Band 2012, Heft 672

In this paper we introduce a general version of the Loewner differential equation which allows us to present a new and unified treatment of both the radial equation introduced in 1923 by K. Loewner and the chordal equation introduced in 2000 by O. Schramm. In particular, we prove that evolution families in the unit disc are in one to one correspondence with solutions to this new type of Loewner equations. Also, we give a Berkson–Porta type formula for non-autonomous weak holomorphic vector fields which generate such Loewner differential equations and study in detail geometric and dynamical properties of evolution families.

In this paper, we construct smooth forward Ricci flow evolutions of singular initial metrics resulting from rotationally symmetric neckpinches on 𝒮 n  + 1 , without performing an intervening surgery. In the restrictive context of rotational symmetry, this construction gives evidence in favor of Perelman's hope for a “canonically defined Ricci flow through singularities”.

Having developed a description of indefinite extrinsic symmetric spaces by corresponding infinitesimal objects in the preceding paper we now study the classification problem for these algebraic objects. In most cases the transvection group of an indefinite extrinsic symmetric space is not semisimple, which makes the classification difficult. We use the recently developed method of quadratic extensions for (𝔥,  K )-invariant metric Lie algebras to tackle this problem. We obtain a one-to-one correspondence between isometry classes of extrinsic symmetric spaces and a certain cohomology set. This allows a systematic construction of extrinsic symmetric spaces and explicit classification results, e.g., if the metric of the embedded manifold or the ambient space has a small index. We will illustrate this by classifying all Lorentzian extrinsic symmetric spaces.

We consider a parabolic version of the mass transport problem, and show that a solution converges to a solution of the original mass transport problem under suitable conditions on the cost function, and initial and target domains.

Let p ( n ) denote the ordinary partition function. Subbarao conjectured that in every arithmetic progression r  ( mod   t ) there are infinitely many integers N  ≡  r  ( mod   t ) for which p ( N ) is even, and infinitely many integers M  ≡  r  ( mod   t ) for which p ( M ) is odd. In the even case the conjecture was settled by Ken Ono. In this paper we prove the odd part of the conjecture which together with Ono's result implies the full conjecture. We also prove that for every arithmetic progression r  ( mod   t ) there are infinitely many integers N  ≡  r  ( mod   t ) such that p ( N ) ≢ 0 ( mod  3), which settles an open problem posed by Scott Ahlgren and Ken Ono.

We define a regularized theta lift from SL 2 to orthogonal groups over totally real fields. It takes harmonic ‘Whittaker forms’ to automorphic Green functions and weakly holomorphic Whittaker forms to meromorphic modular forms on orthogonal groups with zeros and poles supported on special divisors, generalizing Borcherds' work on automorphic products. To prove our results, we use the spectral expansion of the lift and study its relationship with the cohomological theta lift of Kudla and Millson.

In this paper, we study the structure of the reduced C*-algebras and von Neumann algebras associated to the free orthogonal and free unitary quantum groups. We show that the reduced von Neumann algebras of these quantum groups always have the Haagerup approximation property. Combining this result with a Haagerup-type inequality due to Vergnioux, we also show that the reduced C*-algebras always have the metric approximation property.