Mathematisches Institute


Mathematisches Institute

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Freitag E.,Mathematisches Institute
Communications in Number Theory and Physics | Year: 2016

We study a projective Calabi-Yau threefold Y+ which has been constructed in [FS]. It is rigid (h12 = 0) and has Picard number h11 = 2. We construct a pair of divisors D± which give a basis of Pic(Y+) ⊗ℤ and determine all intersection numbers D± · D± · D±.

Galaz-Garcia F.,Mathematisches Institute | Galaz-Garcia F.,Karlsruhe Institute of Technology | Guijarro L.,Autonomous University of Madrid
International Mathematics Research Notices | Year: 2015

We study three-dimensional (3D) Alexandrov spaces with a lower curvature bound, focusing on extending several results on closed 3D manifolds: first, we show that a closed 3D Alexandrov space of positive curvature, with at least one topological singularity, must be homeomorphic to the suspension of RP2; we use this to classify, up to homeomorphism, closed, positively curved Alexandrov spaces of dimension three. Secondly, we classify closed 3D Alexandrov spaces of nonnegative curvature. Thirdly, we study the well-known Poincaré Conjecture in dimension three, in the context of Alexandrov spaces, in the two forms it is usually formulated for manifolds. We first show that the only closed 3D Alexandrov space that is also a homotopy sphere is the three-sphere; then we give examples of closed, geometric, simply connected 3D Alexandrov spaces for five of the eight Thurston geometries, proving along the way the impossibility of getting such examples for the Nil, SL2(R), and Sol geometries. We conclude the paper by proving the analog of the geometrization conjecture for closed 3D Alexandrov spaces. © 2014 The Author(s) 2014. Published by Oxford University Press. All rights reserved. For permissions, please e-mail:

Cuntz J.,Mathematisches Institute
Communications in Mathematical Physics | Year: 2013

A surjective endomorphism or, more generally, a polymorphism in the sense of Schmidt and Vershik [Erg Th Dyn Sys 28(2):633-642, 2008], of a compact abelian group H induces a transformation of L2(H). We study the C*-algebra generated by this operator together with the algebra of continuous functions C(H) which acts as multiplication operators on L2(H). Under a natural condition on the endo- or polymorphism, this algebra is simple and can be described by generators and relations. In the case of an endomorphism it is always purely infinite, while for a polymorphism in the class we consider, it is either purely infinite or has a unique trace. We prove a formula allowing to determine the K-theory of these algebras and use it to compute the K-groups in a number of interesting examples. © 2012 Springer-Verlag Berlin Heidelberg.

Freitag E.,Mathematisches Institute
Communications in Number Theory and Physics | Year: 2011

In two recent papers we described some Siegel modular threefolds which admit a weak Calabi-Yau model. Not all of them admit a projective model. In fact, Bert van Geemen, in a private communication, pointed out a significative example which cannot admit a projective model. A weak Calabi-Yau threefold is projective if, and only if, it admits a Kaehler metric. The purpose of this paper is to exhibit criteria for the projectivity, to treat several examples, and to compute their Hodge numbers. Some of these Calabi-Yau manifolds seem to be new. We obtain a rigid Calabi-Yau manifold with Euler number 4.

Gravejat P.,Ecole Polytechnique - Palaiseau | Hainzl C.,Mathematisches Institute | Lewin M.,Cergy-Pontoise University | Sere E.,University of Paris Dauphine
Archive for Rational Mechanics and Analysis | Year: 2013

Using the Pauli-Villars regularization and arguments from convex analysis, we construct solutions to the classical time-independent Maxwell equations in Dirac's vacuum, in the presence of small external electromagnetic sources. The vacuum is not an empty space, but rather a quantum fluctuating medium which behaves as a nonlinear polarizable material. Its behavior is described by a Dirac equation involving infinitely many particles. The quantum corrections to the usual Maxwell equations are nonlinear and nonlocal. Even if photons are described by a purely classical electromagnetic field, the resulting vacuum polarization coincides to first order with that of full Quantum Electrodynamics. © 2013 Springer-Verlag Berlin Heidelberg.

Metsch K.,Mathematisches Institute
Designs, Codes, and Cryptography | Year: 2011

The famous Dembowski-Wagner theorem gives various characterizations of the classical geometric 2-design PG n-1(n, q) among all 2-designs with the same parameters. One of the characterizations requires that all lines have size q + 1. It was conjectured [2] that this is also true for the designs PG d (n, q) with 2 ≤ d ≤ n - 1. We establish this conjecture, hereby improving various previous results. © 2010 Springer Science+Business Media, LLC.

Deitmar A.,Mathematisches Institute
Selecta Mathematica, New Series | Year: 2012

A general structure theorem on higher order invariants is proven. For an arithmetic group, the structure of the corresponding Hecke module is determined. It is shown that the module does not contain any irreducible submodule. This explains the fact that L-functions of higher order forms have no Euler product. Higher order cohomology is introduced, classical results of Borel are generalized, and a higher order version of Borel's conjecture is stated. © 2012 Springer Basel AG.

Steinerberger S.,Mathematisches Institute
Annales Henri Poincare | Year: 2013

Let {Mathematical expression} be an open, bounded domain and {Mathematical expression} be a partition. Denote the Fraenkel asymmetry by {Mathematical expression} and write {Mathematical expression}with {Mathematical expression}. For N sufficiently large depending only on {Mathematical expression}, there is an uncertainty principle {Mathematical expression}The statement remains true in dimensions {Mathematical expression} for some constant {Mathematical expression}. As an application, we give an (unspecified) improvement of Pleijel's estimate on the number of nodal domains of a Laplacian eigenfunction and an improved inequality for a spectral partition problem. © 2013 Springer Basel.

Blomer V.,Mathematisches Institute | Maga P.,MTA Alfred Renyi Institute of Mathematics
Selecta Mathematica, New Series | Year: 2016

Let F be an L2-normalized Hecke Maaß cusp form for Γ 0(N) ⊆ SL n(Z) with Laplace eigenvalue λF. If Ω is a compact subset of Γ 0(N) \ PGL n/ PO n, we show the bound ‖F|Ω‖∞≪ΩNελFn(n-1)/8-δ for some constant δ= δn> 0 depending only on n. © 2016, Springer International Publishing.

Hong S.,Mathematisches Institute
Differential Geometry and its Application | Year: 2015

Harish-Chandra's volume formula shows that the volume of a flag manifold G/T, where the measure is induced by an invariant inner product on the Lie algebra of G, is determined up to a scalar by the algebraic properties of G. This article explains how to deduce Harish-Chandra's formula from Weyl's law by utilizing the Euler-Maclaurin formula. This approach suggests that there may be an elementary explanation available for the appearance of the power series x/(1-e-x) in the Atiyah-Singer index theorem. © 2014 Elsevier B.V.

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