Hautes Etudes Sci Research Articles

Traveling Waves Near Couette Flow for the 2D Euler Equation

In this paper we reveal the existence of a large family of new, nontrivial and smooth traveling waves for the 2D Euler equation at an arbitrarily small distance from the Couette flow in Hs, with s<3/2, at the level of the vorticity. The speed of these waves is of order 1 with respect to this distance. This result strongly contrasts with the setting of very high regularity in Gevrey spaces [see Bedrossian and Masmoudi (Publ Math Inst Hautes Etudes Sci 122:195–300, 2015)], where the problem exhibits an inviscid damping mechanism that leads to relaxation of perturbations back to nearby shear flows. It also complements the fact that there not exist nontrivial traveling waves in the H32+ neighborhoods of Couette flow [see Lin and Zeng (Arch Ration Mech Anal 200:1075–1097, 2011)].

The Farrell–Jones Conjecture for mapping class groups

We prove the Farrell–Jones Conjecture for mapping class groups. The proof uses the Masur–Minsky theory of the large scale geometry of mapping class groups and the geometry of the thick part of Teichmuller space. The proof is presented in an axiomatic setup, extending the projection axioms in Bestvina et al. (Publ Math Inst Hautes Etudes Sci 122:1–64, 2015). More specifically, we prove that the action of $${{\,\mathrm{Mod(\Sigma )}\,}}$$ on the Thurston compactification of Teichmuller space is finitely $$\mathcal F$$ -amenable for the family $${\mathcal {F}}$$ consisting of virtual point stabilizers.

Volume entropy for minimal presentations of surface groups in all ranks

We study the volume entropy of a class of presentations (including the classical ones) for all surface groups, called minimal geometric presentations. We rediscover a formula first obtained by Cannon and Wagreich (Math Ann 293(2), 239–257, 1992) with the computation in a non published manuscript by Cannon (The growth of the closed surface groups and the compact hyperbolic coxeter groups, 1980). The result is surprising: an explicit polynomial of degree n, the rank of the group, encodes the volume entropy of all classical presentations of surface groups. The approach we use is completely different. It is based on a dynamical system construction following an idea due to Bowen and Series (Inst Hautes Etudes Sci Publ Math 50, 153–170, 1979) and extended to all geometric presentations in Los (J Topol, 7(1), 120–154, 2013). The result is an explicit formula for the volume entropy of minimal presentations for all surface groups, showing a polynomial dependence in the rank \(n>2\). We prove that for a surface group \(G_n\) of rank n with a classical presentation \(P_n\) the volume entropy is \(\log (\lambda _n)\), where \(\lambda _n\) is the unique real root larger than one of the polynomial $$\begin{aligned} x^{n} - 2(n - 1) \sum _{j=1}^{n-1} x^{j} + 1. \end{aligned}$$

On connectivity of Julia sets of transcendental meromorphic maps and weakly repelling fixed points I

It is known that the Julia set of the Newton method of a non-constant polynomial is connected (Mitsuhiro Shishikura, Preprint, 1990, M/90/37, Inst. Hautes Etudes Sci.). This is, in fact, a consequence of a much more general result that establishes the relationship between simple connectivity of Fatou components of rational maps and fixed points which are repelling or parabolic with multiplier 1. In this paper we study Fatou components of transcendental meromorphic functions; that is, we show the existence of such fixed points, provided that immediate attractive basins or preperiodic components are multiply connected.

Harmonic bundle solutions of topological–antitopological fusion in para-complex geometry

In this work we introduce the notion of a para-harmonic bundle, i.e. the generalization of a harmonic bundle [C.T. Simpson, Higgs-bundles and local systems, Inst. Hautes Etudes Sci. Publ. Math. 75 (1992) 5–95] to para-complex differential geometry. We show that para-harmonic bundles are solutions of the para-complex version of metric t t ∗ -bundles introduced in [L. Schäfer, t t ∗ -bundles in para-complex geometry, special para-Kähler manifolds and para-pluriharmonic maps, Differential Geom. Appl. 24 (1) (2006) 60–89]. Further we analyze the correspondence between metric para- t t ∗ -bundles of rank 2 r over a para-complex manifold M and para-pluriharmonic maps from M into the pseudo-Riemannian symmetric space GL ( r , R ) / O ( p , q ) , which was shown in [L. Schäfer, t t ∗ -bundles in para-complex geometry, special para-Kähler manifolds and para-pluriharmonic maps, Differential Geom. Appl. 24 (1) (2006) 60–89], in the case of a para-harmonic bundle. It is proven, that for para-harmonic bundles the associated para-pluriharmonic maps take values in the totally geodesic subspace GL ( r , C ) / U π ( C r ) of GL ( 2 r , R ) / O ( r , r ) . This defines a map Φ from para-harmonic bundles over M to para-pluriharmonic maps from M to GL ( r , C ) / U π ( C r ) . The image of Φ is also characterized in the paper.

Simplical volume of closed locally symmetric spaces of non-compact type

We show that closed, locally symmetric spaces of non-compact type have positive simplicial volume. This gives a positive answer to a question that was first raised by (Gromov in Inst Hautes Etudes Sci Publ Math 56:5, 1982).

Concentration for locally acting permutations

Talagrand (Publ. Math. Inst. Hautes Etudes Sci. 81 (1995) 73) gave a concentration inequality concerning permutations picked uniformly at random from a symmetric group, and this was extended in McDiarmid (Combin. Probab. Comput. 11 (2002) 163) to handle permutations picked uniformly at random from a direct product of symmetric groups. Here we extend these results further, to cover more general permutation groups which act suitably ‘locally’.

Abelian Chern–Simons theory. I. A topological quantum field theory

We give a construction of the Abelian Chern–Simons gauge theory from the point of view of a 2+1-dimensional topological quantum field theory. The definition of the quantum theory relies on geometric quantization ideas that have been previously explored in connection to the non-Abelian Chern–Simons theory [J. Diff. Geom. 33, 787–902 (1991); Topology 32, 509–529 (1993)]. We formulate the topological quantum field theory in terms of the category of extended 2- and 3-manifolds introduced in a preprint by Walker in 1991 and prove that it satisfies the axioms of unitary topological quantum field theories formulated by Atiyah [Publ. Math. Inst. Hautes Etudes Sci. Pans 68, 175–186 (1989)].

A Note on the Shimura Subgroup of J0( p)

Let J 0( p) be the Jacobian of the modular curve X 0( p). The group of connected components of its Néron model is cyclic, and has two special generators. One comes from the cuspidal group, the other from the Shimura subgroup. We compute their ratio, answering a question raised by B. Mazur ( Inst. Hautes Etudes Sci. Publ. Math. 47, 1977, 33-186).