The abstract discusses how sub-Riemannian manifolds can have branching geodesics.
problem The existence of branching geodesics in sub-Riemannian geometry.
method Analyzing the rank discontinuity of normal geodesics and constructing specific examples.
result Sub-Riemannian manifolds can contain branching normal minimizing geodesics.
Paper finds surface groups can deform in reductive symmetric spaces.
problem Finding deformations of discontinuous groups in reductive symmetric spaces.
method Analyzing Zariski dense surface subgroups and their deformations.
result Surface groups of high genus can deform in reductive symmetric spaces.
We consider the action of Anosov subgroups of a semi-simple Lie group on the associated flag manifolds. A systematic approach to construct cocompact domains of discontinuity for this action was given by Kapovich, Leeb and Porti in arXiv:1306.3837. For Δ-Anosov representations, we prove that every cocompact domain of …
The paper studies proper discontinuity of actions on Weyl chamber flow spaces.
problem Properly discontinuous actions on Weyl chamber flow spaces for transverse subgroups.
method Analyzes limit sets and quotient spaces, introduces growth indicators and conformal measures.
result Establishes ergodic dichotomy for Weyl chamber flow and introduces new measures.
The paper classifies fiber structures of discontinuity domains for Anosov representations.
problem Understanding the topology of discontinuity domains for Anosov representations.
method Explicitly working out a smooth version of Fintushel's classification theorem for S1-actions on 4-manifolds. result The action on the fiber is equivalent to a circle action on a Hirzebruch surface.
Every lattice H in a connected semi-simple Lie group G acts properly discontinuously by isometries on the contractible manifold G/K (K a maximal compact subgroup of G). We prove that if H acts on a contractible manifold W and if either 1) the action is properly discontinuous, or 2) W is equipped with a complete Riemann…
Study nearly geodesic immersions in symmetric spaces, leading to explicit fibrations.
problem Understanding geodesic immersions in higher rank symmetric spaces.
method Define and analyze nearly geodesic immersions with a bound on fundamental form.
result Explicit fibrations of domains of discontinuity for certain representations.
Margulis space-times with parabolic holonomy elements are stable under sufficiently small deformations.
problem Stability of Margulis space-times with parabolic holonomy elements
method Combining compactification and partial generalization of earlier work
result Openness result on the number of conjugacy classes of parabolic elements under deformation
Study consumption-investment problem in markets with rank-based returns.
problem Consumption-investment problem in markets with rank-based returns.
method Derives an HJB equation with Neumann boundary conditions for the value function and proves a corresponding verification theorem.
result Explicit solutions for unconstrained, open market constraints, and fully invested cases.
The Tits alternative applies to groups acting on specific CAT(0) spaces.
problem Understanding the structure of groups acting on CAT(0) spaces.
method Proving the Tits alternative for groups acting on visibility CAT(0) spaces with bounded packing property.
result Groups either almost nilpotent or contain a free nonabelian subgroup of rank 2.
New findings on stability and Q-conditions for free group actions in hyperbolic spaces.
problem Understanding stability and Q-conditions for free group actions in hyperbolic spaces.
method Generalization of Minsky's and Bowditch's results to higher dimensions and W_3-extensible representations.
result Equivalence between primitive stability and generalized Q-conditions for F_2 in hyperbolic d-space (d >= 3).
We study the topology and geometry of compact complex manifolds associated to Anosov representations of surface groups and other hyperbolic groups in a complex semisimple Lie group G. These manifolds are obtained as quotients of the domains of discontinuity in generalized flag varieties G/P constructed by Kapovich-…
This survey is based on a series of lectures that we gave at MSRI in Spring 2015 and on a series of papers, mostly written jointly with Joan Porti. Our goal here is to: 1. Describe a class of discrete subgroups Γ<G of higher rank semisimple Lie groups, which exhibit some "rank 1 behavior". 2. Give different character…
Proves nonemptyness of domains for specific group actions.
problem Nonemptyness of domains of proper discontinuity for Anosov groups of affine Lorentzian transformations.
method Proof of nonemptyness of domains of proper discontinuity.
result Proves nonemptyness of domains for Anosov groups of affine Lorentzian transformations.
Study of SU(2,1) character varieties on one-holed torus.
problem Characterize representations of mapping class group on SU(2,1) character variety.
method Explicit description of SU(2,1) character variety, use of Farey graph adaptation, and mapping class group action analysis.
result Description of an open domain of discontinuity for mapping class group action.
