Max Planck Institute for Mathematics

Mirror symmetry (MS) was discovered several years ago in string theory as a duality between families of 3-dimensional Calabi-Yau manifolds (more precisely, complex algebraic manifolds possessing holomorphic volume elements without zeros). The name comes from the symmetry among Hodge numbers. For dual Calabi-Yau manifolds V, W of dimension n (not necessarily equal to 3) one has $$\dim {H^p}(V,{\Omega ^q}) = \dim {H^{n - p}}(W,{\Omega ^q}).$$ .

In this paper, without assuming symmetry, irreducibility, or linearity of the couplings, we prove that a single controller can pin a coupled complex network to a homogenous solution. Sufficient conditions are presented to guarantee the convergence of the pinning process locally and globally. An effective approach to adapt the coupling strength is proposed. Several numerical simulations are given to verify our theoretical analysis.

All languages have demonstratives, but their form, meaning and use vary tremendously across the languages of the world. This book presents the first large-scale analysis of demonstratives from a cross-linguistic and diachronic perspective. It is based on a representative sample of 85 languages. The first part of the book analyzes demonstratives from a synchronic point of view, examining their morphological structures, semantic features, syntactic functions, and pragmatic uses in spoken and written discourse. The second part concentrates on diachronic issues, in particular on the development of demonstratives into grammatical markers. Across languages demonstratives provide a frequent historical source for definite articles, relative and third person pronouns, nonverbal copulas, sentence connectives, directional preverbs, focus markers, expletives, and many other grammatical markers. The book describes the different mechanisms by which demonstratives grammaticalize and argues that the evolution of grammatical markers from demonstratives is crucially distinct from other cases of grammaticalization.

In this book, an extensive circle of questions originating in the classical work of P. L. Chebyshev and A. A. Markov is considered from the more modern point of view. It is shown how results and methods of the generalized moment problem are interlaced with various questions of the geometry of convex bodies, algebra, and function theory. From this standpoint, the structure of convex and conical hulls of curves is studied in detail and isoperimetric inequalities for convex hulls are established; a theory of orthogonal and quasiorthogonal polynomials is constructed; problems on limiting values of integrals and on least deviating functions (in various metrics) are generalized and solved; problems in approximation theory and interpolation and extrapolation in various function classes (analytic, absolutely monotone, almost periodic, etc.) are solved, as well as certain problems in optimal control of linear objects.

We give a sufficient condition for an Ext-finite triangulated category to be saturated. Saturatedness means that every contravariant cohomological functor of finite type to vector spaces is representable. The condition consists in the existence of a strong generator. We prove that the bounded derived categories of coherent sheaves on smooth proper commutative and noncommutative varieties have strong generators, and are hence saturated. In contrast, the similar category for a smooth compact analytic surface with no curves is not saturated. 2000 Math. Subj. Class. Primary 18E30.

Zeta functions of various sorts are all-pervasive objects in modern number theory, and an ever-recurring theme is the role played by their special values at integral arguments, which are linked in mysterious ways to the underlying geometry and often seem to dictate the most important properties of the objects to which the zeta functions are associated. It is this latter property to which the word “applications” in the title refers. In this article we will give a highly idiosyncratic and prejudiced tour of a number of these “applications,” making no attempt to be systematic, but only to give a feel for some of the ways in which special values of zeta functions interrelate with other interesting mathematical questions. The prototypical zeta function is “Riemann’s” (math) and the prototypical result on special values is the theorem that ζ(k) = rational number × π k (k > 0 even), (1) which Euler proved in 1735 and of which we will give a short proof in Section 1. (The “applications” in this case are the role which the rational numbers occurring on the right-hand side of this formula play in the theory of cyclotomic fields, in the construction of p-adic zeta functions, and in the investigation of Fermât’s Last Theorem.)

With few exceptions, current methods for short read mapping make use of simple seed heuristics to speed up the search. Most of the underlying matching models neglect the necessity to allow not only mismatches, but also insertions and deletions. Current evaluations indicate, however, that very different error models apply to the novel high-throughput sequencing methods. While the most frequent error-type in Illumina reads are mismatches, reads produced by 454's GS FLX predominantly contain insertions and deletions (indels). Even though 454 sequencers are able to produce longer reads, the method is frequently applied to small RNA (miRNA and siRNA) sequencing. Fast and accurate matching in particular of short reads with diverse errors is therefore a pressing practical problem. We introduce a matching model for short reads that can, besides mismatches, also cope with indels. It addresses different error models. For example, it can handle the problem of leading and trailing contaminations caused by primers and poly-A tails in transcriptomics or the length-dependent increase of error rates. In these contexts, it thus simplifies the tedious and error-prone trimming step. For efficient searches, our method utilizes index structures in the form of enhanced suffix arrays. In a comparison with current methods for short read mapping, the presented approach shows significantly increased performance not only for 454 reads, but also for Illumina reads. Our approach is implemented in the software segemehl available at http://www.bioinf.uni-leipzig.de/Software/segemehl/.

