Torsion is a measure of how a curve twists out of the plane formed by its tangent and normal vectors. In mathematical terms, torsion is defined for space curves, which are curves that exist in three-dimensional space.

Theorema Egregium, which is Latin for "Remarkable Theorem," is a fundamental result in differential geometry, particularly in the study of surfaces. It was formulated by the mathematician Carl Friedrich Gauss in 1827. The theorem states that the Gaussian curvature of a surface is an intrinsic property, meaning it can be determined entirely by measurements made within the surface itself, without reference to the surrounding space.

In the context of differential geometry and the study of manifolds, "congruence" can refer to a few different concepts based on the specific context in which it is used. However, it is not a standard term that is widely recognized across all branches of mathematics.

In differential geometry, a **translation surface** is a type of surface that can be constructed by translating a polygon in the Euclidean plane. The concept is closely related to flat surfaces and is prevalent in the study of flat geometry, especially in the context of billiards, dynamical systems, and algebraic geometry. ### Definition A translation surface is defined as a two-dimensional surface that is locally Euclidean and has a flat metric.

The Whitehead manifold is an example of a specific type of 3-manifold that is notable in the field of topology. It is particularly interesting because it is an example of a non-trivial manifold that is "homotopy equivalent" to a 3-sphere but is not homeomorphic to any standard manifold.

Yau's conjecture refers to a prediction made by the mathematician Shing-Tung Yau regarding the first eigenvalue of the Laplace operator on compact Riemannian manifolds. Specifically, the conjecture addresses the relationship between the geometry of a manifold and the spectrum of the Laplace operator defined on it.

The Wu–Yang dictionary is a conceptual framework established by Wu and Yang in the context of mathematical physics, particularly in the study of quantum field theory and the relationship between different physical theories. The dictionary helps to connect various physical concepts and structures found in different contexts, such as gauge theories, topological field theories, and string theory. This dictionary serves as a bridge between the theoretical descriptions and the corresponding mathematical structures, facilitating the understanding of how different physical phenomena relate to one another.

The Yamabe invariant is an important concept in differential geometry, particularly in the study of conformal classes of Riemannian metrics. It is named after the Japanese mathematician Hidehiko Yamabe, who contributed significantly to the field. Formally, the Yamabe invariant is defined for a compact Riemannian manifold \( M \) and is associated with the problem of finding a metric in a given conformal class that has constant scalar curvature.

The Gluing Axiom is a principle in the field of set theory and topology, particularly in the context of the definition of sheaves and bundles. It essentially relates to the ability to construct global sections or features from local data.

Differential forms are a foundational concept in differential geometry and calculus on manifolds. They provide a powerful and flexible language for discussing integration and differentiation on different types of geometric objects, particularly in multi-dimensional spaces. Here are the key ideas associated with differential forms: ### Basic Concepts 1. **Definition**: A differential form is a mathematical object that can be integrated over a manifold.

Akbulut cork refers to natural cork produced in the Akbulut region, which is known for its high-quality cork material. Cork is harvested from the bark of cork oak trees, primarily the Quercus suber species, which are predominantly found in Mediterranean regions. The Akbulut cork is recognized for its unique properties, such as being lightweight, buoyant, and resistant to water, fire, and rot.

Conley's fundamental theorem of dynamical systems, often referred to as Conley's theorem, addresses the behavior of dynamical systems, particularly focusing on asymptotic behavior and the presence of invariant sets. The theorem is part of the broader study of dynamical systems and lays the groundwork for understanding the structure of trajectories of these systems.

Atmospheric diffraction is the bending of light waves as they pass through different layers of the atmosphere with varying temperatures and densities. This phenomenon occurs due to the interaction of light with atmospheric particles and the varying refractive index of air caused by changes in temperature, pressure, and humidity. When light waves encounter obstacles or pass through apertures, they can bend around the edges, leading to effects such as the spreading of light and the formation of patterns.

Stunted projective space is a type of topological space that can be defined in the context of algebraic topology. More specifically, it involves modifying the standard projective space in a way that truncates it or "stunts" its structure.

A Lie algebra bundle is a mathematical structure that arises in the context of differential geometry and algebra. It is an extension of the concept of a vector bundle, where instead of focusing solely on vector spaces, we consider fibers that are Lie algebras. #### Components of a Lie Algebra Bundle: 1. **Base Space**: The base space is typically a smooth manifold \( M \). This space serves as the domain over which the bundle is defined.

The Poincaré–Hopf theorem is a fundamental result in differential topology that relates the topology of a compact manifold to the behavior of vector fields defined on it. Specifically, it provides a formula for the Euler characteristic of a manifold in terms of the zeros of a smooth vector field on that manifold. Here's a more detailed breakdown of the theorem’s key concepts: 1. **Setting**: Let \( M \) be a compact, oriented \( n \)-dimensional manifold without boundary.

A diffraction spike is an optical phenomenon commonly observed in photographs of bright light sources, especially stars, taken with telescopes or camera lenses that utilize a diaphragm with sharp edges. When light from these sources enters the lens system, it is diffracted—meaning it bends around the edges of the aperture (the opening through which light passes). The result of this diffraction can create streaks or spikes radiating outward from the bright light source in the image.

Coherent Diffraction Imaging (CDI) is a powerful imaging technique used primarily in the fields of materials science, biology, and nanotechnology. It allows researchers to obtain high-resolution images of the internal structures of samples without the need for lenses, which can often introduce aberrations or restrict resolution.

Convergent Beam Electron Diffraction (CBED) is a technique used in electron microscopy, particularly in the field of crystallography and materials science, to study the crystallographic structure of materials at the atomic level. It is an extension of conventional electron diffraction methods and allows for the collection of detailed information about the symmetry and electronic structure of crystals. In CBED, a focused beam of electrons is converged onto the sample, typically using a high-resolution transmission electron microscope (HRTEM).

Card Shark is a unique video game developed by Nerial and published by Devolver Digital. Released in June 2022, the game combines elements of card games and narrative-driven gameplay. Set in 18th century France, players take on the role of a young servant who gets involved in a world of high-stakes gambling. The gameplay focuses on mastering various card tricks and schemes to cheat opponents, using skill and strategy to outsmart them.