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Exponential Family Mixed Membership Models for Soft~Clustering of Multivariate DataAug 10 2016For several years, model-based clustering methods have successfully tackled many of the challenges presented by data-analysts. However, as the scope of data analysis has evolved, some problems may be beyond the standard mixture model framework. One such ... More

Variable Selection for Latent Class Analysis with Application to Low Back Pain DiagnosisDec 10 2015Feb 05 2018The identification of most relevant clinical criteria related to low back pain disorders may aid the evaluation of the nature of pain suffered in a way that usefully informs patient assessment and treatment. Data concerning low back pain can be of categorical ... More

Products of Differences over Arbitrary Finite FieldsMay 18 2017Nov 13 2018There exists an absolute constant $\delta > 0$ such that for all $q$ and all subsets $A \subseteq \mathbb{F}_q$ of the finite field with $q$ elements, if $|A| > q^{2/3 - \delta}$, then \[ |(A-A)(A-A)| = |\{ (a -b) (c-d) : a,b,c,d \in A\}| > \frac{q}{2}. ... More

Kakeya Configurations in Lie Groups and Homogeneous SpacesMar 04 2013In this paper, we study continuous Kakeya line and needle configurations, of both the oriented and unoriented varieties, in connected Lie groups and some associated homogenous spaces. These are the analogs of Kakeya line (needle) sets (subsets of $\mathbb{R}^n$ ... More

A Second Wave of Expanders over Finite FieldsJan 06 2017This is an expository survey on recent sum-product results in finite fields. We present a number of sum-product or "expander" results that say that if $|A| > p^{2/3}$ then some set determined by sums and product of elements of $A$ is nearly as large as ... More

Riemannian foliations of projective space admitting complex leavesFeb 27 2012Jul 10 2013Motivated by Gray's work on tube formulae for complex submanifolds of complex projective space equipped with the Fubini-Study metric, Riemannian foliations of projective space are studied. We prove that there are no complex Riemannian foliations of any ... More

Inverse-Closedness of a Banach Algebra of Integral Operators on the Heisenberg GroupDec 01 2006Dec 06 2007Let $\mathbb{H}$ be the general, reduced Heisenberg group. Our main result establishes the inverse-closedness of a class of integral operators acting on $L^{p}(\mathbb{H})$, given by the off-diagonal decay of the kernel. As a consequence of this result, ... More

Popular Products and Continued FractionsAug 17 2018Aug 23 2018We prove bounds for the popularity of products of sets with weak additive structure, and use these bounds to prove results about continued fractions. Namely, we obtain a nearly sharp upper bound for the cardinality of Zaremba's set modulo $p$.

Random Manifolds have no Totally Geodesic SubmanifoldsMar 27 2017Jan 18 2018For $n\geq 4$ we show that generic closed Riemannian $n$-manifolds have no nontrivial totally geodesic submanifolds, answering a question of Spivak. An immediate consequence is a severe restriction on the isometry group of a generic Riemannian metric. ... More

Bivariate Gamma Mixture of Experts Models for Joint Insurance Claims ModelingApr 09 2019In general insurance, risks from different categories are often modeled independently and their sum is regarded as the total risk the insurer takes on in exchange for a premium. The dependence from multiple risks is generally neglected even when correlation ... More

On the few products, many sums problemDec 01 2017We prove new results on additive properties of finite sets $A$ with small multiplicative doubling $|AA|\leq M|A|$ in the category of real/complex sets as well as multiplicative subgroups in the prime residue field. The improvements are based on new combinatorial ... More

Gaussian Parsimonious Clustering Models with CovariatesNov 15 2017Dec 10 2018We consider model-based clustering methods for continuous, correlated data that account for external information available in the presence of mixed-type fixed covariates by proposing the MoEClust suite of models. These allow covariates influence the component ... More

Mixed-Membership of Experts Stochastic BlockmodelApr 01 2014Social network analysis is the study of how links between a set of actors are formed. Typically, it is believed that links are formed in a structured manner, which may be due to, for example, political or material incentives, and which often may not be ... More

Mixture of Latent Trait Analyzers for Model-Based Clustering of Categorical DataJan 10 2013Feb 19 2013Model-based clustering methods for continuous data are well established and commonly used in a wide range of applications. However, model-based clustering methods for categorical data are less standard. Latent class analysis is a commonly used method ... More

