), no. 2, 331–405.
  • M. Feigin, M.  Hallnäs, A. Veselov, Quasi-invariant Hermite polynomials and Lassalle-Nekrasov correspondence. Comm. Math. Phys. 386 (2021), no. 1,107–141.

2020

  • F. Cléry, E. Ferapontov, Dispersionless Hirota equations and the genus 3 hyperelliptic divisor. Comm. Math. Phys. 376 (2020), no. 2, 1397–1412.
  • S. Berjawi, E. Ferapontov, B. Kruglikov, V. Novikov, Second-order PDEs in four dimensions with half-flat conformal structure. Proc. A. 476 (2020), no. 2233, 20190642, 11 pp.
  • V. Knibbeler, S. Lombardo, J. Sanders, Hereditary automorphic Lie algebras. Commun. Contemp. Math. 22 (2020), no. 8, 1950076, 32 pp.
  • V. Knibbeler, S. Lombardo, J. Sanders, Automorphic Lie algebras and cohomology of root systems. J. Lie Theory 30 (2020), no. 1, 59–84.
  • A. Harder, A. Thompson, Pseudolattices, del Pezzo surfaces, and Lefschetz fibrations. Trans. Amer. Math. Soc. 373 (2020), no. 3, 2071–2104.
  • C. Doran, A. Harder, A. Novoseltsev, A. Thompson, Calabi-Yau threefolds fibred by high rank lattice polarized K3 surfaces. Math. Z. 294 (2020), no. 1-2,783–815.
  • A. Veselov, Y. Ye, New integrable two-centre problem on sphere in Dirac magnetic field. Lett. Math. Phys. 110 (2020), no. 11, 3105–3119.

2019

  • A. Bolsinov, A. Izosimov, Smooth invariants of focus-focus singularities and obstructions to product decomposition. J. Symplectic Geom. 17 (2019), no. 6, 1613–1648.
  • B. Doubrov, E. Ferapontov, B. Kruglikov, V. Novikov, Integrable systems in four dimensions associated with six-folds in Gr(4,6). Int. Math. Res. Not. IMRN 2019, no. 21, 6585–6613.
  • M. Hay, A. Hone, V. Novikov, J. Wang, Remarks on certain two-component systems with peakon solutions. J. Geom. Mech. 11 (2019), no. 4, 561–573.
  • O. Dumitrescu, E. Postinghel, Positivity of divisors on blown-up projective spaces, II. J. Algebra 529 (2019), 226–267.
  • I. Coskun, A. Prendergast-Smith, Fano manifolds of index n−2 and the cone conjecture. Math. Proc. Cambridge Philos. Soc. 166 (2019), no. 1, 1–31.
  • A. Sorrentino, A. Veselov, Markov numbers, Mather's beta-function and stable norm, Nonlinearity, 32: 6 (2019), 2147-2156. 
  • V. Buchstaber, A. Veselov, Conway topograph, PGL_2(Z)-dynamics and two-valued groups, Russian Math. Surveys, 74:3 (2019), 387-430.
  • E. Fatighenti, L. Rizzi, F. Zucconi, Weighted Fano varieties and infinitesimal Torelli problem, Journal of Geometry and Physics, 139 (2019), 1-16 

2018 

  • H. Ahmadinezhad, T. Okada, Stable rationality of higher dimensional conic bundles. Épijournal Geom. Algébrique 2 (2018), Art. 5, 13 pp.
  • H. Ahmadinezhad, I. Cheltsov, J. Schicho, On a conjecture of Tian. Mathematische Zeitschrift  288 (2018), no. 1-2, 217–241.
  • H. Ahmadinezhad, T. Okada, Birationally rigid pfaffian Fano 3-folds, Algebraic Geometry , (2018) Volume 5, Issue 2, 160-199.
  • B Doubrov, E. Ferapontov, B. Kruglikov, V. Novikov, On integrability in Grassmann geometries: integrable systems associated with fourfolds in Gr(3, 5). Proc. Lond. Math. Soc. 3 (2018), 1-32.
  • M. Feigin, A. Veselov, ∨-systems, holonomy Lie algebras, and logarithmic vector fields. Int. Math. Res. Not. IMRN 2018, no. 7, 2070–2098.

