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Gießen, Germany

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.


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.


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.

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