Joint Modelling of Multiple Network ViewsJan 16 2013Sep 25 2014Latent space models (LSM) for network data were introduced by Hoff et al. (2002) under the basic assumption that each node of the network has an unknown position in a D-dimensional Euclidean latent space: generally the smaller the distance between two ... More

Terahertz nonlinear conduction and absorption saturation in silicon waveguidesMar 19 2015We employ a silicon dielectric waveguide to confine and concentrate terahertz pulses, and observe that the absorption saturates under strong terahertz fields. By comparing the response between lightly-doped and intrinsic silicon waveguides, we confirm ... More

Speech bandwidth extension with WaveNetJul 05 2019Large-scale mobile communication systems tend to contain legacy transmission channels with narrowband bottlenecks, resulting in characteristic "telephone-quality" audio. While higher quality codecs exist, due to the scale and heterogeneity of the networks, ... More

Smooth and Starburst Tidal Tails in the GEMS and GOODS FieldsApr 06 2007GEMS and GOODS fields were examined to z~1.4 for galaxy interactions and mergers. The basic morphologies are familiar: antennae with long tidal tails, tidal dwarfs, and merged cores; M51-type galaxies with disk spirals and tidal arm companions; early-type ... More

Variational Learning in Mixed-State Dynamic Graphical ModelsJan 23 2013Many real-valued stochastic time-series are locally linear (Gassian), but globally non-linear. For example, the trajectory of a human hand gesture can be viewed as a linear dynamic system driven by a nonlinear dynamic system that represents muscle actions. ... More

Prophet Inequalities Made Easy: Stochastic Optimization by Pricing Non-Stochastic InputsDec 09 2016Jul 09 2017We present a general framework for stochastic online maximization problems with combinatorial feasibility constraints. The framework establishes prophet inequalities by constructing price-based online approximation algorithms, a natural extension of threshold ... More

A mixture of experts model for rank data with applications in election studiesJan 27 2009A voting bloc is defined to be a group of voters who have similar voting preferences. The cleavage of the Irish electorate into voting blocs is of interest. Irish elections employ a ``single transferable vote'' electoral system; under this system voters ... More

A robust approach to model-based classification based on trimming and constraintsApr 12 2019In a standard classification framework a set of trustworthy learning data are employed to build a decision rule, with the final aim of classifying unlabelled units belonging to the test set. Therefore, unreliable labelled observations, namely outliers ... More

Bayesian variable selection for latent class analysis using a collapsed Gibbs samplerFeb 27 2014Apr 30 2015Latent class analysis is used to perform model based clustering for multivariate categorical responses. Selection of the variables most relevant for clustering is an important task which can affect the quality of clustering considerably. This work considers ... More

Variable Selection for Latent Class Analysis with Application to Low Back Pain DiagnosisDec 10 2015The identification of most relevant clinical criteria related to low back pain disorders is a crucial task for a quick and correct diagnosis of the nature of pain and its treatment. Data concerning low back pain can be of categorical nature, in form of ... More

Model-based clustering for random hypergraphsAug 15 2018A probabilistic model for random hypergraphs is introduced to represent unary, binary and higher order interactions among objects in real-world problems. This model is an extension of the Latent Class Analysis model, which captures clustering structures ... More

Delayed Dynamical Systems: Networks, Chimeras and Reservoir ComputingAug 14 2018We present a systematic approach to reveal the correspondence between time delay dynamics and networks of coupled oscillators. After early demonstrations of the usefulness of spatio-temporal representations of time-delay system dynamics, extensive research ... More

Scalable parallel physical random number generator based on a superluminescent LEDMar 15 2011We describe an optoelectronic system for simultaneously generating parallel, independent streams of random bits using spectrally separated noise signals obtained from a single optical source. Using a pair of non-overlapping spectral filters and a fiber-coupled ... More

Using Synchronization for Prediction of High-Dimensional Chaotic DynamicsSep 22 2008We experimentally observe the nonlinear dynamics of an optoelectronic time-delayed feedback loop designed for chaotic communication using commercial fiber optic links, and we simulate the system using delay differential equations. We show that synchronization ... More

A Search for Improved Performance in Regular ExpressionsApr 13 2017The primary aim of automated performance improvement is to reduce the running time of programs while maintaining (or improving on) functionality. In this paper, Genetic Programming is used to find performance improvements in regular expressions for an ... More

