Systolic geometry is a branch of differential geometry and topology that primarily studies the relationship between the geometry of a manifold and the topology of the manifold. It focuses on the concept of "systoles," which are defined as the lengths of the shortest non-contractible loops in a given space. More formally, for a given manifold, the systole is the infimum of the lengths of all non-contractible loops.

A **transition system** is a mathematical model used to describe the behavior of a system in terms of states and transitions. It is particularly useful in fields such as computer science, particularly in the study of formal verification, automata theory, and modeling dynamic systems. A transition system is formally defined as a tuple \( T = (S, S_0, \Sigma, \rightarrow) \), where: - \( S \): A set of states.

A **Lie groupoid** is a mathematical structure that generalizes the notion of a Lie group and captures certain aspects of differentiable manifolds and group theory. It provides a framework for studying categories of manifolds where both the "objects" and "morphisms" have smooth structures, and it is particularly useful in the study of differential geometry and mathematical physics. Here are the key components and concepts related to Lie groupoids: ### Components of a Lie Groupoid 1.

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.

A **Banach bundle** is a mathematical structure that generalizes the concept of a vector bundle where the fibers are not merely vector spaces but complete normed spaces, specifically Banach spaces. To understand the definition and properties of a Banach bundle, let’s break it down: 1. **Base Space**: Like any bundle, a Banach bundle has a base space, which is typically a topological space. This is commonly denoted by \( B \).

The Bel-Robinson tensor is a mathematical object in general relativity that is used to describe aspects of the gravitational field in a way that is similar to how the energy-momentum tensor describes matter and non-gravitational fields. Specifically, the Bel-Robinson tensor is an example of a pseudo-tensor that represents the gravitational energy and momentum in a localized manner.

In mathematics, a caustic refers to a curve or surface that is generated by the envelope of light rays refracted or reflected by a surface, such as a lens or mirror. The term is often used in optics, particularly in the study of how light behaves when it interacts with curved surfaces.

The Chern–Weil homomorphism is a fundamental concept in differential geometry and algebraic topology that establishes a connection between characteristic classes of vector bundles and differential forms on manifolds. It provides a way to compute characteristic classes, which are topological invariants that classify vector bundles over a manifold, by using the curvature of connections on those bundles.

A complex manifold is a type of manifold that, in addition to being a manifold in the topological sense, has a structure that allows for the use of complex numbers in its local coordinates. More formally, a complex manifold is defined as follows: 1. **Manifold Structure**: A complex manifold \( M \) is a topological space that is locally homeomorphic to open subsets of \( \mathbb{C}^n \) (for some integer \( n \)).

Diffeology is a branch of mathematics that generalizes the notion of smooth manifolds. It was introduced by Jean-Marie Dufour and his collaborators in the 1980s to provide a more flexible framework for studying smooth structures on spaces that may not have a well-defined manifold structure. In traditional differential geometry, a smooth manifold is defined as a topological space that locally resembles Euclidean space and has a compatible smooth structure.

As of my last update in October 2023, "Diffiety" does not appear to be a widely recognized term in academic or popular culture. It's possible that it could be a misspelling, a new concept, or a niche term that has emerged after my last update.

In mathematics, particularly in the fields of convex analysis and differential geometry, the term "dual curve" can refer to different concepts depending on the context in which it is used. Here are a few interpretations: 1. **Dual Curves in Projective Geometry**: In projective geometry, the duality principle states that points and lines can be interchanged. The dual curve of a given curve can be constructed where each point on the dual curve represents a line tangent to the original curve.

The generalized flag variety is a geometric object that arises in the context of algebraic geometry and representation theory. It can be thought of as a space that parameterizes chains of subspaces of a given vector space, analogous to how a projective space parameterizes lines through the origin in a vector space.

The exterior covariant derivative is a concept that arises in differential geometry, particularly in the context of differential forms on a manifold. It generalizes the idea of a standard exterior derivative, which is a way to differentiate differential forms, by incorporating the notion of a connection (or a covariant derivative) to account for possible curvature in the underlying manifold. ### Key Concepts: 1. **Differential Forms**: - Differential forms are objects in a manifold that can be integrated over submanifolds.

A \( G_2 \) manifold is a specific type of differentiable manifold that admits a particular geometric structure characterized by a special kind of 3-form, which leads to a unique relationship between its differential geometry and algebraic topology. More technically, \( G_2 \) can be understood in the context of the theory of connections and holonomy groups.

The Grassmannian is a fundamental concept in the field of mathematics, particularly in geometry and linear algebra. More formally, the Grassmannian \( \text{Gr}(k, n) \) is a space that parameterizes all \( k \)-dimensional linear subspaces of an \( n \)-dimensional vector space. Here, \( k \) and \( n \) are non-negative integers with \( 0 \leq k \leq n \).

The Hopf conjecture is a statement in differential geometry and topology that concerns the curvature of Riemannian manifolds. More specifically, it was proposed by Heinz Hopf in 1938. The conjecture states that if a manifold is a compact, oriented, and simply connected Riemannian manifold of even dimension, then its total scalar curvature is non-negative.

An inflection point is a point on a curve where the curvature changes sign. In other words, it is a point at which the curve transitions from being concave (curved upwards) to convex (curved downwards), or vice versa. This concept is crucial in calculus and helps in understanding the behavior of functions. In mathematical terms, for a function \( f(x) \): 1. The second derivative \( f''(x) \) exists at the point of interest.

Isothermal coordinates refer to a specific type of coordinate system used in differential geometry, particularly in the study of surfaces and Riemannian manifolds. These coordinates are characterized by their property that the metric induced on the surface can be expressed in a particularly simple form.

The Lanczos tensor, often referred to in the context of numerical linear algebra and more specifically in the Lanczos algorithm, is associated with the process of reducing large symmetric matrices to tridiagonal form. The Lanczos algorithm is used to find the eigenvalues and eigenvectors of large, sparse symmetric matrices, which often arise in various fields like quantum mechanics, structural engineering, and machine learning.