arXiv research

Study geodesics entering a fixed cusp neighborhood multiple times.

We generalize the notion of cusp excursion of geodesic rays by introducing for any $k \geq 1$ the $k^{th}$ excursion in the cusps of a hyperbolic $N$-manifold of finite volume. We show that on one hand, this excursion is at most linear for geodesics that are generic with respect to the hitting measure of a random walk.…

In this paper we study the typical speed of a generic earthquake trajectory leaving compact sets in the moduli space of the once-punctured torus. Mirzakhani showed that the earthquake flow is measurably equivalent to the horocyclic flow, which has been studied extensively. Our main result shows that the earthquake flow…

Stationary measures on hyperbolic surfaces with cusps are singular and stable under quasi-symmetries.

We derive several results that describe the rate at which a generic geodesic makes excursions into and out of a cusp on a finite area hyperbolic surface and relate them to approximation with respect to the orbit of infinity for an associated Fuchsian group. This provides proofs of some well known theorems from metric d…

Motivated by the notion of cusp excursion in geometrically finite hyperbolic manifolds, we define a notion of excursion in any subgroup of a given group, and study its asymptotic distribution for right-angled Artin groups and graph products. In particular, for any irreducible right-angled Artin group we show that with …

For a non-uniform lattice in SL(2,R), we consider excursions in cusp neighborhoods of a random geodesic on the corresponding finite area hyperbolic surface or orbifold. We prove a strong law for a certain partial sum involving these excursions. This generalizes a theorem of Diamond and Vaaler for continued fractions. I…

We study geodesics on the modular surface, comparing WP and hyperbolic metrics.

This paper proves exponential mixing for frame flows on hyperbolic manifolds with cusps.

We construct a Poincaré section for the horocycle flow on the modular surface $SL(2, \R)/SL(2, \Z)$, and study the associated first return map, which coincides with a transformation (the {\it BCZ map}) defined by Boca-Cobeli-Zaharescu. We classify ergodic invariant measures for this map and prove equidistribution of pe…

Given for instance a finite volume negatively curved Riemannian manifold $M$, we give a precise relation between the logarithmic growth rates of the excursions into cusps neighborhoods of the strong unstable leaves of negatively recurrent unit vectors of $M$ and their linear divergence rates under the geodesic flow. As…

Study examines local extrema and crossing statistics in financial markets.

New model captures fast price excursions in finance.

A model-free method analyzes trading strategies using excursion paths.

New method distinguishes stochastic from deterministic signals using excursion counts.

We focus on the problem of estimating and quantifying uncertainties on the excursion set of a function under a limited evaluation budget. We adopt a Bayesian approach where the objective function is assumed to be a realization of a Gaussian random field. In this setting, the posterior distribution on the objective func…

This note re-addresses the Paris barrier options proposed by Yor and collaborators and their valuation using the Laplace transform approach. The notion of Paris barrier options, based on excursion theory and using the Brownian meander, is extended such that their valuation is now possible at any point during their life…

Meta-learning method for estimating time-varying mHealth intervention effects.

The paper analyzes insurance risk with Parisian ruin and capital injection.

The paper extends consistency results for sequential design strategies to vector-valued Gaussian processes.

This work analyzes the gap between off-policy and on-policy policy gradient methods and provides conditions to reduce this gap.

Modeling price dynamics in AMMs with fees using geometric Brownian motion.

New method infers and samples point processes from latent diffusion.

We consider the problem of estimating the set of all inputs that leads a system to some particular behavior. The system is modeled by an expensive-to-evaluate function, such as a computer experiment, and we are interested in its excursion set, i.e. the set of points where the function takes values above or below some p…

A new algorithm learns from failures to optimize under constraints efficiently.

Generative model calibrates 3D battery cathode morphologies from 2D images.

This paper introduces a new method for uncertainty quantification in prediction models.

Study on spin random fields using chaos decomposition for cosmic microwave background modeling.

We study a parsimonious but non-trivial model of the latent limit order book where orders get placed with a fixed displacement from a center price process, i.e.\ some process in-between best bid and best ask, and get executed whenever this center price reaches their level. This mechanism corresponds to the fundamental …

Along with excursions into manifolds with corners, resolution towers of Thom and Whitney stratifications, I show that for a generic gradientlike vector field on a manifold with a Morse function, the stable manifolds give a CW decomposition of the manifold. This has been done before.

Study fractal and regular geometry in deep neural networks.

4-manifolds show every flat 3-manifold as cusp sections.

Characterizes holonomies of convex projective cusps.

The importance of the global financial system cannot be exaggerated. When a large financial institution becomes problematic and is bailed out, that bank is often claimed as "too big to fail". On the other hand, to prevent bank's failure, regulatory authorities adopt the Prompt Corrective Action (PCA) against a bank tha…

We find that cusp densities of hyperbolic knots in the 3-sphere are dense in [0,0.6826...] and those of links are dense in [0,0.853...]. We define a new invariant associated with cusp volume, the cusp crossing density, as the ratio between the cusp volume and the crossing number of a link, and show that cusp crossing d…

The paper classifies different types of cusps on plane curves.

The paper characterizes links in 3D from divides with cusps.

New method reduces memory usage for Bayesian inverse problems on large grids.

The paper explores cusp types in hyperbolic 4-manifolds and their commensurability classes.

New findings on cusped Borel Anosov representations and their properties.

Classifies low-volume hyperbolic 3-manifolds with a maximal cusp.

We prove that non-compact finite volume hyperbolic 3-manifolds that satisfy a mild cohomological condition (infinitesimal rigidity) admit a family of properly convex deformations of their complete hyperbolic structure where the ends become generalized cusps of type 1 or type 2. We also discuss methods for controlling w…

Four hyperbolic 24-cell 4-manifolds with one cusp are identified.

We calculate the asymptotic average rate at which a generic geodesic on a finite area hyperbolic 2-orbifold returns to an embedded disc on the surface, as well as the average amount of time it spends in the disc during each visit. This includes the case where the center of the disc is a cone point.

Study of Eisenstein series linked to hyperbolic cusps.

This paper completes a classification of the types of orientable and non-orientable cusps that can arise in the quotients of hyperbolic knot complements. In particular, $S^2(2,4,4)$ cannot be the cusp cross-section of any orbifold quotient of a hyperbolic knot complement. Furthermore, if a knot complement covers an orb…

We construct here two new examples of non-orientable, non-compact, hyperbolic 4-manifolds. The first has minimal volume $v_m = 4π^2/3$ and two cusps. This example has the lowest number of cusps among known minimal volume hyperbolic 4-manifolds. The second has volume $2\cdot v_m$ and one cusp. It has lowest volume among…

Study Dirac operator on cusped hyperbolic manifolds, finding spectrum properties.