## 5 point comparison

Meeting-ID: 973 8400 7043

For password please contact Stephan Stadler (stadler@mpim-bonn.mpg.de).

We give an iff condition on 5-point metric spaces that can be embedded into ALexandrov space with nonnegative curvature (+ related topics). This is a joint work with Nina Lebedeva.

## Heegaard Floer homology and minimal genus problems

```
Zoom meeting ID: 916 5855 1117
Contact: Barthel, Ozornova, Ray, Teichner
```

## The K-Theory of Varieties

```
Zoom meeting ID: 916 5855 1117
Contact: Barthel, Ozornova, Ray, Teichner
```

## Defect TQFTs and generalised orbifolds

Zoom meeting ID: 916 5855 1117

Contact: Barthel, Ozornova, Ray, Teichner

## Star Scenery in stable homotopy

```
Zoom meeting ID: 916 5855 1117
Contact: Barthel, Ozornova, Ray, Teichner
```

## The Wiles defect for Hecke algebras via local-global arguments

Zoom ID: 919 6497 4060

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

In his work on modularity theorems, Wiles proved a numerical criterion for a map of rings R->T to be an isomorphism of complete intersections. He used this to show that certain deformation rings and Hecke algebras associated to a mod p Galois representation at non-minimal level were isomorphic and complete intersections, provided the same was true at minimal level. In addition to proving modularity theorems, this numerical criterion also implies a connection between the order of a certain Selmer group and a special value of an L-function.

In this talk I will consider the case of a Hecke algebra acting on the cohomology of a Shimura curve associated to a quaternion algebra. In this case, one has an analogous map of ring R->T which is known to be an isomorphism, but in many cases the rings R and T fail to be complete intersections. This means that Wiles' numerical criterion will fail to hold. I will describe a method for precisely computing the extent to which the numerical criterion fails (i.e. the 'Wiles defect"). In particular, I will show that it can be computed entirely from the structure of certain local Galois deformation rings at the primes dividing the discriminant of the quaternion algebra, and thus depends only on local information.

This is joint work with Gebhard Bockle and Chandrashekhar Khare.

## Super J-holomorphic curves

Attendance for max. 20 MPIM guests (2G) in the MPIM lecture hall. For everyone else participation via Zoom only!

Connection link: https://hu-berlin.zoom.us/j/61686623112

Contact: Gaetan Borot (HU Berlin)

J-holomorphic curves or pseudoholomorphic curves are maps from Riemann surfaces to symplectic manifolds satisfying the Cauchy-Riemann equations.

J-holomorphic curves are of great interest because they allow to construct invariants of symplectic manifolds and those invariants are deeply related to topological superstring theory. A crucial step towards Gromov–Witten invariants is the compactification of the moduli space of J-holomorphic curves via stable maps which was first proposed by Kontsevich and Manin. In this talk, I want to report on a supergeometric generalization of J-holomorphic curves and stable maps where the domain is a super Riemann surface. Super Riemann surfaces have first appeared as generalizations of Riemann surfaces with anti-commutative variables in superstring theory. Super J-holomorphic curves couple the equations of classical J-holomorphic curves with a Dirac equation for spinors and are critical points of the superconformal action. The compactification of the moduli space of super J-holomorphic curves via super stable maps might, in the future, lead to a supergeometric generalization of Gromov-Witten invariants. Based on arXiv:2010.15634 [math.DG] and arXiv:1911.05607 [math.DG], joint with Artan Sheshmani and Shing-Tung Yau.

## The specialization principle for p-adic kimberlites, II

Attendance for max. 20 MPIM guests (2G) in the MPIM lecture hall. For everyone else participation via Zoom only!

For zoom details please contact Peter Scholze (scholze@mpim-bonn.mpg.de).

In his master thesis Lourenço proposes and proves a specialization principle for certain formal schemes. Roughly speaking, this says that a formal scheme can be recovered from its special fiber, its rigid generic fiber and the specialization map. Kimberlites were introduced by the speaker as analogues of formal schemes in Scholze's theory of v-sheaves. In forthcoming joint work with Anschütz, Lourenço and Richarz we prove a similar specialization principle for p-adic kimberlites. This result is a key step of our proof of the Scholze-Weinstein conjecture on local models of Shimura varieties.

## The specialization principle for p-adic kimberlites, I

Attendance for max. 20 MPIM guests (2G) in the MPIM lecture hall. For everyone else participation via Zoom only!

For zoom details please contact Peter Scholze (scholze@mpim-bonn.mpg.de).

