In plane geometry, a quadrant refers to one of the four sections created by dividing a Cartesian coordinate plane with the x-axis and y-axis. The axes intersect at the origin (0,0), which is the point where the x and y values are both zero.

"Squaring the circle" is a classic problem in geometry that involves constructing a square with the same area as a given circle using only a finite number of steps with a compass and straightedge. More formally, it requires finding a square whose area is equal to πr², where r is the radius of the circle. The problem has its origins in ancient Greece, where it was one of the three famous problems of antiquity, alongside duplicating the cube and trisecting an angle.

The Volume Conjecture is a mathematical hypothesis related to the field of knot theory and hyperbolic geometry. It proposes a deep connection between the volumes of hyperbolic 3-manifolds and quantum invariants of knots, specifically those derived from a quantum invariant known as the Kauffman polynomial or the colored Jones polynomial.

In geometry, congruence refers to a relationship between two geometric figures in which they have the same shape and size. When two figures are congruent, one can be transformed into the other through a series of rigid motions, such as translations (shifts), rotations, and reflections, without any alteration in size or shape. Congruent figures can include various geometric objects, such as triangles, squares, circles, and polygons.

The term "double wedge" can refer to various concepts depending on the context. Here are a few interpretations: 1. **Mechanical Tool**: In mechanics or woodworking, a double wedge refers to a tool that consists of two wedge shapes often used for splitting or lifting materials. The design allows for more efficient force distribution.

In geometry, particularly in the study of figures in a plane or in space, the **homothetic center** refers to the point from which two or more geometric shapes are related through homothety (also known as a dilation). Homothety is a transformation that scales a figure by a certain factor from a fixed point, which is the homothetic center.

"On the Sphere and Cylinder" is a mathematical work by the ancient Greek philosopher and mathematician Archimedes. Written in the 3rd century BC, the treatise explores the geometric properties of spheres and cylinders, deriving formulas related to their volumes and surface areas. In the text, Archimedes examines the relationships between these shapes, showcasing his groundbreaking methods in geometry.

In the context of geometry, particularly when discussing triangles, "straight lines" generally refer to the sides of a triangle. A triangle is defined by three straight lines that connect three points, known as vertices, in a two-dimensional plane. These straight lines meet the following criteria: 1. **Straightness**: Each side is a straight line segment connecting two vertices. 2. **Consecutive**: Each side is adjacent to two other sides, forming the perimeter of the triangle.

In topology, a cofibration is a specific type of map between topological spaces that satisfies certain conditions. Cofibrations play a crucial role in homotopy theory and the study of fibration and cofibration sequences. They are often defined in terms of the homotopy extension property. ### Definition: A map \( i : A \to X \) is called a **cofibration** if it satisfies the homotopy extension property with respect to any space \( Y \).

Simple homotopy theory is a branch of algebraic topology that provides a way to study the properties of topological spaces through the lens of homotopy equivalence. It is particularly concerned with the study of CW complexes and involves a concept known as simple homotopy equivalence. ### Key Concepts 1. **Homotopy**: In general, homotopy is a relation between continuous functions, where two functions are considered equivalent if one can be transformed into the other through continuous deformation.

The Generalized Poincaré Conjecture extends the classical Poincaré Conjecture, which is a statement about the topology of 3-dimensional manifolds. The original Poincaré Conjecture, proposed by Henri Poincaré in 1904, asserts that any simply connected, closed 3-manifold is homeomorphic to the 3-sphere.

Homotopical connectivity is a concept from algebraic topology, a branch of mathematics that studies topological spaces through the lens of homotopy theory. It provides a way to classify topological spaces based on their "connectedness" in a homotopical sense. In more detail, homotopical connectivity can be understood through the following concepts: 1. **Connectedness**: A topological space is called connected if it cannot be divided into two disjoint open sets.

In topology, the localization of a topological space is a method of constructing a new topological space from an existing one by focusing on a particular subset of the original space. The concept of localization can be understood in several contexts, such as the localization of rings or sheaves, but here I will outline the localization of a topological space itself, particularly in algebraic topology. ### 1.

In category theory, an \( N \)-group is a concept that extends the notion of groups to a more general framework, particularly in the context of higher-dimensional algebra. The term "N-group" can refer to different concepts depending on the specific area of study, but it is commonly associated with the study of higher categories and homotopy theory.

The Novikov conjecture is a significant hypothesis in the field of topology and geometry, particularly concerning the relationships between the algebraic topology of manifolds and their geometric structure. It was proposed by the Russian mathematician Sergei Novikov in the 1970s. At its core, the Novikov conjecture deals with the higher dimensional homotopy theory, specifically the relationship between the homotopy type of a manifold and the groups of self-homotopy equivalences of the manifold.

In mathematics, particularly in the field of algebraic topology, a **simplicial space** is a topological space that is equipped with a simplicial structure. More specifically, a simplicial space is a contravariant functor from the simplex category, which comprises simplices of various dimensions and their face and degeneracy maps, to the category of topological spaces.

Willerton's fish (scientific name: *Sicyopterus williardsoni*) is a species of freshwater fish belonging to the family Gobiidae. It is notable for its unique characteristics, including its small size and adaptations to a specific habitat. Willerton's fish is primarily found in the streams and rivers of tropical regions, often inhabiting areas with rocky substrates and fast-flowing waters.

A **circle bundle** is a specific type of fiber bundle in differential geometry, where the fibers are circles \( S^1 \).

In the context of quantum mechanics and quantum field theory, the term "quantum invariant" generally refers to a property or quantity that remains unchanged under certain transformations or changes in the system. Here are some key points regarding quantum invariants: 1. **Symmetry and Invariance**: Quantum invariants often relate to symmetries in physical systems.

In mathematics, particularly in the fields of algebraic geometry and representation theory, the term "norm variety" has specific meanings depending on the context. However, without a specified context, it might refer to a couple of different concepts related to norms in algebraic settings or varieties in algebraic geometry. 1. **In Algebraic Geometry**: The notion of a "variety" often pertains to a geometric object defined as the solution set of polynomial equations.