Fields are a fundamental concept in abstract algebra, a branch of mathematics that studies algebraic structures. A field is a set equipped with two operations: addition and multiplication, satisfying certain properties. Here are the key properties that define a field: 1. **Closure**: For any two elements \(a\) and \(b\) in the field, both \(a + b\) and \(a \cdot b\) are also in the field.
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Algebraic logic is a branch of mathematical logic that studies logical systems using algebraic techniques and structures. It provides a framework where logical expressions and their relationships can be represented and manipulated algebraically. This area of logic encompasses various subfields, including: 1. **Algebraic Semantics**: This involves modeling logical systems using algebraic structures, such as lattices, Boolean algebras, and other algebraic systems.
Abstract Algebraic Logic (AAL) is a field of study that lies at the intersection of logic, algebra, and category theory. It focuses on the algebraic aspects of various logical systems—particularly non-classical logics—by examining how logic can be understood and represented using algebraic structures. ### Key Concepts in Abstract Algebraic Logic: 1. **Algebraic Structures**: AAL often involves the study of algebras that correspond to logical systems.
Cylindric algebra is a mathematical structure that arises in the study of multi-dimensional logics and is particularly relevant in the fields of model theory and algebraic logic. It is an extension of Boolean algebras to accommodate more complex relationships involving multiple dimensions or "cylindrical" structures. A cylindric algebra can be thought of as an algebraic structure that captures the properties of relations in multiple dimensions, enabling the representation of various logical operations and relations.
A Heyting algebra is a specific type of mathematical structure that arises in the field of lattice theory and intuitionistic logic. Heyting algebras generalize Boolean algebras, which are used in classical logic, by accommodating the principles of intuitionistic logic. ### Definition A Heyting algebra is a bounded lattice \( H \) equipped with an implication operation \( \to \) that satisfies certain conditions.
The Leibniz operator is a differential operator used in the context of calculus, particularly in the formulation of differentiating products of functions. It is named after the mathematician Gottfried Wilhelm Leibniz, who made significant contributions to the development of calculus.
Polyadic algebra is a branch of algebra that extends the concept of traditional algebraic structures, such as groups, rings, and fields, to include operations that involve multiple inputs or arities. In particular, it focuses on operations that can take more than two variables (unlike binary operations, which are the most commonly studied).
Predicate functor logic is a formal system that combines elements of predicate logic with concepts from category theory, specifically functors. To understand it, it's helpful to break down the two main components: 1. **Predicate Logic**: This is an extension of propositional logic that includes quantifiers and predicates. In predicate logic, statements can involve variables and can assert relationships between objects.
Algebraic topology is a branch of mathematics that studies topological spaces with the help of algebraic methods. The fundamental idea is to associate algebraic structures, such as groups or rings, to topological spaces in order to gain insights into their properties. Key concepts in algebraic topology include: 1. **Homotopy**: This concept deals with the notion of spaces being "continuously deformable" into one another.
Cohomology theories are mathematical frameworks used in algebraic topology, geometry, and related fields to study topological spaces and their properties. They serve as tools for assigning algebraic invariants to topological spaces, allowing for deeper insights into their structure. Cohomology theories capture essential features such as connectivity, holes, and other topological characteristics. ### Key Concepts in Cohomology Theories 1.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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 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.
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.
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.
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.
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.
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.
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.
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.
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.
Group cohomology is a mathematical tool used in algebraic topology, group theory, and various other areas of mathematics. It provides a way to study the properties of groups using cohomological methods, which are analogous to those used in homology theory but focus on the algebraic structure associated with groups.
The Hodge–de Rham spectral sequence is a mathematical tool used in algebraic topology and differential geometry, specifically in the context of studying the relationships between differential forms on a smooth manifold and the topology of that manifold. This spectral sequence arises from the filtration provided by the Hodge decomposition theorem in conjunction with the de Rham complex of differential forms. ### Overview 1.
