## Topological abelian groups and the Dold-Kan correspondence

Meeting ID: 974 6894 7515

For passcode see the email or contact Christian Kaiser (kaiser@mpim...).

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The seminar is virtual via Zoom. If you are interested in participating, please contact Stephan Stadler

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## Total curvature and the isoperimetric inequality

The classical isoperimetric inequality states that in Euclidean space spheres provide enclosures of least perimeter for any given volume. According to the Cartan-Hadamard conjecture, this inequality may be generalized to spaces of nonpositive curvature. In this talk we discuss an approach to proving this conjecture via a comparison formula for the total curvature of level sets of functions on manifolds. In particular we will show that the conjecture holds when the variation in the curvature of the ambient space is small. This is joint work with Joel Spruck.

## Area minimizing surfaces for singular configurations of boundary curves

Let g be a nonnegative integer and C be a finite configuration of disjoint Jordan curves in Euclidean space. Then, by a classical result of Douglas, there is an area minimizer among all surfaces of genus at most g which span C. In the talk we will discuss a generalization of this theorem to singular configurations C of possibly non-disjoint or self-intersecting curves. Furthermore, the talk will contain new existence results for regular curve configurations C in general metric ambient spaces.

## The Fontaine-Mazur conjecture in dimension two

The Fontaine-Mazur conjecture predicts which two-dimensional p-adic representations of the absolute Galois group of Q arise from modular forms. In this talk, I will explain this onjecture by some concrete examples and report some recent progress.

Zoom Meeting ID: 943 4202 1275

For password please contact Pieter Moree (moree@mpim...)

## Enumerative Geometry: Old and New

Zoom Meeting ID: 943 4202 1275

For password please contact Pieter Moree (moree@mpim...)

For as long as people have studied geometry, they have counted geometric objects. For example, Euclid's Elements starts with the postulate that there is exactly one line passing through two distinct points in the plane. Since then, the kinds of counting problems we are able to pose and to answer has grown. Today enumerative geometry is a rich subject with connections to many fields, including combinatorics, physics, representation theory, number theory and integrable systems.

## Diffeomorphism groups of discs

The closed n-dimensional disc is the simplest smooth compact n-manifold and yet, despite continuous efforts of geometers and topologists since the beginning of the 60s, its group of diffeomorphisms is still little understood. Over time it has become apparent that, although rooted in geometry and topology, the study of these groups is closely linked to several other areas within mathematics such as the study of the cohomology of arithmetic groups, stable homotopy theory, or the combinatorics of finite graphs.

## (infty,2)-categories

Recent advances, in particular in topology and in algebraic geometry, showed the need for a well-developed language of higher categories. In this talk, I will present my long-term project, joint with Julie Bergner and Martina Rovelli, aimed towards understanding (infty,2)-categories and more generally (infty,n)-categories through the lens of marked simplicial sets.

Zoom Meeting ID: 943 4202 1275

For password please contact Pieter Moree (moree@mpim...)

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## Functor calculus with a view toward smooth embeddings.

Orthogonal calculus was developed in the 1990s as a tool to study functors from the category of real vector spaces to the category of (based) topological spaces. One inputs a functor F, and the calculus outputs a tower of polynomial approximations of F similar to Taylor’s Theorem in differential calculus. The difference between successive polynomial approximations is a topological space built from the derivatives of the inputted functor and can be studied homotopy theoretically.

## Bordered Heegaard Floer homology and Mazur pattern satellites

In this talk, I'll discuss how bordered Heegaard Floer homology has appeared in my work on satellite knots. This talk can be considered to be a companion to Tom Hockenhull's talk from last week.

Zoom Meeting ID: 916 5855 1117

For password see the email or contact: Arunima Ray or Tobias Barthel.

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https://bbb.mpim-bonn.mpg.de/b/rei-xh2-kg6

For password email to rkramer@mpim...

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https://bbb.mpim-bonn.mpg.de/b/rei-xh2-kg6

For password email to rkramer@mpim...

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https://bbb.mpim-bonn.mpg.de/b/rei-xh2-kg6

For password email to rkramer@mpim...

## tba

https://bbb.mpim-bonn.mpg.de/b/rei-xh2-kg6

For password email to rkramer@mpim...

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https://bbb.mpim-bonn.mpg.de/b/rei-xh2-kg6

For password email to rkramer@mpim...

## Divisors

I will review the notion of Cartier divisors briefly and show that we can intersect Cartier divisors with arbitrary cycles in the Chow group. Then, I will show some properties satisfied by this intersection class, and finally discuss some applications (first Chern class and the Gysin map).

https://bbb.mpim-bonn.mpg.de/b/rei-xh2-kg6

For password email to rkramer@mpim...

## Algebraic cycles and rational equivalence

Following chapter 1 of Fulton's book, I will introduce cycles on schemes and define when cycles are rationally equivalent. I will show that rational equivalence behaves well with respect to certain maps between schemes and that these maps therefore descend to the cycle class groups or Chow groups.

https://bbb.mpim-bonn.mpg.de/b/rei-xh2-kg6

For password email to rkramer@mpim...

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