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European Mathematical Society Publishing House
2024-03-29 01:03:40
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https://www.ems-ph.org/meta/jmeta-stream.php?jrn=CMH&vol=93&iss=3&update_since=2024-03-29
Commentarii Mathematici Helvetici
Comment. Math. Helv.
CMH
0010-2571
1420-8946
General
10.4171/CMH
http://www.ems-ph.org/doi/10.4171/CMH
subscribers, moving wall 5 years
European Mathematical Society Publishing House
Zuerich, Switzerland
© Swiss Mathematical Society
93
2018
3
Hurwitz numbers for real polynomials
Ilia
Itenberg
Sorbonne Université, Paris, and École Normale Supérieure, Paris, France
Dimitri
Zvonkine
Université de Versailles, France
Hurwitz numbers, real polynomials, signed count, real enumerative problems
We consider the problem of defining and computing real analogs of polynomial Hurwitz numbers, in other words, the problem of counting properly normalized real polynomials with fixed ramification profiles over real branch points.We show that, provided the polynomials are counted with an appropriate sign, their number does not depend on the order of the branch points on the real line. We study generating series for the invariants thus obtained, determine necessary and sufficient conditions for the vanishing and nonvanishing of these generating series, and obtain a logarithmic asymptotic for the invariants as the degree of the polynomials tends to infinity.
Algebraic geometry
Combinatorics
441
474
10.4171/CMH/440
http://www.ems-ph.org/doi/10.4171/CMH/440
9
10
2018
Engel structures and weakly hyperbolic flows on four-manifolds
Dieter
Kotschick
Universität München, Germany
Thomas
Vogel
Universität München, Germany
Engel structure, hyperbolic flow, four-manifold, even contact structure
We study pairs of Engel structures on four-manifolds whose intersection has constant rank one and which define the same even contact structure, but induce different orientations on it. We establish a correspondence between such pairs of Engel structures and a class of weakly hyperbolic flows. This correspondence is analogous to the correspondence between bi-contact structures and projectively or conformally Anosov flows on three-manifolds found by Eliashberg–Thurston and by Mitsumatsu.
Dynamical systems and ergodic theory
Differential geometry
Global analysis, analysis on manifolds
475
491
10.4171/CMH/441
http://www.ems-ph.org/doi/10.4171/CMH/441
9
10
2018
Kloosterman paths of prime powers moduli
Guillaume
Ricotta
Université de Bordeaux I, Talence, France
Emmanuel
Royer
Université Clermont Auvergne, Aubière, France
Kloosterman sums, moments, random Fourier series, probability in Banach spaces
In [12], the authors proved, using a deep independence result of Kloosterman sheaves, that the polygonal paths joining the partial sums of the normalized classical Kloosterman sums $S(a,b_0;p)/p^{1/2}$ converge in the sense of finite distributions to a specific random Fourier series, as $a$ varies over $(\mathbb{Z}/p\mathbb{Z})^\times$, $b_0$ is fixed in $(\mathbb{Z}/p\mathbb{Z})^\times$ and $p$ tends to infinity among the odd prime numbers. This article considers the case of $S(a,b_0;p^n)/p^{n/2}$, as $a$ varies over $(\mathbb{Z}/p^n\mathbb{Z})^\times$, $b_0$ is fixed in $(\mathbb{Z}/p^n\mathbb{Z})^\times$, $p$ tends to infinity among the odd prime numbers and $n\geq 2$ is a fixed integer. A convergence in law in the Banach space of complex-valued continuous function on $[0,1]$ is also established, as $(a,b)$ varies over $(\mathbb{Z}/p^n\mathbb{Z})^\times\times(\mathbb{Z}/p^n\mathbb{Z})^\times$, $p$ tends to infinity among the odd prime numbers and $n\geq 2$ is a fixed integer. This is the analogue of the result obtained in [12] in the prime moduli case.
