# Motives Seminar

## A series of lectures on current topics

This semester the motives seminar will consist of lectures on topics of interest for our research group and (hopefully) others: motivic homotopy theory, algebraic stacks, enumerative geometry, tropical geometry, ... . As usual, we will meet on Tuesdays, 16:00. We meet in WSC-N-U-4.05 and we will continue to stream the lectures online.

As there is some overlap with talks in Jochen Heinloth's Seminar on Stacks (Mondays, 16:00 in WSC-S-U-3.02) some of the talks will take place there.

### Previous Motives Seminars

Follow the Teaching link at the top of this page to find links to previous motives seminars.

### Program

April 5  Andrés Jaramillo-Puentes, Sabrina Pauli: Enriched tropical intersection I
Lecture 1 Notes (1st half)
Lecture 1 Notes (2nd half)
Tropical geometry has been proven to be a powerful computational tool in enumerative geometry over the complex and real numbers. In these two talks we present one of the first examples of a quadratic refinement of this tool, namely a proof of the quadratically refined Bézout's theorem for tropical curves. In the first talk we recall the necessary notions of enumerative geometry over arbitrary fields valued in the Grothendieck-Witt ring. Then we define tropical curves from a combinatorial and algebraic approach, and their relation with curves defined over the field of Puisseux series over an arbitrary field. We will mention the Viro's patchworking method that served as inspiration to our construction based on the duality of the tropical curves and the refined Newton polytope associated to its defining polynomial. Lastly, we will illustrate an elementary proof of a Bézout theorem for tropical curves.

April 12  Andrés Jaramillo-Puentes, Sabrina Pauli: Enriched tropical intersection II
Lecture 2 Video
Lecture 2 Notes
In this talk we will prove that the quadratically refined multiplicity of an intersection point of two tropical curves can be computed combinatorially. We will use this new approach to prove an enriched version of the Bézout theorem and of the Bernstein–Kushnirenko theorem, both for enriched tropical curves.

April 19  Herman Rohrbach: Introduction to Hermitian K-theory
Lecture 3 Video
Hermitian K-theory is a refinement of algebraic K-theory, which captures information about the quadratic forms on a given geometric or algebraic object. The theory was pioneered by Ernst Witt in the 1930's with what are now known as Witt groups. The theory has many surprising connections, for example to Milnor K-theory (and Galois cohomology, via the Milnor conjecture), surgery theory and fundamental facts in motivic homotopy theory. In this talk, I will give an introduction to Hermitian K-theory using Marco Schlichting's framework of pretriangulated dg categories with duality, and use it to state and sketch the proof of the projective bundle formula for Hermitian K-theory.

April 26  Herman Rohrbach: Atiyah-Segal completion for trivial torus actions
Lecture 4 Video
The Atiyah-Segal completion theorem (1969) gives a way of approximating, for a compact Lie group G, the G-equivariant complex topological K-theory of a compact G-space with the non-equivariant K-theory of an auxiliary topological space, which is closely related to the classifying space of G. An analogue of this theorem for algebraic K-theory has been proven by Amalendu Krishna (2018), but as of yet no such analogue exists for Hermitian K-theory. In this talk, I will prove the basic case of the completion theorem for a scheme with a trivial torus action using the theory of derived completion.

May 3  Dhyan Aranha: Motivic Localisation for Stacks
Lecture 5 Video
We will sketch a general concentration theorem in motivic Borel-Moore homology for derived Artin stacks, and then explain how this implies the (virtual) Atiyah-Bott formula for Deligne-Mumford stacks over a general base with no global resolution hypothesis. If time permits, I'll discuss what we can say about virtual Atiyah-Bott in the Artin case and perhaps mumble something about cosection localization for derived Artin stacks. All this is joint work with Adeel Khan, Aloysha Latyntsev, Hyeonjun Park and Charanya Ravi.

May 10 (moved to May 9-Seminar on Stacks-16:15 in WSC-S-U-3.02.)  Michele Pernice (Pisa): Results about the Chow ring of moduli of stable curves of genus three
Abstract: In this talk, we will discuss some results concerning the Chow ring of $\bar{M}_3$, the moduli stack of stable curves of genus three. In particular, we will describe the main new idea, which consists of enlarging the moduli stack of stable curves by adding curves with worse singularities, like cusps and tacnodes. This will make the geometry of the stack uglier but in turn its Chow ring will be easier to compute. The Chow ring of $\bar{M}_3$ can then be recovered by excising the locus of non-stable curves.

May 17  Marc Levine: Categories of non-$\mathbb{A}^1$-invariant $\mathbb{P}^1$-spectra and the universality of $K$-theory, following Annala and Iwasa. Abstract: We give a report on a recent paper of Annala and Iwasa: Motivic spectra and universality of K-theory, closely following the talk of Iwasa at Oberwolfach. Annala and Iwasa construct categories of $\mathbb{P}^1$-spectra satisfying various conditions, such as orientability, the existence of a Bass fundamental sequence, or a projective bundle formula, and use these to show that algebraic K-theory is the universal theory on schemes satisfying Zariski descent, admitting an action of the Picard stack, and satisfying the projective bundle formula.

May 24 (moved to May 23-Seminar on Stacks-16:15 in WSC-S-U-3.02.)  Marc Levine: Quadratic versions of Atiyah-Bott localization The torus localization theorems of Atiyah-Bott have been proved in the context of equivariant Chow groups by Eddidin-Graham and Krishna, and were recently generalised to stacks by Aranha, Khan, Latyntsev, Park and Ravi (see above). We prove a version of these results (for schemes) in the setting of Witt-sheaf cohomology, where one replaces $\mathbb{G}_m$ with the normaliser of the torus in $\text{SL}_2$. If time permits, we will also describe a similar Witt-sheaf cohomology version of the virtual localisation formula of Graber-Pandharipande.

May 31

June 7

July 5   Lucas Mann (Univ. Bonn) A $p$-adic 6-Functor Formalism in Rigid-Analytic Geometry
Lecture 9 Video
Abstract: I develop a full 6-functor formalism for $p$-torsion étale sheaves in rigid-analytic geometry, which in particular proves $p$-adic Poincaré duality in this setting. The main idea is to introduce a nice category of "solid quasicoherent almost $O_X^+/p$-modules" on every rigid variety $X$ (or even any small $v$-stack $X$) and construct the 6 functors $\otimes$, $\underline{Hom}$, $f^*$, $f_*$, $f_!$ and $f^!$ in this setting. Via a $p$-torsion Riemann-Hilbert correspondence I relate these sheaves to actual étale $\mathbb{F}_p$-sheaves, which ultimately results in previously unknown properties of $\mathbb{F}_p$-cohomology. I expect many applications of this formalism to the $p$-adic Langlands program