2017

  • H. Ahmadinezhad, On pliability of del Pezzo fibrations and Cox rings. J. Reine Angew. Math. 723 (2017) 101-125.
  • A. Bolsinov, Some remarks about Mishchenko-Fomenko subalgebras. J. Algebra 483 (2017), 58-70.
  • V. Knibbeler, S. Lombardo, J.A. Sanders, Higher-dimensional automorphic Lie algebras. Found. Comput. Math. 17 (2017), no. 4, 987-1035.
  • A. Hone, V. Novikov, J.P. Wang, Two-component generalizations of the Camassa-Holm equation. Nonlinearity 30 (2017), no. 2, 622-658. 

  • O. Dumitrescu, E. Postinghel, Vanishing theorems for linearly obstructed divisors. J. Algebra 477 (2017), 312-359.
  • A. Prendergast-Smith, A special Cayley octad, Trans. Amer. Math. Soc. (2017).
  • A. Matsuo, A. Veselov, Universal formula for the Hilbert series of minimal nilpotent orbits. Proc. Amer. Math. Soc. 145 (2017), no. 12, 5123-5130.
  • A. Sergeev, A. Veselov,Orbits and invariants of super Weyl groupoid. Int. Math. Res. Not. IMRN (2017), no. 20, 6149-6167.
  • A. Sergeev, A. Veselov, Symmetric Lie superalgebras and deformed quantum Calogero-Moser problems. Adv. Math. 
304 (2017), 728-768.

2016

  • H. Ahmadinezhad, F. Zucconi, Mori dream spaces and birational rigidity of Fano 3-folds. Adv. Math. 292 (2016) 410-445.
  • H. Ahmadinezhad, A.S. Kaloghiros, Non-rigid quartic 3-folds, (2016)  Compositio Mathematica, 152(5) (2016) 955-983. 
  • E. Ferapontov, M. Pavlov, R. Vitolo, Towards the classification of homogeneous third-order Hamiltonian operators. Int. Math. Res. Not. IMRN 2016, no. 22, 6829–6855.
  • L. Aguirre, G. Felder, A. Veselov, Gaudin subalgebras and wonderful models. Selecta Math. (N.S.) 22 (2016), no. 3, 1057–1071.

2015

  • A. Bolsinov, V. Matveev, T. Mettler, S. Rosemann, Four-dimensional Kahler metrics admitting c-projective vector fields. J. Math. Pures Appl. (9) 103 (2015), no. 3, 619-657.
  • A. Bolsinov, V. Matveev, Local normal forms for geodesically equivalent pseudo-Riemannian metrics. Trans. Amer. Math. Soc. 367 (2015), no. 9, 6719-6749.
  • A. Bolsinov, A. Borisov, I. Mamaev, Geometrisation of Chaplygin’s reducing multiplier theorem. Nonlinearity 28 (2015), no. 7, 2307-2318.
  • E. Ferapontov, V. Novikov, I. Roustemoglou, On integrability of discrete Hirota type equations in 3D. Int. Math. Res. Not. IMRN  (2015), no. 13, 4933-4974.
  • M. Brambilla, O. Dumitrescu, E. Postinghel, On a notion of speciality of linear systems in Pn. Trans. Amer. Math. Soc. 
367 (2015), no. 8, 5447-5473.
  • E. Postinghel, F. Sottile, N. Villamizar, Degenerations of real irrational toric varieties. J. Lond. Math. Soc. 92 (2015), no. 2, 223-241.
  • I. Coskun, A. Prendergast-Smith, Eckhard loci on hypersurfaces. Comm. Algebra 43 (2015), no. 8, 3083-3101.
  • A. Sergeev, A. Veselov, Jack-Laurent symmetric functions. Proc. Lond. Math. Soc. (3) 111 (2015), no. 1, 63-92.
  • K. Schobel, A. Veselov, Separation coordinates, moduli spaces and Stasheff polytopes. Comm. Math. Phys. 337 (2015), no. 3, 1255-1274.