The paper defines infinite Schottky groups and their applications to infinite type surfaces.
problem Understanding group actions on infinite type surfaces.
method Definition and analysis of infinite Schottky groups and their properties.
result Every infinite type Riemann surface can be obtained as a quotient of a region of discontinuity of an infinite Schottky group.
We consider the deformation of a discontinuous group acting on the Euclidean space by affine transformations. A distinguished feature here is that even a `small' deformation of a discrete subgroup may destroy proper discontinuity of its action. In order to understand the local structure of the deformation space of disc…
Cut-DeepONet handles discontinuities and sharp transitions in neural operators.
problem Neural operators struggle with discontinuities and sharp transitions in PDEs.
method Two-stage training framework that explicitly models discontinuities via a lifting strategy and input-dependent discontinuity prediction.
result Cut-DeepONet outperforms state-of-the-art methods on benchmark PDEs with low-resolution datasets.
Study proves existence of equilibrium in incomplete economies with discontinuous volatility.
problem Existence of incomplete Radner equilibrium with nondegenerate endogenous volatility.
method Established existence of solution for Markovian quadratic BSDEs with discontinuous generators using unique continuation and backward uniqueness.
result Existence of incomplete Radner equilibrium with nondegenerate endogenous volatility.
New domains of discontinuity found for Anosov representations.
problem Understanding Anosov representations acting on homogeneous spaces.
method Constructing open domains of discontinuity for Anosov representations acting on specific homogeneous spaces.
result Describes the largest possible open domains of discontinuity for Zariski dense Anosov representations.
As early as 1972, Penrose - in a purely formal way - introduced a "discontinuous coordinate transformation", which relates a continuous representation of the metric of impulsive pp-waves to a discontinuous one. On the basis of the invertibility concept for generalized functions developed recently by the first author, w…
Recovering a large matrix from limited measurements is a challenging task arising in many real applications, such as image inpainting, compressive sensing and medical imaging, and this kind of problems are mostly formulated as low-rank matrix approximation problems. Due to the rank operator being non-convex and discont…
Study the discontinuity of functions not embeddable in Euclidean space.
problem Understanding discontinuity of non-embeddable functions.
method Define a modulus of discontinuity and establish lower bounds.
result Quantified nonembeddability results and topological Tverberg theorem.
Bayesian nonparametric discontinuity design improves causal inference without randomization.
problem Causal inference in non-randomized settings with additional assumptions.
method Bayesian model comparison and Gaussian process regression.
result BNDD can detect discontinuities of any order and spectral features.
Paper tackles low-rank matrix recovery with column ℓ2,0-norm regularization.
problem Low-rank matrix recovery problems with column sparsity constraints.
method Developed alternating majorization-minimization (AMM) methods with extrapolation and hybrid AMM.
result Global convergence analysis and superior performance in matrix completion problems.
Study new symmetries in non-symmetric spaces and discontinuous groups.
problem Analyze symmetries in non-symmetric homogeneous spaces and discontinuous groups.
method Investigate discrete series, discontinuous groups, and analysis on pseudo-Riemannian spaces.
result New insights into symmetries of non-symmetric homogeneous spaces and discontinuous groups.
For the action of the outer automorphism group of the rank two free group on the corresponding variety of PSL(2,C) characters, two domains of discontinuity have been known to exist that are strictly larger than the set of Schottky characters. One is introduced by Bowditch in 1998 (followed by Tan, Wong and Zhang in 200…
We study the Weyl chamber length boundary both of the Hitchin and of the maximal character varieties and determine therein an open set of discontinuity for the action of the mapping class group. This result is obtained as consequence of a canonical decomposition of a geodesic current on a surface of finite type arising…
A new method for unsupervised disentanglement using axis-aligned cliffs.
problem Unsupervised disentanglement of latent factors under nonlinear maps.
method Encouraging axis-aligned discontinuities (cliffs) in the estimated density of factors.
result Cliff method outperforms baselines on disentanglement benchmarks.
New method estimates active subspaces for jump-discontinuous functions.
problem Estimating active subspaces for discontinuous functions like ABMs.
method Extending active subspaces to discontinuous functions, using Gaussian process.
result Identifies important parameters in ABM simulations of refugee movement.
