The paper studies constraint maps with singularities and free boundaries, proving continuity near singularities and optimality.
problem Analyzing the structure of constraint maps with singularities and free boundaries.
method Establish continuity near singularities using a new quantitative unique continuation principle, and investigate the presence of branch points leading to new singularities.
result Topological singularities can only lie in the interior of the contact set in the uniformly convex setting, and the optimality of this result is proven.
Estimates for Schrödinger operators on manifolds with bounded Ricci curvature.
problem Quantifying unique continuation for Schrödinger operators on manifolds with specific curvature conditions.
method Proving quantitative unique continuation estimates for Schrödinger operators on manifolds with Ricci curvature bounded below.
result Upper bound for energy range and constant in terms of Ricci curvature and parameters of relatively dense set.
Paper shows stability of metric reconstruction for orbifolds from spectral data.
problem Determining the metric structure of collapsing orbifolds from spectral data.
method Improved quantitative unique continuation for wave operator on Riemannian manifolds.
result Quantitative stability of inverse problem for Riemannian orbifolds.
Research on unique continuation principles in medical and seismic imaging.
problem Understanding unique continuation principles for inverse problems.
method Integral geometry and fractional calculus methods applied to various imaging problems.
result Developed new techniques for solving inverse problems with partial data.
Stable solution found for manifold topology from boundary data.
problem Determining manifold properties from boundary data and eigenvalues.
method Quantitative stability estimates and unique continuation for the wave operator.
result Eigenvalues and boundary values determine a metric space close to the manifold.
Proves unique continuation for area minimizing currents.
problem Ensuring area minimizing currents match minimal surfaces.
method Analyzes infinite order contact between currents and minimal surfaces.
result Currents and minimal surfaces coincide in a neighborhood.
We discuss the maximum modulus principle, and weak unique continuation, for CR functions on an abstract almost CR manifold M. We investigate these matters under the assumption of weak pseudoconcavity, and obtain sharp results about propagation along Sussmann leaves.
In this paper we extend the results of "A strong minimax property of nondegenerate minimal submanifolds" by White, where it is proved that any smooth, compact submanifold, which is a strictly stable critical point for an elliptic parametric functional, is the unique minimizer in a certain geodesic tubular neighbourhood…
Proves existence and uniqueness of mean curvature flow.
problem Existence and uniqueness of mean curvature flow.
method Heat kernel estimates and contraction mapping principle.
result Continuous dependence of mean curvature flow on initial data.
Unique steady and expanding solitons with spherical links identified.
problem Characterizing steady and expanding Ricci solitons with specific asymptotic symmetries.
method Symmetry principle applied to asymptotically cylindrical and conical GRSs, proving uniqueness for Bryant solitons.
result Bryant steady and expanding solitons are the unique asymptotically cylindrical and conical GRSs with spherical links under certain conditions.
Study on moduli spaces of Seiberg-Witten equations on manifolds with boundary.
problem Analyzing moduli spaces of Seiberg-Witten equations on manifolds with boundary.
method General regularity theorem, strong unique continuation principle, and gluing theorem for Dirac operators; smoothness of restriction map.
result Proves moduli spaces are Hilbert manifolds and have semi-infinite-dimensionality properties.
This paper establishes the existence of a unique nonnegative continuous viscosity solution to the HJB equation associated with a Markovian linear-quadratic control problems with singular terminal state constraint and possibly unbounded cost coefficients. The existence result is based on a novel comparison principle for…
The paper explores unique continuation properties for polyharmonic maps between Riemannian manifolds.
problem Investigating unique continuation principles for polyharmonic maps.
method Analyzing critical points of higher order functionals to prove extensions of known results in harmonic and biharmonic cases.
result Proving extensions of unique continuation principles for k-harmonic maps.
Paper quantifies neural operators' efficiency for solving nonlinear parabolic PDEs.
problem Quantifying the efficiency of neural operators for solving nonlinear parabolic PDEs.
method Deriving approximation rates by transferring PDEs to integral equations and leveraging Picard's iteration.
result Neural operators can efficiently approximate solution operators of nonlinear PDEs without exponential complexity growth.
Quantifies closeness of special Lagrangians under Floer conditions.
problem Estimating closeness of special Lagrangians.
method Floer theoretic conditions leading to quantitative estimates.
result Strong-weak uniqueness theorem for special Lagrangians.
This paper presents quantitative shrinking target results for rotations and interval exchange transformations. To do this a quantitative version of a unique ergodicity criterion of Boshernitzan is established.
Existence and uniqueness of bounded solutions to complex Monge-Ampère flows on Kähler manifolds.
problem Existence and uniqueness of bounded solutions to complex Monge-Ampère flows.
method Proved existence and uniqueness of bounded solutions with specific conditions on the right-hand side.
result Existence and uniqueness of bounded solutions to the complex Monge-Ampère flow on compact Kähler manifolds.
