arXiv research

The sl_3 spider is a diagrammatic category used to study the representation theory of the quantum group U_q(sl_3). The morphisms in this category are generated by a basis of non-elliptic webs. Khovanov- Kuperberg observed that non-elliptic webs are indexed by semistandard Young tableaux. They establish this bijection v…

KMRCD detects outliers in non-elliptical data using kernel trick.

Study on deformations of Spin(7)-structures on manifolds.

A new vine copula mixture model improves clustering accuracy for non-Gaussian data.

Let G be a simple algebraic group. Labelled trivalent graphs called webs can be used to product invariants in tensor products of minuscule representations. For each web, we construct a configuration space of points in the affine Grassmannian. Via the geometric Satake correspondence, we relate these configuration spaces…

Flexible classifier using Mahalanobis distances for non-elliptical distributions.

The paper provides coordinates for ${ m SL}_3$-web diagrams on surfaces.

For a compact Riemannian surface with boundary we study attenuated geodesic transform of functions and differential forms. We generalize several known results on uniqueness and stability of this transform dropping condition of absence of conjugate points.

We show that non-elliptic prime 3-manifolds satisfy integral approximation for the simplicial volume, i.e., that their simplicial volume equals the stable integral simplicial volume. The proof makes use of integral foliated simplicial volume and tools from ergodic theory.

We show in this article that Kähler hyperbolic manifolds satisfy a family of optimal Chern number inequalities and the equality cases can be attained by some compact ball quotients. These present restrictions to complex structures on negatively-curved compact Kähler manifolds, thus providing evidence to the rigidity co…

Copula is a powerful tool to model multivariate data. We propose the modelling of intraday financial returns of multiple assets through copula. The problem originates due to the asynchronous nature of intraday financial data. We propose a consistent estimator of the correlation coefficient in case of Elliptical copula …

We recall a construction of Mackaay, Pan and Tubbenhauer of the algebras $K^ε$ which allow to understand the $sl_3$ homology for links in a local way (i.e. for tangles). Then, by studying the combinatorics of the Kuperberg bracket, we give a large family of non-elliptic webs whose associated projective $K^ε$-modules ar…

In this paper, we prove a quantitative version of the Tits alternative for negatively pinched manifolds $X$. Precisely, we prove that a nonelementary discrete isometry subgroup of $\mathrm{Isom}(X)$ generated by two non-elliptic isometries $g$, $f$ contains a free subgroup of rank $2$ generated by isometries $f^N , h$ …

Let $x$ denote a diffusion process defined on a closed compact manifold. In an earlier article, the author introduced a new approach to constructing admissible vector fields on the associated space of paths, under the assumption of ellipticity of $x$. In this article, this method is extended to yield similar results fo…

Let $u$ denote a solution to a rotationally invariant Hessian equation $F(D^2u)=0$ on a bounded simply connected domain $Ω\subset R^2$, with constant Dirichlet and Neumann data on $\partial Ω$. In this paper we prove that if $u$ is real analytic and not identically zero, then $u$ is radial and $Ω$ is a disk. The fully …

Complex hyperbolic triangle groups are discrete when certain conditions are met.

The abstract manifold cannot have uniformly quasiregular self-maps.

Let $M$ be a closed symplectic manifold of dimension $2n$ with non-ellipticity. We can define an almost Kähler structure on $M$ by using the given symplectic form. Hence, we have a $\G=π_1(M)$-invariant almost Kähler structure on the universal covering, $\ti M$, of $M$. Using Darboux coordinate charts, we globally defo…

We present a definition of indefinite Kasparov modules, a generalisation of unbounded Kasparov modules modelling non-symmetric and non-elliptic (e.g. hyperbolic) operators. Our main theorem shows that to each indefinite Kasparov module we can associate a pair of (genuine) Kasparov modules, and that this process is reve…

New non-trivial Kaehler-Ricci solitons found in infinite dimensional complex space forms.

Define web algebras for annular SL(2) and SL(3) using foam TQFTs.

