The term "shape" can refer to different concepts depending on the context in which it is used: 1. **Geometry**: In mathematics, a shape is the form or outline of an object, defined by its boundaries. Common geometric shapes include circles, squares, triangles, and polygons. Shapes can be two-dimensional (2D) or three-dimensional (3D), with 2D shapes having length and width, and 3D shapes having length, width, and height.

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.

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.

David Spivak is known in the field of mathematics, particularly in the areas of category theory and its applications. He has made contributions to various topics within mathematics, and his work often involves the intersection of algebra, topology, and theoretical computer science. Additionally, Spivak has been involved in educational initiatives and has worked on projects related to the application of mathematical concepts in practical settings.

André Joyal is a Canadian mathematician known for his contributions to category theory, topos theory, and combinatorial set theory. He has worked extensively on the foundational aspects of mathematics, particularly in relation to the interactions between category theory and logic. Joyal is perhaps best known for developing the concept of "quasi-categories," which are a generic notion that generalizes many structures in category theory, particularly in the context of homotopy theory.

As of my last knowledge update in October 2023, there is no widely known individual, concept, or entity specifically called "Myles Tierney." It is possible that it might refer to a private individual, a lesser-known public figure, or a character from a work of fiction.

Picard–Vessiot theory is a framework in differential algebra that generalizes the concepts of Galois theory to the setting of differential equations. It deals with the study of algebraic properties of differential fields—fields equipped with a derivation—and the solutions of linear differential equations.

In the context of abelian groups, the term "height" can refer to a couple of different concepts depending on the specific area of mathematics being considered, such as group theory or algebraic geometry. 1. **In Group Theory**: The height of an abelian group can refer to a measure of the complexity of the group, particularly when it comes to finitely generated abelian groups.

A Barsotti–Tate group is an important concept in the area of algebraic geometry and representation theory, particularly in the study of p-adic representations and finite field extensions. Named after mathematicians Francesco Barsotti and John Tate, these groups are essentially a kind of p-divisible group that has additional structure, allowing them to be classified and understood in terms of their representation theory.

In the context of algebraic groups, the **radical** refers to a specific type of subgroup that is closely related to the structure of the group itself. More formally, the radical of an algebraic group \( G \) is defined as the largest normal solvable subgroup of \( G \). ### Key Concepts: 1. **Algebraic Group**: An algebraic group is a group that is also an algebraic variety, meaning it can be defined by polynomial equations.

In the context of algebraic groups and Lie algebras, a **root datum** is a structured way of encoding certain aspects of the symmetries and properties of these mathematical objects. Specifically, a root datum consists of the following components: 1. **A finite set of roots**: These are usually vectors in a Euclidean space, which can be thought of as directions that reflect the symmetries of the system.

A **subgroup series** in group theory is a sequence of subgroups of a given group \( G \) that is organized such that each subgroup is a normal subgroup of the next one in the series.

The affine group is a mathematical concept that arises in the context of geometry and linear algebra. It is essentially a group that consists of affine transformations, which are a generalization of linear transformations that include translations.

In the context of group theory, particularly in the study of algebraic groups and Lie groups, a diagonal subgroup is typically a subgroup that is constructed from the diagonal elements of a product of groups. For example, consider the direct product of two groups \( G_1 \) and \( G_2 \).

The Five Lemma is a result in the field of homological algebra, particularly in the context of derived categories and spectral sequences. It provides a criterion for when a five-term exact sequence of chain complexes splits. This lemma is commonly used in the study of abelian categories and the derived functor theory.

The Nine Lemma is a result in algebraic topology, specifically in the study of homotopy theory. It deals with examining the relationships between spaces that can be constructed from homotopy types and the existence of certain maps between them. The lemma provides a way to relate complications in a certain diagram of spaces and maps to giving conditions under which some homotopy properties hold true.

The Short Five Lemma is a tool in algebraic topology, specifically in the context of homological algebra and the theory of derived functors. It deals with the properties of cohomology in the setting of a commutative diagram of chain complexes and can be used to derive relationships between cohomology groups of various objects.

A triangulated category is a particular type of category that arises in the context of homological algebra and derived categories. It provides a framework to study homological properties by relating them to geometric intuition through triangles, similar to how one uses exact sequences in abelian categories.

A semimodular lattice is a special type of lattice in the field of order theory and abstract algebra. A lattice \( L \) is a partially ordered set (poset) in which any two elements have a unique supremum (join) and an infimum (meet). The term "semimodular" specifically refers to a certain condition that relates to the structure of the lattice.

In the context of databases and relational algebra, a relation of degree zero is a special case of a relation where there are no attributes (or columns). In relational database terminology, the degree of a relation (or table) refers to the number of attributes it contains. When a relation has a degree of zero, it means that it is essentially an empty set without any data or structure.