TopICS

TopICS is the Topology Intercity Seminar joint between Utrecht University and Radboud University, Nijmegen.

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The schedule for the earlier editions of the seminar can be found here, here, and here.

Fall 2023 Schedule

Tuesday 26 September, 2023

Time and Location: Utrecht BBG119 13:15-15:00
Speaker: Sil Linskens (Bonn)
Title: Parametrized (and) higher semiadditivity
Abstract: I will give a leisurely overview of parametrized and higher semiadditivity. In particular I will motivate this concept by giving a variety of examples. As one such example, I will explain how it gives a conceptual interpretation of definitions in (globally) equivariant algebra and homotopy theory. I will then finish by discussing the close connection between generalized semiadditivity and the construction of transfer maps.

Time and Location: Vrije Universiteit Amsterdam NU Building 9A46, 14:00-16:00
Speaker: Adela Zhang (Copenhagen)
Title: An equivariant Adams spectral sequence for $tmf(2)$
Abstract: In this talk, I will explain how to compute the $C_3$-equivariant relative Adams spectral sequence for the Borelification of $tmf(2)$.This yields an entirely algebraic computation of the 3-local homotopy groups of $tmf$. The final answer is well-known of course – the novelty here is that the rASS is completely determined by it $E_1$-page as a cochain complex of Mackey functors. Explicitly, the input consists of the Hopf algebroid structure on $\mathbb{F}_3 \otimes_{tmf(2)}\mathbb{F}_3$ modulo transfer, which is deduced from the structure maps on the equivariant dual Steenrod algebra, as well as the knowledge of the homotopy group of the underlying $tmf(2)$ along with the $C_3-$action. Then we construct a bifiltration on $tmf(2)$ and use synthetic arguments to deduce the Adams differentials from the associated square of spectral sequences. The rASS degenerates on $E_{12}$ for tridegree reasons and stabilizes to a periodic pattern that essentially lies within a band of slope 1/4. This is joint work with Jeremy Hahn, Andrew Senger, and Foling Zou.

Time and Location: Radboud University Nijmegen, Huygensgebouw HG00.065 15:30-17:30
Speaker: John Greenlees
Title: Rational equivariant cohomology theories for compact Lie groups
Abstract: Several structural questions have emerged at least twice in topology: once in chromatic homotopy theory and once in equivariant topology (completions and localization, fracture squares, Balmer spectra, support, telescope conjecture, sheaves, filtrations, ….). In the chromatic world they arise in hard-core form, and in equivariant topology they reach a benign algebraic manifestation in the rational case. My talk is from this gentler world. The overall project is to build an algebraic model for rational G-equivariant cohomology theories for all compact Lie groups G, and when G is small or abelian this has been done. In general, the model is expected to take the form of a category of sheaves of modules over a sheaf of rings over the space of closed subgroups of G. The talk will focus on structural features of the expected model for general G such as those above, and feature recent joint work with Balchin and Barthel.

Time and Location: Utrecht BBG023 14:00-16:00
Speaker: Jack Davies (Bonn)
Title: Geometric norms on equivariant elliptic cohomology
Abstract: Inspired by the work of Lurie and others, Gepner—Meier define families of equivariant cohomology theories based on oriented elliptic curves. By construction, these equivariant elliptic cohomologies are multiplicative, but only in a naïve equivariant sense—there is no obvious construction of norm maps on these theories. In this talk, I will describe how to use a moduli interpretation of the geometric fixed points of these equivariant theories due to Gepner—Meier, to construct what we call “geometric norms”. Some applications of these geometric norms will also be discussed. This is joint work-in-progress with William Balderrama and Sil Linskens.

Time and Location: Radboud University Nijmegen, Huygensgebouw HG00.308 15:30-17:30
Speaker: Clover May (Trondheim)
Title: The $RO(\Pi)$-graded cohomology of $B_{C_2}O(1)$
Abstract: Classically the Thom isomorphism relates the cohomology of the Thom space of a vector bundle to the cohomology of its base. The Thom isomorphism for equivariant vector bundles fails in $RO(G)$-graded cohomology, even for $G=C_2$. However, Costenoble–Waner developed an $RO(\Pi)$-graded equivariant cohomology theory, extending the usual representation grading $RO(G)$ to representations of the equivariant fundamental groupoid, and they showed the Thom isomorphism holds in this extended grading. Costenoble recently computed the $RO(\Pi)$-graded cohomology of $B_{C_2}U(1)$, the classifying space for complex $C_2$-line bundles. In this talk I will describe these different gradings and talk about work in progress computing the $RO(\Pi)$-graded cohomology of $B_{C_2}O(1)$, the classifying space for real $C_2$-line bundles. This is joint work with Agnés Beaudry, Chloe Lewis, Sabrina Pauli and Elizabeth Tatum.