Topology Seminar
Upcoming Talks
In Fall 2025, the UChicago Algebraic Topology Seminar will meet on Tuesdays at 4:00-5:00PM in Eckhart Hall 206 and will be preceded by a pretalk 3:30-4PM (unless otherwise noted).
To receive emails about the seminar, please email Nikolai Konovalov requesting to be subscribed to our mailing list.
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Alexander Petrov (Clay/MIT)
Torsion in $p$-adic cohomology via power operations
Cohomology of a smooth projective variety over the complex numbers admits a Hodge decomposition, and the de Rham algebra of smooth differential forms on it is canonically formal, as a commutative differential graded algebra. For a variety over a field $k$ of characteristic $p$ the analogous statements are no longer true. The de Rham cohomology algebra is in general not formal as an $E_\infty$-algebra over $k$ as can be seen by considering the Frobenius action on de Rham cohomology. I will discuss a recipe for constructing examples of situations where (logarithmic) de Rham cohomology fails to have a Hodge decomposition, based on the discrepancy between the $E_\infty$-algebra structures on de Rham and Hodge cohomology. Analogous construction also produces examples of smooth schemes over $\mathbf{Z}_p$ whose (logarithmic) prismatic cohomology has non-zero $u$-torsion -- a phenomenon that is specific to cohomology in degrees $\geq p$. This is joint work with Shizhang Li.
If you have any questions, please contact Sanath Devalapurkar, Nikolai Konovalov, Akhil Mathew, Tomer Schlank, or Peter May.