Claude Chevalley wrote that algebra plays the same role to mathematics, as mathematics itself plays with respect to physics. It studies the most general structures occurring in various areas of mathematics, and general procedures of calculations therein. Historically, many such classical structures occurred in number theory and algebraic geometry, in connection with solution of systems of algebraic equations, say in complex numbers, or integers. Below we list some of the most active research directions in Saint Petersburg.
The theory of linear algebraic groups is the modern stage of the theory of classical groups, and Lie theory. It studies matrix groups, primarily simple groups (such as linear, symplectic, orthogonal, unitary), and their analogues. Saint Petersburg is one of the leading world centers in the structure theory of algebraic groups over arbitrary fields an commutative rings, an especially in the study of exceptional groups.
This theory occurred in early 1960s in the works of Grothendieck, Bass, and MIlnor, later fantastic progress was achieved in the works of Quillen, Suslin, and other classics. Algebraic K-theory relates to a ring a new type of invariants, the values of K-functors, that measure to what extend the answers to classical problems of linear algebra differ from the known answers for fields.
Historically, algebraic geometry occurred as the study of systems of algebraic equations, and the structure of their solutions. Over a field, such solutions form what is known as an algebraic variety. Starting with 1950s, in the context of some classical problems of number theory, and other important applications, one started to systematically study solutions of algebraic systems over arbitrary commutative rings, and related algebraic and geometric structures, such as schemes, motives, algebraic spaces, etc.
Homological algebra associates with various situations new invariants, such as homology and cohomology groups, etc., measuring the deviation from classical or expected answers, for various natural problems. The subfield most prominently represented in Saint Petersburg is the ring cohomology, historically closely related to representation theory of finite groups and associative algebras.
Among the people working in this field are A.I. Generalov, M.A. Antipov and Yu.V. Volkov.
Groups are one of the paramount classical structures studied in algebra. The main direction represented in Saint Petersburg is the study of concrete groups, occurring in arithmetic and geometric contexts, such as finite groups, arithmetic groups, and various groups of geometric origin presented by generators and relations (such as Coxeter groups, braid groups, etc.).
Algebraic number theory studies the structure of rings similar to the ring of integers, as well as further rings occurring thereby, such as the rings of p-adic integers. Apart from the purely algebraic methods, this theory extensively uses also various powerful analytic methods (complex and harmonic analysis, etc.) and geometric methods (algebraic geometry, arithmetic geometry, geometry of numbers, etc.).