Infinitesimal cohomology is a concept from the field of algebraic geometry and is particularly associated with the study of formal schemes and deformation theory. It provides a way to study the local behavior of schemes using a "cohomological" approach that incorporates infinitesimal neighborhoods. In more detailed terms, infinitesimal cohomology typically arises in contexts involving the study of deformations of algebraic objects.
Koszul cohomology is a concept from algebraic topology and homological algebra that arises in the context of differential graded algebras and the study of the algebraic invariants associated with topological spaces or algebraic varieties. It is named after Jean-Pierre Serre and Jean Koszul, who developed the foundational ideas related to this cohomology theory.
Kähler differentials are a concept from algebraic geometry and commutative algebra. They arise in the context of the study of a ring \( R \) and its associated differentials with respect to a base field or a base ring. Specifically, Kähler differentials provide a way to study the infinitesimal behavior of functions and their properties on schemes.
Lie algebra cohomology is a mathematical concept that arises in the study of Lie algebras, which are algebraic structures used extensively in mathematics and physics to describe symmetries and conservation laws. Cohomology, in this context, refers to a homological algebra framework that helps in analyzing the structure and properties of Lie algebras.
Cohomology theories are mathematical frameworks used in algebraic topology, algebraic geometry, and other areas to study the properties of topological spaces and algebraic structures. Here’s a list of notable cohomology theories, each with unique properties and applications: 1. **Singular Cohomology**: The most fundamental cohomology theory for topological spaces, using singular simplices. It is defined for any topological space and provides multiplicative structures.
Local cohomology is a concept in algebraic geometry and commutative algebra that extends the notion of ordinary cohomology to study the local behavior of a module over a ring, particularly with respect to a specified ideal. It is particularly useful for understanding the properties of sheaves and modules around points in a space or in relation to certain subvarieties.
Monsky–Washnitzer cohomology is a type of cohomology theory developed in the context of the study of schemes, particularly over fields of positive characteristic. It is named after mathematicians Paul Monsky and Michiel Washnitzer, who introduced the concept in 1970s. This cohomology theory is specifically designed to work with algebraic varieties defined over fields of characteristic \( p > 0 \) and offers a way to analyze their geometric and topological properties.
Motivic cohomology is a concept in algebraic geometry and topology that generalizes classical cohomology theories to the framework of algebraic varieties. It is particularly influential in the study of algebraic cycles, motives, and the relationship between algebraic geometry and topology. ### Background Motivic cohomology was introduced in the context of the theory of motives, which aims to unify various cohomological approaches to algebraic varieties.
Nonabelian cohomology is a branch of mathematics that studies the cohomological properties of nonabelian structures, particularly in the context of group theory and algebraic geometry. It generalizes classical cohomology theories to contexts where the groups involved do not necessarily obey the commutative property, hence the term "nonabelian.
P-adic cohomology is a branch of mathematics that studies the properties of algebraic varieties and schemes over p-adic fields using cohomological methods. It is particularly important in number theory, algebraic geometry, and arithmetic geometry, as it provides tools to understand the relationships between algebraic structures and their properties over p-adic numbers.
In the context of cohomology, a pullback is a construction that allows you to take a cohomology class on a target space and "pull it back" to a cohomology class on a domain space via a continuous map. This is particularly common in algebraic topology and differential geometry. ### Formal Definition Let \( f: X \to Y \) be a continuous map between two topological spaces \( X \) and \( Y \).
Quantum cohomology is a branch of mathematics that combines concepts from algebraic geometry, symplectic geometry, and quantum physics. It arises in the study of certain moduli spaces and has applications in various fields, including string theory, mathematical physics, and enumerative geometry. At a high level, quantum cohomology seeks to extend classical cohomology theories, particularly for projective varieties, to incorporate quantum effects, which can be thought of as counting curves under certain conditions.
Sheaf cohomology is a fundamental concept in algebraic geometry and topology that provides a way to study the properties of sheaves on topological spaces or schemes. It serves as a powerful tool for capturing global sections of sheaves and understanding their finer structures. ### Key Concepts 1.