We improve our previous results on indefinite Kasparov modules, which provide a generalisation of unbounded Kasparov modules modelling non-symmetric and non-elliptic (e.g. hyperbolic) operators. In particular, we can weaken the assumptions that are imposed on indefinite Kasparov modules. Using a new theorem by Lesch an…

Let $M$ be a compact Riemannian manifold, $π:\widetilde{M}\rightarrow M$ be the universal covering and $ω$ be a smooth $2$-form on $M$ with $π^*ω$ cohomologous to zero. Suppose the fundamental group $π_1(M)$ satisfies certain radial quadratic (resp. linear) isoperimetric inequality, we show that there exists a smooth $…

Flow cytometry is a high-throughput technology used to quantify multiple surface and intracellular markers at the level of a single cell. This enables to identify cell sub-types, and to determine their relative proportions. Improvements of this technology allow to describe millions of individual cells from a blood samp…

We investigate when the Chevalley-Eilenberg differential of a complex Lie algebroid on a manifold with boundary admits a Hodge decomposition. We introduce the concepts of Cauchy-Riemann structures, elliptic and non-elliptic boundary points and Levi-forms, which we use to define the notion of q-convexity. We show that t…

We study the small-time fluctuations for diffusion processes which are conditioned by their initial and final positions, under the assumptions that the diffusivity has a sub-Riemannian structure and that the drift vector field lies in the span of the sub-Riemannian structure. In the case where the endpoints agree and t…

In this paper we describe a new method for analyzing the Laplacian on asymptotically hyperbolic spaces, which was introduced recently by the author. This new method in particular constructs the analytic continuation of the resolvent for even metrics (in the sense of Guillarmou), and gives high energy estimates in strip…

We describe explicitly the moduli spaces $M^{pst}_g(S,E)$ of polystable holomorphic structures $E$ with $\det E\cong K$ on a rank 2 vector bundle $E$ with $c_1(E)=c_1(K)$ and $c_2(E)=0$ for all minimal class VII surfaces $S$ with $b_2(S)=1$ and with respect to all possible Gauduchon metrics $g$. These surfaces $S$ are …

Study flat connections on hypersurfaces of 4-manifolds with parallel spinors.

We prove the linear stability of slowly rotating Kerr black holes as solutions of the Einstein vacuum equation: linearized perturbations of a Kerr metric decay at an inverse polynomial rate to a linearized Kerr metric plus a pure gauge term. We work in a natural wave map/DeTurck gauge and show that the pure gauge term …

This paper is devoted to study the optimal portfolio problem. Harry Markowitz's Ph.D. thesis prepared the ground for the mathematical theory of finance. In modern portfolio theory, we typically find asset returns that are modeled by a random variable with an elliptical distribution and the notion of portfolio risk is d…

In this paper we propose a problem-driven scenario generation approach to the single-period portfolio selection problem which use tail risk measures such as conditional value-at-risk. Tail risk measures are useful for quantifying potential losses in worst cases. However, for scenario-based problems these are problemati…

The paper constructs bases for cluster varieties using ${ m SL}_3$-webs and laminations.

Big data sets must be carefully partitioned into statistically similar data subsets that can be used as representative samples for big data analysis tasks. In this paper, we propose the random sample partition (RSP) data model to represent a big data set as a set of non-overlapping data subsets, called RSP data blocks,…

Prevents sensitive data generation in diffusion models using labeled and unlabeled data.

Study reveals Data Shapley's inconsistent performance in data selection tasks.

PRRO generates synthetic tabular data that improves SL performance and class distribution.

Defines data science as a natural ecosystem with challenges and missions.

Differences in data size per class, also known as imbalanced data distribution, have become a common problem affecting data quality. Big Data scenarios pose a new challenge to traditional imbalanced classification algorithms, since they are not prepared to work with such amount of data. Split data strategies and lack o…

Synthetic data enhances analytics but requires careful volume management.

New test ensures quality of shared data in machine learning.

Differential privacy allows quantifying privacy loss resulting from accessing sensitive personal data. Repeated accesses to underlying data incur increasing loss. Releasing data as privacy-preserving synthetic data would avoid this limitation, but would leave open the problem of designing what kind of synthetic data. W…

Paper creates fair synthetic data ensuring equal predictions across sensitive attributes.

DPA preserves data distribution in reduced dimensions.

Efficient synthetic data generation improves model performance on tabular data.

For most problems in science and engineering we can obtain data sets that describe the observed system from various perspectives and record the behavior of its individual components. Heterogeneous data sets can be collectively mined by data fusion. Fusion can focus on a specific target relation and exploit directly ass…

DAERNN models censored data using neural networks with data augmentation.

Data preprocessing techniques are devoted to correct or alleviate errors in data. Discretization and feature selection are two of the most extended data preprocessing techniques. Although we can find many proposals for static Big Data preprocessing, there is little research devoted to the continuous Big Data problem. A…