Vortex Lattice Formation in Dipolar Bose-Einstein Condensates via Rotation of the PolarizationJun 20 2019The behaviour of a harmonically trapped dipolar Bose-Einstein condensate with its dipole moments rotating at angular frequencies lower than the transverse harmonic trapping frequency is explored in the co-rotating frame. We obtain semi-analytical solutions ... More

On Estimation of Parameter Uncertainty in Model-Based ClusteringOct 02 2015Mixture models are a popular tool in model-based clustering. Such a model is often fitted by a procedure that maximizes the likelihood, such as the EM algorithm. At convergence, the maximum likelihood parameter estimates are typically reported, but in ... More

Motor Insurance Accidental Damage Claims Modeling with Factor Collapsing and Bayesian Model AveragingOct 10 2017Accidental damage is a typical component of motor insurance claim. Modeling of this nature generally involves analysis of past claim history and different characteristics of the insured objects and the policyholders. Generalized linear models (GLMs) have ... More

Variable selection and updating in model-based discriminant analysis for high dimensional data with food authenticity applicationsOct 14 2009Oct 07 2010Food authenticity studies are concerned with determining if food samples have been correctly labeled or not. Discriminant analysis methods are an integral part of the methodology for food authentication. Motivated by food authenticity applications, a ... More

Bayesian nonparametric Plackett-Luce models for the analysis of preferences for college degree programmesNov 21 2012Aug 01 2014In this paper we propose a Bayesian nonparametric model for clustering partial ranking data. We start by developing a Bayesian nonparametric extension of the popular Plackett-Luce choice model that can handle an infinite number of choice items. Our framework ... More

Latent Space Modeling of Multidimensional Networks with Application to the Exchange of Votes in Eurovision Song ContestMar 13 2018The Eurovision Song Contest is a popular TV singing competition held annually among country members of the European Broadcasting Union. In this competition, each member can be both contestant and jury, as it can participate with a song and/or vote for ... More

Node-specific effects in latent space modelling of multidimensional networksJul 10 2018Observed multidimensional network data can have different levels of complexity, as nodes may be characterized by heterogeneous individual-specific features. Also, such characteristics may vary across the networks. This article discusses a novel class ... More

Bayesian model averaging in model-based clustering and density estimationJun 30 2015We propose Bayesian model averaging (BMA) as a method for postprocessing the results of model-based clustering. Given a number of competing models, appropriate model summaries are averaged, using the posterior model probabilities, instead of being taken ... More

Experimental Observations of Group Synchrony in a System of Chaotic Optoelectronic OscillatorsJan 16 2013Mar 27 2013We experimentally demonstrate group synchrony in a network of four nonlinear optoelectronic oscillators with time-delayed coupling. We divide the nodes into two groups of two each, by giving each group different parameters and by enabling only inter-group ... More

Multi-marginal optimal transport and multi-agent matching problems: uniqueness and structure of solutionsOct 27 2012We prove uniqueness and Monge solution results for multi-marginal optimal transportation problems with a certain class of surplus functions; this class arises naturally in multi-agent matching problems in economics. This result generalizes a seminal result ... More

Regularity properties of optimal transportation problems arising in hedonic pricing modelsOct 12 2011We study a form of optimal transportation surplus functions which arise in hedonic pricing models. We derive a formula for the Ma-Trudinger-Wang curvature of these functions, yielding necessary and sufficient conditions for them to satisfy \textbf{(A3w)}. ... More

Classical and minimal models of the moduli space of curves of genus twoAug 24 2004This semi-expository paper discusses the log minimal model program as applied to the moduli space of curves, especially in the case of curves of genus two. Log canonical models for these moduli spaces can often be constructed using the techniques of Geometric ... More

Query-driven PAC-Learning for ReasoningJun 24 2019We consider the problem of learning rules from a data set that support a proof of a given query, under Valiant's PAC-Semantics. We show how any backward proof search algorithm that is sufficiently oblivious to the contents of its knowledge base can be ... More

Classification of simple weight modules with finite-dimensional weight spaces over the Schrödinger algebraSep 05 2013We classify simple weight modules with finite-dimensional weight spaces over the (centrally extended complex) Schr\"odinger algebra in (1+1)-dimensional space-time. Our arguments use the description of lowest weight modules by Dobrev, Doebner and Mrugalla; ... More

