Higher Topos Theory (AM-170) (Annals of Mathematics Studies)

When a shock hits an obstacle, shock reflection-diffraction configurations take shape. To understand the fundamental issues involved, such as the structure and transition criteria of different configuration patterns, it is essential to establish the global existence, regularity, and structural stability of shock reflection-diffraction solutions. This involves dealing with several core difficulties in the analysis of nonlinear PDEs—mixed type, free boundaries, and corner singularities—that also arise in fundamental problems in diverse areas such as continuum mechanics, differential geometry, mathematical physics, and materials science.

Presenting recently developed approaches and techniques, which will be useful for solving problems with similar difficulties, this book opens up new research opportunities. The proof of the product formula will appear in a sequel volume.

This book covers categories, homological algebra and sheaves in a systematic and exhaustive manner starting from scratch, and continues with full proofs to an exposition of the most recent results in the literature, and sometimes beyond. The authors present the general theory of categories and functors, emphasising inductive and projective limits, tensor categories, representable functors, ind-objects and localization.

Then they study homological algebra including additive, abelian, triangulated categories and also unbounded derived categories using transfinite induction and accessible objects. Finally, sheaf theory as well as twisted sheaves and stacks appear in the framework of Grothendieck topologies.

Account Options Sign in. Higher category theory is generally regarded as technical and forbidding, but part of it is considerably more tractable: In Higher Topos Theory , Jacob Lurie presents the foundations of this theory, using the language of weak Kan complexes introduced by Boardman and Vogt, and shows how existing theorems in algebraic topology can be reformulated and generalized in the theory's new language.

The result is a powerful theory with applications in many areas of mathematics. Jacob Lurie is associate professor of mathematics at Massachusetts Institute of Technology.

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Seminar on Differential Geometry. This collection of papers constitutes a wide-ranging survey of recent developments in differential geometry and its interactions with other fields, especially partial differential equations and mathematical physics. This area of mathematics was the subject of a special program at the Institute for Advanced Study in Princeton during the academic year ; the papers in this volume were contributed by the speakers in the sequence of seminars organized by Shing-Tung Yau for this program.

Both survey articles and articles presenting new results are included. The theory of graded sets is extended in three directions. However, this category fails to be closed under some important categorical constructions. This problem is dealt with by embedding into a larger category of -graded sets.

by Lurie, Jacob

Next, it is shown show that most constructions make sense with replaced by a small symmetric monoidal category. In particular, we have a symmetric monoidal category of -graded sets. These foundations are used for two further developments: This material will be exposed elsewhere. References [Enhancements On Off] What's this? Retrieve articles in St.

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