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We study the function $$\mbox{KVol} : (X,\omega)\mapsto \mbox{Vol} (X,\omega) \sup_{\alpha,\beta} \frac{\mbox{Int} (\alpha,\beta)}{l_g (\alpha) l_g (\beta)}$$ defined on the moduli spaces of translation surfaces. More precisely, let $\mathcal T_n$ be the Teichm\"uller discs of the original Veech surface $(X_n,\omega_n)$ arising from right-angled triangle with angles $(\pi/2,\pi/n,(n-2)\pi/2n)$ by the unfolding construction for $n\geq 5$. For $n \equiv 1 \mod 2$ and any $(X,\omega)\in \mathcal T_n$, we establish the (sharp) bounds $$ \frac{n}{2} \cot \frac{\pi}{n} \leq \mbox{KVol}(X,\omega) \leq \frac{n}{2} \cot \frac{\pi}{n} \cdot \frac1{\sin \frac{2\pi}{n}}.$$ The lower bound is uniquely realized at $(X_n,\omega_n)$., Comment: New version with the first author added and completely different methods. We focus on the $n=2m+1$ case, the $n=4m$ case is dealt with in a forthcoming paper by the first author. 30pages, 15 figures

We review the different notions about translation surfaces which are necessary to understand McMullen's classification of $GL_2^+(\mathbb{R})$-orbit closures in genus two. In Section 2 we recall the different definitions of a translation surface, in increasing order of abstraction, starting with cutting and pasting plane polygons, ending with Abelian differentials. In Section 3 we define the moduli space of translation surfaces and explain its stratification by the type of zeroes of the Abelian differential, the local coordinates given by the relative periods, its relationship with the moduli space of complex structures and the Teichm\H{u}ller geodesic flow. In Part II we introduce the $GL_2^+(\mathbb{R})$-action, and define the related notions of Veech group, Teichm\H{u}ller disk, and Veech surface. In Section 9 we explain how McMullen classifies $GL_2^+(\mathbb{R})$-orbit closures in genus $2$: you have orbit closures of dimension $1$ (Veech surfaces, of which a complete list is given), $2$ (Hilbert modular surfaces, of which again a complete list is given), and $3$ (the whole moduli space of complex structures). In the last section we review some recent progress in higher genus., Comment: The final version of this paper will appear in the book: Surveys in Geometry I, ed. A. Papadopoulos, Springer Nature, 2021

This paper considers a multi-patch model, where each patch follows a logistic law, and patches are coupled by asymmetrical migration terms. First, in the case of perfect mixing, i.e when the migration rate tends to infinity, the total population follows a logistic equation with a carrying capacity which in general is different from the sum of the n carrying capacities, and depends on the migration terms. Second, we determine, in some particular cases, the conditions under which fragmentation and asymmetrical migration can lead to a total equilibrium population greater or smaller than the sum of the carrying capacities. Finally, for the three-patch model, we show numerically the existence of at least three critical values of the migration rate for which the total equilibrium population equals the sum of the carrying capacities., Comment: 29 pages, 7 figures, 1 table. Section 5 added in version 2

We study the quantity $\mbox{KVol}$ defined as the supremum, over all pairs of closed curves, of their algebraic intersection, divided by the product of their lengths, times the area of the surface. The surfaces we consider live in the stratum $\mathcal{H}(2)$ of translation surfaces of genus $2$, with one conical point. We provide an explicit sequence $L(n,n)$ of surfaces such that $\mbox{KVol}(L(n,n)) \longrightarrow 2$ when $n$ goes to infinity, $2$ being the conjectured infimum for $\mbox{KVol}$ over $\mathcal{H}(2)$., Comment: 8 pages, 3 figures. arXiv admin note: text overlap with arXiv:2007.10847

The setting is a square-tiled surface X. We study the quantity KVol, defined as the supremum over all pairs of closed curves, of their algebraic intersection divided by the product of their length, times the volume of X (so as to make it scaling-invariant). We give a hyperbolic-geometric construction to compute KVol in a family of Teichm\H{u}ller disks of square-tiled surfaces., Comment: 25 pages, 15 figures

We define Dirichlet type series associated with homology length spectra of Riemannian, or Finsler, manifolds, or polyhedra, and investigate some of their analytical properties. As a consequence we obtain an inequality analogous to Gromov's classical intersystolic inequality, but taking the whole homology length spectrum into account rather than just the systole.