Spencer cohomology is a mathematical framework used in the study of differential operators and the cohomology of various algebraic and geometric structures. It is a cohomology theory primarily associated with the analysis of differential equations, particularly in the context of differential forms and sheaf theory on smooth manifolds.
Weil cohomology theory is a set of tools and concepts in algebraic geometry and number theory developed by André Weil to study the properties of algebraic varieties over fields, particularly over finite fields and more generally over local fields. It was introduced as a way to provide a cohomology theory that would capture essential topological and algebraic features of varieties and is particularly characterized by its application to counting points on varieties over finite fields.
Witt vector cohomology is a tool in algebraic geometry and number theory that utilizes Witt vectors to study the cohomological properties of schemes in the context of p-adic cohomology theories. Witt vectors are a generalization of the notion of numbers in a ring, particularly for fields of characteristic \( p \), and they allow the construction of an effective cohomology theory that preserves useful algebraic properties. ### Key Concepts 1.
Étale cohomology is a cohomological theory in algebraic geometry that provides a means to study the properties of algebraic varieties over fields, particularly in the context of fields that are not algebraically closed. It was developed in the mid-20th century, notably by Alexander Grothendieck, and is part of the broader framework of schemes in modern algebraic geometry.
Čech cohomology is a mathematical tool used in algebraic topology to study the properties of topological spaces. Named after the Czech mathematician Eduard Čech, this cohomology theory is particularly useful for analyzing spaces that may not be well-behaved in a classical sense.
Double torus knots and links are concepts from the field of knot theory, which is a branch of topology. In topology, knots are considered as embeddings of circles in three-dimensional space, and links are collections of such embeddings. ### Double Torus A double torus is a surface that is topologically equivalent to two tori (the plural of torus) connected together. It's often visualized as the shape of a "figure eight" or a surface with two "holes.
A knot is a unit of speed equal to one nautical mile per hour, commonly used in maritime and air navigation. To convert knots to more familiar units like miles per hour (mph) or kilometers per hour (km/h): - **62 knots** is approximately equal to: - 71.4 miles per hour (mph) - 113.0 kilometers per hour (km/h) So, 62 knots is a measure of speed often used at sea or in aviation contexts.
A knot is a unit of speed equal to one nautical mile per hour. When you refer to "63 knots," it indicates a speed of 63 nautical miles per hour. To provide some context, converting knots to other units: - 1 knot is approximately equal to 1.15 miles per hour (mph). - 63 knots is roughly equal to 72.5 mph. Knots are commonly used in maritime and aviation contexts to measure speed.
A knot is a unit of speed used in maritime and air navigation, equivalent to one nautical mile per hour. To understand what 74 knots means in other units: - **In miles per hour (mph)**: 1 knot is approximately equal to 1.15078 miles per hour. Therefore, 74 knots is about 85.3 mph. - **In kilometers per hour (km/h)**: 1 knot is approximately equal to 1.852 kilometers per hour.
The term "figure-eight knot" in mathematics refers to a specific type of knot that is recognized in knot theory, which is a branch of topology. The figure-eight knot is one of the simplest and most well-known non-trivial knots, and it is often represented as can be visualized as a loop that crosses over itself to form a pattern resembling the numeral "8".
In mathematics, particularly in the field of knot theory, a **stevedore knot** refers to a specific type of knot that is categorized as a nontrivial knot. Knot theory is a branch of topology that studies mathematical knots, which are embeddings of a circle in three-dimensional space, essentially investigating their properties and classifications. The stevedore knot is typically recognized for its distinct shape and characteristics, separating it from trivial knots (which can be untangled without cutting the string).
The Three-twist knot, also known as the trefoil knot, is one of the simplest and most well-known types of nontrivial knots in topology. It can be visualized as a loop with three twists in it, and it is often represented as a closed curve that can be drawn in three-dimensional space without self-intersecting, yet cannot be untangled into a simple loop without cutting it.