Cohomology classes of interval positroid varieties and a conjecture of LiuOct 08 2014Jul 24 2018To each finite subset of $\mathbb{Z}^2$ (a diagram), one can associate a subvariety of a complex Grassmannian (a diagram variety), and a representation of a symmetric group (a Specht module). Liu has conjectured that the cohomology class of a diagram ... More

Decorated CospansFeb 03 2015Aug 11 2015Let $\mathcal C$ be a category with finite colimits, writing its coproduct $+$, and let $(\mathcal D, \otimes)$ be a braided monoidal category. We describe a method of producing a symmetric monoidal category from a lax braided monoidal functor $F: (\mathcal ... More

On Isolated Umbilic PointsDec 09 2018Counter-examples to the famous conjecture of Caratheodory, as well as the bound on umbilic index proposed by Hamburger, are constructed with respect to Riemannian metrics that are arbitrarily close to the flat metric on Euclidean 3-space. In particular, ... More

Instantons on cylindrical manifolds and stable bundlesOct 11 2000Nov 09 2001Let Sigma be a smooth complex curve, and let S be the product ruled surface Sigma \times CP^1. We prove a correspondence conjectured by Donaldson between finite energy U(2)-instantons over the cylinder Sigma \times S^1 \times R, and rank 2 holomorphic ... More

The Algebra of Open and Interconnected SystemsSep 17 2016Herein we develop category-theoretic tools for understanding network-style diagrammatic languages. The archetypal network-style diagrammatic language is that of electric circuits; other examples include signal flow graphs, Markov processes, automata, ... More

Corrigendum to: Equivariant embeddings of rational homology balls [arXiv:1707.00928]Jun 11 2019There was an unfortunate oversight in a remark in our paper "Equivariant embeddings of rational homology balls", Q. J. Math. 69 (2018), no. 3, 1101--1121 [arXiv:1707.00928]. Correcting this yields an interesting result which we omitted to observe in that ... More

Beyond Equilibria: Mechanisms for Repeated Combinatorial AuctionsSep 30 2009We study the design of mechanisms in combinatorial auction domains. We focus on settings where the auction is repeated, motivated by auctions for licenses or advertising space. We consider models of agent behaviour in which they either apply common learning ... More

Cubic fourfolds, K3 surfaces, and rationality questionsJan 21 2016Jul 17 2016This is a survey of the geometry of complex cubic fourfolds with a view toward rationality questions. Topics include classical constructions of rational examples, Hodge structures and special cubic fourfolds, associated K3 surfaces and their geometric ... More

Induction and restriction on representations of dihedral groupsMay 07 2018We study the algebras generated by restriction and induction operations on complex modules over dihedral groups. In the case where the orders of all dihedral groups involved are not divisible by four, we describe the relations, a basis, the center, and ... More

Convexity and multi-dimensional screening for spaces with different dimensionsAug 18 2011We study the principal-agent problem. We show that $b$-convexity of the space of products, a condition which appears in a recent paper by Figalli, Kim and McCann \cite{fkm}, is necessary to formulate the problem as a maximization over a convex set. We ... More

Universal graph Schubert varietiesFeb 25 2019Apr 19 2019We consider the loci of invertible linear maps $f : \mathbb{C}^n \to {(\mathbb{C}^n)}^*$ together with pairs of flags $(E_\bullet, F_\bullet)$ in $\mathbb{C}^n$ such that the various restrictions $f : F_j \to E_i^*$ have specified ranks. Identifying an ... More

Incidence category of the Young lattice, injections between finite sets, and KoszulityJul 01 2016Oct 04 2018We describe the Gabriel quiver with defining relations of the category of injections between finite sets, show that it is quadratic self-dual, and construct linear resolutions for its simple modules.

Abelian Analytic Torsion and Symplectic VolumeAug 12 2012Aug 17 2015This article studies the abelian analytic torsion on a closed, oriented, Sasakian three-manifold and identifies this quantity as a specific multiple of the natural unit symplectic volume form on the moduli space of flat abelian connections. This identification ... More

Localization in Abelian Chern-Simons TheoryAug 08 2012Jul 18 2014Chern-Simons theory on a closed contact three-manifold is studied when the Lie group for gauge transformations is compact, connected and abelian. A rigorous definition of an abelian Chern-Simons partition function is derived using the Faddeev-Popov gauge ... More

Koszulity of some path categoriesFeb 22 2015Jun 06 2015We prove Koszulity of certain linear path categories obtained from connected graphs with some infinite directed walk. These categories can be viewed as locally quadratic dual to preprojective algebras.

