Study geodesics entering a fixed cusp neighborhood multiple times.
problem Understanding geodesics entering a specific cusp neighborhood multiple times.
method Investigate reciprocal geodesics entering a fixed cusp neighborhood a fixed number of times.
result Characterized the class of reciprocal geodesics entering a fixed cusp neighborhood a fixed number of times.
Study shows how geodesics behave in hyperbolic manifolds and proves measure singularity.
problem Understanding geodesic behavior and measure singularity in hyperbolic manifolds.
method Introduced kth excursion for geodesics, analyzed hitting and Lebesgue measures. result Proved hitting and Lebesgue measures on hyperbolic space are mutually singular.
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.
problem Understanding stationary measures on hyperbolic surfaces with cusps.
method Analyzing exponential decay of cusp excursions and proving quasi-symmetry stability.
result Stationary measures on hyperbolic surfaces with cusps are quasi-symmetrically stable and singular.
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…
Study excursions of geodesics in RAAGs and graph products, finding a log(n) max excursion.
problem Understanding geodesic excursions in RAAGs and graph products.
method Defined and studied excursions in subgroups of groups, focusing on right-angled Artin groups and graph products.
result Max excursion of generic geodesics in flat tends to log(n) for irreducible RAAGs.
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.
problem Comparing geodesics on the modular surface under different metrics.
method Lift WP geodesics to the universal cover, analyze geometric properties, and compare deviations.
result WP and hyperbolic geodesics fellow-travel in the thick part of the universal cover.
This paper proves exponential mixing for frame flows on hyperbolic manifolds with cusps.
problem Establishing exponential mixing for frame flows on geometrically finite hyperbolic manifolds with cusps.
method Symbolic coding of geodesic flow, Dolgopyat's method, large deviation property, combinatorics of cusp excursions, renewal theorem.
result Frame flows for geometrically finite hyperbolic manifolds of arbitrary dimensions are exponentially mixing.
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.
problem Understanding local extrema and crossing statistics in financial markets.
method Excursion set theory, numerical computation, theoretical prediction, clustering of geometrical measures, cross-correlation, Singular Value Decomposition.
result Excursion sets reveal statistical coherency and sensitivity to crises in financial markets.
New model captures fast price excursions in finance.
problem Capturing fast price excursions in financial models.
method Heston model with fast-reversion limit.
result Model shows significant hitting probabilities for barrier options.
A model-free method analyzes trading strategies using excursion paths.
problem Analyzing risk and return for dynamic trading strategies without probabilistic assumptions.
method Pathwise analysis of trading signals using δ-excursions.
result Continuous paths can be uniquely decomposed into δ-excursions.
New method distinguishes stochastic from deterministic signals using excursion counts.
problem Distinguishing between stochastic and deterministic signals in discrete time series.
method Excursion and crossing theorems for continuous semimartingales, comparing empirical excursion counts to theoretical expectation.
result A robust data-driven diffusion test that classifies signals based on log-log slope deviation.
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.
problem Complex mHealth data and uncertain randomization probabilities.
method DR-WCLS meta-learning procedure for causal excursion effects.
result More efficient and consistent estimates of causal excursion effects.
The paper analyzes insurance risk with Parisian ruin and capital injection.
problem Analyzing insurance risk with Parisian ruin and capital injection.
method Using fluctuation and excursion theory of spectrally negative Levy processes.
result Distributional identities and ruin probabilities are derived.
The paper extends consistency results for sequential design strategies to vector-valued Gaussian processes.
problem Estimating excursion sets of vector-valued Gaussian processes.
method Clarifying the connection between continuous Gaussian processes and Gaussian measures in Banach spaces, extending concepts and properties from scalar-valued settings to vector-valued settings.
result Consistency results for sequential design strategies can be applied 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.
problem The gap between off-policy and on-policy policy gradient methods and conditions to reduce it.
method Theoretical analysis and empirical evidence of conditions to reduce the on-off gap.
result Conditions to reduce the on-off gap between off-policy and on-policy policy gradient methods.
Modeling price dynamics in AMMs with fees using geometric Brownian motion.
problem Understanding price dynamics in AMMs with transaction fees.
method Geometric Brownian motion, local times, excursion theory.
result Derivation of time-changed representation and limiting behavior of AMM prices.
New method infers and samples point processes from latent diffusion.
problem Modeling point processes with latent diffusion.
method Itô's excursion theory for inference and sampling.
result Proposes a new method to infer and sample point processes.
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.
problem Optimizing under unknown constraints with limited failure tolerance.
method Excursion search controls risk as a function of a failure budget.
result The algorithm achieves lower regret and uses failures budget more efficiently.
