CAREER: Higher Brauer Groups and Topological Azumaya Algebras

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This award is focused on algebraic geometry and algebraic topology, two areas of modern mathematics. Algebraic geometry has ancient origins with many connections to real-world problems. Its goal is to understand the geometry of solutions sets of polynomial equations, equations of central importance in various disciplines, such as theoretical physics, cryptography, and number theory. Algebraic topology on the other hand developed more recently, in the 19th century, and aims to study a general notion of shape, less rigid than the idea of shape studied in geometry. It has found striking applications in the last decade, for instance to robotics and to the analysis of large data sets, and a recent revolution in the foundations of algebraic topology has broadened its applicability to other fields of pure mathematics. The proposal of the PI will bring the considerable machinery and insight of algebraic topology to bear on several well-known questions and conjectures in algebraic geometry. Some of these questions will be investigated jointly with students as part of undergraduate research projects in the Mathematical Computing Laboratory at UIC, which the PI founded in 2015 with David Dumas and Jan Verschelde. This integration will provide valuable opportunities for students to engage in research and impact ongoing research.The PI will work on several projects at the border of algebraic geometry and algebraic topology. Three projects aim to use various topological methods to understand the Brauer group and Azumaya algebras as well as the role of higher algebraic structures in algebraic geometry. (1) The PI will study (higher) algebraic representatives of étale cohomology classes, generalizing the connection between the Picard and Brauer groups and low-degree cohomology groups. (2) The PI will explore separate applications of motivic homotopy theory and persistent homology to the period-index conjecture with the aim of finding algebraic counterexamples to the conjecture. (3) The PI will explore the Hochschild-Kostant-Rosenberg theorem in characteristic p, specifically focusing on the question of whether or not the local-global spectral sequence for Hochschild homology degenerates at the E_2-page for smooth projective surfaces in characteristic 2.