Weak approximation and rationally connected varieties over function fields of curvesAug 16 2010This is a survey of weak approximation over complex function fields, touching on the Koll'ar-Miyaoka-Mori theorem, places of good and bad reduction, the special case of rational surfaces, rationally simply connected varieties, and applications to low-degree ... More

A representation-theoretic interpretation of positroid classesDec 01 2016A positroid is the matroid of a real matrix with nonnegative maximal minors, a positroid variety is the closure of the locus of points in a complex Grassmannian whose matroid is a fixed positroid, and a positroid class is the cohomology class Poincar\'e ... More

A point-line incidence identity in finite fields, and applicationsJan 15 2016Feb 20 2016Let $E \subseteq \mathbb{F}_q^2$ be a set in the 2-dimensional vector space over a finite field with $q$ elements. We prove an identity for the second moment of its incidence function and deduce a variety of existing results from the literature, not all ... More

An example related to the Erdos-Falconer question over arbitrary finite fieldsMay 13 2019There exists an infinite family of examples of subsets of $\mathbb{F}_q^2$ with $q^{4/3}$ elements whose distance sets are not the whole of $\mathbb{F}_q$.

Complete Characterization of Stability of Cluster Synchronization in Complex Dynamical NetworksJul 15 2015Dec 03 2015Synchronization is an important and prevalent phenomenon in natural and engineered systems. In many dynamical networks, the coupling is balanced or adjusted in order to admit global synchronization, a condition called Laplacian coupling. Many networks ... More

Robustness of Optimal Synchronization in Real NetworksJun 20 2011Experimental studies of synchronization properties on networks with controlled connection topology can provide powerful insights into the physics of complex networks. Here, we report experimental results on the influence of connection topology on synchronization ... More

Symmetries, Cluster Synchronization, and Isolated Desynchronization in Complex NetworksSep 25 2013Synchronization is of central importance in power distribution, telecommunication, neuronal, and biological networks. Many networks are observed to produce patterns of synchronized clusters, but it has been difficult to predict these clusters or understand ... More

Dynamic synchronization of a time-evolving optical network of chaotic oscillatorsDec 07 2010We present and experimentally demonstrate a technique for achieving and maintaining a global state of identical synchrony of an arbitrary network of chaotic oscillators even when the coupling strengths are unknown and time-varying. At each node an adaptive ... More

Multiresolution network modelsAug 26 2016Jul 05 2018Many existing statistical and machine learning tools for social network analysis focus on a single level of analysis. Methods designed for clustering optimize a global partition of the graph, whereas projection based approaches (e.g. the latent space ... More

Multiresolution network modelsAug 26 2016Sep 08 2016Social networks exhibit two key topological features: global sparsity and local density. That is, the overall propensity for interaction between any two randomly selected actors is infinitesimal, but for any given individual there is massive variability ... More

Modeling the social media relationships of Irish politicians using a generalized latent space stochastic blockmodelJul 16 2018D\'ail \'Eireann is the principal chamber of the Irish parliament. The 31st D\'ail \'Eireann is the principal chamber of the Irish parliament. The 31st D\'ail was in session from March 11th, 2011 to February 6th, 2016. Many of the members of the D\'ail ... More

Multiresolution network modelsAug 26 2016Nov 22 2016Social networks exhibit two key topological features: global sparsity and local density. That is, the overall propensity for interaction between any two randomly selected actors is infinitesimal, but for any given individual there is massive variability ... More

Clustering in networks with the collapsed Stochastic Block ModelMar 14 2012Nov 08 2012An efficient MCMC algorithm is presented to cluster the nodes of a network such that nodes with similar role in the network are clustered together. This is known as block-modelling or block-clustering. The model is the stochastic blockmodel (SBM) with ... More

Investigation of Parameter Uncertainty in Clustering Using a Gaussian Mixture Model Via Jackknife, Bootstrap and Weighted Likelihood BootstrapOct 02 2015Feb 28 2018Mixture models are a popular tool in model-based clustering. Such a model is often fitted by a procedure that maximizes the likelihood, such as the EM algorithm. At convergence, the maximum likelihood parameter estimates are typically reported, but in ... More

