Project Details
Description
Calculus is one of the most important, successful developments, and perhaps the most applied area of mathematics. Since calculus is revered for its place in applied mathematics, one rarely considers its foundations. The foundational perspective broadens our view and understanding of calculus: why it works and why it behaves the way it does. A foundational description of calculus provides a way of transporting ideas from calculus such as differentiation to settings where there is no sensible notion of limit. As such, this allows for the construction and study of generalized models of differential calculus in a wide variety of fields, of which classical multivariable differential calculus is but one example.
The theory of differential categories uses category theory to study the foundations of differential calculus in a variety of interesting settings from commutative algebra, differential geometry, and other areas of pure mathematics. Most recently, the theory of differential categories has gained interest and popularity in theoretical computer science due to its applications in differentiable programming, machine learning, and various other areas.
The main theoretical objective of this research proposal is to expand upon the theory of both forward differential categories and reverse differential categories. This will answer important open questions that remain for (reverse) differential categories, such as completing the map of differential categories and constructing cofree reverse differential categories, as well as pushing the overall field into new and exciting directions, such as reverse differential geometry and fixed point operators. The main practical objective of this research program is to use reverse differential categories for the foundations of automatic differentiation and develop novel models for machine learning. In particular, we will make use of reverse tangent categories to provide new models of machine learning based on smooth manifolds and other suitable models of differential geometry.
The theory of differential categories uses category theory to study the foundations of differential calculus in a variety of interesting settings from commutative algebra, differential geometry, and other areas of pure mathematics. Most recently, the theory of differential categories has gained interest and popularity in theoretical computer science due to its applications in differentiable programming, machine learning, and various other areas.
The main theoretical objective of this research proposal is to expand upon the theory of both forward differential categories and reverse differential categories. This will answer important open questions that remain for (reverse) differential categories, such as completing the map of differential categories and constructing cofree reverse differential categories, as well as pushing the overall field into new and exciting directions, such as reverse differential geometry and fixed point operators. The main practical objective of this research program is to use reverse differential categories for the foundations of automatic differentiation and develop novel models for machine learning. In particular, we will make use of reverse tangent categories to provide new models of machine learning based on smooth manifolds and other suitable models of differential geometry.
| Status | Active |
|---|---|
| Effective start/end date | 15/11/23 → 14/11/28 |