Department of Algebra and Number Theory

The Department of Algebra was created in the middle of 1930's. B. N. Delone was the first head. The list of people working at the department in the late 1930's and 1940's includes: O. Yu. Schmidt, S. A. Chunikhin, I. M. Gelfand, A. I. Malcev.

Starting from 1946 I. R. Shafarevich is working at the department, being its head from 1960 to 1995. Many people in the department are his pupils: A. I. Lapin (1950 and 1957–1969), A. I. Kostrikin (1956–2000; starting from 1977 he was also the head of the Chair of Algebra at the Moscow State University), S. P. Demushkin (1959–1975), A. B. Zhizhchenko (1959–1965), Yu. I. Manin (since 1960), A. N. Tyurin (1963–2002), V. A. Demyanenko (1967–1969), A. N. Parshin (since 1968, starting from 1995 he is the head of the department), S. Yu. Arakelov (PhD student from 1971 to 1974), V. V. Nikulin (1987–2002, out-of-staff member since 2002), V. A. Kolyvagin (1988–2004, out-of-staff member from 2004 to 2011), V. A. Abrashkin (1996–2002, out-of-staff member since 2002), Vik. S. Kulikov (PhD student from 1974 to 1977, then a member since 1997).

The following people have been working at the department: S. P. Novikov (1960–1975), F. A. Bogomolov (PhD student from 1970 to 1973, employee from 1973 to 1993, out-of-staff member until 2011), M. M. Kapranov (1986–1990), S. A. Stepanov (1987–2000), A. T. Fomenko (1998–2001).

Now the following people also work at the department: A. I. Bondal (since 1994), D. O. Orlov (since 1996), D. V. Osipov (since 1999), V. L. Popov (since 2002), D. B. Kaledin (since 2002), A. G. Kuznetsov (since 2002), R. V. Mikhailov (since 2004), V. V. Shokurov (since 2004), S. O. Gorchinskiy (since 2007), C. A. Shramov (since 2008), I. D. Shkredov (since 2010), A. I. Efimov (since 2010).

In 2009 the Department of Algebra was united with the Department of Number Theory. The following people have thus entered the department: G. I. Arkhipov, M. M. Grinenko (out-of-staff member since 2011), M. A. Korolev, V. V. Przyjalkowski, A. V. Pukhlikov, I. S. Rezvyakova.

In 2012 the Department of Algebraic Geometry was created on the base of the Department of Algebra and Number Theory. The following people are the members of the new department: D. O. Orlov (head of the department), A. I. Bondal, M. M. Grinenko, A. I. Efimov, D. B. Kaledin, A. G. Kuznetsov, V. V. Przyjalkowski, A. V. Pukhlikov, V. V. Shokurov, C. A. Shramov.

Algebraic number theory, Galois theory, Lie groups and Lie algebras, theory of algebraic groups, algebraic geometry (especially the category theory of coherent sheaves, birational geometry, invariant theory), arithmetic of algebraic varieties, algebraic and differential topology, mathematical physics, combinatorial group theory, homological algebra, representations of groups, mirror symmetry, theory of adeles.

1. Algebraic number theory and Galois theory

• Construction of a general reciprocity law (I. R. Shafarevich, A. I. Lapin), solution of the inverse problem of the Galois theory for solvable groups (I. R. Shafarevich).
• Description of p-extensions of local and global fields (I. R. Shafarevich, S. P.  Demushkin, H. Koch), solution of the problem of class field tower (E. S. Golod and I. R. Shafarevich), the structure of the Galois group for local fields (V. A. Abrashkin).
• Theory of Euler systems (V. A. Kolyvagin).

2. Lie groups and Lie algebras

• Semi-simple subgroups of Lie groups, nilmanifolds (A. I. Malcev).
• Theory of infinite-dimensional representations of classical Lie groups (I. M. Gelfand and M. A. Naimark).
• Solution of the weak Burnside problem for an arbitrary prime exponent (A. I. Kostrikin).
• Classification of simple Lie algebras in the positive characteristic (A. I. Kostrikin and I. R. Shafarevich).
• Integral lattices and orthogonal decompositions of Lie algebras (A. I. Kostrikin).
• Theory of Kac–Moody Lorentz algebras (V. A. Gritsenko, V. V. Nikulin).