Model-based clustering in networks with Stochastic Community FindingMay 09 2012Oct 28 2012In the model-based clustering of networks, blockmodelling may be used to identify roles in the network. We identify a special case of the Stochastic Block Model (SBM) where we constrain the cluster-cluster interactions such that the density inside the ... More

Affine embeddings of a reductive groupDec 17 2010We classify affine varieties with an action of a connected, reductive algebraic group such that the group is isomorphic to an open orbit in the variety. This is accomplished by associating a set of one-parameter subgroups of the group to the variety, ... More

Arboreal singularities and loose Legendrians IFeb 13 2019Arboreal singularities are an important class of Lagrangian singularities. They are conical, meaning that they can be understood by studying their links, which are singular Legendrian spaces in $S^{2n-1}_{\text{std}}$. Loose Legendrians are a class of ... More

Mid-infrared time-resolved photoconduction in black phosphorusSep 07 2016Black phosphorus has attracted interest as a material for use in optoelectronic devices due to many favorable properties such as a high carrier mobility, field-effect, and a direct bandgap that can range from 0.3 eV in its bulk crystalline form to 2 eV ... More

Abelian fibrations and rational points on symmetric productsSep 14 1999Given a variety over a number field, are its rational points potentially dense, i.e., does there exist a finite extension over which rational points are Zariski dense? We study the question of potential density for symmetric products of surfaces. Contrary ... More

Geometry of equivariant compactifications of $G^n_a$Feb 11 1999Equivariant compactifications of reductive groups can be described by combinatorial data. On the other hand, equivariant compactifications of the additive group G^n_a are more complicated in at least two respects. First, they often admit moduli. Second, ... More

The homeostatic dynamics of feeding behaviour identify novel mechanisms of anorectic agentsFeb 22 2019May 09 2019Better understanding of feeding behaviour will be vital in reducing obesity and metabolic syndrome, but we lack a standard model that captures the complexity of feeding behaviour. We construct an accurate stochastic model of rodent feeding at the bout ... More

The MASSIVE Survey - I. A Volume-Limited Integral-Field Spectroscopic Study of the Most Massive Early-Type Galaxies within 108 MpcJul 03 2014Oct 13 2014Massive early-type galaxies represent the modern-day remnants of the earliest major star formation episodes in the history of the universe. These galaxies are central to our understanding of the evolution of cosmic structure, stellar populations, and ... More

The MASSIVE Survey II: Stellar Population Trends Out to Large Radius in Massive Early Type GalaxiesApr 09 2015Jun 17 2015We examine stellar population gradients in ~100 massive early type galaxies spanning 180 < sigma* < 370 km/s and M_K of -22.5 to -26.5 mag, observed as part of the MASSIVE survey (Ma et al. 2014). Using integral-field spectroscopy from the Mitchell Spectrograph ... More

Adaptive synchronization of coupled chaotic oscillatorsJul 22 2009Oct 05 2009We experimentally demonstrate and numerically simulate a new adaptive method to maintain synchronization between coupled nonlinear chaotic oscillators, when the coupling between the systems is unknown and time-varying (e.g., due to environmental parameter ... More

Embedding pointed curves in K3 surfacesJan 30 2013We analyze morphisms from pointed curves to K3 surfaces with a distinguished rational curve, such that the marked points are taken to the rational curve, perhaps with specified cross ratios. This builds on work of Mukai and others characterizing embeddings ... More

A note on the history of the four-colour conjectureJan 13 2012Feb 21 2012The four-colour conjecture was brought to public attention in 1854, most probably by Francis or Frederick Guthrie. This moves back by six years the date of the earliest known publication.

Extremal rays and automorphisms of holomorphic symplectic varietiesJun 26 2015We survey recent results on ample cones and birational contractions of holomorphic symplectic varieties of K3 type, focusing on explicit constructions and concrete examples.