On a surface with a Finsler metric, we investigate the asymptotic growth of the number of closed geodesics of length less than $L$ which minimize length among all geodesic multicurves in the same homology class. An important class of surfaces which are of interest to us are hyperbolic surfaces.

We prove that if a time-periodic Tonelli Lagrangian on a closed manifold $M$ satisfies a strong version of the Differentiability Problem for Mather's $\beta$-function, then the Legendre transforms of rational homology classes are dense in the first cohomology of $M$, which is a first step towards Ma\~n\'e's conjecture.

Given a closed, oriented surface M, the algebraic intersection of closed curves induces a symplectic form Int(.,.) on the first homology group of M. If M is equipped with a Riemannian metric g, the first homology group of M inherits a norm, called the stable norm. We study the norm of the bilinear form Int(.,.), with respect to the stable norm., Comment: 30 pages, 8 figures (submitted to Manuscripta Mathematica)

We prove that for a generic Tonelli Lagrangian on a configuration space of dimension two, there exists an open dense subset of cohomology classes, whose Aubry set consists of exactly one hyperbolic periodic orbit., Comment: This paper has been withdrawn by the author due to a fatal flaw in the proof of the main results. Actually the error comes from the reference [Mt03]. In Theorem 1 of said reference an equality is claimed in the case where the base manifold is a closed, oriented surface. The proof of this equality uses a result by Boyland and Gol\'e. This result is incorrectly cited

We review the author's results on Mather's $\beta$ function : non-strict convexity of $\beta$ when the configuration space has dimension two, link between the size of the Aubry set and the differentiability of $\beta$, correlation between the rationality of the homology class and the differentiability of $\beta$, equality of the Mather set and the Aubry set for a large number of cohomology classes when the configuration space has dimension two, link beween the differentiability of $\beta$ and the integrability of the system. Ma\~{n}\'e's conjectures are discussed in Chapters 6 and 7. A short list of open problems is given at the end of each chapter. In Appendix A we prove a theorem which extends Theorem 5 of reference [Mt09]. In Appendix B we discuss a geometrical problem which arises from Chapter 3, but may be of independant interest., Comment: M\'emoire d'habilitation, 53 pages, 5 figures

We prove Ma\~n\'e's conjectures in the context of codimension one Aubry-Mather theory

Let L be a time-periodic Lagrangian on a two-torus. Then the beta-function of L is differentiable at least in k directions at any k-irrational homology class, for k= 0, 1, 2., Comment: New version which focuses on the two-torus case

We prove that Ma\~n\'e's conjecture, as stated in {\em Lagrangian flows: the dynamics of globally minimizing orbits}, Bol. Soc. Brasil. Mat. (N.S.) 28 (1997), no. 2, 141--153, contains another conjecture of Ma\~n\'e, stated in {\em Generic properties and problems of minimizing measures of Lagrangian systems} Nonlinearity 9 (1996) 273-310.

In this article we study the differentiability of Mather's $\beta$-function on closed surfaces and its relation to the integrability of the system., Comment: 20 pages, 1 figure. Major revision of the paper

We study autonomous Tonelli Lagrangians on closed surfaces. We aim to clarify the relationship between the Aubry set and the Mather set, when the latter consists of periodic orbits which are not fixed points. Our main result says that in that case the Aubry set and the Mather set almost always coincide., Comment: Revised and expanded version. New proof of Lemma 2.3 (formerly Lemma 14)

We study the stable norm on the first homology of a closed, non-orientable surface equipped with a Riemannian metric. We prove that in every conformal class there exists a metric whose stable norm is polyhedral. Furthermore the stable norm is never strictly convex if the first Betti number of the surface is greater than two.

If the $\beta$-function of a time-periodic Lagrangian on a manifold $M$ has a vertex at a $k$-irrational homology class $h$, then $2k \leq \dim M$. Furthermore if $\dim M =2$ $h$ is rational.

We prove the existence of $C^{1}$ critical subsolutions of the Hamilton-Jacobi equation for a time-periodic Hamiltonian system. We draw a consequence for the Minimal Action functional of the system., Comment: To appear in Ergodic Theory and Dynamical Systems

We study Lagrangian systems on a closed manifold. We link the differentiability of Mather's beta-function with the topological complexity of the complement of the Aubry set. As a consequence, when the dimension of the manifold is less than or equal to two, the differentiability of the beta-function at a given homology class is forced by the irrationality of the homology class. As an application we prove the two-dimensional case of a conjecture by Ricardo Mane., Comment: 17 pages, 2nd version