We study the complexity of functions computable by deep feedforward neural networks with piecewise linear activations in terms of the symmetries and the number of linear regions that they have. Deep networks are able to sequentially map portions of each layer's input-space to the same output. In this way, deep models compute functions that react equally to complicated patterns of different inputs. The compositional structure of these functions enables them to re-use pieces of computation exponentially often in terms of the network's depth. This paper investigates the complexity of such compositional maps and contributes new theoretical results regarding the advantage of depth for neural networks with piecewise linear activation functions. In particular, our analysis is not specific to a single family of models, and as an example, we employ it for rectifier and maxout networks. We improve complexity bounds from pre-existing work and investigate the behavior of units in higher layers.

The detection of differentially methylated regions (DMRs) is a necessary prerequisite for characterizing different epigenetic states. We present a novel program, metilene, to identify DMRs within whole-genome and targeted data with unrivaled specificity and sensitivity. A binary segmentation algorithm combined with a two-dimensional statistical test allows the detection of DMRs in large methylation experiments with multiple groups of samples in minutes rather than days using off-the-shelf hardware. metilene outperforms other state-of-the-art tools for low coverage data and can estimate missing data. Hence, metilene is a versatile tool to study the effect of epigenetic modifications in differentiation/development, tumorigenesis, and systems biology on a global, genome-wide level. Whether in the framework of international consortia with dozens of samples per group, or even without biological replicates, it produces highly significant and reliable results.

This paper contains an attempt to formulate rigorously and to check predictions in enumerative geometry of curves following from Mirror Symmetry.

We study the complexity of functions computable by deep feedforward neural networks with piecewise linear activations in terms of the symmetries and the number of linear regions that they have. Deep networks are able to sequentially map portions of each layer's input-space to the same output. In this way, deep models compute functions that react equally to complicated patterns of different inputs. The compositional structure of these functions enables them to re-use pieces of computation exponentially often in terms of the network's depth. This paper investigates the complexity of such compositional maps and contributes new theoretical results regarding the advantage of depth for neural networks with piecewise linear activation functions. In particular, our analysis is not specific to a single family of models, and as an example, we employ it for rectifier and maxout networks. We improve complexity bounds from pre-existing work and investigate the behavior of units in higher layers.

