The term "worldly cardinal" isn't widely recognized in common discourse or established literature, so it could refer to different concepts depending on context. However, it seems to suggest two distinct meanings: 1. **Religious Context**: In a more traditional sense, a "cardinal" typically refers to a high-ranking official in the Roman Catholic Church, a cardinal is a member of the clergy who is appointed by the Pope and is eligible to participate in papal elections.

T. M. Scanlon, or Thomas M. Scanlon, is an American philosopher known for his work in moral philosophy and political philosophy. He has made significant contributions to the understanding of moral reasoning, contractualism, and the nature of rights and obligations.

Alexander S. Kechris is a prominent mathematician known for his contributions to set theory and its connections to other areas of mathematics, particularly in model theory and descriptive set theory. He has published numerous research papers and has co-authored influential texts, including works on the structure of the real line and on the foundations of set theory. Kechris is known for his rigorous approach to mathematics and has made significant contributions to the understanding of topological groups and their classifications.

Frederick Rowbottom does not appear to be a widely recognized public figure or topic based on my knowledge up to October 2021. It's possible that he could be a private individual, a lesser-known historical figure, or a character in a specific context.

Gottlob Frege (1848–1925) was a German philosopher, logician, and mathematician, widely regarded as one of the founding figures of modern logic and analytic philosophy. His work primarily focused on the philosophy of language, mathematics, and logic, and he made significant contributions to the foundations of mathematics.

Itay Neeman is a mathematician known for his work in the fields of model theory, set theory, and descriptive set theory. His research often involves topics like the interaction between logic and other areas of mathematics, including analysis and topology. He has produced a number of important results and publications in these areas.

Keith Devlin is a British mathematician, author, and educator known for his work in mathematics communication and mathematics education. He is a prominent advocate for the importance of mathematics in everyday life and has been involved in various efforts to enhance public understanding of mathematics. Devlin has written numerous books and articles, including works aimed at general audiences as well as those focused on mathematics education for teachers and students.

"Leo Harrington" typically refers to a theorem or a concept in mathematical logic, specifically related to set theory. The Leo-Harrington principle is a powerful result in model theory, dealing with possible extensions of structures in certain set-theoretic contexts. It is named after mathematicians Raymond Leo and Philip Harrington, who contributed to this area of mathematics.

In group theory, the term "component" can refer to various concepts depending on the context. However, one common usage pertains to the component of a group element in a topological or algebraic sense. 1. **Connected Components in Topological Groups**: In the context of topological groups, the component of a group element \( g \) refers to the connected component of the identity element that contains \( g \).

In geometry, a line is a fundamental concept that represents a straight one-dimensional figure that extends infinitely in both directions. It has no thickness, width, or curvature, and is typically defined by at least two points. Lines can be described using a variety of properties: 1. **Definition**: A line is determined by any two distinct points on it.

In geometry, the term "parallel" refers to two or more lines or planes that are the same distance apart at all points and do not meet or intersect, no matter how far they are extended. This property is fundamental in understanding the behavior of lines within Euclidean geometry. ### Key Properties of Parallel Lines: 1. **Equidistant**: Parallel lines maintain a constant distance from each other, meaning the distance between them remains consistent along their entire length.

In geometry, a "slab" typically refers to a three-dimensional shape that is essentially a thick, flat object bounded by two parallel surfaces. This can be visualized as a rectangular prism with very small height relative to its length and width, resembling a sheet or a plate. In a more formal mathematical context, particularly in the study of convex geometry, a slab can be defined by two parallel hyperplanes in higher-dimensional spaces.

A spherical shell is a three-dimensional hollow structure that is shaped like a sphere. It is typically defined as the space between two concentric spherical surfaces — an outer surface and an inner surface. The shell has a certain thickness, which is the difference between the radii of the outer and inner surfaces. Key characteristics of a spherical shell include: 1. **Outer Radius (R_outer)**: The radius of the outer surface of the shell.

Bredon cohomology is a type of cohomology theory that is particularly useful in the context of spaces with group actions. It was introduced by Glen Bredon in the 1960s and is designed to study topological spaces with an additional structure of a group action, often leading to insights in equivariant topology.

De Rham cohomology is a mathematical concept from the field of differential geometry and algebraic topology that studies the topology of smooth manifolds using differential forms. It provides a bridge between analysis and topology by utilizing the properties of differential forms and their relationships through the exterior derivative. ### Key Concepts 1. **Differentiable Manifolds**: A differentiable manifold is a topological space that is locally similar to Euclidean space and has a well-defined notion of differentiability.

Elliptic cohomology is a branch of algebraic topology that generalizes classical cohomology theories using the framework of elliptic curves and modular forms. It is an advanced topic that blends ideas from algebraic geometry, number theory, and homotopy theory. ### Key Features 1.

Lie algebra cohomology is a mathematical concept that arises in the study of Lie algebras, which are algebraic structures used extensively in mathematics and physics to describe symmetries and conservation laws. Cohomology, in this context, refers to a homological algebra framework that helps in analyzing the structure and properties of Lie algebras.

In the context of cohomology, a pullback is a construction that allows you to take a cohomology class on a target space and "pull it back" to a cohomology class on a domain space via a continuous map. This is particularly common in algebraic topology and differential geometry. ### Formal Definition Let \( f: X \to Y \) be a continuous map between two topological spaces \( X \) and \( Y \).

Witt vector cohomology is a tool in algebraic geometry and number theory that utilizes Witt vectors to study the cohomological properties of schemes in the context of p-adic cohomology theories. Witt vectors are a generalization of the notion of numbers in a ring, particularly for fields of characteristic \( p \), and they allow the construction of an effective cohomology theory that preserves useful algebraic properties. ### Key Concepts 1.