arXiv research

New rigidity result for fat bundles with equal vertical curvatures.

Paper classifies fibers of fat Riemannian submersions with non-negative curvature.

This article deals with fat bundles. Berard-Bergery classified all homogeneous bundles of that type. We ask a question of a possibility to generalize his description in the case of arbitrary G-structures over homogeneous spaces. We obtain necessary conditions for the existence of such bundles. These conditions yield a …

This work is devoted to new constructions of symplectically fat fiber bundles. The latter are constructed in two ways: using the Kirwan map and expressing the fatness condition in terms of the isotropy representation related to the G-structure over some homogeneous spaces.

The aim of the present paper is to investigate new classes of symplectically fat fibre bundles. We prove a general existence theorem for fat vectors with respect to the canonical invariant connections. Based on this result we give new proofs of some constructions of symplectic structures. This includes twistor bundles …

We prove that a compact 4-manifold which supports a circle-invariant fat SO(3)-bundle is diffeomorphic to either S^4 or CP^2-bar. The proof involves studying the resulting Hamiltonian circle action on an associated symplectic 6-manifold. Applying our result to the twistor bundle of Riemannian 4-manifolds shows that S^4…

The paper proves a stronger Petersen--Wilhelm conjecture for principal bundles.

Alan Weinstein showed that certain characteristic numbers of any Riemannian submersion with totally geodesic fibers and positive vertizontal curvatures are nonzero. In this paper we explicitly compute these invariants in terms of Chern and Pontrjagin numbers of the bundle. This allows us to show that many bundles do no…

The study constructs metrics with positive 2nd Ricci curvature on various manifolds.

Narasihman and Ramanan proved that an arbitrary connection in a vector bundle over a base space B can be obtained as the pull-back (via a correctly chosen classifying map from B into the appropriate Grassmannian) of the universal connection in the universal bundle over the Grassmannian. The purpose of this paper is to …

We solve the problem posed by Boyer and Galicki about the existence of K-contact simply connected manifolds with no Sasakian structure. Although the result lies in the framework of metric contact geometry, our methods come from contact and symplectic geometry and are based on the method of fat bundles developed by Ster…

Constructs fat, shellable 3-spheres with specific $f$-vectors.

Paper introduces fat CW complexes including all closed manifolds.

Introduces fat Lie theory for Lie groupoids and algebroids.

The article proves the existence of horizontal immersions into fat distributions and contact structures.

Improved uniform convergence bound with fat-shattering dimension reduces sample complexity gap.

A classic problem in physics is the origin of fat tailed distributions generated by complex systems. We study the distributions of stock returns measured over different time lags $τ.$ We find that destroying all correlations without changing the $τ= 1$ d distribution, by shuffling the order of the daily returns, causes…

Godin introduced the categories of open closed fat graphs $Fat^{oc}$ and admissible fat graphs $Fat^{ad}$ as models of the mapping class group of open closed cobordism. We use the contractibility of the arc complex to give a new proof of Godin's result that $Fat^{ad}$ is a model of the mapping class group of open-close…

Study horizontal discs in fat distributions, proving their existence.

New example disproves complex contact theory for fat distributions with Reeb directions.

Estimates fat-shattering dimension of aggregated function classes.

We study closed non-positively curved Riemannian manifolds $M$ which admit `fat $k$-flats': that is, the universal cover $\tilde M$ contains a positive radius neighborhood of a $k$-flat on which the sectional curvatures are identically zero. We investigate how the fat $k$-flats affect the cardinality of the collection …

New bounds on inscribed triangles in arbitrary planar domains.

The h-principle fails for prelegendrians in fat distributions of corank 2.

In these lecture notes we discuss a body of work in which Morse theory is used to construct various homology and cohomology operations. In the classical setting of algebraic topology this is done by constructing a moduli space of graph flows, using homotopy theoretic methods to construct a virtual fundamental class, an…

This study empirically re-examines fat tails in stock return distributions by applying statistical methods to an extensive dataset taken from the Korean stock market. The tails of the return distributions are shown to be much fatter in recent periods than in past periods and much fatter for small-capitalization stocks …

Geometrically, Kostant's Convexity Theorem is extended to submetries with a fat section.

