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Trace and Künneth formulas for singularity categories and applications
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- Topological $K$-theory
- Algebraic number theory: local and $p$-adic fields
Bertrand Toën, Gabriele Vezzosi
Journal:

Compositio Mathematica / Volume 158 / Issue 3 / March 2022

Published online by Cambridge University Press:

03 June 2022, pp. 483-528

Print publication:

March 2022
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We present an $\ell$-adic trace formula for saturated and admissible dg-categories over a base monoidal differential graded (dg)-category. Moreover, we prove Künneth formulas for dg-category of singularities and for inertia-invariant vanishing cycles. As an application, we prove a categorical version of Bloch's conductor conjecture (originally stated by Spencer Bloch in 1985), under the additional hypothesis that the monodromy action of the inertia group is unipotent.

Edited by Jácome (Jay) Armas
Published online:

05 August 2021

Print publication:

26 August 2021
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The holy grail of theoretical physics is to find the theory of everything that combines all the forces of nature, including gravity. This book addresses the question: how far are we from such discovery? Over the last few decades, multiple roads to finding a quantum theory of gravity have been proposed but no obvious description of nature has emerged in this domain. What is to be made of this situation? This volume probes the state-of-the art in this daunting quest of theoretical physics by collecting critical interviews with nearly forty leading theorists in this field. These broad-ranging conversations give important insights and candid opinions on the various approaches to quantum gravity, including string theory, loop quantum gravity, causal set theory and asymptotic safety. This unique, readable overview provides a gateway into cutting edge research for students and others who wish to engage with the open problem of quantum gravity.

Index Theory and Non-Commutative Geometry II. Dirac Operators and Index Bundles
Moulay-Tahar Benameur, James L. Heitsch
Journal:

Journal of K-Theory / Volume 1 / Issue 2 / April 2008

Published online by Cambridge University Press:

30 November 2007, pp. 305-356

Print publication:

April 2008
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When the index bundle of a longitudinal Dirac type operator is transversely smooth, we define its Chern character in Haefliger cohomology and relate it to the Chern character of the K—theory index. This result gives a concrete connection between the topology of the foliation and the longitudinal index formula. Moreover, the usual spectral assumption on the Novikov-Shubin invariants of the operator is improved.

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