Scientific career
One of the goals of Hodge theory is to measure and describe what happens if you introduce extra ``parameters'' in the equations. In our example, one might want to vary the radius of the circle. A related question is to describe the subsets of a given variety which themselves are given by supplementary polynomial equations; these are called ``subvarieties, or more generally ``algebraic cycles''. The theory has been named after Sir William Hodge (19031975) who boosted research in this area after he formulated a conjecture. In fact it is one of the millenium prize questions posed by the Clay Foundation.
Hodge theory has links to other fields. It has for instance been discovered that Hodge theory is a natural tool to attack the so called ``mirror conjecture'', a conjecture posed by mathematical physicists. See for instance this book.
 The local Torelli theorem I, Complete intersections, Math. Ann 217 (1974), 116.
 The local Torelli theorem II, Cyclic branched coverings, Ann. Sc. Norm. Pisa, Cl. Sc. Ser IV, II (1976) 321339.
 (with J. SIMONIS) A secant formula, Quart. J. math. 27 (1976) 182189.
 On two types of surfaces with vanishing geometric genus, Inv. Math. 32 (1976) 3347.
 On certain examples of surfaces with p_{g} =0 due to Burniat, Nagoya Math. J., 66 (1977) 109119.
 (with D. LIEBERMAN, R. WILSKER) A theorem of local Torellitype, Math. Ann. 231 (1977) 3945.
 Holomorphic automorphisms of compact Kähler surfaces and their induced actions in cohomology, Inv. Math. 52 (1979) 143148.
 On automorphisms of compact Kähler surfaces, in Journées de géométrie algébrique d'Angers (juillet 1979) SijthoffNoordhoff (1980) 249267.
 (with E. LOOIJENGA) Torelli theorems of Kähler K3surfaces, Comp. Math. 42 (1981) 145186.
 (with F. OORT) A Campedelli surface with torsion group Z/ 2Z , Indag. Math. 3 (1981) 399407.
 (with W. BARTH) Automorphisms of Enriques surfaces, Inv. Math. 73 (1983) 383411.
 (with F. BEUKERS) A family of K3surfaces and ζ(3), J. für reine u. angew. Math. 351 (1984) 4254.
 A criterion for flatness of Hodge bundles over curves and geometric applications, Math. Ann. 268 (1984) 119.
 Some applications of the Lefschetz fixed point theorems in (complex) algebraic geometry, in The Lefschetz Centennial Conference, I: Proc. Alg. geometry, Cont. Math. 58 (part 1) (1985) AMS, Providence, R.I., p.213221.
 Monodromy and PicardFuchs equations for families of K3surfaces and elliptic curves, Ann. ENS. 19 (1986) 583607.
 On Arakelov's finiteness theorem for higher dimensional varieties. Rend. Sem. Mat. Univ. Politec. Torino, 1986.
 Some remarks on Reider's article "On the infinitesimal Torelli theorem for certain irregular surfaces", Math. Ann. 281, 315324 (1988).
 (With J. STIENSTRA) A pencil of K3surfaces related to Apéry's recurrence for ζ(3) and Fermi surfaces for potential zero. Arithmetic of Complex Manifolds, Proc. Erlangen 1988, Springer Lect. Notes Math. 1399, pp. 110127, 1989.
 Rigidity for variations of Hodge structure and Arakelovtype finiteness theorems, Comp. Math. 75, 113126 (1990).
 Maximal rank nonrigid variations of Hodge structure of weight one and two, in Complex algebraic varieties. Proc. Bayreuth 1990, Springer Lecture Notes 1507, 157162, Berlin etc., Springer Verlag (1992).
 (with J. TOP, M. van der VLUGT) The Hasse zeta function of a K3 surface related to the number of words of weight 5 in the Melas codes. J. reine angew. Math. 432 (1992) 151176.
 (With U. PERSSON) Homeomorphic nondiffeomorphic surfaces with small invariants. Manuscripta Math. 79 (1993) 173182.
 (With U. PERSSON) Some aspects of the topology of algebraic surfaces. In: Israel Mathematical Conference Proceedings (IMCP) AMS (1995).
 (With U. PERSSON, X, GIAO), Geography of Spin surfaces, Topology 35 (1996) 845862. (publised version)
 Lawson homology for varieties with small Chow groups and the induced filtration of the Griffiths group, Math. Zeitschrift 234 (2000) 209223 (pdf)
 With J. STEENBRINK) Degeneration of the Leray spectral sequence for certain geometric quotients math. AG/0112093 Moscou Math. Journal 3 (2003) 111.
 (With J. STEENBRINK) Monodromy of variations of Hodge structure, Acta Applicanda Math. 75 (2003) 183194.
 (With J. STEENBRINK) Hodge Number Polynomials for Nearby and Vanishing Cohomology, pp. 597611 (2006) in Algebraic Cycles and Motives. Also as preprint available.
 (With J. STEENBRINK) Hodge Theory: the search for purity, in Proceedings of the Trieste Singularity Summer School and Workshop ICTP, Trieste, Italy 15 August  3 September 2005, edited by JeanPaul Brasselet, World Scientific, Singapore etc., 556583 (2007).
 (With F. GALLUZZI et G. LOMBARDO) Automorphs of indefinite binary quadratic forms and K3surfaces with Picard number 2. Rendiconti Mat. Torino, 68 (2010) 5777, preprint ArXiv mathAG/0804.0725
 Blochtype conjectures and an example of a threefold of general type. Contemporary Mathematics 12 (2010) 587605.
 (With Morihiko SAITO) Lowest Weights in Cohomology of Variations of Hodge Structure. Nagoya Math. J. 206 (2012), 124. Here for the complete version and here arxiv:0708.0130v2 for a first version.
 (with V. SRINIVAS and S. MÜLLERSTACH) Abelian varieties and theta functions as invariants for compact Riemannian manifolds; constructions inspired by superstring theory. Journal de mathématiques pures et appliquées 97 (2012) (pdf); see also the video presentation on the occasion of Alberto Contes 70th birthday.
Teaching: some course notesPhDcourses
Third year (all in french)
Magistère:
