- Scholarship in 2021 at University of Illinois Urbana-Champaign
About Levi's work
Levi's work focuses on motivic homotopy theory, specifically on the norm structures on motivic spectra. Classical algebraic topology studies cohomology theories via the category of spectra. These classical methods are not immediately available for schemes, which also admit a wide variety of cohomology theories. Morel-Voevodsky developed a algebro-geometric analog of spaces and spectra, known as motivic spaces and motivic spectra, which played an essential role in the resolution of major conjectures like the Bloch-Kato conjecture.
One can straightforwardly transfer the
classical notion of commutative ring spectra to the category of motivic
spectra. However, a further enhancement of this structure, known as a
norm structure, that takes into account the algebro-geometric data is
possible and was developed by Bachmann-Hoyois. It characterizes a
multiplicative transfer map along finite etale maps. Many well-known
examples of motivic spectra admits a norm structure. However, we
currently do not have a good way of constructing such norm structures.
Levi's work aims to understand obstructions to equipping a commutative
motivic ring spectrum with a compatible norm structure.