## "Pentagramma Mirificum". Hirzebruch lecture by Sergey Fomin on Friday, November 8, University Club Bonn

**Pentagramma Mirificum** (the miraculous pentagram) is a beautiful geometric construction studied by Napier and Gauss. Its algebraic description yields the simplest instance of cluster transformations, a remarkable family of recurrences which arise in diverse mathematical contexts, from representation theory and enumerative combinatorics to theoretical physics and classical geometry (Euclidean, spherical, or hyperbolic).

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## Boundary-adapted arithmetic random waves

## General quantization scheme for symplectic manifolds which admit Kähler structures

We will consider arbitrary holomorphic families of Kähler structures on a fixed prequantizable symplectic manifold and provided the corresponding family of Kähler quantisations for level k form a vector bundle over the chosen family of Kähler structures, we will for large enough k construct a large family of Hitchin connections in this bundle. We will also consider a natural family of Hermitian structures on the bundle of quantum spaces and understand which (unique) Hitchin connection is compatible with a given one of these Hermitian structures.

## Lattice points, Ehrhart theory, and the relation to volumes

Continuing from the previous session, we will define the valuation for exponential sums, a discrete analogue of exponential integrals, and obtain another version of Brion's theorem. We continue to Ehrhart theory, showing polynomiality of lattice counts for polytopes with fixed cones of feasible directions. Finally, we relate exponential sums and exponential integrals via an Euler-Maclaurin type formula.

## Automorphisms and Periods of Cubic Fourfolds

Cubic fourfold plays a central role in algebraic geometry because of its close relation to hyper-Kahler geometry. In this talk I will start with this relation and discuss the Hodge theory and moduli theory for cubic fourfolds. Then I will introduce an application of the above theories, namely, a complete classification of the symplectic automorphism groups of smooth cubic fourfolds. This work can be regarded as a higher dimensional analogue of Mukai's celebrated result on classification of finite symplectic automorphism groups of K3 surfaces. This is a joint work with Radu Laza.

## New guests at the MPIM

## Splitting theorems for configuration spaces

## Formal groups

## Trisections in CP^2, I

In the first hour, I will describe the standard trisection of CP^2 and how to put complex curves in CP^2 into bridge position with respect to this trisection. Time permitting, I will also present trisections of projective surfaces obtained as a branched covers over these curves. In the second hour, I will describe how to compute the homology and intersection form from a trisection diagram.

## Some recent results of free boundary minimal hypersurfaces

In this talk, I will introduce some recent developments of free boundary minimal hypersurfaces, including compactness, generic finiteness and index estimates. As an application, we also give the existence of infinitely many minimal hypersurfaces with non-empty free boundary under some weak assumptions. Part of the work is joint with Q.Guang, M.Li and X.Zhou.

## Trisections in CP^2, II

In the first hour, I will describe the standard trisection of CP^2 and how to put complex curves in CP^2 into bridge position with respect to this trisection. Time permitting, I will also present trisections of projective surfaces obtained as a branched covers over these curves. In the second hour, I will describe how to compute the homology and intersection form from a trisection diagram.

## Resonant spaces for volume-preserving Anosov flows

## Cohomology of arithmetic groups - A specific -highly non trivial- example

## Mapping class groups of highly connected manifolds, II

The classical mapping class group Γ(g) of a surface of genus g shares many features with its higher dimensional analogue Γ(g,n)—the group of isotopy classes of diffeomorphisms of #ᵍ(Sⁿ x Sⁿ)—but some aspects become easier in high dimensions. This enabled Kreck in the 70’s to describe Γ(g,n) for n>2 in terms of an arithmetic group and the group of exotic spheres. His answer, however, left open two extension problems which were later understood in some dimensions, but remained unsettled in most cases.

## Mapping class groups of highly connected manifolds, I

The classical mapping class group Γ(g) of a surface of genus g shares many features with its higher dimensional analogue Γ(g,n)—the group of isotopy classes of diffeomorphisms of #ᵍ(Sⁿ x Sⁿ)—but some aspects become easier in high dimensions. This enabled Kreck in the 70’s to describe Γ(g,n) for n>2 in terms of an arithmetic group and the group of exotic spheres. His answer, however, left open two extension problems which were later understood in some dimensions, but remained unsettled in most cases.

## Specializations of the Burau representation at roots of unity

The Burau representation and its specialisations at roots of unity are closely related to monodromy representations associated to families of cyclic coverings of a fixed degree of the projective line. If the number of ramifications is small relative to the degree, Deligne and Moscow constructed non arithmetic monodromy using these families. We show

## Quasi Isometries

## Estimates for the Artin conductor

By using geometry of numbers, Minkowski showed that there exists a constant C such that if D_K is the discriminant of a number field K, then |D_K|>C^[K:Q]. In 1978, from the existence of infinite class field towers, Martinet constructed sequences of number fields of growing degree and bounded root discriminant.

It is natural to ask if it is possible to extends the previous results to the Artin conductor.

## Trisections and branched covers of S^4, I

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