3. Algebraic geometry

• Geometry of algebraic varieties: pencils of elliptic curves (I. R. Shafarevich), the Gauss–Manin connection (Yu. I. Manin), finiteness theorems for families of curves (A. N. Parshin and S. Yu. Arakelov).
• Theory of vector bundles: classification and Torelli type theorems for vector bundles over algebraic curves, the problem of a bundle of quadrics, vector bundles over an infinite-dimensional projective space (A. N. Tyurin).
• Theory of algebraic K3 surfaces and manifolds with a trivial canonical class: Torelli theorem (I. I. Piatetski-Shapiro, I. R. Shafarevich), structure of K3 surfaces in positive characteristic (A. N. Rudakov, I. R. Shafarevich), group of automorphisms, topological classification (V. V. Nikulin), surjectivity of the period map (V. S. Kulikov), classification of complex manifolds with trivial canonical class (F. A. Bogomolov).
• Solution of three-dimensional Lьroth problem (V. A. Iskovskikh, Yu. I. Manin).
• Flat and projective structures on Riemann surfaces (A. N. Tyurin).
• Theory of stable vector bundles on algebraic varieties (F. A. Bogomolov).
• Arithmetic groups in hyperbolic spaces and integral lattices (V. V. Nikulin).
• Smooth invariants of algebraic surfaces (V. Y. Pidstrigach, A. N. Tyurin).
• Derived categories of coherent sheaves on algebraic varieties and equivalences between them for varieties with ample and anti-ample canonical class, as well as for abelian varieties (M. M. Kapranov, A. I. Bondal, D. O. Orlov).
• Derived categories of coherent sheaves on a symplectic resolution of an arbitrary singularity (D. B. Kaledin).
• Theorem on integral kernel for an equivalence between derived categories of coherent sheaves on possibly singular projective varieties (D. O. Orlov, V. A. Lunts).
• Theory of homological projective duality (A. Kuznetsov).
• Derived categories of coherent sheaves on isotropic Grassmannians (A. G. Kuznetsov, A. E. Polishchuk).
• Prym varieties, their difference with Jacobians, and applications to the three-dimensional birational geometry (V. V. Shokurov).
• Minimal model program and its applications to higher-dimensional geometry. Moduli of polarized log pairs and positivity of the module part in the adjunction formula (V. V. Shokurov).
• Application of unramified Brauer group to the unirationality problem for algebraic varieties (F. A. Bogomolov).
• Topology of algebraic surfaces: Chisini conjecture for generic projections of algebraic surfaces onto projective plane, counterexamples in deformation theory, description of components of the Hurwitz spaces of coverings of algebraic curves (Vik. S. Kulikov).
• Birational geometry of Fano varieties: description of the structures of a rationally connected fibration on Fano double spaces of index 2 and dimension 5 and above, computation of the group of birational automorphisms and the proof of non-rationality (A. V. Pukhlikov).
• Invariant theory: algebraic groups as groups of automorphisms of algebras, solution of the problem of rationality of the function field on a connected semisimple algebraic group over the subfield of central functions, description of Cayley groups (V. L. Popov).
• Applications of algebraic geometry and Tannakian categories to the differential Galois theory, parametrized Picard–Vessiot extensions (S. O. Gorchinskiy, A. I. Ovchinnikov).
• Deformation quantization of algebraic varieties over a field of positive characteristic. Noncommutative analogues of the Cartier morphism and the Frobenius map for cyclic homology (D. B. Kaledin).

4. Arithmetic of algebraic varieties

• Diophantine equations of degree three (B. N. Delone and D. K. Faddeev).
• Arithmetic of elliptic curves and abelian varieties: theory of principal homogeneous spaces (I. R. Shafarevich), unboundedness of rank over function fields (A. I. Lapin), boundedness of p-torsion of elliptic curves (Yu. I. Manin), estimates for the torsion of elliptic curves (V. A. Demyanenko), canonical heights of abelian varieties (A. N. Parshin), l-adic representations of Galois groups associated with abelian varieties, the group of points of finite order (F. A. Bogomolov), proof of the nonexistence of smooth abelian schemes over Z (V. A. Abrashkin).
• Finiteness theorems in Diophantine geometry: proof of the Mordell conjecture on rational points over function fields (Yu. I. Manin), method of ramified coverings (A. N. Parshin), finiteness of the Tate–Shafarevich group for modular curves (V. A. Kolyvagin).
• Arithmetic surfaces (Arakelov geometry).
• Arithmetic of rational and cubic 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 and its applications to class field theory, vector bundles and the theory of algebraic groups (A. N. Parshin).
• Theory of adeles: measure theory and harmonic analysis on adelic spaces of two-dimensional schemes (D. V. Osipov, A. N. Parshin), symbols and reciprocity laws (D. V. Osipov), adelic resolutions for sheaves (S. O. Gorchinskiy, D. V. Osipov).

5. Algebraic and differential topology

• Theory of cohomological operations. Description of complex cobordisms. Classification of simply connected smooth manifolds of dimension ≷ 4. Proof of topological invariance of Pontryagin classes. Theory of foliations on smooth manifolds. Foundations of Hermitian K-theory (S. P. Novikov).
• Theory of derived functors for non-additive functors (R. V. Mikhailov, L. Breen).
• Functorial methods in the unstable homotopy theory (R. V. Mikhailov).

