Asymptotic dimension is a concept from geometric topology and metric geometry that provides a way to measure the "size" or "dimension" of a metric space in a manner that is sensitive to the space's large-scale structure. It was introduced by the mathematicians J. M. G. B. Connes and more extensively developed by others in the context of spaces that arise in analysis, algebra, and topology.

The Curve Complex is a mathematical structure used in the field of low-dimensional topology, particularly in the study of surfaces. It provides a combinatorial way to study the mapping class group of a surface, which is the group of isotopy classes of homeomorphisms of the surface.

Flexagon is a term that can refer to a few different concepts, depending on the context. However, it is most commonly recognized in the following ways: 1. **In Mathematics**: A flexagon is a type of flexible polygonal structure that can be manipulated to reveal different faces.

A Vandermonde polynomial is a type of polynomial that arises in various areas of mathematics, particularly in interpolation and number theory.

Vieta's formulas are a set of relations in algebra that relate the coefficients of a polynomial to sums and products of its roots. They are particularly useful in the context of polynomial equations.

The term "free factor complex" often arises in the context of group theory, particularly in the study of free groups and their actions. A free group is a group that has a basis such that every element can be uniquely expressed as the product of finitely many basis elements and their inverses.

This is an unsolvable problem because there's no way to extract large amounts of value from a person without cruelty, and even with cruelty, there's a limit to how much you can extract.

A Følner sequence is a concept from the field of mathematical analysis, particularly in ergodic theory and group theory. It is named after the mathematician Ernst Følner. A Følner sequence provides a way to study the asymptotic behavior of actions of groups on sets and is often used in the context of amenable groups.

A geometric group action is a specific type of action by a group on a geometric space, which can often be thought of in terms of symmetries or transformations of that space. More formally, if we have a group \( G \) and a geometric object (often a topological space or manifold) \( X \), a geometric group action is defined when \( G \) acts on \( X \) in a way that respects the structure of \( X \).

A "graph of groups" is a combinatorial and algebraic structure that can be used to study groups, particularly in the context of group theory and geometric topology. It is a way to construct larger groups from smaller ones by specifying how they are connected through a graph. ### Components of a Graph of Groups: 1. **Graph**: A graph \( G \) consists of vertices (also called nodes) and edges connecting them.

Price fixing would require every single company selling something to cooperate. And if they do, another person sick of them can start a new company and undercut them. Price fixing would also destroy any goodwill towards these companies and many customers wouldn't patronize them even if they lowered their prices again.

The Grushko theorem is a result in the field of group theory, particularly concerning free groups and their subgroups. It provides a criterion to establish whether a given group is free and helps characterize the structure of free groups.

The Haagerup property, also known as being "exact," refers to a specific geometric property of certain groups or von Neumann algebras in the context of functional analysis and noncommutative geometry. It is named after Danish mathematician Uffe Haagerup, who first introduced the concept in the context of von Neumann algebras.

In mathematics, "outer space" typically refers to a certain type of geometric space associated with free groups and their actions. The most common reference is to "Outer space" denoted as \( \mathcal{O}(F_n) \), which is the space of marked metric graphs that correspond to the free group \( F_n \) of rank \( n \).

The term "Rips machine" could refer to several things, but in a common context, it often relates to a "Rips" machine used for a specific purpose in various industries. Here are some possibilities: 1. **Rips Software**: In computational topology, Rips complexes are used to study metric spaces. A machine or software that implements Rips complexes allows researchers to analyze the structure and properties of data using topological methods.

Stallings' theorem concerns the structure of finitely generated groups in relation to their ends. In topology, the "ends" of a space can intuitively be understood as the number of "directions" in which the space can be infinitely extended. For groups, ends are related to how a group's Cayley graph behaves at infinity.

Intersection theory is a branch of algebraic geometry that studies the intersection of subvarieties within algebraic varieties. It provides a framework for counting the number of points at which varieties intersect, understanding their geometric properties, and understanding how these intersections behave under various operations. Here are the main concepts involved in intersection theory: 1. **Subvarieties**: In algebraic geometry, a variety can be thought of as a solution set to a system of polynomial equations.

The triangle inequalities are fundamental properties of triangles related to the lengths of their sides. They state that, for any triangle with sides of lengths \(a\), \(b\), and \(c\), the following inequalities must hold: 1. \(a + b > c\) 2. \(a + c > b\) 3.