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394
Homotopical Algebraic Geometry I: Topos theory
, 2002
"... This is the first of a series of papers devoted to lay the foundations of Algebraic Geometry in homotopical and higher categorical contexts. In this first part we investigate a notion of higher topos. For this, we use Scategories (i.e. simplicially enriched categories) as models for certain kind of ..."
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Cited by 29 (18 self)
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This is the first of a series of papers devoted to lay the foundations of Algebraic Geometry in homotopical and higher categorical contexts. In this first part we investigate a notion of higher topos. For this, we use Scategories (i.e. simplicially enriched categories) as models for certain kind of ∞categories, and we develop the notions of Stopologies, Ssites and stacks over them. We prove in particular, that for an Scategory T endowed with an Stopology, there exists a model
Compact moduli of plane curves
, 2004
"... We construct a compactification Md of the moduli space of plane curves of degree d. We regard a plane curve C ⊂ P 2 as a surfacedivisor pair (P 2, C) and define Md as a moduli space of pairs (X, D) where X is a degeneration of the plane. We show that, if d is not divisible by 3, the stack Md is smo ..."
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Cited by 27 (6 self)
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We construct a compactification Md of the moduli space of plane curves of degree d. We regard a plane curve C ⊂ P 2 as a surfacedivisor pair (P 2, C) and define Md as a moduli space of pairs (X, D) where X is a degeneration of the plane. We show that, if d is not divisible by 3, the stack Md is smooth and the degenerate surfaces X can be described explicitly.
Smashing Subcategories And The Telescope Conjecture  An Algebraic Approach
 Invent. Math
, 1998
"... . We prove a modified version of Ravenel's telescope conjecture. It is shown that every smashing subcategory of the stable homotopy category is generated by a set of maps between finite spectra. This result is based on a new characterization of smashing subcategories, which leads in addition to a cl ..."
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Cited by 25 (6 self)
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. We prove a modified version of Ravenel's telescope conjecture. It is shown that every smashing subcategory of the stable homotopy category is generated by a set of maps between finite spectra. This result is based on a new characterization of smashing subcategories, which leads in addition to a classification of these subcategories in terms of the category of finite spectra. The approach presented here is purely algebraic; it is based on an analysis of pureinjective objects in a compactly generated triangulated category, and covers therefore also situations arising in algebraic geometry and representation theory. Introduction Smashing subcategories naturally arise in the stable homotopy category S from localization functors l : S ! S which induce for every spectrum X a natural isomorphism l(X) ' X l(S) between the localization of X and the smash product of X with the localization of the sphere spectrum S. In fact, a localization functor has this property if and only if it preserv...
Formalized mathematics
 TURKU CENTRE FOR COMPUTER SCIENCE
, 1996
"... It is generally accepted that in principle it’s possible to formalize completely almost all of presentday mathematics. The practicability of actually doing so is widely doubted, as is the value of the result. But in the computer age we believe that such formalization is possible and desirable. In c ..."
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Cited by 23 (0 self)
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It is generally accepted that in principle it’s possible to formalize completely almost all of presentday mathematics. The practicability of actually doing so is widely doubted, as is the value of the result. But in the computer age we believe that such formalization is possible and desirable. In contrast to the QED Manifesto however, we do not offer polemics in support of such a project. We merely try to place the formalization of mathematics in its historical perspective, as well as looking at existing praxis and identifying what we regard as the most interesting issues, theoretical and practical.
A Bost–Connes–Marcolli system for Shimura varieties, in preparation
"... We construct a Quantum Statistical Mechanical system (A, σt) analogous to the BostConnesMarcolli system of [CM04] in the case of Shimura varieties. Along the way, we define a new BostConnes system for number fields which has the “correct” symmetries and the “correct ” partition function. We give ..."
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Cited by 21 (0 self)
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We construct a Quantum Statistical Mechanical system (A, σt) analogous to the BostConnesMarcolli system of [CM04] in the case of Shimura varieties. Along the way, we define a new BostConnes system for number fields which has the “correct” symmetries and the “correct ” partition function. We give a formalism that applies to general Shimura data (G, X). The object of this series of papers is to show that these systems have phase transitions and spontaneous symmetry breaking, and to classify their KMS states, at least for low temperature.