arXiv research

New representation theory for surface groups to SO0(2,3).

The paper defines almost strict domination for representations and connects it to anti-de Sitter 3-manifolds.

In a previous paper by Deroin-Tholozan, the authors construct a map $\mathbfΨ_ρ$ from the Teichmüller space of $S$ to itself and prove that, when $M$ has sectional curvature $\leq -1$, the image of $\mathbfΨ_ρ$ lies (almost always) in the domain $\mathrm{Dom}(ρ)$ of Fuchsian representations stricly dominating $ρ$. Here…

New framework for ranking distributions using variable fractional parameters.

Study domination between non-Fuchsian surface group representations and anti-de Sitter geometry.

This study finds a special class of representations that dominate others in a complex hyperbolic group.

Dominant representations found via Fock-Goncharov coordinates.

New method finds Fuchsian representations dominating others in surface group representations.

This note removes technical assumptions and characterizes relatively dominated representations.

The paper proves conditions for non-uniform expansion in partially hyperbolic systems.

Anosov representations give a higher-rank analogue of convex cocompactness in a rank-one Lie group which shares many of its good geometric and dynamical properties; geometric finiteness in rank one may be seen as a controlled weakening of convex cocompactness to allow for isolated failures of hyperbolicity. We introduc…

We prove that a representation from the fundamental group of a closed surface of negative Euler characteristic with values in the isometry group of a Riemannian manifold of sectional curvature bounded by -1 can be dominated by a Fuchsian representation. Moreover, we prove that the domination can be made strict, unless …

The study shows how energy density of harmonic maps dominates in $n$-Fuchsian fibers, leading to unique minimal surfaces.

New EI strategies using OWA and SSD for excess return.

New method assesses multivariate stochastic dominance using Optimal Transport.

Study max- and min-stability under first-order stochastic dominance, finding new functional characterizations.

New insights into Bartnik mass from improvability of dominant energy scalar.

We prove that any nonabelian, non-Fuchsian representation of a surface group into PSL(2,R) is the holonomy of a folded hyperbolic structure on the surface. Using similar ideas, we establish that any non-Fuchsian representation rho of a surface group into PSL(2,R) is strictly dominated by some Fuchsian representation j,…

We generalize a well-known existence and uniqueness result for equivariant harmonic maps due to Corlette, Donaldson, and Labourie to a non-compact infinite energy setting and analyze the asymptotic behaviour of the harmonic maps. When the relevant representation is Fuchsian and has hyperbolic monodromy, our constructio…

New GCNs solve graph embedding problems efficiently and interpretably.

Study connects Lie groups to specific Riemannian manifolds.

We show that many Lorentzian manifolds of dimension >2 do not admit a spacelike codimension-one foliation, and that almost every manifold of dimension >2 which admits a Lorentzian metric at all admits one which satisfies the dominant energy condition and the timelike convergence condition. These two seemingly unrelated…

The paper proves a quadratic formality for Sasakian manifolds' representation varieties.

We derive properties of the cdf of random variables defined as saddle-type points of real valued continuous stochastic processes. This facilitates the derivation of the first-order asymptotic properties of tests for stochastic spanning given some stochastic dominance relation. We define the concept of Markowitz stochas…

Suppose an investor aims at Delta hedging a European contingent claim $h(S(T))$ in a jump-diffusion model, but incorrectly specifies the stock price's volatility and jump sensitivity, so that any hedging strategy is calculated under a misspecified model. When does the erroneously computed strategy super-replicate the t…

In this paper we introduce the notion of almost flatness for (stably) relative bundles on a pair of topological spaces and investigate basic properties of it. First, we show that almost flatness of topological and smooth sense are equivalent. This provides a construction of an almost flat stably relative bundle by usin…

Develops theory of Anosov representations for Fuchsian groups, showing stability and analytical properties.

The paper finds representations of surface groups in SO(4,1) with specific curvature properties.

Investigates the effects of nondominated sets of probability measures in robust models of finance.

In a system containing a large number of interacting stochastic processes, there will typically be many non-zero correlation coefficients. This makes it difficult to either visualize the system's inter-dependencies, or identify its dominant elements. Such a situation arises in Foreign Exchange (FX) which is the world's…

Framework for real-time win probability and player ability in sports.

Study on surface group representations in PU(2,1) leading to convex-cocompact examples.

The Mean-Variance Criterion is equivalent to Second-order Stochastic Dominance under symmetric Elliptical distributions.

This paper shows how to approximate CAT(-1) representations by Fuchsian ones.

We consider filtration consistent nonlinear expectations in probability spaces satisfying only the usual conditions and separability. Under a domination assumption, we demonstrate that these nonlinear expectations can be expressed as the solutions to Backward Stochastic Differential Equations with Lipschitz continuous …

We study, using Mean Curvature Flow methods, 2+1 dimensional cosmologies with a positive cosmological constant and matter satisfying the dominant and the strong energy conditions. If the spatial slices are compact with non-positive Euler characteristic and are initially expanding everywhere, then we prove that the spat…

An important research direction in machine learning has centered around developing meta-learning algorithms to tackle few-shot learning. An especially successful algorithm has been Model Agnostic Meta-Learning (MAML), a method that consists of two optimization loops, with the outer loop finding a meta-initialization, f…

We call a given American option representable if there exists a European claim which dominates the American payoff at any time and such that the values of the two options coincide in the continuation region of the American option. This concept has interesting implications from a probabilistic, analytic, financial, and …

The object of investigation are Lie groups considered as almost contact B-metric manifolds of the lowest dimension three. It is established a correspondence of all basic-class-manifolds of the Ganchev-Mihova-Gribachev classification of the studied manifolds and the explicit matrix representation of Lie groups. Some kno…

Study geometric properties of loss functions to understand neural network performance.

The paper optimizes portfolios by selecting financial ratios via PCA for better value investment.

For a compact almost complex 4-manifold $(M,J)$, we study the subgroups $H^{\pm}_J$ of $H^2(M, \mathbb{R})$ consisting of cohomology classes representable by $J$-invariant, respectively, $J$-anti-invariant 2-forms. If $b^+ =1$, we show that for generic almost complex structures on $M$, the subgroup $H^-_J$ is trivial. …

Real-valued word representations have transformed NLP applications; popular examples are word2vec and GloVe, recognized for their ability to capture linguistic regularities. In this paper, we demonstrate a {\em very simple}, and yet counter-intuitive, postprocessing technique -- eliminate the common mean vector and a f…

Almost-flat manifolds were defined by Gromov as a natural generalisation of flat manifolds and as such share many of their properties. Similarly to flat manifolds, it turns out that the existence of a spin structure on an almost-flat manifold is determined by the canonical orthogonal representation of its fundamental g…

We provide a characterization in terms of Fatou closedness for weakly closed monotone convex sets in the space of $\mathcal{P}$-quasisure bounded random variables, where $\mathcal{P}$ is a (possibly non-dominated) class of probability measures. Applications of our results lie within robust versions the Fundamental Theo…

We study the curvature of almost Hermitian manifolds and their special analogues via intrinsic torsion and representation theory. By deriving different forumlae for the skew-symmetric part of the star-Ricci curvature, we find that some of these contributions are dependent on the approach used, and for the almost Hermit…

Large neural networks learn low-dimensional representations that balance complexity and regularity.

Let $Γ$ be a nonuniform lattice acting on real hyperbolic n-space. We show that in dimension greater than or equal to 4, the volume of a representation is constant on each connected component of the representation variety of $Γ$ in SO(n,1). Furthermore, in dimensions 2 and 3, there is a semialgebraic subset of the repr…