6. Mathematical physics

• Solution of the periodic problem for the KdV equation by methods of algebraic geometry (S. P. Novikov).
• Classification of instantons (V. G. Drinfeld, Yu. I. Manin).
• Models of classical field theory: supergeometry, the Yang–Mills theory, and string theory (Yu. I. Manin, M. M. Kapranov).
• Description of instantons on noncommutative spaces and noncommutative twistor transform (A. Kapustin, A. G. Kuznetsov, D. O. Orlov).
• Homological mirror symmetry and the category of D-branes for Landau–Ginzburg models (D. O. Orlov). Homological mirror symmetry for curves of genus greater than one (A. I. Efimov) and del Pezzo surfaces (D. O. Orlov).

7. Combinatorial group theory and applications

• Theory of central series for groups (R. V. Mikhailov).
• Description of homotopy groups of spheres in terms of group theory (R. V. Mikhailov, J. Wu).

8. Representation theory

• Moduli space of representations of Lie algebras in positive characteristic (I. R. Shafarevich, A. N. Rudakov).
• Classification and character theory for irreducible representations with finite weight of discrete Heisenberg groups (A. N. Parshin, S. A. Arnal).

Among the members of the department there are recipients of the Fields Medal (S. P. Novikov), the Lenin Prize (I. R. Shafarevich, Yu. I. Manin, S. P. Novikov), the State USSR Prize (A. I. Kostrikin, S. A. Stepanov), the Lomonosov Prize (A. I. Kostrikin), the Alexander von Humboldt Prize (A. N. Parshin), the European Mathematical Society Prize (A. G. Kuznetsov), Russian Federation President Prize in Science and Innovation for Young Scientists (A. G. Kuznetsov) and others.

Department members have been repeatedly invited to the International Congresses of Mathematicians as speakers:

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),

D. B. Kaledin (Hyderabad, 2010).

The Department of Algebra and Number Theory has many relations with Russian and foreign mathematicians. Department visitors include:

V. Alexeev, A. N. Andrianov, R. V. Bezrukavnikov, A. A. Beilinson, N. A. Vavilov, B. B. Venkov, A. M. Vershik, V. A. Voevodsky, V. E. Voskresensky, S. V. Vostokov, V. Ginzburg, V. A. Gritsenko, V. I. Guletskii, A. S. Dzhumadildaev, N. V. Durov, Yu. L. Ershov, Yu. G. Zarhin, M. M. Kapranov, A. A. Klyachko, M. L. Kontsevich, V. A. Lunts, S. A. Merkulov, I. A. Panin, A. A. Panchishkin, F. V. Petrov, V. P. Platonov, A. E. Polishchuk, Yu. G. Prokhorov, A. A. Rosly, A. N. Skorobogatov, A. L. Smirnov, S. G. Tankeev, A. S. Tikhomirov, N. A. Tyurin, L. D. Faddeev, V. M. Kharlamov, I. A. Cheltsov, V. I. Janchevsky, W. Baily, L. Bers, L. Breen, F. Campana, J. W. S. Cassels, A. Corti, P. Deligne, H. Esnault, G. van der Geer, D. Gieseker, Ph. Griffiths, M. Harris, M. Hazewinkel, F. Hirzebruch, R. Holzapfel, E. Kaehler, E. Kani, L. Katzarkov, B. Keller, H. Koch, S. Lang, R. P. Langlands, R. MacPherson, Y. Miyaoka, D. Mumford, M. S. Narasimhan, A. Neeman, H. Opolka, T. Pantev, I. B. Passi, G. Prasad, M. Raghunathan, M. Reid, N. Schappacher, T. Shioda, J.-P. Serre, C. S. Seshadri, J. Tate, A. Todorov, J.-L. Verdier, E. Vieweg, M. Wodzicki, G. Wuestholz, D. Zagier, E.-W. Zink, T. Zink, and many others.

The department actively cooperates with many institutes and universities, including:

Moscow State University, Novosibirsk State University, Independent University of Moscow, St. Petersburg Department of the Steklov Mathematical Institute, St. Petersburg State University, Yaroslavl State Pedagogical University, ETH (Switzerland), Harish-Chandra Research Institute (India), IAS (U.S.A.), IHES (France), IPMU (Japan), ICTP (Italy), London Imperial College (UK), MPIM (Germany), POSTECH (South Korea), Punjab University (India), RIMS (Japan), TIFR (India), University of Durham (UK), University of Edinburgh (UK), University of Liverpool (UK), University of Warwick (UK), University of Vienna (Austria).