CAREER: Higher Brauer Groups and Topological Azumaya Algebras

Project: Research project

Project Details

Description

Overview
The PI proposes to explore three new ideas in algebraic geometry and algebraic topology. The first is to use higher Morita-theoretic versions of Azumaya algebras, which are to be special types of 1En-algebras, to obtain higher algebraic representatives of higher etale cohomology classes. The second is to use persistent homology and AJ-homotopy theory to study the period-index conjecture on division algebras. The third is to use characteristic p methods to give an algebraic proof of the theorem of Deligne-Griffiths-Morgan-Sullivan on formality of de Rham cohornology of smooth projective complex varieties.
Azurnaya algebras are geometric objects (sheaves of algebras) which look locally like matrix algebras. Over a field k, Azurnaya algebras are central simple algebras, exactly the matrix algebras Mn(D) where D is a finite dimensional division algebra with center k. The Brauer group of a space classifies Azumaya algebras up to tensoring with endomorphism algebras of vector bundles. This group plays a fundamental role in the arithmetic of elliptic curves and of number fields. In algebraic geometry, the Brauer group was used by Artin and Mumford to construct one of the earliest exam­ ples of a non-rational unirational variety. Classes in the Brauer group also obstruct the existence of a universal family of vector bundles on moduli spaces of vector bundles. In mathematical physics, Brauer groups appear in consideration of mirror symmetry for Calabi-Yau manifolds, as well in the study of certain 2d topological quantum field theories.
The PI's research, beginning with his Ph.D. thesis and continuing with his work with Ben Williams, has established fundamental new tools for formulating and attacking problems about Azurnaya algebras on topological spaces. That work has led to the resolution of a major outstanding problem on algebraic Azumaya algebras as well as the startling discovery that, at least in topology, the natural period-index conjecture is false. Another main thread of the PI's research began with a paper on Azumaya algebras over ring spectra with David Gepner. This paper represents the first step in a much broader program to relate higher algebraic techniques arising in algebraic topology to etale cohornology. Additionally, that paper led to one of the fundamental insights in the PI's recent work with Barthel and Gepner answering a major outstanding question in the K-theory of ring spectra, which asked whether Grothendieck-Quillen devissage works in that setting.
First, the PI will study higher versions of Azurnaya algebras and Brauer groups, aiming to find non-classical representatives for etale cohomology classes for fields. These are supposed to be higher dimensional topological quantum field theories, and this work, joint with David Gepner, aims at connecting the arithmetic, geometry, and mathematical physics together.
Second, the PI will set out to disprove the period-index conjecture for threefolds using two new approaches, the study of Postnikov towers in A1-homotopy theory and the use of persistent homology as a way method of finding interesting non-complete intersections in projective space.
Third, the PI will explore an idea to use results of Deligne and Illusie on degeneration of the Hodge spectral sequence in characteristic p to prove formality of the algebraic de Rharn complex.

Broader Impacts
The broader impacts resulting from the proposed research include professional training for graduate students in the Midwest via a series of summer schools as well as undergraduate outreach via the Mathemat
StatusActive
Effective start/end date2/15/2111/30/21

Funding

  • National Science Foundation (DMS-2120005)

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