A cluster ensemble is a pair <img src="http://smf.emath.fr.ez.statsbiblioteket.dk:2048/Publications/php/images/fd5a01f65ba5d4ecea558b576254358e.png" alt="(X, A)" style="border-width: 0px" title="(X, A)"/> of positive spaces (i.e. varieties equipped with positive atlases), coming with an action of a symmetry group <img src="http://smf.emath.fr.ez.statsbiblioteket.dk:2048/Publications/php/images/07710b5c43702a8bb7b9104eacc6ba71.png" alt="" style="border-width: 0px"/>. The space <img src="http://smf.emath.fr.ez.statsbiblioteket.dk:2048/Publications/php/images/f5bffcc138953e427e2d7aeb93be32ae.png" alt="A" style="border-width: 0px" title="A"/> is closely related to the spectrum of a cluster algebra [<a href="http://smf.emath.fr.ez.statsbiblioteket.dk:2048/en/Publications/AnnalesENS/4_42/html/ens_ann-sc_42_865-930.php#BIB12">12</a>]. The two spaces are related by a morphism <img src="http://smf.emath.fr.ez.statsbiblioteket.dk:2048/Publications/php/images/0772d6b83c60821483a69726bb329b91.png" alt="p: A X" style="border-width: 0px" title="p: A X"/>. The space <img src="http://smf.emath.fr.ez.statsbiblioteket.dk:2048/Publications/php/images/f5bffcc138953e427e2d7aeb93be32ae.png" alt="A" style="border-width: 0px" title="A"/> is equipped with a closed <img src="http://smf.emath.fr.ez.statsbiblioteket.dk:2048/Publications/php/images/c81e728d9d4c2f636f067f89cc14862c.png" alt="2" style="border-width: 0px" title="2"/>-form, possibly degenerate, and the space <img src="http://smf.emath.fr.ez.statsbiblioteket.dk:2048/Publications/php/images/246ad06d783bb284abae53bc2131c571.png" alt="X" style="border-width: 0px" title="X"/> has a Poisson structure. The map <img src="http://smf.emath.fr.ez.statsbiblioteket.dk:2048/Publications/php/images/83878c91171338902e0fe0fb97a8c47a.png" alt="p" style="border-width: 0px" title="p"/> is compatible with these structures. The dilogarithm together with its motivic and quantum avatars plays a central role in the cluster ensemble structure. We define a non-commutative <img src="http://smf.emath.fr.ez.statsbiblioteket.dk:2048/Publications/php/images/7694f4a66316e53c8cdd9d9954bd611d.png" alt="q" style="border-width: 0px" title="q"/>-deformation of the <img src="http://smf.emath.fr.ez.statsbiblioteket.dk:2048/Publications/php/images/246ad06d783bb284abae53bc2131c571.png" alt="X" style="border-width: 0px" title="X"/>-space. When <img src="http://smf.emath.fr.ez.statsbiblioteket.dk:2048/Publications/php/images/7694f4a66316e53c8cdd9d9954bd611d.png" alt="q" style="border-width: 0px" title="q"/> is a root of unity the algebra of functions on the <img src="http://smf.emath.fr.ez.statsbiblioteket.dk:2048/Publications/php/images/7694f4a66316e53c8cdd9d9954bd611d.png" alt="q" style="border-width: 0px" title="q"/>-deformed <img src="http://smf.emath.fr.ez.statsbiblioteket.dk:2048/Publications/php/images/246ad06d783bb284abae53bc2131c571.png" alt="X" style="border-width: 0px" title="X"/>-space has a large center, which includes the algebra of functions on the original <img src="http://smf.emath.fr.ez.statsbiblioteket.dk:2048/Publications/php/images/246ad06d783bb284abae53bc2131c571.png" alt="X" style="border-width: 0px" title="X"/>-space. The main example is provided by the pair of moduli spaces assigned in [<a href="http://smf.emath.fr.ez.statsbiblioteket.dk:2048/en/Publications/AnnalesENS/4_42/html/ens_ann-sc_42_865-930.php#BIB7">7</a>] to a topological surface <img src="http://smf.emath.fr.ez.statsbiblioteket.dk:2048/Publications/php/images/5dbc98dcc983a70728bd082d1a47546e.png" alt="S" style="border-width: 0px" title="S"/> with a finite set of points at the boundary and a split semisimple algebraic group <img src="http://smf.emath.fr.ez.statsbiblioteket.dk:2048/Publications/php/images/dfcf28d0734569a6a693bc8194de62bf.png" alt="G" style="border-width: 0px" title="G"/>. It is an algebraic-geometric avatar of higher Teichmüller theory on <img src="http://smf.emath.fr.ez.statsbiblioteket.dk:2048/Publications/php/images/5dbc98dcc983a70728bd082d1a47546e.png" alt="S" style="border-width: 0px" title="S"/> related to <img src="http://smf.emath.fr.ez.statsbiblioteket.dk:2048/Publications/php/images/dfcf28d0734569a6a693bc8194de62bf.png" alt="G" style="border-width: 0px" title="G"/>. We suggest that there exists a duality between the <img src="http://smf.emath.fr.ez.statsbiblioteket.dk:2048/Publications/php/images/f5bffcc138953e427e2d7aeb93be32ae.png" alt="A" style="border-width: 0px" title="A"/> and <img src="http://smf.emath.fr.ez.statsbiblioteket.dk:2048/Publications/php/images/246ad06d783bb284abae53bc2131c571.png" alt="X" style="border-width: 0px" title="X"/> spaces. In particular, we conjecture that the tropical points of one of the spaces parametrise a basis in the space of functions on the Langlands dual space. We provide some evidence for the duality conjectures in the finite type case.

To what extent the growth dynamics of tumors is controlled by nutrients, biomechanical forces and other factors at different stages and in different environments is still largely unknown. Here we present a biophysical model to study the spatio-temporal growth dynamics of two-dimensional tumor monolayers and three-dimensional tumor spheroids as a complementary tool to in vitro experiments. Within our model each cell is represented as an individual object and parametrized by cell-biophysical and cell-kinetic parameters that can all be experimentally determined. Hence our modeling strategy allows us to study which mechanisms on the microscopic level of individual cells may affect the macroscopic properties of a growing tumor. We find the qualitative growth kinetics and patterns at early growth stages to be remarkably robust. Quantitative comparisons between computer simulations using our model and published experimental observations on monolayer cultures suggest a biomechanically-mediated form of growth inhibition during the experimentally observed transition from exponential to sub-exponential growth at sufficiently large tumor sizes. Our simulations show that the same transition during the growth of avascular tumor spheroids can be explained largely by the same mechanism. Glucose (or oxygen) depletion seems to determine mainly the size of the necrotic core but not the size of the tumor. We explore the consequences of the suggested biomechanical form of contact inhibition, in order to permit an experimental test of our model. Based on our findings we propose a phenomenological growth law in early expansion phases in which specific biological small-scale processes are subsumed in a small number of effective parameters.

