I am member of the Geometry group 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 .
I gave a graduate lecture course on Singularity Theory in Trinity Term, 2010. The lecture notes are available
here.
I gave Part C 3.1a Algebraic Geometry lectures in Michaelmas 2010 and 2012, here are the lecture notes.
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
When I started my PhD studies under the supervision of András Szenes, I though I would never work again on singularities. But we returned to singularities and developed a new formula for the A_d ( or Morin) singularities using localization techniques on the moduli of k-jets of germs (C,0)->(C^n,0).
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 this algebra and gives a geometric description of the generators as follows. The set of invariant jets fibers over X with fiber over x isomorphic to the invariant jets of germs f:(C,0) -> (X,x) under the k-jets of reparametrizations of (C,0). Using the construction of in our paper with Szenes, we can embed the quotient J_k(1,n)/J_k(1,1) into a Grassmannian of the sum of symmetric powers of the standard representaion of GL(n). The invariant jets then can be identified with the global sections of the tautological line bundle over the Grassmannian, due to the main result of this paper saying that the boundary components of the image in the Grassmannian have codimension at least two. Higher Demailly invariants are also studied, and a similar geometric description is given.
This is the second main application of our quotient construction. The Green-Griffiths conjecture form 1979 says that any projective hypersurface X of high degree has a proper subvariety Y, such that any entire holomorphic curve f:c \to X sits in Y. Following the strategy of Demailly/Siu/Diverio/Merker/Rousseau we prove the positivity of an intersection number on the space of map-germs from C to X modulo the reparametrizations. This is a non-reductive quotient, and in the first half of the paper we use localization on the smooth compactification of the quotient constructed as a tower of fibrations by Demailly to prove the positivity for degree d>n^{7n}. In the second part we use our new description of this quotient to prove the Green-Griffiths conjecture for d>n^6, modulo a conjecture on Thom polynomials of Morin singularities.
This paper summarizes our recent work on a new construction of quotients of algebraic varieties by non-reductive group actions. Let W be a representation of GL(n), typically a direct sum of symmetric powers of the standard representation. Partial flag manifolds of W have an induced GL(n) action, and a generic point has a non-reductive stabiliser. Surprisingly, the reverse is also true, namely, any non-reductive unipotent, graded subgroup of GL(n) is the stabiliser of some point in some partial flag manifold.
This is a survey paper based on my IMPANGA lectures held in Warsaw, Poland, 2011.
This is my PhD thesis.
Here are the slides of some recent talks:
From singularities of maps to non-reductive quotients , Oxford, February 2009
Thom polynomials of singularities and the Green-Griffiths conjecture , Oxford, March 2010
Thom polynomials of singularities via equivariant localization , Sao Carlos, Brazil, July 2010
Thom polynomials and the Green-Griffiths conjecture , Strasbourg, June 2010
Two talks given at Rencontre Hyperbolicite, Paris, November 2010 Talk1, Talk2
Invariants for non-reductive group actions , Warwick, October 2011
Mirror symmetry, Langlands duality and the Hitchin system, lecture course by Tamas Hausel in Oxford. These notes were taken by Geordie Williamson , Michael Groechenig and myself.
Hungarian water polo -- Gods of water
Hungary won 9 olympic gold medals, three consecutive titles in Sydney, Athen, Beijing.
Hungarian Water Polo Association