## 2-torsion of class groups of multiquadratic fields

The narrow class group of a number field is one of the most fundamental and, yet mysterious, objects in arithmetic. Its study was initiated by Gauss in 1801, using the language of quadratic forms. In his dissertation Gauss reported what still is one of the very few explicit results about the class group, namely an explicit description of the 2-torsion of the class group and the dual class group of a quadratic number field.

## New guests at the MPIM

## Singularities old and new, II

In the early days of glory, geometric topology scored many astonishing achievements in the theory of manifolds and submanifolds, a few samples of which will be mentioned.

## Singularities old and new, I

In the early days of glory, geometric topology scored many astonishing achievements in the theory of manifolds and submanifolds, a few samples of which will be mentioned.

## Condensed abelian groups and solid completion

I will describe a modification of the notion of topological abelian group, called "condensed abelian group". This modification has favorable categorical properties, and in particular it is possible to define a very well-behaved completion functor, the "solid completion", which is suitable for non-archimedean geometry in the style of Tate and Huber. This is joint work with Peter Scholze.

## Mordell Weil rank jumps and the Hilbert property

Let X be an elliptic surface with a section defined over a number field. Specialization theorems by N\'eron and Silverman imply that the rank of the Mordell-Weil group of special fibers is at least equal to the MW rank of the generic fiber. We say that the rank jumps when the former is strictly large than the latter. In this talk, I will discuss rank jumps for elliptic surfaces fibred over the projective line. If the surface admits a conic bundle we show that the subset of the line for which the rank jumps is not thin in the sense of Serre. This is joint work with Dan Loughran.

## Fibered, homotopy ribbon knots and the Poincaré Conjecture, II

A homotopy 4-sphere that is built without 1-handles can be encoded as a $n$-component link with an integral

Dehn surgery to $\#^n(S^1\times S^2)$. I'll describe a program to prove that such spheres are smoothly standard

## Fibered, homotopy ribbon knots and the Poincaré Conjecture, I

A homotopy 4-sphere that is built without 1-handles can be encoded as a $n$-component link with an integral

Dehn surgery to $\#^n(S^1\times S^2)$. I'll describe a program to prove that such spheres are smoothly standard

## New guests at the MPIM

## Stein open subsets and topological pseudoconvexity

We will examine the interplay between Eliashberg's theory for embedding Stein domains and Freedman's theory of topological 4-manifolds. The speaker showed (J. Top. 2013) that an open subset $U$ of a complex surface $X$ is smoothly isotopic to a Stein open subset iff the induced almost-complex structure on $U$ is homotopic to a Stein structure. Using Casson handles, he showed (J. Symp. Geom. 2005) that $U$ is topologically isotopic to a Stein open subset iff it is homeomorphic to the interior of a 2-handlebody.

## 4-manifolds and topological modular forms

Physics predicts the existence of a novel class of smooth invariants of 4-manifolds valued in the ring of topological modular forms. In this talk, we will describe general properties of this new invariant and consider some examples.

## Knot cobordisms, torsion, and Floer homology

Given a cobordism between two knots in the 3-sphere, we present an inequality involving torsion orders of the knot Floer homology of the knots, and the number of critical points of index zero, one, and two of the cobordism. In particular, the torsion order gives a lower bound on the number of minima appearing in a slice disk of a knot. This has a number of topological applications: The torsion order gives lower bounds on the bridge index and the band-unlinking number of a knot, and on the fusion number of a ribbon knot.

## Homotopy 4-spheres and generalized square knots

The Smooth 4-dimensional Poincaré Conjecture (S4PC) asserts that every homotopy 4-sphere is diffeomorphic to the standard 4-sphere $S^4$. We prove a special case of the S4PC: If $X$ is a homotopy 4-sphere that can be built with two 2-handles and two 3-handles, and such that one component of the 2-handle attaching link $L$ is a generalized square knot $T(p,q) \# T(-p,q)$, then $X$ is diffeomorphic to $S^4$. This is joint work with Jeffrey Meier.

## Homotopy versus isotopy for spheres with duals in 4-manifolds

David Gabai's smooth 4-dimensional "Light Bulb Theorem'' says that in the absence of involutions in the fundamental group of the ambient 4-manifold, homotopy implies isotopy for embedded 2-spheres which have a common geometric dual. In joint work with Peter Teichner we extend his result to orientable 4-manifolds with arbitrary fundamental group by showing that an invariant of Mike Freedman and Frank Quinn gives the complete obstruction to the "homotopy implies isotopy'' question. The invariant takes values in an F2-vector space generated by the involutions in the fundamental group.

## Embedding spheres and the Kervaire-Milnor invariant

I will explain the definition of the Kervaire-Milnor invariant of an immersed 2-sphere in a 4-manifold, which obstructs the sphere being homotopic to an embedding. I will describe two instances in 4-manifold classification where this invariant appears, one in the sphere embedding theorem and one in stable classification.

## New lower bounds on the 4-genera of knots

A knot is slice if it bounds an embedded disc in the 4-ball. There are (at least) two natural generalizations of sliceness: one might weaken either 'disc' to 'small genus surface' or 'the 4-ball' to 'any 4-manifold that is simple in some sense.' In this talk, I'll discuss joint work with Jae Choon Cha and Mark Powell that gives new evidence that these two approaches measure very different things. Our tools include Casson-Gordon style representations of knot groups, L2 signatures of 3-manifolds, and the notion of a minimal generating set for a module.

## Instanton and Heegaard Floer homologies of surgeries on torus knots

A Floer homology is an invariant of a closed, oriented 3-manifold $Y$ that arises as the homology of a chain complex whose generators are either the set of solutions to a differential equation or the intersection points between Lagrangian manifold, and its differential arises as the count of solutions of a differential equation on $Y \times \mathbb{R}$. The Instanton Floer chain complex is generated by flat connections on a principal $SU(2)$-bundle, and the differential counts solutions to the Yang-Mills equation (known as instantons).

## Hidden Algebraic Structure in Topology

Which 4d TQFTs and 4-manifold invariants detect the Gluck twist? Guided by questions like this, we will look for new invariants of smooth 4-manifolds and knotted surfaces in 4-manifolds.

## Intersection forms of definite manifolds bounded by lens spaces

We find some restrictions on the intersection forms of smooth definite manifolds bounded by rational homology spheres which are rationally cobordant to lens spaces. As a first consequence, we show that several natural maps to the rational homology cobordism group have infinite rank cokernels. Further, we show that there is no $n$ such that every lens space smoothly embeds in $n$ copies of the complex projective plane. This is joint work with Paolo Aceto and Daniele Celoria.

## Families of diffeomorphisms of 4-manifolds via graph surgery

In this talk, I will explain a method to construct families of diffeomorphisms of a 4-manifold by using a 4D analogue of Goussarov-Habiro's theory of graph surgery in 3D. Our graph surgery would produce lots of potentially nontrivial elements of homotopy groups of diffeomorphism groups of 4-manifolds. I will discuss about their applications to 4D analogue of the Smale conjecture and the 4D light bulb theorem for 3-disk.

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