Transforms curves and surfaces for efficient geometric analysis.
problem Efficiently analyzing and comparing curves and surfaces.
method Square root velocity transformation for curves and intrinsic comparison for surfaces.
result Fundamental geometric properties of curves under the transformation.
Solves optimal liquidation problem for stock price following geometric Brownian motion.
problem Optimal liquidation problem for stock price process following geometric Brownian motion.
method Functional analysis tools; working in terms of cash.
result Explicit solution to the problem, extending to stochastic drift.
Survey on Finsler manifolds with weighted Ricci curvature, focusing on geometric analysis.
problem Analysis of Finsler manifolds with weighted Ricci curvature.
method Nonlinear geometric analysis based on the Bochner inequality.
result Gradient estimates, functional inequalities, and isoperimetric inequalities.
Geometric analysis improves convergence of variational inference.
problem Challenges in analyzing convergence of variational inference due to non-convexity and non-smoothness.
method Exploits exponential family structure and Bregman divergences to geometrically analyze the optimization landscape.
result Establishes non-asymptotic convergence rates for gradient descent algorithms.
Survey on manifold ends with new heat kernel estimates.
problem Analyzing geometric properties on manifolds with ends.
method Constructing manifolds with ends and analyzing their heat kernel estimates.
result Found manifolds with ends that have different heat kernel estimates.
Survey article on solving geometric problems with calculus and harmonic analysis.
problem Solving geometric problems using functional calculus and harmonic analysis.
method Combining methods from functional calculus and real-variable harmonic analysis.
result Recent geometric problems have been resolved by these methods.
Poincare function counts geometric moduli, solving Arnold's conjecture.
problem Counting moduli in differential-geometric problems.
method Derives Poincare function properties and solves Arnold's conjecture.
result Derives new formulae for differential invariants and classification problems.
Paper analyzes IRL problem and provides sample complexity analysis.
problem Finding a reward function for a given optimal policy.
method Geometric analysis and L1-regularized SVM formulation.
result Sample complexity of O(n^2 log(nk)) for recovering reward function.
The paper connects harmonic mappings to Ricci flow using geometric analysis.
problem Global geometry of harmonic mappings and Ricci solutions.
method Geometric analysis, focusing on subharmonic functions.
result Results on connections between manifold geometry and subharmonic functions.
Harmonic gauge simplifies geometric analysis of Riemannian metrics.
problem Analyzing the Hilbert-Einstein functional and its stability.
method Developed a harmonic gauge to eliminate divergence terms and induce elliptic structure.
result Positivity of curvature operator implies spectral stability of the functional.
The paper contrasts different convergence notions in geometric analysis.
problem Exploring discrepancies between various convergence concepts in geometric analysis.
method Examples and proof of a theorem requiring specific bounds on warping functions.
result Warped product manifolds can have different convergence limits even if the warping functions converge in L p L^p L p . New Morse theory for shapes at distances.
problem Understanding shapes at distances from a reference point.
method Defining Morse functions and using non-smooth analysis, geometric measure theory.
result Homotopy type changes at critical values, with one cell added per critical point.
New cyclicity measures defined in weighted Besov spaces, with stability and geometric analysis.
problem Characterizing cyclicity in weighted Besov spaces.
method Defining cyclicity indices based on potential theory and capacity, studying stability under perturbations, and linking zero set structure to cyclicity.
result Novel invariants and conditions for cyclicity in various function spaces.
Geometric theory developed for ultradifferentiable functions and their wavefront sets.
problem Developing a geometric theory for ultradifferentiable functions.
method Using Dyn'kin's Theorem and Bony's Theorem, the ultradifferentiable wavefront set is defined and its properties are proven.
result Microlocal elliptic regularity theorem for ultradifferentiable vector bundles is proven.
Geometric analysis of nonlinear dynamics applied to financial time series.
problem Understanding dynamic properties of financial time series.
method Nonparametric filtering method to estimate vector fields and their derivatives from nonlinear oscillation models.
result Vector fields and their derivatives provide insights into the dynamic properties of financial time series.
The chapter explores globally hyperbolic spacetimes using topology and functional analysis.
problem Understanding global hyperbolicity in spacetimes.
method Foundational tools from order theory and topology, geometric analysis, and connections to physics.
result A connection between global hyperbolicity and geodesic completeness of space-like surfaces.
Study of maximal hypersurfaces in spacetimes, proving uniqueness results.
problem Uniqueness of compact maximal hypersurfaces in stably causal spacetimes.
method Analysis of a distinguished function on maximal hypersurfaces under first order conditions of the spacetime.
result Several uniqueness results on compact maximal hypersurfaces in stably causal spacetimes.
