The Cotangent complex is a fundamental construction in algebraic geometry and homotopy theory, especially within the context of derived algebraic geometry. It can be seen as a tool to study the deformation theory of schemes and their morphisms.

The Whitehead product is a concept from algebraic topology, specifically in the context of algebraic K-theory and homotopy theory. It is named after the mathematician G. W. Whitehead and plays a significant role in the study of higher homotopy groups and the structure of loop spaces. In general, the Whitehead product is a binary operation that can be defined on the homotopy groups of a space.

The Atiyah–Hirzebruch spectral sequence is an important tool in algebraic topology, specifically in the computation of homotopy groups and cohomology theories. It provides a way to calculate the homology or cohomology of a space using a spectral sequence that is associated with a specific filtration. The original context for the spectral sequence primarily relates to complex vector bundles and characteristic classes.

The Milnor conjecture, proposed by John Milnor in the 1950s, is a statement in the field of algebraic topology, particularly concerning the nature of the relationship between the topology of smooth manifolds and algebraic invariants known as characteristic classes. The conjecture specifically relates to the Milnor's "h-cobordism" theorem and the properties of the "stable" smooth structures on high-dimensional manifolds.

Knot theory is a branch of mathematics that studies knots, their properties, and the various ways they can be manipulated and classified. Here is a list of topics within knot theory: 1. **Basic Concepts** - Knots and links: Definitions and examples - Open and closed knots - Tangles - Reidemeister moves - Knot diagrams 2. **Knot Invariants** - Fundamental group - Knot polynomials (e.g.

The term "Ribbon category" could refer to different concepts depending on the context in which it is used. However, it is often associated with specific types of user interface design, data visualization, or organizational structures. Below are a few interpretations: 1. **User Interface Design**: In software applications, a "ribbon" refers to a graphical control element in the form of a set of toolbars placed on several tabs.

Esakia duality is a correspondence between two categories: the category of certain topological spaces (specifically, spatial modal algebras) and the category of certain algebraic structures known as frame homomorphisms. This duality is named after the mathematician Z. Esakia, who developed the theory in the context of modal logic and topological semantics.

A **convex cap** typically refers to a mathematical concept used in various fields, including optimization and probability theory. However, the term might also be context-specific, so I’ll describe its uses in different areas: 1. **Mathematics and Geometry**: In geometry, a convex cap can refer to the convex hull of a particular set of points, which is the smallest convex set that contains all those points.

Verdier duality is a concept from the field of algebraic geometry and consists of a duality theory for sheaves on a topological space, particularly in the context of schemes and general sheaf theory. It is named after Jean-Louis Verdier, who developed this theory in the context of derived categories. At its core, Verdier duality provides a way to define a duality between certain categories of sheaves.

In category theory, the concept of an **end** is a particular construction that arises when dealing with functors from one category to another. Specifically, an end is a way to "sum up" or "integrate" the values of a functor over a category, similar to how an integral works in calculus but in a categorical context.

Ind-completion is a concept from the field of category theory, specifically related to the completion of a category with respect to a certain type of structure or property. In mathematical contexts, "ind-completion" often refers to a way of completing a category by formally adding certain limits or colimits.

A string diagram is a visual representation used in various fields, most prominently in mathematics and physics, particularly in category theory and string theory. The term may be interpreted in different contexts, but here are the two primary uses: 1. **String Diagrams in Category Theory**: - In category theory, string diagrams are a way to visualize morphisms (arrows) and objects (points) within a category.

In category theory, a coproduct is a generalization of the concept of a disjoint union of sets, and more broadly, it can be thought of as a way to combine objects in a category. The coproduct of a collection of objects provides a means of "merging" these objects while preserving their individual identities.

The Monster group, denoted as \( \mathbb{M} \) or sometimes \( \text{Mon} \), is the largest of the 26 sporadic simple groups in group theory, a branch of mathematics that studies algebraic structures known as groups. It was first discovered by Robert Griess in 1982 and has a rich structure that connects various areas of mathematics, including number theory, geometry, and mathematical physics.

A *free-by-cyclic group* is a specific type of group that can be thought of as a combination of two structures: a free group and a cyclic group. More formally, a free-by-cyclic group is a group \( G \) that can be expressed in the form: \[ G = F \rtimes C \] where \( F \) is a free group and \( C \) is a cyclic group.

In the context of group theory, specifically in the study of automorphisms of algebraic structures, a **fully irreducible automorphism** generally refers to a certain type of automorphism of a free group or a free object in category theory.

The Gromov boundary is a concept in geometric topology, particularly in the study of metric spaces, especially those that are geodesic and hyperbolic. It is used to analyze the asymptotic behavior of spaces and to understand their large-scale geometry. More formally, the Gromov boundary can be defined for a proper geodesic metric space. A metric space is considered proper if every closed ball in the space is compact.

The Švarc–Milnor lemma is a result in differential geometry and algebraic topology, particularly concerning the relationship between the topology of a space and the geometry of its covering spaces. It is named after mathematicians David Švarc and John Milnor.