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The ADHM construction, which stands for Atiyah-Drinfeld-Hitchin-Manin construction, is a mathematical framework used in theoretical physics and geometry, particularly in the study of instantons in gauge theory. It is a method for constructing solutions to the self-duality equations of gauge fields in four-dimensional Euclidean space, which are fundamental in the study of Yang-Mills theory.

Abstract differential geometry is a branch of mathematics that studies geometric structures on manifolds in a more general and abstract setting, primarily using concepts from differential geometry and algebraic topology. It emphasizes the intrinsic properties of geometric objects without necessarily attributing them to any specific coordinate system or representation. Some key features of abstract differential geometry include: 1. **Smooth Manifolds**: Abstract differential geometry focuses on smooth manifolds, which are spaces that locally resemble Euclidean space and possess a differentiable structure.

In the context of differential geometry, acceleration refers to the derivative of the tangent vector along a curve.

The Affine Grassmannian is a mathematical object that arises in the fields of algebraic geometry and representation theory, particularly in relation to the study of loop groups and their associated geometric structures. It can be understood as a certain type of space that parametrizes collections of subspaces of a vector space that can be defined over a given field, typically associated with the field of functions on a curve.

In differential geometry, an **affine bundle** is a generalization of the concept of a vector bundle. While a vector bundle provides a way to associate a vector space to each point in a base manifold, an affine bundle allows for a more general structure, specifically associating an affine space to each point of the manifold.

An **affine connection** is a mathematical concept used primarily in differential geometry and the theory of manifolds. It provides a way to define a notion of parallel transport, which allows one to compare vectors at different points on a manifold. The affine connection also enables the definition of derivatives of vector fields along curves in a manifold.

Affine curvature is a concept from differential geometry, particularly in the study of affine differential geometry, which focuses on the properties of curves and surfaces that are invariant under affine transformations (linear transformations that preserve points, straight lines, and planes). In more detail, affine curvature pertains to the curvature of an affine connection, which is a way to define parallel transport and consequently, the notion of curvature in a space that doesn't necessarily have a metric (length) structure like Riemannian geometry.

Affine differential geometry is a branch of mathematics that studies the properties and structures of affine manifolds, which are manifolds equipped with an affine connection. Unlike Riemannian geometry, which relies on the notion of a metric to define geometric properties like lengths and angles, affine differential geometry primarily focuses on the properties that are invariant under affine transformations, such as parallel transport and affine curvature.

In the context of mathematics, particularly in geometry and algebraic geometry, an **affine focal set** typically refers to a specific type of geometric construction related to curves and surfaces in affine space. While the term isn't universally standard, it can often involve the study of points that share certain properties regarding curvature, tangency, or other geometric relationships. One common interpretation is related to **focal points** or **focal loci** which pertain to conic sections or more general curves.

Affine geometry is a branch of geometry that studies the properties of figures that remain invariant under affine transformations, which include linear transformations and translations. In the context of curves, affine geometry focuses on characteristics that do not change when a curve is subjected to such transformations.

An affine manifold is a type of manifold that is equipped with an additional structure that allows for the concepts of affine geometry to be applied. More specifically, an affine manifold is a manifold where the transition functions between charts are affine transformations. ### Key Characteristics of Affine Manifolds: 1. **Manifold Structure**: An affine manifold is a differentiable manifold, meaning it has a smooth structure and local charts that give it a topological and differentiable structure.

An affine sphere is a concept from differential geometry that relates to a certain class of surfaces in affine geometry. Specifically, an affine sphere is a surface in an affine space (a geometric setting that generalizes the properties of Euclidean spaces without the need for a fixed origin or notion of distance) that has the property that the one-parameter family of tangent planes at each point has a constant affine mean curvature. To elaborate, the affine mean curvature is a measure of how the surface bends in space.

Alexandrov's soap bubble theorem is a result in geometric measure theory that deals with the existence of minimal surfaces. Specifically, it states that any simply connected, compact surface with a boundary can be realized as the boundary of a minimizer of area among all surfaces that enclose a given volume.

An Alexandrov space is a type of metric space that satisfies certain curvature bounds. Named after the Russian mathematician P. S. Alexandrov, these spaces generalize the concept of curvatures in a way that allows for the study of geometric properties in situations where traditional Riemannian concepts might not apply.

An **almost-contact manifold** is a type of differentiable manifold equipped with a structure that is somewhat analogous to that of contact manifolds, but not quite as strong.

Analytic torsion is a concept in mathematical analysis, particularly in the fields of differential geometry and topology, relating to the behavior of certain types of Riemannian manifolds. It arises in the context of studying the spectral properties of differential operators, especially the Laplace operator.

Anti-de Sitter space (AdS) is a spacetime geometry that arises in the context of general relativity and is characterized by a constant negative curvature. It is one of the classical solutions to Einstein's field equations and is commonly used in theoretical physics, particularly in theories of gravity and in the study of gauge/gravity duality, particularly in the context of string theory and the holographic principle.

An Arithmetic Fuchsian group is a type of Fuchsian group, which is a group of isometries of the hyperbolic plane. To understand Arithmetic Fuchsian groups, it's helpful to break down the components of the term: 1. **Fuchsian Groups**: These are groups of isometries of the hyperbolic plane, which means they consist of transformations that preserve the hyperbolic metric.

An arithmetic group is a type of group that arises in the context of number theory and algebraic geometry, particularly in the study of algebraic varieties over number fields or bipartite rings. The term often refers to groups of automorphisms of algebraic structures that preserve certain arithmetic properties or structures. A common example is the **arithmetic fundamental group of a variety**, which captures information about its algebraic and topological structure.

Arthur Besse does not appear to be a widely recognized term, individual, or concept, as of my last update in October 2021. It's possible that it could refer to a private individual or a less known entity not widely covered in publicly available information.