A twist knot, also known as a twisted knot, is a type of knot characterized by the intertwining of two or more strands. This type of knot can be used in various applications, including climbing, boating, crafting, and more. The twisting action creates friction, which helps secure the knot. Twist knots can vary in complexity and construction, with some being relatively simple and others more intricate.
Homology theory is a branch of algebraic topology that studies topological spaces through the use of algebraic structures, primarily by associating a sequence of abelian groups or modules, called homology groups, to a topological space. These groups encapsulate information about the space's shape, connectivity, and higher-dimensional features.
Borel–Moore homology is a homological algebraic concept that arises in the study of algebraic varieties, particularly in the context of algebraic geometry and algebraic topology. It is a form of homology theory designed to handle locally compact topological spaces, with particular application to appropriate classes of varieties, such as quasi-projective varieties or complex algebraic varieties.
"Bump and hole" is a term that can have different meanings depending on the context, but it is often associated with construction, civil engineering, or road maintenance, referring to an issue related to road surfaces. When roads develop bumps and holes (or potholes), it can lead to uneven driving surfaces that can be dangerous for vehicles and pedestrians. In a more technical sense, "bump" refers to elevated areas on a surface, while "hole" refers to depressions.
Cellular homology is a tool in algebraic topology that allows for the computation of homology groups of a topological space by using a cellular structure derived from a CW-complex. A CW-complex is a kind of topological space that is built up from basic building blocks called cells, which are homeomorphic to open disks in Euclidean space, glued together in a specific way.
In algebraic topology, the cohomology ring is an important algebraic structure associated with a topological space. It is formed from the cohomology groups of the space, which provide algebraic invariants that help in understanding the topological properties of spaces.
Compactly supported homology is a version of homology theory that focuses on the study of spaces where the singular chains are required to have compact support. This concept is particularly useful in various areas of mathematics, including algebraic topology and differential geometry. ### Key Concepts: 1. **Homology**: Homology is a tool used in algebraic topology to study topological spaces by associating sequences of abelian groups (or modules) to them.
In mathematics, the term "continuation map" can refer to different concepts depending on the context, particularly in the realms of topology, functional analysis, and other areas of mathematics related to the study of continuous mappings and their properties. Here are a few interpretations: 1. **Topological Continuation Map**: In topology, a continuation map may refer to a function that extends (or continues) a function defined on a smaller space to a larger space while preserving certain properties, like continuity.
In category theory, a **cyclic category** typically refers to a category that captures the idea of cycles or circular structures. It can be viewed as a specialized type of category that includes objects and morphisms that relate to cyclical processes or relationships.
The Eilenberg-Moore spectral sequence is a mathematical construct used in the field of algebraic topology and homological algebra. It arises in the context of homotopical algebra, particularly when dealing with fibred categories and the associated homotopy theoretic situations.
The Eilenberg–Steenrod axioms are a set of axioms in algebraic topology that characterize (reduced) singular homology and cohomology theories. Formulated by Samuel Eilenberg and Norman Steenrod in the mid-20th century, these axioms provide a rigorous framework for what constitutes a generalized homology or cohomology theory. They serve as a foundation for the study of topological spaces through algebraic means.
The Excision Theorem is a fundamental result in algebraic topology, particularly in the context of singular homology. It addresses how the homology groups of a topological space can be affected by the removal of a "nice" subspace.
Graph homology is a concept in algebraic topology that extends the ideas of homology from topological spaces to combinatorial structures known as graphs. Essentially, it assigns algebraic invariants to graphs that capture their topological properties, allowing one to study and classify graphs in a way that is analogous to how homology groups classify topological spaces. ### Key Elements of Graph Homology 1. **Graphs**: A graph consists of vertices and edges connecting pairs of vertices.
The Hodge conjecture is a fundamental statement in algebraic geometry and topology that relates the topology of a non-singular projective algebraic manifold to its algebraic cycles. Formulated by W.V. Hodge in the mid-20th century, the conjecture suggests that certain classes of cohomology groups of a projective algebraic variety have a specific geometric interpretation.
