Property (RD) for Hecke pairs

As the first step towards developing noncommutative geometry over Hecke C*-algebras, we study property (RD) (Rapid Decay) for Hecke pairs. When the subgroup H in a Hecke pair (G,H) is finite, we show that the Hecke pair (G,H) has (RD) if and only if G has (RD). This provides us with a family of examples of Hecke pairs with property (RD). We also adapt Paul Jolissant's works in 1989 to the setting of Hecke C*-algebras and show that when a Hecke pair (G,H) has property (RD), the algebra of rapidly decreasing functions on the set of double cosets is closed under holomorphic functional calculus of the associated (reduced) Hecke C*-algebra. Hence they have the same K_0-groups.Comment: A short note added explaining other methods to prove that the subalgebra of rapidly decreasing functions is smooth. This is the final version as published. The published version is available at: springer.co

Quantum Symmetries and Strong Haagerup Inequalities

In this paper, we consider families of operators $\{x_r\}_{r \in \Lambda}$ in a tracial C$^\ast$-probability space $(\mathcal A, \phi)$, whose joint $\ast$-distribution is invariant under free complexification and the action of the hyperoctahedral quantum groups $\{H_n^+\}_{n \in \N}$. We prove a strong form of Haagerup's inequality for the non-self-adjoint operator algebra $\mathcal B$ generated by $\{x_r\}_{r \in \Lambda}$, which generalizes the strong Haagerup inequalities for $\ast$-free R-diagonal families obtained by Kemp-Speicher \cite{KeSp}. As an application of our result, we show that $\mathcal B$ always has the metric approximation property (MAP). We also apply our techniques to study the reduced C$^\ast$-algebra of the free unitary quantum group $U_n^+$. We show that the non-self-adjoint subalgebra $\mathcal B_n$ generated by the matrix elements of the fundamental corepresentation of $U_n^+$ has the MAP. Additionally, we prove a strong Haagerup inequality for $\mathcal B_n$, which improves on the estimates given by Vergnioux's property RD \cite{Ve}

Uniformizing the Stacks of Abelian Sheaves

Elliptic sheaves (which are related to Drinfeld modules) were introduced by Drinfeld and further studied by Laumon--Rapoport--Stuhler and others. They can be viewed as function field analogues of elliptic curves and hence are objects "of dimension 1". Their higher dimensional generalisations are called abelian sheaves. In the analogy between function fields and number fields, abelian sheaves are counterparts of abelian varieties. In this article we study the moduli spaces of abelian sheaves and prove that they are algebraic stacks. We further transfer results of Cerednik--Drinfeld and Rapoport--Zink on the uniformization of Shimura varieties to the setting of abelian sheaves. Actually the analogy of the Cerednik--Drinfeld uniformization is nothing but the uniformization of the moduli schemes of Drinfeld modules by the Drinfeld upper half space. Our results generalise this uniformization. The proof closely follows the ideas of Rapoport--Zink. In particular, analogies of $p$-divisible groups play an important role. As a crucial intermediate step we prove that in a family of abelian sheaves with good reduction at infinity, the set of points where the abelian sheaf is uniformizable in the sense of Anderson, is formally closed.Comment: Final version, appears in "Number Fields and Function Fields - Two Parallel Worlds", Papers from the 4th Conference held on Texel Island, April 2004, edited by G. van der Geer, B. Moonen, R. Schoo

Nonlinear spectral calculus and super-expanders

Nonlinear spectral gaps with respect to uniformly convex normed spaces are shown to satisfy a spectral calculus inequality that establishes their decay along Cesaro averages. Nonlinear spectral gaps of graphs are also shown to behave sub-multiplicatively under zigzag products. These results yield a combinatorial construction of super-expanders, i.e., a sequence of 3-regular graphs that does not admit a coarse embedding into any uniformly convex normed space.Comment: Typos fixed based on referee comments. Some of the results of this paper were announced in arXiv:0910.2041. The corresponding parts of arXiv:0910.2041 are subsumed by the current pape

Red Queen Coevolution on Fitness Landscapes

Species do not merely evolve, they also coevolve with other organisms. Coevolution is a major force driving interacting species to continuously evolve ex- ploring their fitness landscapes. Coevolution involves the coupling of species fit- ness landscapes, linking species genetic changes with their inter-specific ecological interactions. Here we first introduce the Red Queen hypothesis of evolution com- menting on some theoretical aspects and empirical evidences. As an introduction to the fitness landscape concept, we review key issues on evolution on simple and rugged fitness landscapes. Then we present key modeling examples of coevolution on different fitness landscapes at different scales, from RNA viruses to complex ecosystems and macroevolution.Comment: 40 pages, 12 figures. To appear in "Recent Advances in the Theory and Application of Fitness Landscapes" (H. Richter and A. Engelbrecht, eds.). Springer Series in Emergence, Complexity, and Computation, 201

Model-selection-based approach for calculating cellular multiplicity of infection during virus colonization of multi-cellular hosts

The cellular multiplicity of infection (MOI) is a key parameter for describing the interactions between virions and cells, predicting the dynamics of mixed-genotype infections, and understanding virus evolution. Two recent studies have reported in vivo MOI estimates for Tobacco mosaic virus (TMV) and Cauliflower mosaic virus (CaMV), using sophisticated approaches to measure the distribution of two virus variants over host cells. Although the experimental approaches were similar, the studies employed different definitions of MOI and estimation methods. Here, new model-selection-based methods for calculating MOI were developed. Seven alternative models for predicting MOI were formulated that incorporate an increasing number of parameters. For both datasets the best-supported model included spatial segregation of virus variants over time, and to a lesser extent aggregation of virus-infected cells was also implicated. Three methods for MOI estimation were then compared: the two previously reported methods and the best-supported model. For CaMV data, all three methods gave comparable results. For TMV data, the previously reported methods both predicted low MOI values (range: 1.04-1.23) over time, whereas the best-supported model predicted a wider range of MOI values (range: 1.01-2.10) and an increase in MOI over time. Model selection can therefore identify suitable alternative MOI models and suggest key mechanisms affecting the frequency of coinfected cells. For the TMV data, this leads to appreciable differences in estimated MOI values.This work was supported by grant BFU2012-30805 (SFE) and by 'Juan de la Cierva' postdoctoral contract JCI-2011-10379 (MPZ) from the Spanish Secretaria de Estado de Investigacion, Desarrollo e Innovacion. 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