Hasok Chang is a philosopher of science, particularly known for his work in the philosophy of physics and the history of science. He is a professor at the University of Cambridge and has written extensively on topics such as scientific realism, the nature of scientific knowledge, and the interactions between science and society. His research also often emphasizes the importance of historical context in understanding scientific concepts and practices.
In equivariant cohomology, the localization theorem relates the equivariant cohomology of a space to the data at fixed points under a group action.
Plücker embedding is a mathematical construction that embeds a projective space into a higher-dimensional projective space. Specifically, it is most commonly associated with the embedding of the projective space \( \mathbb{P}^n \) into \( \mathbb{P}^{\binom{n+1}{2} - 1} \) using the concept of the lines in \( \mathbb{P}^n \).
A Poisson manifold is a particular type of differentiable manifold equipped with a Poisson bracket, which is a bilinear operation that satisfies certain algebraic properties.
Regular homotopy is a concept from algebraic topology, specifically in the field of differential topology. It relates to the study of two smooth maps from one manifold to another and the idea of deforming one map into another through smooth transformations. In a more precise sense, let \( M \) and \( N \) be smooth manifolds.
The Seifert conjecture is a conjecture in the field of topology, specifically dealing with the properties of certain types of manifolds known as Seifert fibered spaces. It was proposed by the mathematician Herbert Seifert in the late 1950s. The conjecture posits that: **Every Seifert fibered manifold (which is a type of 3-manifold) has an incompressible surface.
The Serre–Swan theorem is a fundamental result in algebraic topology and differential geometry that establishes a profound connection between vector bundles and sheaves of modules.
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.
In mathematics, particularly in the field of differential geometry, a **submanifold** is a subset of a manifold that itself has the structure of a manifold, often with respect to the topology and differential structure induced from the larger manifold.
In differential geometry, the tangent bundle is a fundamental construction that enables the study of the properties of differentiable manifolds. It provides a way to associate a vector space (the tangent space) to each point of a manifold, facilitating the analytical treatment of curves, vector fields, and differential equations. ### Definition: For a differentiable manifold \( M \), the tangent bundle \( TM \) is defined as the collection of all tangent spaces at each point of \( M \).
Bordism is a concept in algebraic topology that relates to the classification of manifolds based on their "bordism" relation, which can be thought of as a way of determining whether two manifolds can be connected by a "bordism," or a higher-dimensional manifold that has the given manifolds as its boundary.
The Whitney topology is a specific topology that can be defined on the space of smooth maps (or differential functions) between two smooth manifolds, typically denoted as \(C^\infty(M, N)\), where \(M\) and \(N\) are smooth manifolds. The Whitney topology can also refer to the topology on a space of curves in a manifold, particularly when discussing the space of embeddings of one manifold into another.
The Gauss separation algorithm, often referred to in the context of numerical methods, relates to the separation of variables, particularly in the context of solving partial differential equations (PDEs) or systems of equations. However, it seems there might be a confusion, as "Gauss separation algorithm" is not a widely recognized or standard term in mathematics or numerical analysis.
Thessaly is a geographical and historical region in central Greece. It is bordered by Macedonia to the north, Epirus to the west, and Boeotia to the south. Thessaly is known for its fertile plains, which are among the most productive agricultural areas in Greece, making it a significant center for agriculture, particularly for wheat, corn, and livestock. Historically, Thessaly was important in ancient Greek times and was characterized by city-states such as Pharsalus and Larissa.
The Neovius surface refers to a specific type of mathematical surface that has properties useful in the study of differential geometry and topology. It is named after the Finnish mathematician A.F. Neovius, who studied the surface and its properties. The Neovius surface is typically characterized by its complex structure, including features like cusps and self-intersections, making it interesting from the perspectives of both geometry and mathematical physics.
The Reilly formula is a method used to estimate the probable maximum loss (PML) of a particular asset or group of assets in the context of insurance and risk management. The formula helps organizations estimate potential losses from catastrophic events like natural disasters, based on historical data, exposure factors, and other variables. While there may be variations or specific interpretations of the Reilly formula in different contexts, the general aim is to provide a statistical approach to understand potential risks and losses.
Ricci curvature is a geometric concept that arises in the study of Riemannian and pseudo-Riemannian manifolds within the field of differential geometry. It measures how much the shape of a manifold deviates from being flat in a particular way, focusing on how volumes are distorted by the curvature of the space. To define Ricci curvature, we start with the Riemann curvature tensor, which encapsulates all the geometrical information about the curvature of a manifold.
The Riemann curvature tensor is a fundamental object in differential geometry and mathematical physics that measures the intrinsic curvature of a Riemannian manifold. It provides a way to describe how the geometry of a manifold is affected by its curvature. Specifically, it captures how much the geometry deviates from being flat, which corresponds to the geometry of Euclidean space.
The Schouten–Nijenhuis bracket is an important tool in differential geometry and algebraic topology, particularly in the study of multivector fields and their relations to differential forms and Lie algebras. It generalizes the Lie bracket of vector fields to multivector fields, which are generalized objects that can be thought of as skew-symmetric tensors of higher degree. ### Definition 1. **Multivector Fields**: Let \( V \) be a smooth manifold.
The second fundamental form is a mathematical object used in differential geometry that provides a way to describe how a surface bends in a higher-dimensional space. Specifically, it is associated with a surface \( S \) embedded in a higher-dimensional Euclidean space, such as \(\mathbb{R}^3\).