This article gives an up-to-date account of the theory of discrete group actions on non-Riemannian homogeneous spaces. As an introduction of the motifs of this article, we begin by reviewing the current knowledge of possible global forms of pseudo-Riemannian manifolds with constant curvatures, and discuss what kind of …
The first known example of a complete Riemannian manifold whose isoperimetric profile is discontinuous is given.
Paper analyzes error in stochastic approximation for discontinuous functions.
problem Estimating expected error in discontinuous stochastic approximation.
method Uses finite differences and O(n−1/5) error estimate for discontinuous functions. result Achieves error estimate of O(n−1/5) for discontinuous stochastic representation. Bayesian emulator tackles models with discontinuities efficiently.
problem Uncertainty quantification of models with multiple discontinuities.
method TENSE framework, designed correlation structures, single emulator object.
result Efficient emulation of complex models with multiple discontinuities.
We characterise completely when limit sets, as parametrised by Cannon-Thurston maps, move discontinuously for a sequence of algebraically convergent quasi-Fuchsian groups.
We show that there is a complete connected 2-dimensional Riemannian manifold with discontinuous isoperimetric profile, answering a question of Nardulli and Pansu.
Volunteer labor can temporarily yield lower benefits to charities than its costs. In such instances, organizations may wish to defer volunteer donations to a later date. Exploiting a discontinuity in blood donations' eligibility criteria, we show that deferring donors reduces their future volunteerism. In our setting, …
We develop a general term structure framework taking stochastic discontinuities explicitly into account. Stochastic discontinuities are a key feature in interest rate markets, as for example the jumps of the term structures in correspondence to monetary policy meetings of the ECB show. We provide a general analysis of …
Study proves rigid spectral properties of planets with metric discontinuities.
problem Establishing spectral rigidity for spherically symmetric planets with discontinuities.
method Novel trace formula applied to two wave types in spherically symmetric manifolds with boundary and interior interfaces.
result Spectral rigidity of spherically symmetric planets with discontinuities is proven.
In this paper, we present a method for the accurate estimation of the derivative (aka.~sensitivity) of expectations of functions involving an indicator function by combining a stochastic algorithmic differentiation and a regression. The method is an improvement of the approach presented in [Risk Magazine April 2018]. T…
We study the properly discontinuous and isometric actions on the unit sphere of infinite dimensional Hilbert spaces and we get some new examples of Hilbert manifold with costant positive sectional curvature. We prove some necessary conditions for a group to act isometrically and properly discontinuously and in the case…
This work models overnight rates with jumps and discontinuities, extending classical short-rate models.
problem Capturing the jump behavior and discontinuities in overnight rates for accurate modeling.
method Developed a term structure modeling framework based on overnight rates, accommodating stochastic discontinuities.
result Simple specifications can capture the jump behavior of overnight rates, and explicit valuation formulas are provided.
Geometric methods for surface group representations in higher rank SL(2m+1,R).
problem Representations of surface groups in higher rank SL(2m+1,R).
method Para-complex and pseudo-Riemannian geometric techniques.
result One-to-one correspondence between Higgs bundles and isotropic P-alternating surfaces.
The study explores deformations of discrete subgroups in non-compact homogeneous spaces.
problem Addressing the proper discontinuity of discrete subgroups in non-compact homogeneous spaces.
method Classification results for deformations of standard discontinuous groups in pseudo-Riemannian homogeneous spaces.
result Conditions for local rigidity and Zariski-dense deformations in standard quotients.
Paper analyzes SGHMC for non-convex optimization with discontinuous gradients.
problem Training neural networks with ReLU activation.
method Non-asymptotic convergence analysis of SGHMC with discontinuous gradients.
result Explicit upper bounds for expected excess risk in non-convex optimization.
This paper tackles discontinuous neural networks for better approximation of piecewise continuous functions.
problem Limitation of neural networks in approximating piecewise continuous functions due to discontinuities.
method Proposes a decoupled two-step procedure to train a discontinuous deep neural network model.
result Provides approximation guarantees for the proposed model in piecewise continuous function spaces.
In the present paper, we prove that no infinite group acts isometrically, effectively, and properly discontinuously on a certain class of Lorentzian manifolds that are not necessarily homogeneous.
Framework uses deep learning and statistical models to solve PDEs with discontinuous coefficients.
problem Solving PDEs with discontinuous coefficients.
method Two-stage physics-informed deep learning and statistical mixture models.
result Framework achieves adaptability and accurate parameter identification.