Related papers: A Course on Derived Categories
Firstly, we compare the bounded derived categories with respect to the pure-exact and the usual exact structures, and describe bounded derived category by pure-projective modules, under a fairly strong assumption on the ring. Then, we study…
Categories of W*-bimodules are shown in an explicit and algebraic way to constitute an involutive W*-bicategory.
These are course notes I wrote for my Fall 2013 graduate topics course on geometric structures, taught at ICERM. The notes rework many of proofs in William P. Thurston's beautiful but hard-to-understand paper, "Shapes of Polyhedra". A…
We prove that the adjoint equivariant derived category of a reductive group $G$ is equivalent to the appropriately defined monoidal center of the torus-equivariant version of the Hecke category. We use this to give new proofs, independent…
We study aisles in the derived category of a hereditary abelian category. Given an aisle, we associate a sequence of subcategories of the abelian category by considering the different homologies of the aisle. We then obtain a sequence,…
Gradient-based techniques are becoming increasingly critical in quantitative fields, notably in statistics and computer science. The utility of these techniques, however, ultimately depends on how efficiently we can evaluate the derivatives…
We sketch the construction of a derived enhancement of the reciprocity isomorphism of class field theory. Details will appear in a forthcoming joint paper of the authors with A. Raksit.
We classify the thick subcategories of discrete derived categories. To do this we introduce certain generating sets called arc-collections which correspond to configurations of non-crossing arcs on a geometric model. We show that every…
These are notes from elementary lectures given in the summer of 2013 at the YMSC center at Tsinghua University in Beijing.
This short paper discusses continually updated causal abstractions as a potential direction of future research. The key idea is to revise the existing level of causal abstraction to a different level of detail that is both consistent with…
These are the lecture notes for an advanced Ph.D. level course I taught in Spring'02 at the C.N. Yang Institute for Theoretical Physics at Stony Brook. The course primarily focused on an introduction to stochastic calculus and derivative…
It is known that finite crossed modules provide premodular tensor categories. These categories are in fact modularizable. We construct the modularization and show that it is equivalent to the module category of a finite Drinfeld double.
A near-group category is an additively semisimple category with a product such that all but one of the simple objects is invertible. We classify braided structures on near-group categories, and give explicit numerical formulas for their…
We define a notion of categorical first order deformations for (enhanced) triangulated categories. For a category $\mathcal{T}$, we show that there is a bijection between $\operatorname{HH}^2(\mathcal{T})$ and the set of categorical…
We study derived categories of coherent sheaves on abelian varieties. We give a criterion for the equivalence of the derived categories on two abelian varieties. We describe the autoequivalence group for the derived category of coherent…
We investigate the correspondence between generalized persistence modules and graded modules in the case the indexing set has a monoid action. We introduce the notion of an action category over a monoid graded ring. We show that the…
The notes were prepared for a series of talks that I gave in Hagen in late June and early July 2003, and, with some changes, in the University of La Lagu\~{n}a, the Canary Islands, in September, 2003. They aim (i) to revisit some oldish…
These are the notes corresponding to the course given at the IAS-Park City graduate summer school in July 2007.
We propose a solution to the "curvature problem" from arXiv:1505.03698 and arXiv:0905.3845 for infinitesimal deformations. Let $k$ be a field, $A$ a dg algebra over $k$ and $A_n = A[t]/(t^{n+1})$ a cdg algebra over $R_n = k[t]/(t^{n+1})$,…
A brief introduction to exterior differential systems for graduate students familiar with manifolds and differential forms. For complete files, see https://github.com/Ben-McKay/introduction-to-exterior-differential-systems