### A classification of 2-type curves in the Minkowski space ${\mathbb{E}}_{1}^{n}$.

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A cluster ensemble is a pair $(\mathcal{X},\mathcal{A})$ of positive spaces (i.e. varieties equipped with positive atlases), coming with an action of a symmetry group $\Gamma $. The space $\mathcal{A}$ is closely related to the spectrum of a cluster algebra [12]. The two spaces are related by a morphism $p:\mathcal{A}\to \mathcal{X}$. The space $\mathcal{A}$ is equipped with a closed $2$-form, possibly degenerate, and the space $\mathcal{X}$ has a Poisson structure. The map $p$ is compatible with these structures. The dilogarithm together with its motivic and quantum avatars plays a central role...

O’Grady showed that certain special sextics in ${\mathbb{P}}^{5}$ called EPW sextics admit smooth double covers with a holomorphic symplectic structure. We propose another perspective on these symplectic manifolds, by showing that they can be constructed from the Hilbert schemes of conics on Fano fourfolds of degree ten. As applications, we construct families of Lagrangian surfaces in these symplectic fourfolds, and related integrable systems whose fibers are intermediate Jacobians.

We study multiple Bernoulli series associated to a sequence of vectors generating a lattice in a vector space. The associated multiple Bernoulli series is a periodic and locally polynomial function, and we give an explicit formula (called wall crossing formula) comparing the polynomial densities in two adjacent domains of polynomiality separated by a hyperplane. We also present a formula in the spirit of Euler-MacLaurin formula. Finally, we give a decomposition formula for the Bernoulli series describing...

Let Σ be a closed oriented Riemann surface of genus at least 2. By using symplectic chain complex, we construct a volume element for a Hitchin component of Hom(π₁(Σ),PSLₙ(ℝ))/PSLₙ(ℝ) for n > 2.

This article is concerned with moduli spaces of connections on bundles on Riemann surfaces, where the structure group of the bundle may vary in different regions of the surface. Here we will describe such moduli spaces as complex symplectic manifolds, generalising the complex character varieties of Riemann surfaces.