Gelfand–Fuks cohomology is a concept in the field of mathematics that arises from the study of infinite-dimensional Lie algebras and their representations. It provides a powerful tool for analyzing and understanding the structure of these algebras, particularly in the context of the theory of differential operators and the geometry of manifolds. The cohomology theory was developed by Israel Gelfand and Sergei Fuks in the 1960s.

Galois cohomology is a branch of mathematics that studies objects known as "cohomology groups" in the context of Galois theory, which is a part of algebra concerned with the symmetries of polynomial equations. To understand Galois cohomology, we start with a few key ideas: 1. **Galois Groups**: A Galois group is a group associated with a field extension, representing the symmetries of the roots of polynomials.

The term "Factor system" can refer to various concepts depending on the context, including mathematics, economics, and systems theory. Here are a few interpretations: 1. **Mathematics**: In mathematics, a factor system typically refers to a collection of factors that can be used to break down numbers or algebraic expressions into their constituent parts. For example, in number theory, factorization involves expressing a number as a product of its prime numbers.

Elliptic cohomology is a branch of algebraic topology that generalizes classical cohomology theories using the framework of elliptic curves and modular forms. It is an advanced topic that blends ideas from algebraic geometry, number theory, and homotopy theory. ### Key Features 1.

Dolbeault cohomology is a mathematical concept that arises in the field of complex differential geometry and algebraic geometry. It provides a way to study the properties of complex manifolds by using differential forms. In essence, Dolbeault cohomology is a specific kind of cohomology theory that is particularly suited to complex manifolds. While ordinary cohomology deals with real-valued differential forms, Dolbeault cohomology focuses specifically on complex-valued differential forms.

Deligne cohomology is a cohomology theory that generalizes the classical notions of singular cohomology by incorporating additional structures, specifically those related to sheaf theory and algebraic geometry. It was introduced by Pierre Deligne in the context of his work on the Weil conjectures and arithmetic geometry.

De Rham cohomology is a mathematical concept from the field of differential geometry and algebraic topology that studies the topology of smooth manifolds using differential forms. It provides a bridge between analysis and topology by utilizing the properties of differential forms and their relationships through the exterior derivative. ### Key Concepts 1. **Differentiable Manifolds**: A differentiable manifold is a topological space that is locally similar to Euclidean space and has a well-defined notion of differentiability.

Crystalline cohomology is a cohomology theory in algebraic geometry and arithmetic geometry that is particularly useful for studying schemes over fields of characteristic \( p \). Developed primarily by Pierre Deligne in the 1970s, it is related to several important concepts in both algebraic geometry and number theory.

Cohomology with compact support is a concept in algebraic topology and differential geometry that generalizes the notion of cohomology by focusing on those cochains that vanish outside of compact sets. This has important implications for the study of properties of spaces when dealing with functions or forms that are localized in compact subsets.

Cohomology of a stack is a concept that extends the idea of cohomology from algebraic topology and algebraic geometry to the realm of stacks, which are sophisticated objects that generalize schemes and sheaves. Stacks allow one to systematically handle problems involving moduli spaces, particularly when there are nontrivial automorphisms or when the objects involved have "geometric" or "categorical" structures.

Cohomology is a fundamental concept in algebraic topology and other fields of mathematics that studies the properties of spaces through algebraic invariants. It provides a way to associate a sequence of abelian groups or vector spaces to a topological space, which can help in understanding its structure and features.

Coherent sheaf cohomology is a concept in algebraic geometry and sheaf theory, dealing with the study of coherent sheaves on algebraic varieties. Coherent sheaves are a generalization of vector bundles and are important because they allow for the treatment of sections and their relationships in a more general setting.

Chromatic homotopy theory is a branch of algebraic topology that studies stable homotopy groups of spheres and related phenomena through the lens of chromatic filtration. It originated from attempts to better understand the relationship between stable homotopy theory and complex-oriented cohomology theories, particularly in the context of the stable homotopy category.

In the context of mathematics, particularly in the study of Lie groups and Lie algebras, a **Cartan pair** refers to a specific structure that arises in the theory of semisimple Lie algebras.

Brown–Peterson cohomology is a homology theory in algebraic topology that is particularly focused on stable homotopy and complex cobordism. Introduced by Ronald Brown and F. P. Peterson in the context of stable homotopy theory, it serves as a tool for studying the cohomological properties of topological spaces, especially with respect to the stable homotopy category.

Bredon cohomology is a type of cohomology theory that is particularly useful in the context of spaces with group actions. It was introduced by Glen Bredon in the 1960s and is designed to study topological spaces with an additional structure of a group action, often leading to insights in equivariant topology.

Bivariant theory is a concept in algebraic topology and homotopy theory that studies the relationships between different homological or homotopical invariants using a bivariant framework. It essentially generalizes classical invariant theory (like cohomology and homology) to consider pairs of spaces or pairs of morphisms, allowing for a more nuanced and flexible understanding of how different spaces can interact.

BRST quantization is a formalism used in the field of quantum field theory to handle systems with gauge symmetries. It is named after the physicists Bonora, Reisz, Sirlin, and Tyutin, who contributed to its development. BRST stands for Becchi-Rouet-Stora-Tyutin, referring to the key researchers who formulated the method. The motivation for BRST quantization arises from the challenges associated with quantizing gauge theories.

André–Quillen cohomology is a concept in algebraic geometry and homological algebra that provides a way to study deformations of algebraic structures, particularly in the context of algebraic varieties and schemes. It was introduced by the mathematicians Michèle André and Daniel Quillen in the context of their work on deformation theory.

Alexander–Spanier cohomology is a cohomology theory used in algebraic topology that serves to study topological spaces. It extends the notion of singular cohomology, providing a way to compute topological invariants of spaces whether or not they are nice enough to have a smooth structure. It was introduced by John W. Alexander and Paul Spanier. ### Definition and Basic Ideas 1.