Department of Algebra

 Staff
 History of the Department of Algebra
 Research Fields
 Main results
 Awards to members of the Department
 International activity

In 1946, I. R. Shafarevich joined the Department, and from 1960 until 1995 I. R. Shafarevich was the Head of the Department. Almost all the subsequent new members of the Department are followers of I. R. Shafarevich. Among these members are A. I. Lapin (1950 and from 1957 to 1969), A. I. Kostrikin (from 1956 to 2000; since 1977 he was also the Head of Department of High Algebra, Moscow State University), S. P. Demushkin (from 1959 to 1975), A. B. Zhizhchenko (from 1959 to 1965), Yu. I. Manin (since 1960), A. N. Tyurin (since 1963), V. A. Dem'yanenko (from 1967 to 1969), A. N. Parshin (from 1968 to the present time, since 1995 he is the Head of the Department), S. Yu. Arakelov (post graduate student from 1971 to 1974), V. V. Nikulin (since 1987), V. A. Kolyvagin (since 1988), V. A. Abrashkin (since 1996), and V. S. Kulikov (post graduate student from 1974 to 1977, and the member of the Department from 1997).

At different times, the staff of the Department included S. P. Novikov (from 1960 to 1975), M. M. Kapranov (from 1986 to 1990), and S. A. Stepanov (from 1987 to 2000).

The staff of the Department also includes F. A. Bogomolov (post graduate student from 1970 to 1973, and the member of the Department from 1973), A. I. Bondal (since 1994), D. O. Orlov (since 1996), A. T. Fomenko (since 1998), and V. L. Popov (since 2002).

Since 1990th, some members of the Department have positions at foreign research institutions but keep scientific relations with the Department. Among these members are Yu. I. Manin (Max Planck Institute für Mathematik, Bonn) and F. A. Bogomolov (Courant Institute for Mathematical Sciences, New York).

1. Algebraic number theory and Galois theory

• The construction of a general reciprocity law (I. R. Shafarevich, A. I. Lapin), the solution of the inverse problem of Galois theory for solvable groups (I. R. Shafarevich).
• Description of p-extension of local and global fields: I. R. Shafarevich; Kh. Kokh; E. S. Golod and I. R. Shafarevich (solution of the problem of p-class-field tower); S. P. Demushkin; V. A. Abrashkin.
• Theory of Euler Systems (V. A. Kolyvagin).

2. The theory of Lie groups and algebras

• Semi-simple subgroups of Lee groups, nilmanifolds (A. I. Maltsev).
• Theory of infinite dimensional representations of classical Lee groups (I. M. Gelfand and M. A. Naimark).
• Solution of restricted Burnside problem for arbitrary prime power (A. I. Kostrikin).
• Classification of simple Lie algebras of finite characteristic (A. I. Kostrikin and I. R. Shafarevich).
• Integral lattices and orthogonal decompositions (A. I. Kostrikin).
• Theory of Lorentzian Kac-Moody algebras (V. A. Gritsenko and V. V. Nikulin).

3. Algebraic geometry

• The geometry of algebraic varieties:
  I. R. Shafarevich (Bundle over elliptic curves);
  Yu. I. Manin (Gauss–Manin connection);
  A. N. Parshin and S. Yu. Arakelov (Theorems on finiteness of families of curves of algebraic varieties).
• Theory of vector bundles: the classification and Torelli theorems for vector bundles over algebraic curves; vector bundles over an infinite-dimensional projective space (A. N. Tyurin).
• Theory of algebraic K3 surfaces and varieties with the trivial canonical bundle:
  I. I. Pyatetskii-Shapiro and I. R. Shafarevich (Torelli theorem);
  A. N. Rudakov and I. R. Shafarevich (Surfaces of type K3 over fields of finite characteristic);
  V. V. Nikulin (Groups of automorphisms; topological classification);
  V. S. Kulikov (Surjectivity of the period mappings);
  F. A. Bogomolov (Classification of varieties with a numerically trivial canonical class).
• Three-dimensional quartics and the Luroth problem (V. A. Iskovskikh and Yu. I. Manin).
• Projective structures on Riemann surfaces (A. N. Tyurin).
• Theory of stable vector bundles on algebraic surfaces (F. A. Bogomolov).
• Topology of algebraic varieties: V. S. Kulikov (Proof of the Chisini conjecture); A. B. Zhizhchenko, V. V. Nikulin.
• Arithmetic groups in integral hyperbolic lattices (V. V. Nikulin).
• The smooth structure invariants of algebraic surfaces (V. Ya. Pidstrigach and A. N. Tyurin).
• The derived categories of coherent sheaves on algebraic varieties and equivalences between them (M. M. Kapranov, A. I. Bondal, D. O. Orlov).

