Related papers: Derived Categories and Birational Geometry
This is a draft of an article to appear in the October 2022 issue of the Notices of the AMS. In this survey article we explore a fascinating area called descriptive combinatorics and its recently discovered connections to distributed…
These lecture notes for the IAS/Park City Graduate Summer School in Geometric Combinatorics (July 2004) provide an overview of root systems, generalized associahedra, and the combinatorics of clusters. Lectures 1-2 cover classical material:…
This is a survey on the topic explained in the title, for the proceedings on the K-theory 1997 summer institute in Seattle.
This is a survey of M-regularity for coherent sheaves on abelian varieties and its applications, based on lectures given by the second author at the Seattle conference, in August 2005, and at the Luminy GAC workshop in October 2005. Section…
Categorial methods for generating new local algebras from old ones are presented. A direct proof of the differential structure of the prolongations of a manifold is proposed.
This document is an informal bibliography of the papers dealing with distributed approximation algorithms. A classic setting for such algorithms is bounded degree graphs, but there is a whole set of techniques that have been developed for…
Lecture notes of a course on birational geometry (taught at College de France, Winter 2011, with the support of Fondation Sciences Math\'ematiques de Paris). Topics covered: introduction into the subject, contractions and extremal rays,…
The development of mathematics has been characterized by the increasing interconnectivity of seemingly separate disciplines. Such interplay has been facilitated by a massive development in formalism; category theory has provided a common…
This version corrects a wrong proof of Proposition 6.3.2 and simplifies the exposition in Section 6.
Notes from lectures given at the Autumn School on Algebraic and Arithmetic Geometry at the Johannes Gutenberg-Universit\"at Mainz in October 2017.
Awfully idiosyncratic lecture notes from CMI summer school in arithmetic geometry July 31-August 4, 2006. Does not include: rationality problems, techniques of the minimal model problem and much of the rest. Includes: Lecture 0: geometry…
We explore connections between birational anabelian geometry and abstract projective geometry. One of the applications is a proof of a version of the birational section conjecture.
In this paper, we describe a general theory of "spaces with structure sheaves." Specializations of this theory include the classical theory of schemes, the theory of Deligne-Mumford stacks, and their derived generalizations.
This paper will appear in the Proceedings of the 1995 Santa Cruz Summer Institute. The paper is a survey of recent developments in the theory of toric varieties, including new constructions of toric varieties and relations to symplectic…
We prove descent theorems for semiorthogonal decompositions using techniques from derived algebraic geometry. Our methods allow us to capture more general filtrations of derived categories and even marked filtrations, where one descends not…
This is the text of a series of five lectures given by the author at the "Second Annual Spring Institute on Noncommutative Geometry and Operator Algebras" held at Vanderbilt University in May 2004. It is meant as an overview of recent…
On August 5, 2005 in the American Mathematical Society Summer Institute on Algebraic Geometry in Seattle and later in several conferences I gave lectures on my analytic proof of the finite generation of the canonical ring for the case of…
Moduli spaces of sheaves and Hilbert schemes of points have experienced a recent resurgence in interest in the past several years, due largely to new techniques arising from Bridgeland stability conditions and derived category methods. In…
This is the fifth article in the Derived Langlands series which consists of one monograph and four articles. In this article I describe the Hopf algebra and Positive Selfadjoint Hopfalgebra (PSH) aspects to classification of a number of new…
This dissertation presents a multifaceted look into the structural decomposition of permutation classes. The theory of permutation patterns is a rich and varied field, and is a prime example of how an accessible and intuitive definition…