In this expository article, we discuss various monotonicity formulas for parabolic and elliptic operators and explain how the analysis of the function spaces and the geometry of the underlining spaces are intertwined. After briefly discussing some of the well-known analytical applications of monotonicity for parabolic …
Local regularization improves geometric estimates from noisy data.
problem Improving geometric understanding from noisy, perturbed data.
method Local regularization of noisy point clouds to define similarity.
result Locally regularized similarity leads to better geometric recovery.
Enhanced tree-based classifiers use derivatives and geometry for better function classification.
problem Improving classification of high-dimensional time series data.
method Integrates Functional Data Analysis with tree-based ensemble techniques, leveraging derivative and geometric features.
result Significant improvements over traditional approaches in function classification.
This article is based upon lectures given at the 2013 IAS/Park City Mathematics Institute summer program in geometric analysis.
New geometric proofs and interpretations of scattering diagrams and theta functions.
problem Analyzing the asymptotic behavior of Maurer-Cartan elements for differential graded Lie algebras.
method Asymptotic analytic approach and differential geometric proofs.
result Alternative proofs of consistent completion of scattering diagrams and geometric interpretations of theta functions.
An algorithm for clustering data from group-invariant subspaces.
problem Clustering data from a union of group-invariant subspaces.
method Sparse Sub-module Clustering (SSmC) based on group-sparse self-representation.
result General conditions for identifying group-invariant subspaces.
Geometric tempering fails for Langevin dynamics, proving convergence limits.
problem Proving convergence and limitations of geometric tempering for Langevin dynamics.
method Theoretical investigation of geometric tempering using Langevin dynamics.
result Geometric tempering can lead to exponential time convergence and poor functional inequalities.
Study n n n -superharmonic functions and their geometric applications.
problem Asymptotic behavior of n n n -superharmonic functions at isolated singularities. method Using Wolff potential, n n n -capacity estimates, and Adams-Moser-Trudinger inequality. result Strong n n n -capacity lower bound estimate for geometric applications. Unified treatment of stability problems in geometry and analysis.
problem Spherical closeness of hypersurfaces under geometric constraints.
method Estimate relating distance to geodesic spheres with norms of traceless Hessian operator.
result Unified treatment of stability problems in geometry and analysis.
Develops local elliptic regularity for geometrically-natural operators with low regularity coefficients.
problem Local elliptic regularity for operators with low regularity coefficients in Sobolev-type spaces.
method Rescaling estimates and multiplication results for function spaces.
result Unified set of interior estimates and regularity inference for operators with Sobolev-type coefficients.
Proof shows cones minimize certain geometric functionals.
problem Minimizing cones over spheres in geometric functionals.
method Proof by foliation analysis of cone leaves.
result Cone minimizes functionals for S k i m e s S l \mathbb{S}^k imes \mathbb{S}^l S k im es S l . This paper explores geometric and analytic aspects of Lojasiewicz inequalities on vector bundles.
problem Understanding growth and stability conditions for real-analytic functions over vector bundles.
method Outline theory of functionals and variational problems over vector bundles, explore applications to real-analytic functionals.
result Describes the energy functional on $S^{n-1$ as a functional over a vector bundle.
Combining the tools of geometric analysis with properties of Jordan angles and angle space distributions, we derive a spherical and a Euclidean Bernstein theorem for minimal submanifolds of arbitrary dimension and codimension, under the condition that the Gauss image is contained in some geometrically defined closed re…
Geometric study of linear neural networks identifies pure and spurious critical points.
problem Understanding the landscape of loss functions in linear neural networks.
method Geometric properties of functional spaces and parameterization analysis.
result Different phenomena cause the absence of bad local minima in linear networks, depending on the architecture and loss function.
The paper applies math and physics to language models, introducing entropy and geometric concepts.
problem Understanding and improving language models to approximate intelligent language.
method Formal definitions, functional analysis, topology, thermodynamics, and set theory.
result Entropy function reveals key obstacles for LLMs and offers insights into language models.
Geometric wave propagator defined on Riemannian manifolds.
problem Wave equation on Riemannian manifolds.
method Geometric approach to constructing propagator as oscillatory integral.
result Explicit small time asymptotic expansion of subprincipal symbol.
Combines topological and geometric approaches to data analysis.
problem Understanding when and how geometric objects intersect.
method Connects topological and geometric concepts of curvature.
result Reconceptualizes curvature and links it to hyperconvexity.
