## What's happening at MPIM

## Short talks by new postdocs

## Short talks by new postdocs

## Short talks by new postdocs

## A glance at the conjecture of Birch and Swinnerton-Dyer

The talk plans to present an introduction to the eponymous conjecture predicting a mysterious link between rational points on an elliptic curve defined over rational numbers,

and analytic properties of the associated Hasse-Weil L-function. Some recent progress will be discussed.

## K-theory of minimal fields in spectra

## Invariants defined by configuration space integrals

## Manifolds, algebra, and homotopy

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## Dehn twist along Brieskorn spheres along their positive definite fillings

Given any smooth 4-manifold bounding a Seifert manifold, the Seifert action on its boundary can be used to define their boundary Dehn twists. If the given 4-manifold is simply-connected, this Dehn twist is always topologically isotopic to the identity, but usually not smoothly isotopic, making it a very nice potential example of exotic diffeomorphisms. For some choices of small enough Brieskorn spheres and some choices of their fillings, it has been shown that their boundary Dehn twists are infinite-order exotic. In this talk, we prove that for any Brieskorn homology sphere bounding a positive-definite 4-manifold, their boundary Dehn twists are always infinite-order exotic, using both the equivariant Seiberg-Witten homology of Baraglia-Hekmati and the family Seiberg-Witten theory. This is a joint work with JungHwan Park and Masaki Taniguchi.

## On Selmer groups of cyclic twist families of elliptic curves over global function fields [NT seminar]

Let $K = \mathbb{F}_q(t)$ be the global function field of characteristic coprime to 2 and 3. Let $E$ be a non-isotrivial elliptic curve over $K$. Fix a prime number $l$ that is coprime to the characteristic of $K$, and that the primitive $l$-th roots of unity is contained in the constant field of $K$. Let $L$ be a cyclic $\mathbb{Z}/l\mathbb{Z}$ geometric Galois extension over $K$. We will explore two approaches - a probabilistic approach and a geometric approach - to compute a lower bound on the probability that the rank of $E(L)$ is equal to the rank of $E(K)$. If time allows, we will also explore new geometric insights that can be obtained from comparing the two approaches.

## Contributed talk: Refined $TC^-$ over $ku$ and derived q-Hodge complexes

As a consequence of his proof of rigidity of the category of localising motives, Efimov has constructed refinements of localising invariants. Such refined invariants often contain a lot more information than the original ones. For example, the refined $TC^-$ of the rational numbers is not a rational spectrum; it contains very subtle p-complete information as well. In this talk, we'll explain how to compute it after base change to $ku$. The computation involves a surprising connection to q-de Rham cohomology, and in partcular, to the question of whether there exists a "q-Hodge filtration" on q-de Rham cohomology. This is joint work with Samuel Meyer.

## Contributed talk: Smooth and proper categories in analytic geometry

In this talk, I will initially introduce the definition of smooth and proper category, and then explore how it can be used in algebraic and analytic geometry. This definition fits very well within the algebraic setting, as it allows for the characterization of smooth and proper algebraic varieties by examining their category of quasi-coherent sheaves. In the analytic context, such a characterization is more challenging to achieve; during the talk, I will aim to explain the difficulties and some results that we can obtain in this setting.

## Contributed talk: Norm, Assembly and Coassembly

A major open problem in topology are (rational) injectivity results about assembly in $K$- and $L$-theory, e.g. the Borel and Novikov conjectures.

Malkiewich showed that given a ring R and finite group $G$ there exists a so called coassembly map going from the K-theory of the group ring $R[G]$ to the homotopy fixed points of the K-theory spectrum of $R$ equipped with the trivial action. Moreover the composition of the assembly map with this coassembly map agrees with the norm map. This implies that for a finite group G, the assembly map is rationally and $K(n)$-locally injective. In recent work with Alex M\"uller and Holger Reich, we generalize this fact to arbitrary additive invariants of either stable or Poincaré categories, while also allowing for twisted coefficients. In particular, Malkiewich's theorem is true more generally for $THH$, $TC$, Grothendieck-Witt and $L$-theory. The fundamental insight is that the content of the theorem is fundamentally about the interaction of the algebraic theory of G-actions with the algebraic theory of $E_\infty$-monoids. The general nature of this approach allows one to ask the question if there exists a generalization to more general groups, using the framework of dualizable categories.

## A glimpse into Koszul duality for factorization algebras

A fundamental technique in mathematics is to start with an object we understand and deform it to produce a new object with more interesting behavior. Trying to formalize this process leads to the rich fields of deformation theory and Koszul duality. A natural place to apply this technique is to study QFTs, in particular the factorization algebra formalism. Deformation theory and Koszul duality of locally constant factorization algebras is well understood and has remarkable geometric interpretations in terms of configuration spaces of manifolds and Poincaré duality. In this talk, we extend some of the known results on Koszul duality for locally constant factorization algebras to general factorization algebras in an attempt to develop a setting to study deformations of factorization algebras.

## Final Discussion Round of the QFT Workshop

We meet for a final gathering of the QFT Workshop in the MPIM Lecture Hall: There will be no official program, so it is very much appreciated if some participants would like to (informally) share their progress or some interesting ideas, observations and open questions that came up during their stay.

## On the frequency of primes preserving dynamical irreducibility of polynomials

In this talk we address an open question in arithmetic dynamics regarding the frequency of primes modulo which all the iterates of a polynomial remain irreducible. More precisely, for a class of integer polynomials $f$, which in particular includes all quadratic polynomials, we show that, under some natural conditions, the set of primes $p$ such that all iterates of $f$ are irreducible modulo $p$ is of relative density zero. Our results rely on a combination of analytic (Selberg's sieve) and Diophantine (finiteness of solutions to certain hyperelliptic equations) tools, which we will briefly describe. Joint wok with Laszlo M\’ e rai and Igor Shparlinski (2021, 2024).

## Hecke L-values, definite Shimura sets and mod l non-vanishing

We outline mod l non-vanishing of Hecke L-values in self-dual families over imaginary quadratic fields. This includes the vanishing of the mu-invariant of Rubin’s p-adic L-function. (Joint with W. He, S. Kobayashi and K. Ota.)

## Configuration spaces with mice diagrams

I will first give an introduction to the Fulton-MacPherson (also called Axelrod-Singer) compactification of the configuration space of n ordered marked points on R^d. Then I will bring your attention to the (almost trivial) observation that the "node smoothing" procedure still works if we formally substitute a finite number of "screen"s -- which are copies of R^ds -- by arbitrary smooth manifolds that (1) are framed; (2) have an "end" that looks like R^d near infinity. I will explain the connection of this observation to the little (d+1)-disk operad action on the classifying space of framed, smooth d-disk bundles (these terminologies will be introduced in the talk) and, if time permits, also its connection to the Lie bracket in graph homology. This is based on joint work in progress with Robin Koytcheff and Sander Kupers.

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