Generative model calibrates 3D battery cathode morphologies from 2D images.
problem Calibrate 3D morphologies of all-solid-state battery cathodes from 2D microscopy images.
method Combining GANs with excursion sets of Gaussian random fields.
result Calibrated digital twins enable systematic exploration of morphological scenarios.
This paper introduces a new method for uncertainty quantification in prediction models.
problem Quantifying uncertainty in high-stakes applications like medicine and finance.
method Confidence sets for outcome excursions, focusing on identifying subsets of features where outcomes exceed a threshold.
result Theoretical guarantees for the probability that confidence sets contain the true feature subset, both asymptotically and for finite sample sizes.
Study on spin random fields using chaos decomposition for cosmic microwave background modeling.
problem Modeling polarization of Cosmic Microwave Background using spin random fields.
method Explicit Wiener-Itô chaos decomposition of area measures of level sets.
result Reveals a clear difference between high frequency regime and zero spin case.
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.
problem Investigate geometric properties of neural networks.
method Analyze boundary volumes of excursion sets for different activations.
result Hausdorff dimension increases with depth for non-regular activations.
4-manifolds show every flat 3-manifold as cusp sections.
problem Realizing flat 3-manifolds as cusp sections of hyperbolic 4-manifolds.
method Transitive action on cusps, dense flat metrics realization.
result Existence of many cusp-transitive 4-manifolds.
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…
Characterizes holonomies of convex projective cusps.
problem Understanding holonomies in strictly convex projective geometry.
method Complete characterization of holonomies for strictly convex and round cusps, building families of generalized cusps.
result Produces the first example of generalized cusps with non-virtually nilpotent fundamental group.
New method reduces memory usage for Bayesian inverse problems on large grids.
problem Solving large-scale linear inverse problems with Gaussian process priors.
method Implicit representation of posterior covariance matrices, sequential disintegrations of Gaussian measures.
result Significant reduction in uncertainty for high-density regions estimation.
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.
problem Investigating various types of cusps on plane curves.
method Examining criteria for (n,n+1) cusps with differential conditions and relations to evolutes of fronts. result Complete classifications for (4,5)-cusps. The paper characterizes links in 3D from divides with cusps.
problem Characterizing links in 3D from divides with cusps.
method Defines and characterizes divides with cusps and their associated links.
result Every strongly invertible link and 2-periodic link can be described as the link of a divide with cusps.
The paper explores cusp types in hyperbolic 4-manifolds and their commensurability classes.
problem Understanding cusp types in hyperbolic 4-manifolds and their commensurability.
method Criteria for commensurability classes containing specific cusp types.
result Infinitely many examples of commensurability classes without certain cusp types.
The paper classifies cusp types in hyperbolic knot complements.
problem Classifying cusp types in hyperbolic knot complements.
method Analyzing quotients of hyperbolic knot complements.
result All cusp types arise in the quotients of link complements.
New findings on cusped Borel Anosov representations and their properties.
problem Characterizing and understanding cusped Borel Anosov representations.
method Analyzing representations of lattices in PGL2(R) to PGLd(R). result Cusped Borel Anosov representations with specific properties are Hitchin representations.
Cusped hyperbolic 3-manifolds can have up to 4 cusps under certain group actions.
problem Understanding the maximum number of cusps in hyperbolic 3-manifolds under group actions.
method Analyzing the isometry group actions on cusps of hyperbolic 3-manifolds.
result Constructing a family of manifolds with no upper bound on the number of cusps for k=2. Classifies low-volume hyperbolic 3-manifolds with a maximal cusp.
problem Identifying hyperbolic 3-manifolds with minimal volume and maximal cusps.
method Maximal cusp volume classification and analysis of low-volume manifolds.
result Figure-8 knot complement is unique in certain volume and filling categories.
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…
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.
Four hyperbolic 24-cell 4-manifolds with one cusp are identified.
problem Identifying hyperbolic 24-cell 4-manifolds with one cusp.
method Analyzing hyperbolic geometry and isometry.
result Found four one-cusped hyperbolic 4-manifolds of minimum volume.
Study of isometries on hyperbolic 3-manifold cusps.
problem Understanding transitivity in hyperbolic 3-manifold actions.
method Analyzing multiply transitive actions of isometries on cusps.
result Proved a conjecture about the maximum transitivity and upper bounds on cusps.
Computes cusp cobordism groups for Morse functions on manifolds.
problem Understanding the cusp cobordism groups of Morse functions.
method Employed Levine's cusp elimination technique and created pairs of cusps along fold lines.
result Both unoriented and oriented cusp cobordism groups are cyclic of order two in even dimensions and infinite order in odd dimensions.