Algebraic geometry, mirror symmetry, arithmetic of quadratic forms, hyperbolic reflection groups, hyperbolic Kac–Moody algebras.

I was born 1950 in Kirov, Russia. 1965–1967, I was a student of Physical-Mathematical school No. 18 under MGU. 1967–1972, I was a student of Mechanical-Mathematical Department of MGU. 1972–1975, I was a graduate student of MIAN of Steklov under supervision by I.R. Shafarevich.

In 1977, I defended PhD thesis "Finite automorphism groups of Kahlerian K3 surfaces" (was published in 1979 in Proceed. of Moscow Math. Soc.). A general theory of such groups was constructed (especially of finite symplectic automorphism groups), and the classification of Abelian finite symplectic automorphism groups of K3 was given.

In 1979, in the paper "Integral symmetric bilinear forms and some of their geometric applications", I developed the discriminant forms technique for integral symmetric bilinear forms which became very useful in applicaitons. As geometric applications, 1) another approach to finite symplectic automorphism groups of K3 was given; 2) calculation of Milnor quadratic forms of 2-dimensional quasi-homogeneous singularities of functions was given in terms of their resolution, in application to 14 exceptional Arnolds singularities it gave an approach to Arnolds duality for them which was a first example of mirror symmetry; 3) a description of connected components of moduli of real polarized K3 surfaces was given. It is my the most cited paper (more than 100 citations by AMS Math Review).

In papers 1979–1984, I described K3 surfaces with finite automoprhism groups which is equivalent (by Global Torelli Theorem) to description of hyperbolic integral symmetric bilinear forms with automorphism groups generated by 2-reflections up to finite index. It was proved, in particular, that their number is finite in essential. For the rank 4 it was done by Vinberg.

In papers 1980–1981, I generalized above results to arbitrary arithmetic hyperbolic (in Lobachevsky spaces) reflection groups. Finiteness of the number of maximal such groups was proved in dimensions at least 10. Using the developed by me methods, later E.B. Vinberg, M.N. Prokhorov and A.G. Khovansky proved that the dimension of reflection groups in Lobachevsky spaces, with fundamental chambers of finite volume, is absolutely bounded.

In my further papers, the above methods were generalized and applied to different algebraic varieties and related polyhedra (e.g. to Mori polyhedra and nef polyhedra), to hyperbolic Kac–Moody algebras, different types of real algebraic varieties. I shall give more concrete results below.

In 1983–2008 papers, to different types of real algebraic varieties: to curves, surfaces, to description of connected components of moduli of K3 surfaces with different conditions on Picard lattices.

In 1984–2004 papers, to K3 and Enriques surfaces (automorphism groups), to Del Pezzo surfaces with log-terminal singularities, to algebraic surfaces with nef anti-canonical class, to algebraic surfaces with finite polyhedral Mori cone (finiteness results), to 3-dimensional Fano and Calabi–Yau manifolds (bounds on Picard number). Later, to some of these results, some other approaches were found which use Mori theory.

In 1995–2002 papers (most of them together with V.A. Gritsenko), to description and construction of Lorentzian (hyperbolic, generalized) Kac–Moody algebras with denominator identities which are automorphic forms (Borcherds algebras).

In 2003–2011 papers (many of them together with Carlo Madonna) methods of integral symmetric bilinear forms and discriminant forms were applied to description of cases when the moduli space of coherent sheaves with given Mukai vector on a K3 surface is isomorphic to the K3 surface itself which gives an algebraic cycle on the product of K3 with itself (or its self-correspondence).

In 2007–2011 papers, using results by Long, Maclachlan, Reid for 2-dimensional case and Agol for 3-dimensional case, and my old finiteness results and methods of 1981–1982 for arithmetic hyperbolic reflection groups in dimensions greater than 9, these finiteness results were generalized to remaining dimensions 2–8. Moreover, good bounds on ground fields were obtained which gives a hope for complete enumeration of maximal arithmetic hyperbolic refection groups.

In my last 2013 paper, the suggested by me method of 1979 to description of finite symplectic automorphism groups of K3 surfaces was generalized and specialized to concrete Niemeier lattices. This gives, in particular, description of finite symplectic automorphism groups of Kahlerian K3 surfaces together with their non-singular rational curves. This completes the classical results by Mukai, Xiao, Kondo, Hashimoto which were obtained after my 1979 paper.

1. V. Alexeev, V. V. Nikulin, Del Pezzo and $K3$ surfaces, MSJ Memoirs, 15, Mathematical Society of Japan, Tokyo, 2006 , xvi+149 pp.    
2. V. A. Gritsenko, V. V. Nikulin, “Automorphic forms and Lorentzian Kac-Moody algebras. II”, Internat. J. Math., 9:2 (1998), 201–275          
3. V. A. Gritsenko, V. V. Nikulin, “Automorphic forms and Lorentzian Kac-Moody algebras. I”, Internat. J. Math., 9:2 (1998), 153–199          
4. V. V. Nikulin, “On the classification of arithmetic groups generated by reflections in Lobachevsky spaces”, Math. USSR-Izv., 18:1 (1982), 99–123            
5. V. V. Nikulin, “Konechnye gruppy avtomorfizmov kelerovykh poverkhnostei tipa $K_3$”, Tr. MMO, 38, Izd-vo Mosk. un-ta, M., 1979, 75–137      
6. V. V. Nikulin, “Integral symmetric bilinear forms and some of their applications”, Math. USSR-Izv., 14:1 (1980), 103–167            

1. V. V. Nikulin, On the classification of hyperbolic root systems of rank three, Trudy Mat. Inst. Steklova, 230, ed. I. R. Shafarevich, E. F. Mishchenko, 2000, 256 с.
   http://mi.mathnet.ru/book243

• Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
• Division of Pure Mathematics of the Department of Mathematical Science of the University of Liverpool
• Moscow Mathematical Society