Related papers: The universal property of derived geometry
This is an invited contribution to the 2nd edition of the Encyclopedia of Mathematical Physics, that provides a very short survey of derived symplectic geometry. Derived symplectic geometry studies symplectic structures on derived stacks.…
We introduce a notion of global dimension for a triangulated category relative to a compact silting object. We prove that the finiteness of this dimension is an intrinsic property of the triangulated category itself and, therefore,…
Variational analysis presents a unified theory encompassing in particular both smoothness and convexity. In a Euclidean space, convex sets and smooth manifolds both have straightforward local geometry. However, in the most basic hybrid case…
This work is the first in a series laying the foundations of derived geometry in the $C^{\infty}$ setting, and providing tools for the construction and study of moduli spaces of solutions of Partial Differential Equations that arise in…
A different proof to a known criterion of derived equivalence implying birationality is given. Derived equivalent smooth projective curves over an algebraically closed field are proved to be isomorphic. A different proof of derived…
We establish universal approximation theorems for infinite-dimensional geometric rough paths, i.e., we show that continuous functions on the space of infinite-dimensional weakly geometric H\"older continuous rough paths can be approximated…
We prove smoothness in the dg sense of the bounded derived category of finitely generated modules over any finite-dimensional algebra over a perfect field, hereby answering a question of Iyama. More generally, we prove this statement for…
This is a book on derived foliations, that are a generalisation of classical foliations in the context of derived geometry. The text starts with the basic definitions and constructions, then explore foliated cohomology (with crystal…
We study the properties of the set where a generalized function of bounded variation has infinite approximate limit, highlighting in this way the main geometric difference with functions of bounded variation. To this aim we prove a new…
We prove that the dg category of perfect complexes on a smooth, proper Deligne-Mumford stack over a field of characteristic zero is geometric in the sense of Orlov, and in particular smooth and proper. On the level of triangulated…
Distance Geometry is based on the inverse problem that asks to find the positions of points, in a Euclidean space of given dimension, that are compatible with a given set of distances. We briefly introduce the field, and discuss some open…
We define the Deligne Mumford orbifold axiomatically by a universal mapping property, show that this universal mapping property is equivalent to an infinitessimal universal mapping property, and use the latter to give an existence proof.
By a theorem due to the first author, the bounded derived category of a finite-dimensional algebra over a field embeds fully faithfully into the stable category over its repetitive algebra. This embedding is an equivalence iff the algebra…
Some notions of algebraic geometry can be defined for arbitrary varieties of algebras. This leads to universal algebraic geometry. The main idea of the presented theory is to consider interactions between algebra, logic and geometry in…
We introduce a novel formulation for geometry on discrete points. It is based on a universal differential calculus, which gives a geometric description of a discrete set by the algebra of functions. We expand this mathematical framework so…
In this article we prove some previously announced results about metric ultraproducts of finite simple groups. We show that any non-discrete metric ultraproduct of alternating or special linear groups is a geodesic metric space. For more…
A geometric grid class consists of those permutations that can be drawn on a specified set of line segments of slope \pm1 arranged in a rectangular pattern governed by a matrix. Using a mixture of geometric and language theoretic methods,…
In this paper we investigate homologically finite-dimensional objects in the derived category of a given small dg-enhanced triangulated category. Using these we define reflexivity, hfd-closedness, and the Gorenstein property for…
The notion of geometric version of an infinitely divisible law is introduced. Concepts parallel to attraction and partial attraction are developed and studied in the setup of geometric summing of random variables.
These notes explore some aspects of formal derived geometry related to classical field theory. One goal is to explain how many important classical field theories in physics -- such as supersymmetric gauge theories and supersymmetric…