Homological connectivity is a concept from algebraic topology and homological algebra that relates to how well-connected a topological space or algebraic object is in terms of its homological properties. It can involve examining the relationships between different homology groups of a space. In a more specific context, homological connectivity can refer to the lowest dimension in which the homology groups of a space are nontrivial.
Homology is a concept in mathematics, specifically in algebraic topology, that provides a way to associate a sequence of algebraic structures, such as groups or rings, to a topological space. This construction helps to analyze the shape or structure of the space in a more manageable form.
A homology sphere is a topological space that behaves like a sphere in terms of its homological properties, even if it is not actually a sphere in the classical sense. More formally, an \( n \)-dimensional homology sphere is a manifold that is homotopy equivalent to the \( n \)-dimensional sphere \( S^n \), and, importantly, it has the same homology groups as \( S^n \).
Hurewicz's theorem is a result in algebraic topology that pertains to the relationship between the homology and homotopy groups of a space. It specifically addresses the connection between the homology of a space and its fundamental group, particularly for spaces with certain properties.
K-homology is a cohomology theory in the field of algebraic topology that provides a way to study topological spaces using tools from K-theory. It is a variant of K-theory where one considers the behavior of vector bundles and their generalizations over spaces. K-homology is mainly applied in the framework of noncommutative geometry and has connections to several areas such as differential geometry, the theory of operator algebras, and index theory.
The Kan-Thurston theorem is a result in the field of topology and geometric group theory, particularly concerning the relationships between 3-manifolds and the algebraic properties of groups. More specifically, it is related to the conjecture regarding the recognition of certain types of 3-manifolds and the structures of groups that can be associated with them.
Khovanov homology is a mathematical invariant associated with knots and links in three-dimensional space. It was introduced by Mikhail Khovanov in 1999 as a categorification of the Jones polynomial, which is a well-known knot invariant.
The Kirby–Siebenmann class is a concept in the field of algebraic topology, particularly within the study of manifolds and their embeddings. It is named after mathematicians Robion Kirby and Louis Siebenmann, who introduced it in their work on the topology of high-dimensional manifolds. In particular, the Kirby–Siebenmann class arises in the context of the study of manifold structures and their differentiability.
The Mayer–Vietoris sequence is a fundamental tool in algebraic topology, particularly in the study of singular homology and cohomology theories. It provides a way to compute the homology or cohomology of a topological space from that of simpler subspaces.
Morse homology is a tool in differential topology and algebraic topology that studies the topology of a smooth manifold using the critical points of smooth functions defined on the manifold. It relates the topology of the manifold to the critical points of a Morse function, which is a smooth function where all critical points are non-degenerate (i.e., each critical point has a Hessian that is non-singular).
Poincaré duality is a fundamental theorem in algebraic topology that describes a duality relationship between certain topological spaces, particularly manifolds, and their cohomology groups. Named after the French mathematician Henri Poincaré, the theorem specifically applies to compact, oriented manifolds.
Polar homology is an algebraic concept that arises in the study of commutative algebra and algebraic geometry, particularly in the context of the theory of Gröbner bases and polynomial ideals. Polar homology can be thought of as a homology theory that is related to the structure of a polynomial ring, considering the "polar" aspects of a given polynomial or collection of polynomials.
The Pontryagin product is a way to define a multiplication operation on the cohomology ring of a topological group, specifically in the context of homotopy theory and algebraic topology. Named after the mathematician Lev Pontryagin, this product provides a rich algebraic structure that captures important information about the topological properties of the space.
In the context of algebraic topology, particularly in homology theory, the term "pushforward" refers to a specific kind of construction related to the behavior of homology classes under continuous maps between topological spaces.
Reduced homology is a variant of standard homology theory in algebraic topology, typically applied to topological spaces. It is particularly useful for spaces that are not simply connected or that have certain types of singularities, as it helps to simplify some aspects of their homological properties.
