## 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. Motivated by renewed interest in these groups in relation to moduli spaces of manifolds, I will recall Kreck’s description of Γ(g,n) and explain how to resolve the remaining extension problems completely for n odd.

## 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. Motivated by renewed interest in these groups in relation to moduli spaces of manifolds, I will recall Kreck’s description of Γ(g,n) and explain how to resolve the remaining extension problems completely for n odd.

## 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

that if the number of ramifications is sufficiently high relative to the degree, then the monodromy is an arithmetic group. The proof uses the existence of a large number of unipotent elements in the monodromy group in the relevant case.

## 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.

In this talk, we will review the previous results and in particular let us see the existence of irreducible Artin characters of growing degree with bounded root conductors.

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

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

## New guests at the MPIM

## Γ-spaces, PAMs, homology theories and operads

The main goal of the talk is to see how partial abelian monoids give

rise to spectra and homology theories. The classical approach uses

Γ-spaces, so we will discuss the connection between them and PAMs. It

turns out that there is also a strong relation to the tensor product of

PAMs from the first talk. Moreover, we have already seen that the little

n-cubes give rise to a PAM-ring. This idea will be generalised for an

arbitrary operad, using the language of Γ-spaces.

## Representation theory of symmetric groups, linear groups and symmetric polynomials, II

## Volumes and lattice counts in polyhedra

In the second session, we use the formalism exposed in the first talk to various valuations. The exponential valuation allows to compute the volume of polytopes through Brion's theorem, and allows to give a meaning to the notion of volume for polyhedra without lines. Next, lattice points count of polyhedra is considered, where a discrete analog of the exponential valuation is defined via generating functions. We recover then Brion's theorem for lattice point count.

## From ODEs to topological recursion: the case of Hurwitz numbers -- CANCELLED --

I will first explain the general theory due to Bergere, Eynard and myself to associate, to any finite order ODE, a kernel K(x,x') and a collection of (W_n)_{n > 0} satisfying loop equations. Under some assumptions on the semiclassical expansions of these quantities, this implies these expansions are computed by the topological recursion.

Then, following the third paper of Alexandrov-Chapuy-Eynard-Harnad, I will describe the ODE that generating series of Hurwitz numbers satisfy, and show that in this case, the coefficients of the semiclassical expansion of W_n are generating series of Hurwitz numbers with fixed topology and ramification profile of length n over \infty.

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## Curves on K3 surfaces

Bogomolov and Mumford proved that every complex projective K3 surface contains a rational curve. Since then, a lot of progress has been made by Bogomolov, Chen, Hassett, Li, Liedtke, Tschinkel and others, towards the stronger statement that any such surface in fact contains infinitely many rational curves. In this talk I will present joint work with Xi Chen and Christian Liedtke completing the remaining cases of this conjecture, reproving some of the main previously known cases more conceptually and extending the result to arbitrary genus.

## Tropical and logarithmic strata of abelian differentials

## Cohomology of arithmetic groups - the general scenario

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