Differential geometry of singular spaces

Thread starter Korybut
Start date Nov 18, 2023
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Differential geometry Reference

Nov 18, 2023

Korybut

TL;DR Summary: Reference request

Hello!

Reading the book "Differential geometry of Singular Spaces and Reduction of symmetry" by J. Sniatycki https://www.cambridge.org/core/book...-of-symmetry/7D73498C35A5975594605428DA8F9267
I found that not every definition and statement is clear to me and alternative source of information is highly desirable. Can someone provide additional reference to this subject? Many thanks in advance

1. What is differential geometry of singular spaces?

Differential geometry of singular spaces is a branch of mathematics that studies the properties of geometric objects, such as curves and surfaces, that have singular points or regions where the usual rules of calculus do not apply. It combines techniques from differential geometry, topology, and algebraic geometry to understand the behavior of these spaces.

2. What are singular points in a geometric object?

Singular points in a geometric object are points where the object fails to be smooth or differentiable. This can happen when the tangent space is not well-defined, or when the curvature becomes infinite. These points are important to study because they can reveal interesting information about the geometry of the object.

3. What are some applications of differential geometry of singular spaces?

Differential geometry of singular spaces has many applications in various fields, including physics, engineering, and computer graphics. It is used to model and analyze complex surfaces and shapes, such as those found in nature or in man-made structures. It also has applications in the study of phase transitions and critical phenomena in statistical mechanics.

4. How is differential geometry of singular spaces related to algebraic geometry?

Differential geometry of singular spaces and algebraic geometry are closely related fields, as they both study geometric objects using algebraic techniques. In particular, algebraic geometry provides powerful tools for studying singularities in geometric objects, and many results in differential geometry of singular spaces have been obtained using algebraic methods.

5. What are some open problems in differential geometry of singular spaces?

There are many open problems in differential geometry of singular spaces, as it is a rapidly growing field with many challenging questions. Some current research topics include the classification of singularities, the study of moduli spaces of singular spaces, and the development of new techniques for understanding the geometry of these spaces. Additionally, there are many connections to other areas of mathematics, such as symplectic geometry and representation theory, that are still being explored.