Relative Contact Homology (RCH) is a modern invariant in symplectic and contact topology, developed as a tool for studying contact manifolds. It serves as a means of categorifying certain notions from classical contact topology and provides insights into the geometry and topology of contact manifolds when compared to other invariants.
Relative homology is a concept in algebraic topology that extends the notion of homology groups to pairs of spaces. Specifically, if we have a topological space \( X \) and a subspace \( A \subseteq X \), the relative homology groups \( H_n(X, A) \) provide information about the structure of \( X \) relative to the subspace \( A \).
Simplicial volume is a notion from the field of topology and geometry, particularly in the study of manifolds. It essentially provides a way to measure the "size" of a topological manifold in a geometric sense. The concept is closely associated with the study of manifolds and their geometric structures, especially in the context of algebraic topology.
Singular homology is an important concept in algebraic topology, which provides a way to associate a sequence of abelian groups or vector spaces (called homology groups) to a topological space. These groups encapsulate information about the space's structure, such as its number of holes in various dimensions. ### Key Concepts: 1. **Simplices**: The building blocks of singular homology are simplices, which are generalizations of triangles.
The Steenrod problem, named after mathematician Norman Steenrod, refers to a question in the field of algebraic topology concerning the properties and structure of cohomology operations. Specifically, it deals with the problem of determining which cohomology operations can be represented by "natural" cohomology operations on spaces, particularly focusing on the stable homotopy category.
Stratifold is a computational tool used in the field of genomics and molecular biology to predict and analyze the folding structures of proteins. It applies algorithms rooted in statistical mechanics and machine learning to assess how proteins fold into their three-dimensional shapes based on their amino acid sequences. Understanding protein folding is crucial for deciphering biological functions and the development of pharmaceuticals, as misfolded proteins can lead to various diseases.
The Toda–Smith complex is a construction in algebraic topology, specifically in the study of spectra and homotopy theory. It is named after the mathematicians Hirosi Toda and Michael Smith, who contributed to the understanding of stable homotopy types and complex structures. More precisely, the Toda–Smith complex can be constructed from a simplicial set that illustrates certain relationships and equivalences in stable homotopy categories.
Homotopy theory is a branch of algebraic topology that studies the properties of topological spaces through the concept of homotopy, which is a mathematical equivalence relation on continuous functions. The main focus of homotopy theory is to understand the ways in which spaces can be transformed into each other through continuous deformation.
In homotopy theory, a branch of topology, theorems often deal with properties of spaces and maps (functions between spaces) that remain invariant under continuous deformations, such as stretching and bending, but not tearing or gluing.
A **2-group** is a concept in group theory, a branch of mathematics. In particular, a 2-group is a group in which every element has an order that is a power of 2.
Adams filtration is a concept in homotopy theory, particularly in the study of stable homotopy groups of spheres and related areas. It is named after the mathematician Frank Adams, who developed this theory in the mid-20th century. Adams filtration is associated with the idea of understanding the stable homotopy category through a hierarchical structure that helps in studying and organizing the stable homotopy groups of spheres.
Adams spectral sequences are a sophisticated tool used in algebraic topology and homotopy theory, particularly in the study of stable homotopy groups of spheres and related objects. They are named after Frank Adams, who developed the theory in the 1960s. Here's an overview of the key concepts associated with Adams spectral sequences: 1. **Spectral Sequences**: These are mathematical constructs used to compute homology or cohomology groups in a systematic way.
A¹ homotopy theory is a branch of algebraic topology that is concerned with the study of homotopy theories in the context of algebraic varieties over a field, particularly a field with a non-Archimedean valuation or more generally over a base scheme. It is primarily developed in the framework of stable and unstable homotopy types, where the concepts of homotopy can be adapted to the settings of algebraic geometry.
Bousfield localization is a technique in homotopy theory, a branch of algebraic topology, that focuses on constructing new model categories (or topological spaces) from existing ones by inverting certain morphisms (maps). The concept was introduced by Daniel Bousfield in the context of stable homotopy theory, but it has since found applications in various areas of mathematics.
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