Autumn 2025

• • Dec 022025

  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.

• • Nov 182025

  Sanath Devalapurkar (UChicago)

  Local Tate duality for ring spectra

  Tate duality is a form of Poincare duality for the absolute Galois group of a $p$-adic local field $F$, which says that from the perspective of étale cohomology, $\mathrm{Spec}(F)$ behaves like a $2$-manifold. This was refined by Bhatt-Lurie to show that under this duality, the syntomic cohomology of $\mathscr{O}_F$ forms a Lagrangian subspace of $\mathrm{H}^*_{\text{ét}}(F)$. An equicharacteristic generalization was proved by Kato, where it was shown that ''n-local fields'' behave like closed $(n+1)$-manifolds. I will describe some joint work with Jeremy Hahn and John Rognes, where we define the notion of a ''higher local number ring'' and prove a version of Tate duality, which says that higher local number rings of height $n-1$ behave like closed $(n+1)$-manifolds. In particular, I will sketch two key ingredients of our argument, namely duality at the level of $\mathrm{THH}$, and a motivically filtered lift of the transfer map on $\mathrm{THH}$. Both use the theory of finite Hopf algebras over perfect fields.

• • Nov 112025

  Andrew Senger (University of Maryland)

  Synthetic $E_\infty$-rings

  This talk is 5:00pm-6:00pm. The talk will be at Kent Chem Lab 102

  In this talk, I will introduce the category of synthetic $E_\infty$-rings, a categorification of the even filtration of Hahn-Raksit-Wilson, and discuss some of its basic properties. As an application, I will explain how the category of synthetic $E_\infty$-rings can be used to give the even filtration a universal property as a left adjoint. I will also explain how this technology can be used to further our understanding of the motivic filtration on $THH$.

  This is joint work with Devalapurkar, Hahn, and Raksit.

• • Nov 042025

  Natalia Pacheco-Tallaj (MIT)

  Twisted string bordism in $7$ dimensions with applications to anomaly cancellation

  In upcoming joint work with I. Basile, C. Krulewski and G. Leone, we use homotopic and topological techniques to study anomaly cancellation in 6d supergravity theories. In this talk, I will give a quick overview of anomalies of quantum field theories and their classification using twisted bordism groups. Then, I will overview how we leverage the unstable classification of vector bundles over $\mathbb{CP}^2$ to construct explicit generators for $7$-dimensional twisted string bordism groups, and I will show how to construct a complete family of index-theoretic invariants to detect the torsion in these groups.

  Slides

• • Oct 282025

  Natalie Stewart (Harvard)

  Norms for compact Lie groups

  This talk is 3:30pm-4:30pm

  Hill-Hopkins-Ravanel norms give a lift of tensor-induction from the category of $H$-spectra to $G$-spectra for $H \subseteq G$ a subgroup of a finite group, and they were crucial to their resolution of the Kervaire Invariant one problem in all but one dimension; analogously, the influential Angelveit-Blumberg-Gerhardt-Hill-Lawson-Mandell perspective on (twisted) topological Hochschild homology expresses it as a norm from finite subgroups of the circle group. Following this, Blumberg-Hill-Mandell issued a series of conjectures concerning equivariant factorization homology as a technique for constructing norms along closed subgroup inclusions between compact Lie groups (extending the above examples) and computing their geometric fixed points. Following work-in-progress with Juran, I will sketch a resolution of these conjectures.

• • Oct 212025

  Daniel Pomerleano (UMB)

  A Fontaine-Lafaille structure in symplectic topology

  The small quantum connection on a Fano manifold is one of the simplest objects in enumerative geometry. Nevertheless, the poles of the connection have a very rich structure. I will sketch the construction of a (mod $p$) Fontaine-Lafaille structure on its Fourier-Laplace dual and explain some implications of this structure for the monodromy of this connection (e.g. bounds on the size of the Jordan blocks of the connection). This is joint work with Paul Seidel.

• • Oct 142025

  Francis Baer (Wayne State University)

  Composition methods in the unstable Adams spectral sequence

  The stem-wise computation of unstable homotopy groups of spheres is a foundational problem in algebraic topology which has seen little progress in the past several decades. In this talk we will discuss an approach to the problem which combines the composition methods of Toda with the unstable Adams spectral sequence via computer automation. A survey will be given of our results as well as their applications to more tangible problems of a geometric nature.

• • Oct 072025

  Sarah Petersen (University of Colorado Boulder)

  Splittings of equivariant and motivic truncated Brown-Peterson cooperations algebras

  In the 1980's, Mahowald and Kane used integral Brown-Gitler spectra to decompose $BP \langle 1 \rangle \wedge BP \langle 1 \rangle$ as a sum of finitely generated $BP \langle 1 \rangle$-module spectra. This splitting, along with an analogous decomposition of $ko \wedge ko$, led to a great deal of progress in stable homotopy computations and a complete understanding of $v_1$-periodicity in the stable homotopy groups of spheres. In this talk, I will discuss joint work with Guchuan Li, Jackson Morris, and Elizabeth Tatum in which we construct analogues of Mahowald and Kane's splittings in equivariant and motivic homotopy theory. I will then describe the $E_1$-pages of the analogous $BP \langle 1 \rangle$-based Adams spectral sequences in these settings and how this approach gives excellent access to $v_1$-periodicity in equivariant and motivic stable stems.