Number theory
Probability theory and stochastic processes
493
532
10.4171/CMH/442
http://www.ems-ph.org/doi/10.4171/CMH/442
9
10
2018
Non-arithmetic ball quotients from a configuration of elliptic curves in an Abelian surface
Martin
Deraux
Université Grenoble Alpes, Gières, France
Orbifold uniformization, ball quotients, non-arithmetic lattices, complex reflections, affine crystallographic groups
We construct some non-arithmetic ball quotients as branched covers of a quotient of an Abelian surface by a finite group, and compare them with lattices that previously appear in the literature. This gives an alternative construction, which is independent of the computer, of some lattices constructed by the author with Parker and Paupert.
Topological groups, Lie groups
Algebraic geometry
Group theory and generalizations
533
554
10.4171/CMH/443
http://www.ems-ph.org/doi/10.4171/CMH/443
9
10
2018
Finite-dimensional representations constructed from random walks
Anna
Erschler
École Normale Supérieure, Paris, France
Narutaka
Ozawa
Kyoto University, Japan
Orthogonal representations, harmonic cocycle, random walks, transition probabilities, amenable groups, Shalom’s property, Kazhdan’s property $T$ , Central Limit theorem
Given a 1-cocycle $b$ with coefficients in an orthogonal representation, we show that every finite dimensional summand of $b$ is cohomologically trivial if and only if $\| b(X_n) \|^2/n$ tends to a constant in probability, where $X_n$ is the trajectory of the random walk $(G,\mu)$. As a corollary, we obtain sufficient conditions for $G$ to satisfy Shalom's property $H_{\mathrm{FD}}$. Another application is a convergence to a constant in probability of $\mu^{*n}(e) -\mu^{*n}(g)$, $n\gg m$, normalized by its average with respect to $\mu^{*m}$, for any finitely generated infinite amenable group without infinite virtually abelian quotients. Finally, we show that the harmonic equivariant mapping of $G$ to a Hilbert space obtained as an $U$-ultralimit of normalized $\mu^{*n}- g \mu^{*n}$ can depend on the ultrafilter $U$ for some groups.
Probability theory and stochastic processes
Topological groups, Lie groups
Abstract harmonic analysis
555
586
10.4171/CMH/444
http://www.ems-ph.org/doi/10.4171/CMH/444
9
10
2018
Connections on equivariant Hamiltonian Floer cohomology
Paul
Seidel
Massachusetts Institute of Technology, Cambridge, USA
Floer homology, Gauss–Manin connection, topological quantum field theory
We construct connections on $S^1$-equivariant Hamiltonian Floer cohomology, which differentiate with respect to certain formal parameters.
Differential geometry
Algebraic topology
Manifolds and cell complexes
587
644
10.4171/CMH/445
http://www.ems-ph.org/doi/10.4171/CMH/445
9
10
2018
Complete minimal submanifolds with nullity in Euclidean spheres
Marcos
Dajczer
IMPA, Rio de Janeiro, Brazil
Theodoros
Kasioumis
University of Ioannina, Greece
Andreas
Savas-Halilaj
Leibniz Universität Hannover, Germany
Theodoros
Vlachos
University of Ioannina, Greece
Minimal submanifolds, index of relative nullity, real analytic set, Omori–Yau maximum principle
In this paper we investigate $m$-dimensional complete minimal submanifolds in Euclidean spheres with index of relative nullity at least $m–$2 at any point. These are austere submanifolds in the sense of Harvey and Lawson [19] and were initially studied by Bryant [3]. For any dimension and codimension there is an abundance of non-complete examples fully described by Dajczer and Florit [7] in terms of a class of surfaces, called elliptic, for which the ellipse of curvature of a certain order is a circle at any point. Under the assumption of completeness, it turns out that any submanifold is either totally geodesic or has dimension three. In the latter case there are plenty of examples, even compact ones. Under the mild assumption that the Omori–Yau maximum principle holds on the manifold, a trivial condition in the compact case, we provide a complete local parametric description of the submanifolds in terms of 1-isotropic surfaces in Euclidean space. These are the minimal surfaces for which the standard ellipse of curvature is a circle at any point. For these surfaces, there exists a Weierstrass type representation that generates all simply connected ones.
Differential geometry
645
660
10.4171/CMH/446
http://www.ems-ph.org/doi/10.4171/CMH/446
9
10
2018