In spite of physics terms in the title, this paper is purely mathematical. Its purpose is to introduce triangulated categories related to singularities of algebraic varieties and establish a connection of these categories with D-branes in Landau-Ginzburg models.

We present an effective unified theory based on noncommutative geometry for the standard model with neutrino mixing, minimally coupled to gravity. The unification is based on the symplectic unitary group in Hilbert space and on the spectral action. It yields all the detailed structure of the standard model with several predictions at unification scale. Besides the familiar predictions for the gauge couplings as for GUT theories, it predicts the Higgs scattering parameter and the sum of the squares of Yukawa couplings. From these relations, one can extract predictions at low energy, giving in particular a Higgs mass around 170 GeV and a top mass compatible with present experimental value. The geometric picture that emerges is that space-time is the product of an ordinary spin manifold (for which the theory would deliver Einstein gravity) by a finite noncommutative geometry F. The discrete space F is of KO-dimension 6 modulo 8 and of metric dimension 0, and accounts for all the intricacies of the standard model with its spontaneous symmetry breaking Higgs sector.

A synergistic combination of two next-generation sequencing platforms with a detailed comparative BAC physical contig map provided a cost-effective assembly of the genome sequence of the domestic turkey (Meleagris gallopavo). Heterozygosity of the sequenced source genome allowed discovery of more than 600,000 high quality single nucleotide variants. Despite this heterozygosity, the current genome assembly (∼1.1 Gb) includes 917 Mb of sequence assigned to specific turkey chromosomes. Annotation identified nearly 16,000 genes, with 15,093 recognized as protein coding and 611 as non-coding RNA genes. Comparative analysis of the turkey, chicken, and zebra finch genomes, and comparing avian to mammalian species, supports the characteristic stability of avian genomes and identifies genes unique to the avian lineage. Clear differences are seen in number and variety of genes of the avian immune system where expansions and novel genes are less frequent than examples of gene loss. The turkey genome sequence provides resources to further understand the evolution of vertebrate genomes and genetic variation underlying economically important quantitative traits in poultry. This integrated approach may be a model for providing both gene and chromosome level assemblies of other species with agricultural, ecological, and evolutionary interest.

Some time ago B. Feigin, V. Retakh and I had tried to understand a remark of J. Stasheff [S1] on open string theory and higher associative algebras [S2]. Then I found a strange construction of cohomology classes of mapping class groups using as initial data any differential graded algebra with finite-dimensional cohomology and a kind of Poincaré duality.

The theta correspondence has been an important tool in studying cycles in locally symmetric spaces of orthogonal type. In this paper we establish for the orthogonal group O(p,2) an adjointness result between Borcherds's singular theta lift and the Kudla-Millson lift. We extend this result to arbitrary signature by introducing a new singular theta lift for O(p,q). On the geometric side, this lift can be interpreted as a differential character, in the sense of Cheeger and Simons, for the cycles under consideration.

This monograph is devoted to the spectral theory of the Sturm- Liouville operator and to the spectral theory of the Dirac system. In addition, some results are given for nth order ordinary differential operators. Those parts of this book which concern nth order operators can serve as simply an introduction to this domain, which at the present time has already had time to become very broad. For the convenience of the reader who is not familiar with abstract spectral theory, the authors have inserted a chapter (Chapter 13) in which they discuss this theory, concisely and in the main without proofs, and indicate various connections with the spectral theory of differential operators.

Current genomic screens for noncoding RNAs (ncRNAs) predict a large number of genomic regions containing potential structural ncRNAs. The analysis of these data requires highly accurate prediction of ncRNA boundaries and discrimination of promising candidate ncRNAs from weak predictions. Existing methods struggle with these goals because they rely on sequence-based multiple sequence alignments, which regularly misalign RNA structure and therefore do not support identification of structural similarities. To overcome this limitation, we compute columnwise and global reliabilities of alignments based on sequence and structure similarity; we refer to these structure-based alignment reliabilities as STARs. The columnwise STARs of alignments, or STAR profiles, provide a versatile tool for the manual and automatic analysis of ncRNAs. In particular, we improve the boundary prediction of the widely used ncRNA gene finder RNAz by a factor of 3 from a median deviation of 47 to 13 nt. Post-processing RNAz predictions, LocARNA-P's STAR score allows much stronger discrimination between true- and false-positive predictions than RNAz's own evaluation. The improved accuracy, in this scenario increased from AUC 0.71 to AUC 0.87, significantly reduces the cost of successive analysis steps. The ready-to-use software tool LocARNA-P produces structure-based multiple RNA alignments with associated columnwise STARs and predicts ncRNA boundaries. We provide additional results, a web server for LocARNA/LocARNA-P, and the software package, including documentation and a pipeline for refining screens for structural ncRNA, at http://www.bioinf.uni-freiburg.de/Supplements/LocARNA-P/.