Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros.

The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscates and Cassini ovals. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the inflection points, and the points at infinity. More advanced questions involve the topology of the curve and relations between the curves given by different equations.

One key achievement of this abstract algebraic geometry is Grothendieck's scheme theory which allows one to use sheaf theory to study algebraic varieties in a way that is very similar to its use in the study of differential and analytic manifolds. This is obtained by extending the notion of the point: In classical algebraic geometry, a point of an affine variety may be identified, through Hilbert's Nullstellensatz, with a maximal ideal of the coordinate ring, while the points of the corresponding affine scheme are all prime ideals of this ring. This means that a point of such a scheme may be either a usual point or a subvariety. This approach also enables a unification of the language and the tools of classical algebraic geometry, mainly concerned with complex points, and of algebraic number theory. Wiles' proof of the longstanding conjecture called Fermat's Last Theorem is an example of the power of this approach.

Diophantine geometry consists of finding integer solutions to integer polynomial equations in several variables. It dates from the Greeks, as, for example, the Pythagorean theorem. Some conjectures have held up for long periods, such as Fermat's Last Theorem, about 300 years. The proof of this theorem is an incarnation of the successors to Diophantine geometry and arithmetic geometry. To produce the solutions, it is more interesting to understand its arithmetic complexity, measured by its height, as well as its linear incarnations through Gallois representations. Correctness Fermat's Last Theorem is less interesting than the fact that it is implied by the Shimura-Tanyiama conjecture. The latter is what was proved by Wiles and Taylor in 1994. This conjecture is deeply linked to the arithmetical theory of elliptical curves of the second half of the 20th century. It involves a ubiquitous notion in mathematics such as the theory of deformation whose origin goes back to algebraic geometry.

Commutative algebra is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent examples of commutative rings include polynomial rings; rings of algebraic integers, including the ordinary integers and p-adic integers.

Commutative algebra is the main technical tool in the local study of schemes.

The notion of localization of a ring (in particular the localization with respect to a prime ideal, the localization consisting in inverting a single element and the total quotient ring) is one of the main differences between commutative algebra and the theory of non-commutative rings. It leads to an important class of commutative rings, the local rings that have only one maximal ideal. The set of the prime ideals of a commutative ring is naturally equipped with a topology, the Zariski topology. All these notions are widely used in algebraic geometry and are the basic technical tools for the definition of scheme theory, a generalization of algebraic geometry introduced by Grothendieck.

Many other notions of commutative algebra are counterparts of geometrical notions occurring in algebraic geometry. This is the case of Krull dimension, primary decomposition, regular rings, Cohen–Macaulay rings, Gorenstein rings and many other notions.

Group theory studies symmetries. In the first approach, a symmetry is an operation on a certain object and symmetries that preserve particular types of patterns form an algebraic structure that can be thought of as a group in a precise sense. More specifically, a group is a set of elements provided with an operation that combines two given elements to produce a third element in the set. If this operation is associative, admitting inverses and identity, we have the notion of group. While the parentage of group theory is credited to Évariste Galois, the parentage of ring theory is generally attributed to Emmy Noether nearly a century later. Since then, Group Theory and Ring Theory are so deeply related these days that it is difficult to find anyone who can say they work in just one of these two broad areas. The interests of these broad areas are as varied as, representation theory (algebras and groups), algebraic geometry (invariant theory, differential operators), arithmetic (orders, Brauer group), homology, functional analysis (algebra of operators) , Lie theory, geometry groups, K-theory, group algebras, varieties of algebras, identities (polynomials and groups). At IM-UFRJ, Algebra began to consolidate itself as a Research area with the arrival of the "descendants" of the Chicago school, a major highlight in the 60's and 70's, leaving marks on the development of most of the areas mentioned above.