I am a tutor in mathematics in Christ Church College and a member of the Geometry group in the Mathematical Institute in Oxford, working with Frances Kirwan . Before this I was an EPSRC Research Fellow in Oxford and a William Seggie Brown Research Fellow at Edinburgh University. I was a PhD student of András Szenes at Budapest University of Technology .Here is my CV .
College teaching: I am a tutor in Christ Church College, on sabbatical leave in Michaelmas 2015 and Hilary 2016.
I was lecturing and organising Part C Algebraic Geometry course in Oxford in Michaelmas 2010,2012,2013 and 2014. Here are my lecture notes.
I gave a graduate lecture course on Singularity Theory in Trinity Term, 2010. The lecture notes are available
here.
MSc students: Matthew Carr 2015, Temitope Ajileye 2015
DPhil student (joint with Frances Kirwan): Joshua Jackson 2014-
Algebraic geometry and topology, group actions, invariant theory, singularity theory of maps, hyperbolic varieties. In a joint project with Frances Kirwan and Tom Hawes in Oxford I am working on non-reductive group actions in algebraic geometry. We construct quotient spaces through generalising Mumford's reductive geometric invariant theory for non-reductive group actions and study the topology of these non-reductive moduli spaces with applications.
Joint with L. Fehér, and R. Rimányi.
My interest in singularities and Thom polynomials goes back to my undergraduate years and this work. Using the restriction equations of Rimányi we compute the Thom-polynomials of A_3 singularities in any codimension. Following a long and entertaining computation we arrived to the conjecture of this paper, with a long messy proof never published. Proper proof was given later by Pragacz.
Thom polynomials of Morin singularities
Joint with András Szenes
This paper is based on my PhD thesis .We develop a new iterated residue formula for the A_d (or Morin) singularities using equivariant localization on the set of k-jets of germs (C^n,0)->(C^k,0) with one-dimensional local algebra. This set in turn fibres over the non-reductive quotient of k-jets of map germs (C,0)->(C^n,0) by the group of polynomial reparametrisations of (C,0) and the Thom polynomial can be interpreted as the integral of the equivariant duals of the fibres over this non-reductive quotient.
Towards the Green-Griffiths-Lang conjecture via equivariant localisation
The Green-Griffiths-Lang conjecture says that for every complex projective algebraic variety X of general type there exists a proper algebraic subvariety of X containing all nonconstant entire holomorphic curves f:C->X. Using equivariant localisation on the Demailly-Semple jet differentials bundle we give an affirmative answer to this conjecture for generic projective hypersurfaces X in P^{n+1} of degree deg(X) > n^{9n}.
We construct a compactification of the invariant jet differentials bundle over complex manifolds motivated by the test curve model of Morin singularities and we develop an iterated residue formula using equivariant localisation for tautological integrals over it. We show that the polynomial GGL conjecture for a generic projective hypersurface of degree deg(X)>2n^{10} follows from a positivity conjecture for Thom polynomials of Morin singularities.
The Demailly invariant jet differentials are differential operators acting on germs of holomorphic curves f:C -> X which are invariant under the reparametrization of C. They play a crucial role in the strategy worked out by Demailly/Siu to prove the Green-Griffiths conjecture, and a major unsolved problem is the description of these invariants. This paper proves the finite generation of a certain subalgebra and gives a geometric description of the generators. Higher Demailly invariants are also studied, and a similar geometric description is given.
Let G be a reductive group over an algebraically closed field k, H an observable subgroup normalised by a maximal torus of G and X an affine k-variety acted on by G. The Popov-Pommerening conjecture from 1985 says that the invariant algebra k[X]^H is finitely generated. We prove the conjecture for block regular subgroups of linear algebraic groups and give a geometric description of a generating set of the invariant algebra.We give a partial geometric affirmative answer to the conjecture for general regular subgroups.
Let X be a projective variety acted on by a unipotent group U.Assume that X has a bit of extra symmetry in that the U-action extends to a C^* extension \hat{U} of U. We prove that if we are ready to shift the linearisation of the \hat{U} action with a character of \hat{U} then we can guarantee finite generation of the invariant algebra O(X)^\hat{U} and there is a (semi)stable locus X^ss=X^s where the map to Proj of the invariants is a geometric quotient.Moreover, X^ss is given by Hilbert-Mumford type criteria.
Let U be a unipotent group which is graded in the sense that it has an extension H by the multiplicative group of the complex numbers such that all the weights of the adjoint action on the Lie algebra of U are strictly positive. We study embeddings of H in a general linear group G which possess Grosshans-like properties. More precisely, suppose H acts on a projective variety X and its action extends to an action of G which is linear with respect to an ample line bundle on X. Then, provided that we are willing to twist the linearisation of the action of H by a suitable (rational) character of H, we find that the H-invariants form a finitely generated algebra and hence define a projective variety X//H; moreover the natural morphism from the semistable locus in X to X//H is surjective, and semistable points in X are identified in X//H if and only if the closures of their H-orbits meet in the semistable locus. A similar result applies when we replace X by its product with the projective line; this gives us a projective completion of a geometric quotient of a U-invariant open subset of X by the action of the unipotent group U.
This is a survey paper based on my IMPANGA lectures given in Warsaw, Poland, 2011.
We take a new look at the curvilinear Hilbert scheme of points on a smooth projective variety X as a projective completion of the non-reductive quotient of holomorphic map germs from the complex line into X by polynomial reparametrisations. Using an algebraic model of this quotient coming from global singularity theory we develop an iterated residue formula for tautological integrals over curvilinear Hilbert schemes.
We generalise the method of the previous paper to compute tautological intersection numbers on general monomial geometric subsets of Hilbert schemes of points on snonsingular varieties. As an application we develop iterated residue formulae for counts of curves with given singularities on surfaces and more generally counts of hypersurfaces with given set of singularities on smooth projective varieties.In particular, we derive a new formula for counts of nodal curves in sufficiently ample linear systems.
This is my PhD Thesis
Slides of some recent talks:
Non-reductive GIT and curve counting Oxford Algebraic Geometry Seminar and Global Singularity Theory and Curves Workshop in Lausanne May 2016
Non-reductive GIT, curve counting and hyperbolicity Kahler Geometry Workshop, Institute Fourier Grenoble, May 2016
Reductive vs non-reductive group actions in algebraic geometry , Non-reductive GIT and applications 2 talks given at QGM Aarhus, April 2016
GIT for graded unipotent groups and applications Oxford Algebraic Geometry Seminar January 2016
Nonreductive GIT and tautologial integrals on curvilinear Hilbert
schemes , NS@50 Chennai, October 2015 and EPFL Lausanne September 2015
Tautologial integrals on curvilinear Hilbert
schemes , New techniques in GIT, Berlin, September 2015
Moduli of jets of holomorphic curves , CIRM Luminy, March 2012
IMPANGA Lecture 1 , Lecture 2, Lecture 3 Warsaw, January 2011
Thom polynomials and the Green-Griffiths conjecture , Strasbourg, June 2010
Non-reductive GIT and invariant jet differentials Talk1, Talk2 Rencontre Hyperbolicite, Paris, November 2010
Thom polynomials of singularities via equivariant localization , Sao Carlos, Brazil, July 2010
Thom polynomials of singularities and the Green-Griffiths conjecture , Oxford, March 2010
From singularities of maps to non-reductive quotients , Oxford, February 2009
Hungarian water polo -- Gods of water
Hungary won 9 olympic gold medals, three consecutive titles in Sydney, Athen, Beijing.
Hungarian Water Polo Association