Geometric structures on quaternionic unit ball for slice regular Möbius transformations.
problem No new problem introduced.
method Introducing Hermitian, Riemannian, and Kähler-like structures on quaternionic unit ball using regular Möbius transformations.
result Geometric structures are natural generalizations of complex setup and solve problems not achieved by other geometries.
GGA improves untrustworthy prediction detection in neural networks without retraining.
problem Susceptibility of neural networks to untrustworthy predictions, especially adversarial attacks and out-of-distribution data.
method Geometric Gradient Analysis (GGA) analyzes the geometry of neural network loss landscapes based on saliency maps.
result GGA outperforms existing methods in detecting untrustworthy predictions, including adversarial and out-of-distribution data.
Novel approach to financial derivatives pricing using rough path theory.
problem No-arbitrage conditions in financial markets necessitating precise integration methods.
method Developed a polynomial-based approximation class for rough path functionals, extending to non-geometric rough paths.
result Motivated a hypothesis for payoff functionals in financial markets, facilitating analysis.
New method for high-fidelity shape representations from raw data.
problem Creating accurate shape representations from raw data.
method A simple loss function encouraging neural network to vanish on input point cloud and have unit norm gradient.
result Our method produces high-fidelity, smooth, and natural zero level set surfaces.
Study of geometric analysis on asymmetric metric spaces, including heat flow and Sobolev spaces.
problem Analysis of geometric properties on asymmetric metric measure spaces.
method Introduction of upper gradients, q q q -Laplacian, and q q q -heat flow in asymmetric settings. result Extension of concepts from symmetric to asymmetric metric measure spaces.
The nonlinear sigma model with gravitino exhibits symmetries and conservation laws.
problem Exploring symmetries and conservation laws in a nonlinear sigma model with gravitino.
method Geometric analysis of rescaled conformal transformations, super Weyl transformations, and diffeomorphisms.
result The model possesses degenerate super symmetry leading to geometric interpretations of energy-momentum tensor and supercurrent.
Motivated by the supersymmetric extension of Liouville theory in the recent physics literature, we couple the standard Liouville functional with a spinor field term. The resulting functional is conformally invariant. We study geometric and analytic aspects of the resulting Euler-Lagrange equations, culminating in a blo…
A new geometric framework embeds correlation matrices into Euclidean space for scalable brain network analysis.
problem Inefficient and unstable analysis of functional brain networks in high-dimensional contexts.
method Diffeomorphic transformations to embed correlation matrices into Euclidean space, preserving manifold properties.
result Improved computational speed and enhanced accuracy compared to conventional manifold-based approaches.
Stability of biharmonic maps in critical dimension proven.
problem Stability of biharmonic maps between manifolds in critical dimension.
method Generalization of Morse stability theory to biharmonic maps, development of strong energy quantization method.
result Strong energy quantization in a wide class of problems in geometric analysis.
SRCA reduces high-dimensional data to lower dimensions while preserving geometric structures.
problem High-dimensional datasets with underlying geometric structures.
method Spherical Rotation Component Analysis (SRCA) incorporating geometric loss functions.
result SRCA provides a low-rank spherical representation of data with general theoretic guarantees.
GN algorithm solves batched bandit for nondegenerate functions near-optimally.
problem Batched bandit learning for nondegenerate functions.
method Introduces Geometric Narrowing (GN) algorithm with a O ~ ( A + d T ) \widetilde{\mathcal{O}} ( A_{+}^d \sqrt{T} ) O ( A + d T ) regret bound and O ( log log T ) \mathcal{O} (\log \log T) O ( log log T ) batches. result GN achieves near optimal regret with minimal number of batches.
Magnitude of manifolds linked to Riesz energies and beta functions.
problem Magnitude invariant and its geometric significance.
method Relating magnitude invariant to Brylinski's beta function and pseudodifferential analysis.
result Precise relation between magnitude invariant and beta function for closed manifolds.
Corrected a false lemma in Cimasoni's work on linking theory.
problem A false lemma in Cimasoni's geometric construction of the Conway potential function.
method Presented counterexamples and a detailed proof of the corrected lemma.
result The lemma is false and its correction has significant consequences for subsequent works.
Introduces q-paths for generalizing geometric annealing paths in machine learning.
problem Limited applicability of existing path methods in machine learning.
method Develops a family of paths derived from a generalized mean, including geometric and arithmetic mixtures.
result Empirical gains in Bayesian inference and generative model evaluation.