2007, Burban, Igor, Kreussler, Bernd

In this paper we introduce the notion of a gemetric associative r-matrix attached to a genus one fibration with a section and irreducible fibres. It allows us to study degenerations of solutions of the classical Yang-Baxter equation using the approach of Polishchuk. We also calculate certain solutions of the classical, quantum and associative Yang-Baxter equations obtained from moduli spaces of (semi-)stable vector bundles on Weierstraß cubic curves.

2007, Kesseböhmer, Marc, Stratmann, Bernd O.

In this paper we give non-trivial applications of the thermodynamic formalism to the theory of distribution functions of Gibbs measures (devil’s staircases) supported on limit sets of finitely generated conformal iterated function systems in R. For a large class of these Gibbs states we determine the Hausdorff dimension of the set of points at which the distribution function of these measures is not a-Hölder-differentiable. The obtained results give significant extensions of recent work by Darst, Dekking, Falconer, Li, Morris, and Xiao. In particular, our results clearly show that the results of these authors have their natural home within thermodynamic formalism.

2007, Huisken, Gerhard

The current article arose from a lecture1 given by the author in October 2005 on the work of R. Hamilton and G. Perelman on Ricci-flow and explains central analytical ingredients in geometric parabolic evolution equations that allow the application of these flows to geometric problems including the Uniformisation Theorem and the proof of the Poincare conjecture. Parabolic geometric evolution equations of second order are nonlinear extensions of the ordinary heat equation to a geometric setting, so we begin by reminding the reader of the linear heat equation and its properties. We will then introduce key ideas in the simpler equations of curve shortening and 2-d Ricci-flow before discussing aspects of three-dimensional Ricci-flow.

2008, Maubach, Stephan, Poloni, Pierre-Marie

A polynomial automorphism F is called shifted linearizable if there exists a linear map L such that LF is linearizable. We prove that the Nagata automorphism N:=(X−YΔ−ZΔ2,Y+ZΔ,Z) where Δ=XZ+Y2 is shifted linearizable. More precisely, defining L(a,b,c) as the diagonal linear map having a,b,c on its diagonal, we prove that if ac=b2, then L(a,b,c)N is linearizable if and only if bc≠1. We do this as part of a significantly larger theory: for example, any exponent of a homogeneous locally finite derivation is shifted linearizable. We pose the conjecture that the group generated by the linearizable automorphisms may generate the group of automorphisms, and explain why this is a natural question.

2008, Chen, Xia, Mörters, Peter

We determine the precise asymptotics of the logarithmic upper tail probability of the total intersection local time of p independent random walks in Zd under the assumption p(d−2)>d. Our approach allows a direct treatment of the infinite time horizon.

2008, Skalski, Adam, Zacharias, Joachim

Noncommutative topological entropy estimates are obtained for ‘finite range’ endomorphisms of Cuntz algebras, generalising known results for the canonical shift endomorphisms. Exact values are computed for a class of polynomial endomorphisms related to branching function systems introduced and studied by Bratteli, Jorgensen and Kawamura.

2007, Fuchs, Dmitry, Tabachnikov, Serge

[no abstract available]

2008, Benson, Dave, Iyengar, Srikanth B., Krause, Henning

We classify localising subcategories of the stable module category of a finite group that are closed under tensor product with simple (or, equivalently all) modules. One application is a proof of the telescope conjecture in this context. Others include new proofs of the tensor product theorem and of the classification of thick subcategories of the finitely generated modules which avoid the use of cyclic shifted subgroups. Along the way we establish similar classifications for differential graded modules over graded polynomial rings, and over graded exterior algebras.

2009, Haak, Bernhard H., Jacob, Birgit

Volterra observations systems with scalar kernels are studied. New sufficient conditions for admissibility of observation operators are developed and some examples are discussed.

2008, Chelkak, Dmitry, Korotyaev, Evgeny

The matrix-valued Weyl-Titchmarsh functions M(¸) of vector-valued Sturm-Liouville operators on the unit interval with the Dirichlet boundary conditions are considered. The collection of the eigenvalues (i.e., poles of M(¸)) and the residues of M(¸) is called the spectral data of the operator. The complete characterization of spectral data (or, equivalently, N £ N Weyl-Titchmarsh functions) corresponding to N £ N self-adjoint square-integrable matrix-valued potentials is given, if all N eigenvalues of the averaged potential are distinct.