In his master thesis Lourenço proposes and proves a specialization principle for certain formal schemes. Roughly speaking, this says that a formal scheme can be recovered from its special fiber, its rigid generic fiber and the specialization map. Kimberlites were introduced by the speaker as analogues of formal schemes in Scholze's theory of v-sheaves. In forthcoming joint work with Anschütz, Lourenço and Richarz we prove a similar specialization principle for p-adic kimberlites. This result is a key step of our proof of the Scholze-Weinstein conjecture on local models of Shimura varieties.

## Visions for the Future of Physics

## The $K(π,1)$ conjecture for affine Artin groups

Zoom ID: 972 9372 3147

For password please contact Victoriya Ozornova (viktoriya.ozornova@mpim-bonn.mpg.de)

We introduce and outline a proof of the classical $K(π,1)$ conjecture,

recently given in the case of affine Artin groups (G. Paolini, M. Salvetti, ”Proof

of the $K(π,1)$ conjecture for affine Artin groups”, Inven. Math, 224, 2 (2021).

## Fixed subgroups of automorphisms of RAAGs

Meeting ID: 931 7291 0947

For password please contact Christian Kaiser (kaiser@mpim-bonn.mpg.de).

If G is the fundamental group of a closed surface, standard Nielsen-Thurston theory shows that fixed subgroups of automorphisms of G are either trivial, infinite cyclic, or fundamental groups of subsurfaces. In sharp contrast, understanding automorphisms of free groups and their fixed points has proved a much more arduous task, being the subject of a large body of work in the 70s and 80s before being completely resolved by Bestvina and Handel in the early 90s. Comparatively little attention has been given to automorphisms of other groups. Aiming to fill this gap, we study a large class of non-positively curved groups containing all right-angled Artin and Coxeter groups and show that all their "untwisted" automorphisms have particularly nice fixed subgroups: they are finitely generated, quasi-isometrically embedded, and admit finite classifying spaces of non-positive curvature.

## Integrability of Lie Algebroids

Meeting-ID: 994 2805 4844

For passcode contact Christian Kaiser (kaiser@mpim...).

## Lie groupoids and Lie algebroids: representations and equivalences

Meeting-ID: 994 2805 4844

For passcode contact Christian Kaiser (kaiser@mpim...).

## 2021 Fields Medal Symposium: Peter Scholze

**Location: ** online

For the schedule, more detailed information and registration instruction please see here:

http://www.fields.utoronto.ca/activities/21-22/fieldsmedalsym

The Zoom link will be sent in your registration confirmation email.

Description

The 2021 Fields Medal Symposium will honour Peter Scholze (Fields Medal 2018) and explore the current and potential impact

of his work.

The Scientific Program is intended for a wide audience, including graduate students, mathematicians in other research areas, and scientists who use mathematics in an important way.

## 2021 Fields Medal Symposium: Peter Scholze

**Location: ** online

For the schedule, more detailed information and registration instruction please see here:

http://www.fields.utoronto.ca/activities/21-22/fieldsmedalsym

The Zoom link will be sent in your registration confirmation email.

Description

The 2021 Fields Medal Symposium will honour Peter Scholze (Fields Medal 2018) and explore the current and potential impact

of his work.

The Scientific Program is intended for a wide audience, including graduate students, mathematicians in other research areas, and scientists who use mathematics in an important way.

## 2021 Fields Medal Symposium: Peter Scholze

**Location: ** online

For the schedule, more detailed information and registration instruction please see here:

http://www.fields.utoronto.ca/activities/21-22/fieldsmedalsym

The Zoom link will be sent in your registration confirmation email.

Description

The 2021 Fields Medal Symposium will honour Peter Scholze (Fields Medal 2018) and explore the current and potential impact

of his work.

The Scientific Program is intended for a wide audience, including graduate students, mathematicians in other research areas, and scientists who use mathematics in an important way.

## 2021 Fields Medal Symposium: Peter Scholze

**Location: ** online

http://www.fields.utoronto.ca/activities/21-22/fieldsmedalsym

The Zoom link will be sent in your registration confirmation email.

Description

of his work.

## 2021 Fields Medal Symposium: Peter Scholze

**Location: ** online

http://www.fields.utoronto.ca/activities/21-22/fieldsmedalsym

The Zoom link will be sent in your registration confirmation email.

Description

## Relative quantum cohomology and its application

For password please contact Christian Kaiser (kaiser@mpim-bonn.mpg.de).

This talk will take place via Zoom only.

Quantum cohomology has played an important role in the modern development of algebraic geometry and symplectic geometry. In this talk, we will begin with a quick review of quantum cohomology and its classical application: Kontsevich's formula for rational plane curves. Then we will give a construction of relative quantum cohomology ring which encodes the information of relative Gromov-Witten invariants. At last, we will talk about one enumerative application of relative quantum cohomology ring. This is based on joint work with Honglu Fan and Fenglong You.

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