We propose a random walk model of asset returns where the parameters depend on market stress. Stress is measured by, e.g., the value of an implied volatility index. We show that model parameters including standard deviations and correlations can be estimated robustly and that all distributions are approximately normal.…

The literature of heavy tails (typically) starts with a random walk and finds mechanisms that lead to fat tails under aggregation. We follow the inverse route and show how starting with fat tails we get to thin-tails when deriving the probability distribution of the response to a random variable. We introduce a general…

We define "fat" train tracks and use them to give a combinatorial criterion for the Hempel distance of Heegaard splittings for closed orientable 3-manifolds. We apply this criterion to 3-manifolds obtained from surgery on knots in the three sphere.

Optimal portfolios for fat-tailed risks using a new tail risk measure.

Elliptical processes generalize Gaussian and Student-t models with fat tails and computational efficiency.

The book examines statistical issues with fat-tailed distributions and proposes remedies.

We prove that a Kleinian group $G$ acting upon $\mathbb{H}^{n}$ admits a non-constant $G$-automorphic function, even if it has torsion elements, provided that the orders of the elliptic (torsion) elements are uniformly bounded. This is accomplished by developing a technique for mashing distinct fat triangulations while…

We study the differential geometric consequences of our previous result on the existence of fat triangulations, in conjunction with a result of Cheeger, Müller and Schrader, regarding the convergence of Lipschitz-Killing curvatures of piecewise-flat approximations of smooth Riemannian manifolds. A further application t…

Study finds sales forecasters overreact to extreme news.

Starting from an exact relationship between news, threshold and price return distributions in the stationary state, I discuss the ability of the Ghoulmie-Cont-Nadal model of traders to produce fat-tailed price returns. Under normal conditions, this model is not able to transform Gaussian news into fat-tailed price retu…

I report a new statistical distribution formulated to confront the infamous, long-standing, computational/modeling challenge presented by highly skewed and/or leptokurtic ("fat- or heavy-tailed") data. The distribution is straightforward, flexible and effective. Even when working with far fewer data points than are rou…

Given a hyperbolic surface, the set of all closed geodesics whose length is minimal form a graph on the surface, in fact a so-called fat graph, which we call the systolic graph. We study which fat graphs are systolic graphs for some surface (we call these admissible). There is a natural necessary condition on such grap…

Given a smooth closed manifold M with a family {L_i} of closed submanifolds, we consider the free loop space LM and the spaces PM(L_i,L_j) of open strings (paths g:[0,1]->M with g(0) in L_i, and g(1) in L_j). We construct string topology operations resulting in an open-closed TQFT on the family (h_*(LM),h_*(PM(L_i,L_j)…

New method models fat-tailed distributions with anisotropic tail-adaptive flows.

Risk and uncertainty will always be a matter of experience, luck, skills, and modelling. Leverage is another concept, which is critical for the investor decisions and results. Adaptive skills and quantitative probabilistic methods need to be used in successful management of risk, uncertainty and leverage. The author ex…

Large deviations for fat tailed distributions, i.e. those that decay slower than exponential, are not only relatively likely, but they also occur in a rather peculiar way where a finite fraction of the whole sample deviation is concentrated on a single variable. The regime of large deviations is separated from the regi…

In this paper we introduce an efficient fat-tail measurement framework that is based on the conditional second moments. We construct a goodness-of-fit statistic that has a direct interpretation and can be used to assess the impact of fat-tails on central data conditional dispersion. Next, we show how to use this framew…

It is well known that the distribution of returns from various financial instruments are leptokurtic, meaning that the distributions have "fatter tails" than a Normal distribution, and have skew toward zero. This paper presents a graceful micro-level explanation for such fat-tailed outcomes, using agents whose private …

The aim of this paper is to propose a heterogeneous agent model of stock markets that develop complicated endogenous price fluctuations. We find occurrences of non-stationary chaos, or speculative bubble, are caused by the heterogeneity of traders' strategies. Furthermore, we show that the distributions of returns gene…

We study the problems related to the estimation of the Gini index in presence of a fat-tailed data generating process, i.e. one in the stable distribution class with finite mean but infinite variance (i.e. with tail index $α\in(1,2)$). We show that, in such a case, the Gini coefficient cannot be reliably estimated usin…

New learning rule for quantum measurement classes overcomes uniform convergence issues.