Langlands duality for finite-dimensional representations of quantum affine algebras

We describe a correspondence (or duality) between the q-characters of finite-dimensional representations of a quantum affine algebra and its Langlands dual in the spirit of q-alg/9708006 and 0809.4453. We prove this duality for the Kirillov-Reshetikhin modules and their irreducible tensor products. In the course of the proof we introduce and construct "interpolating (q,t)-characters" depending on two parameters which interpolate between the q-characters of a quantum affine algebra and its Langlands dual.Comment: 40 pages; several results and comments added. Accepted for publication in Letters in Mathematical Physic

Discrete symmetries and model-independent patterns of lepton mixing

In the context of discrete flavor symmetries, we elaborate a method that allows one to obtain relations between the mixing parameters in a model-independent way. Under very general conditions, we show that flavor groups of the von Dyck type, that are not necessarily finite, determine the absolute values of the entries of one column of the mixing matrix. We apply our formalism to finite subgroups of the infinite von Dyck groups, such as the modular groups, and find cases that yield an excellent agreement with the best fit values for the mixing angles. We explore the Klein group as the residual symmetry of the neutrino sector and explain the permutation property that appears between the elements of the mixing matrix in this case.Comment: 22 pages, 12 figure

A MOSAIC of methods: Improving ortholog detection through integration of algorithmic diversity

Ortholog detection (OD) is a critical step for comparative genomic analysis of protein-coding sequences. In this paper, we begin with a comprehensive comparison of four popular, methodologically diverse OD methods: MultiParanoid, Blat, Multiz, and OMA. In head-to-head comparisons, these methods are shown to significantly outperform one another 12-30% of the time. This high complementarity motivates the presentation of the first tool for integrating methodologically diverse OD methods. We term this program MOSAIC, or Multiple Orthologous Sequence Analysis and Integration by Cluster optimization. Relative to component and competing methods, we demonstrate that MOSAIC more than quintuples the number of alignments for which all species are present, while simultaneously maintaining or improving functional-, phylogenetic-, and sequence identity-based measures of ortholog quality. Further, we demonstrate that this improvement in alignment quality yields 40-280% more confidently aligned sites. Combined, these factors translate to higher estimated levels of overall conservation, while at the same time allowing for the detection of up to 180% more positively selected sites. MOSAIC is available as python package. MOSAIC alignments, source code, and full documentation are available at http://pythonhosted.org/bio-MOSAIC

Robust forward simulations of recurrent hitchhiking

Evolutionary forces shape patterns of genetic diversity within populations and contribute to phenotypic variation. In particular, recurrent positive selection has attracted significant interest in both theoretical and empirical studies. However, most existing theoretical models of recurrent positive selection cannot easily incorporate realistic confounding effects such as interference between selected sites, arbitrary selection schemes, and complicated demographic processes. It is possible to quantify the effects of arbitrarily complex evolutionary models by performing forward population genetic simulations, but forward simulations can be computationally prohibitive for large population sizes ($> 10^5$). A common approach for overcoming these computational limitations is rescaling of the most computationally expensive parameters, especially population size. Here, we show that ad hoc approaches to parameter rescaling under the recurrent hitchhiking model do not always provide sufficiently accurate dynamics, potentially skewing patterns of diversity in simulated DNA sequences. We derive an extension of the recurrent hitchhiking model that is appropriate for strong selection in small population sizes, and use it to develop a method for parameter rescaling that provides the best possible computational performance for a given error tolerance. We perform a detailed theoretical analysis of the robustness of rescaling across the parameter space. Finally, we apply our rescaling algorithms to parameters that were previously inferred for Drosophila, and discuss practical considerations such as interference between selected sites

A Fourier-Mukai Transform for Stable Bundles on K3 Surfaces

We define a Fourier-Mukai transform for sheaves on K3 surfaces over \C, and show that it maps polystable bundles to polystable ones. The role of ``dual'' variety to the given K3 surface $X$ is here played by a suitable component $\hat X$ of the moduli space of stable sheaves on $X$. For a wide class of K3 surfaces $\hat X$ can be chosen to be isomorphic to $X$; then the Fourier-Mukai transform is invertible, and the image of a zero-degree stable bundle $F$ is stable and has the same Euler characteristic as $F$.Comment: Revised version, 15 pages AMSTeX with AMSppt.sty v. 2.1

A discrete linearizability test based on multiscale analysis

In this paper we consider the classification of dispersive linearizable partial difference equations defined on a quad-graph by the multiple scale reduction around their harmonic solution. We show that the A_1, A_2 and A_3 linearizability conditions restrain the number of the parameters which enter into the equation. A subclass of the equations which pass the A_3 C-integrability conditions can be linearized by a Mobius transformation

Langlands duality for representations of quantum groups

We establish a correspondence (or duality) between the characters and the crystal bases of finite-dimensional representations of quantum groups associated to Langlands dual semi-simple Lie algebras. This duality may also be stated purely in terms of semi-simple Lie algebras. To explain this duality, we introduce an "interpolating quantum group" depending on two parameters which interpolates between a quantum group and its Langlands dual. We construct examples of its representations, depending on two parameters, which interpolate between representations of two Langlands dual quantum groups.Comment: 37 pages. References added. Accepted for publication in Mathematische Annale