Log minimal model program for the moduli space of stable curves: The first flipJun 20 2008We give a geometric invariant theory (GIT) construction of the log canonical model $\bar M_g(\alpha)$ of the pairs $(\bar M_g, \alpha \delta)$ for $\alpha \in (7/10 - \epsilon, 7/10]$ for small $\epsilon \in \mathbb Q_+$. We show that $\bar M_g(7/10)$ ... More

Proof of the Caratheodory ConjectureAug 06 2008Jul 31 2013A well-known conjecture of Caratheodory states that the number of umbilic points on a closed convex surface in ${\mathbb E}^3$ must be greater than one. In this paper we prove this for $C^{3+\alpha}$-smooth surfaces. The Conjecture is first reformulated ... More

On area-stationary surfaces in certain neutral Kaehler 4-manifoldsNov 22 2006We study surfaces in TN that are area-stationary with respect to a neutral Kaehler metric constructed on TN from a riemannian metric g on N. We show that holomorphic curves in TN are area-stationary, while lagrangian surfaces that are area-stationary ... More

A Neutral Kaehler Metric on Space of Time-like Lines in Lorentzian 3-spaceAug 31 2006Nov 24 2006We study the neutral K\"ahler metric on the space of time-like lines in Lorentzian ${\Bbb{E}}^3_1$, which we identify with the total space of the tangent bundle to the hyperbolic plane. We find all of the infinitesimal isometries of this metric, as well ... More

Corelations are the prop for extraspecial commutative Frobenius monoidsJan 11 2016Jan 30 2016Just as binary relations between sets may be understood as jointly monic spans, so too may equivalence relations on the disjoint union of sets be understood as jointly epic cospans. With the ensuing notion of composition inherited from the pushout of ... More

A class of Ramsey-extremal hypergraphsAug 28 2016In 1991, McKay and Radziszowski proved that, however each 3-subset of a 13-set is assigned one of two colours, there is some 4-subset whose four 3-subsets have the same colour. More than 25 years later, this remains the only non-trivial classical Ramsey ... More

Applying Deep Learning To Airbnb SearchOct 22 2018Oct 24 2018The application to search ranking is one of the biggest machine learning success stories at Airbnb. Much of the initial gains were driven by a gradient boosted decision tree model. The gains, however, plateaued over time. This paper discusses the work ... More

Einstein Metrics Adapted to Contact Structures on 3-ManifoldsDec 05 2000The Newman-Penrose-Perjes formalism is applied to smooth contact structures on riemannian 3-manifolds. In particular it is shown that a contact 3-manifold admits an adapted riemannian metric if and only if it admits a metric with a divergence-free, constantly ... More

Rationality of complete intersections of two quadricsMar 21 2019Apr 19 2019We study rationality problems for smooth complete intersections of two quadrics. We focus on the three-dimensional case, with a view toward understanding the invariants governing the rationality of a geometrically rational threefold over a non-closed ... More

Immersed disks, slicing numbers and concordance unknotting numbersNov 26 2013Nov 04 2015We study three knot invariants related to smoothly immersed disks in the four-ball. These are the four-ball crossing number, which is the minimal number of normal double points of such a disk bounded by a given knot; the slicing number, which is the minimal ... More

A characterisation of the Z^n + Z(δ) lattice and definite nonunimodular intersection formsFeb 11 2008We prove a generalisation of Elkies' theorem to nonunimodular definite forms (and lattices). Combined with inequalities of Froyshov and of Ozsvath and Szabo, this gives a simple test of whether a rational homology 3-sphere may bound a definite four-manifold. ... More

The Price of Uncertain Priors in Source CodingNov 22 2018We consider the problem of one-way communication when the recipient does not know exactly the distribution that the messages are drawn from, but has a "prior" distribution that is known to be close to the source distribution, a problem first considered ... More

Geodesic Flow on Global Holomorphic Sections of TS^2Feb 23 2006We study the geodesic flow on the global holomorphic sections of the bundle $\pi:{TS}^2\to {S}^2$ induced by the neutral K\"ahler metric on the space of oriented lines of ${\Bbb{R}}^3$, which we identify with ${TS}^2$. This flow is shown to be completely ... More

Nonabelian localization for U(1) Chern-Simons theoryMar 29 2009This article studies the nonabelian localization results of Beasley and Witten, and considers the analogue of these results when the gauge group is U(1). It compares these results with results of Manoliu on abelian Chern-Simons theory, showing that the ... More

Clearing Markets via BundlesJan 13 2014Jan 17 2014We study algorithms for combinatorial market design problems, where a set of heterogeneous and indivisible objects are priced and sold to potential buyers subject to equilibrium constraints. Extending the CWE notion introduced by Feldman et al. [STOC ... More