2014

  • A. Bolsinov, D. Tsonev, On a new class of holonomy groups in pseudo-Riemannian geometry. J. Differential Geom. 97 (2014), no. 3, 377-394.
  • A. Bolsinov, A. Izosimov, Singularities of bi-Hamiltonian systems. Comm. Math. Phys. 331 (2014), no. 2, 507-543.
  • E. Ferapontov, B. Kruglikov, Dispersionless integrable systems in 3D and Einstein-Weyl geometry. J. Differential Geom. 97 (2014), no. 2, 215-254.
  • F. Baronio, M. Conforti, A. Degasperis, S. Lombardo, M. Onorato, S. Wabnitz, Vector rogue waves and baseband modulation instability in the defocusing regime. Phys. Rev. Lett. 113 (3) (2014), 034101.
  • I. Coskun, A. Prendergast-Smith, Fano manifolds of index n-1 and the cone conjecture. Int. Math. Res. Not. IMRN  (2014), no. 9, 2401-2439.

External

  • EPSRC EP/W026554/1: Dr Prendergast-Smith (PI), £80,635.
  • EPSRC EP/V005545/1: Dr Thompson (PI), £238,263.
  • Isaac Newton Institute for the workshop `Integrable systems and applications’, September 2022, £22,000.
  • Russian Science Foundation Grant: Dr Bolsinov, £250,000.

Past funding 

  • EPSRC EP/S004130/1: Dr Postinghel (PI), £203,354.
  • LMS scheme 4 grant for Dr Pavlov's visit in January 2020, £792.
  • EPSRC EP/N031369/1: Prof Ferapontov (PI), £280,335.
  • Heilbronn Institute focused research grant: Dr Ahmadinezhad £5,485.
  • IAS semester theme on Geometry: Dr Ahmadinezhad (PI) £25,000.
  • INdAM Marie Curie Fellow Dr Casati, 2016-2018: £9,000.
  • LMS Regional Meetings and Workshop: Prof Veselov £5,000.
  • LMS scheme 2 grant for Prof Vitolo: Prof Ferapontov £970.
  • Institute of Advanced Studies (IAS) grant for Prof Vitolo: Prof Ferapontov £750.
  • Royal Society mobility grant: Dr Salagean, £6,000.
  • GRAnD mini-CDT: Dr Abban (PI).
  • EPSRC EP/V048619/1: Dr Abban (PI), £202,211.
  • EPSRC EP/T015896/1: Dr Abban (PI), £234,599.
  • EPSRC EP/T001968/1: Dr Abban (CoI), £170,892.
  • EPSRC EP/V048546/1: Prof Lombardo (PI), £153,559.
  • EPSRC EP/V050451/1: Dr Novikov(Co-I), £77,538.

Seminars

Events coordinated by the Centre

Visitors

  • Prof Boris Kruglikov (Tromso), September 2022
  • Prof Don Zagier (Bonn), August 2022
  • Dr Ziquan Zhuang (MIT), July 2019
  • Prof Alexander Odesski (Brock), June 2019
  • Prof Wadim Zudilin (Nijmegen), April 2019
  • Prof Alberto de Sole (Rome), April 2019
  • Prof Alexander Kuznetsov (Moscow), March 2019
  • Prof Gerard van der Geer (Amsterdam), March 2019
  • Prof Boris Doubrov (Moscow), March 2019
  • Dr Maksym Fedorchuk (Boston), January 2019
  • Dr Igor Krylov (Seoul), January 2019
  • Prof Raffaele Vitolo (Lecce), January 2019
  • Prof Dmitri Markushevich (Lille), November 2018
  • Dr Christian Urech (Imperial College), June 2018