4. The arithmetic of algebraic manifolds

• Diophantine equations of the third degree (B. N. Delone and D. K. Faddeev).
• The Arithmetic of elliptic curves and abelian varieties:
  I. R. Shafarevich (Theory of principal homogeneous spaces);
  A. I. Lapin (The rank unboundedness over function spaces);
  Yu. I. Manin (Boundedness for the p-torsion of elliptic curves);
  V. A. Demyanenko (Estimates of torsion of elliptic curves);
  A. N. Parshin (Invariant altitudes of abelian varieties);
  F. A. Bogomolov (l-adic Galois representations associated with abelian varieties; groups of points of finite order);
  V. A. Abrashkin (Proof of absence of smooth Abelian schemes over Z).
• Finiteness theorems of Diophantine geometry:
  Yu. I. Manin (Proof of the Mordell conjecture on rational points over function fields);
  A. N. Parshin (Method of ramified coverings);
  V. A. Kolyvagin (Finiteness of the Shafarevich group for modular curves).
• Arithmetics surfaces (Arakelov geometry).
• Arithmetics of rational and qubic surfaces (Yu. I. Manin, V. A. Iskovskikh).
• Theory of p-adic L-functions and modular forms (Yu. I. Manin).
• Theory of n-dimensional local fields with applications to class field theory and theory of algebraic groups (A. N. Parshin).

5. Algebraic and differential topology

• Theory of cohomological operations. Description of complex cobordisms. Classification of smooth simply connected varieties in the case when their dimension > 4. Proof of topological invariance of the Pontryagin characteristic classes. Smooth foliation theory. Foundation of Hermite K-theory. (S. P. Novikov).

6. Mathematical physics

• Solving the periodic problem for the KdV equation by methods of algebraic geometry (S. P. Novikov).
• Classification of instantons (V. G. Drinfel'd and Yu. I. Manin).
• Investigation of models of classical field theory: supergeometry, Yang–Mills fields, string theory (Yu. I. Manin; M. M. Kapranov).
• Mirror symmetry (Yu. I. Manin; D. O. Orlov).
• Instantons on non-commutative spaces and a non-commutative twistor transform (A. Kapustin; A. Kuznetsov; D. Orlov).

A large number of the invited speakers on International Mathematical Congresses consist of the members of the Department:

I. R. Shafarevich (Stokholm, 1962; Nice, 1970),

A. I. Kostrikin (Stokholm, 1962; Nice, 1970),

S. P. Novikov (Stokholm, 1962; Moscow, 1966; Nice, 1970),

Yu. I. Manin (Stokholm, 1962; Nice, 1970; Helsinki, 1978; Berkeley, 1986),

A. N. Parshin (Nice, 1970),

S. Yu. Arakelov (Vancouver, 1974),

F. A. Bogomolov (Helsinki, 1978),

V. V. Nikulin (Berkeley, 1986),

V. L. Popov (Berkeley, 1986),

V. A. Kolyvagin (Kioto, 1990),

A. I. Bondal (Pekin, 2002),

D. O. Orlov (Pekin, 2002).

The Department of Algebra maintains stable and many-sided relations with mathematicians of other countries. Among the visitors to the Department of algebra there were the following scientists:

E. Kaehler, J. Tate, S. Lang, L. Bers, D. Mumford, P. Deligne, J.-P. Serre, R. MacPherson, Ph. Griffiths, D. Gieseker, M. Harris, J.-L. Verdier, M. Reid, H. Esnault, E. Vieweg, W. Baily, L. Breen, G. van der Geer, H. Koch, E.-W. Zink, T. Zink, R. Holzapfel, H. Opolka, M. Hazewinkel, M. Wodzicki, N. Schappacher, F. Hirzebruch, D. Zagier, J. W. S. Cassels, A. Todorov, M. S. Narasimhan, C. S. Seshadri, G. Prasad, M. Raghunathan, T. Shioda, Y. Miyaoka, F. Campana.