New method proves uniqueness in mean curvature flow.
problem Proving uniqueness in mean curvature flow.
method Arguments from [CM2] to get stronger uniqueness.
result Stronger effective version of uniqueness of blowups.
We prove that any properly oriented C2,1 isometric immersion of a positively curved Riemannian surface M into Euclidean 3-space is uniquely determined, up to a rigid motion, by its values on any curve segment in M. A generalization of this result to nonnegatively curved surfaces is presented as well under suitable…
We study the growth rate of harmonic functions in two aspects: gradient estimate and frequency. We obtain the sharp gradient estimate of positive harmonic function in geodesic ball of complete surface with nonnegative curvature. On complete Riemannian manifolds with non-negative Ricci curvature and maximal volume growt…
The paper proves a comparison principle for complex Monge-Ampère flows and solves a uniqueness problem.
problem Proving uniqueness of weak solutions to the pluripotential Cauchy-Dirichlet problem.
method Proving a comparison principle for the pluripotential complex Monge-Ampère flows.
result Proves the uniqueness of the weak solution to the pluripotential Cauchy-Dirichlet problem.
A hybrid model for Bayesian optimization handles mixed variables using MCTS for categorical and GP for continuous.
problem Optimizing functions with mixed variable types (continuous, integer, categorical).
method Merges MCTS for categorical and GP for continuous variables, integrates UCTS search strategy, and dynamically selects kernels.
result Hybrid models outperform traditional methods in Bayesian optimization.
Paper proves weak unique continuation for harmonic functions on RCD spaces but finds counterexample for strong uniqueness.
problem Unique continuation of harmonic functions on RCD spaces, especially strong uniqueness.
method Establishes weak unique continuation theorem and provides counterexample for strong uniqueness.
result Found counterexample for strong unique continuation in RCD(K,N) spaces for N≥4 and K∈R.
Supervised topic models utilize document's side information for discovering predictive low dimensional representations of documents. Existing models apply the likelihood-based estimation. In this paper, we present a general framework of max-margin supervised topic models for both continuous and categorical response var…
We prove the two theorems of the title, settling two long standing questions in the local theory of singular minimal hypersurfaces. The sharpness of either result is with respect to its hypothesis on the size of the allowable singular sets. The proofs of both theorems rely heavily on the author's recent regularity and …
This paper proves properties of uniformly hyperbolic sets and constructs Markov partitions.
problem Establishing properties of uniformly hyperbolic sets and constructing Markov partitions.
method Backward graph transform, spectral decomposition, shadowing lemma, Markov partitions construction.
result Explicit bounds and Hölder continuity for the coding map.
We give soft, quantitatively optimal extensions of the classical Sphere Theorem, Wilking's connectivity principle and Frankel's Theorem to the context of k-th Ricci curvature. The hypotheses are soft in the sense that they are satisfied on sets of metrics that are open in the C2-topology.
New principles prove precompactness of domains with lower Ricci curvature bound.
problem Proving precompactness of domains with lower Ricci curvature bound.
method Quantitative Hopf-Rinow theorem and doubling property.
result New precompactness principles applicable to incomplete Riemannian manifolds.
This paper bounds the volume of singular and critical sets for elliptic equations with Hölder coefficients.
problem Bounding the volume of singular and critical sets for elliptic equations with Hölder coefficients.
method Proves explicit bounds for (n−2)-dimensional Minkowski estimates of singular and critical sets using Hölder continuity and new almost monotonicity formula. result Optimal improvement on Cheeger-Naber-Valtorta's volume estimates on each quantitative stratum.
This paper is concerned with the determination of credit risk premia of defaultable contingent claims by means of indifference valuation principles. Assuming exponential utility preferences we derive representations of indifference premia of credit risk in terms of solutions of Backward Stochastic Differential Equation…
Study biharmonic hypersurfaces in spheres, proving unique continuation theorem.
problem Characterize biharmonic hypersurfaces in spheres.
method Prove CMC Unique Continuation Theorem for biharmonic hypersurfaces of spheres.
result Supports the conjecture that biharmonic submanifolds of Euclidean spheres must be of constant mean curvature.
Study on heat equation and eigenfunctions on RCD spaces, proving unique continuation.
problem Unique continuation for caloric functions and eigenfunctions on RCD spaces.
method Establish weak unique continuation theorem for caloric functions and eigenfunctions on compact RCD(K,2) spaces.
result Existence of non-trivial eigenfunctions and caloric solutions vanishing up to infinite order at one point.
Unique continuation result for expanding Ricci solitons.
problem Unique continuation of expanding Ricci solitons.
method Optimal relative integral convergence rate, relative entropy.
result Well-defined relative entropy for expanding solitons.
Qlib aims to integrate AI into quantitative investment.
problem Challenges in applying AI to quantitative investment.
method Design and develop Qlib to accommodate AI-driven workflow.
result Qlib realizes the potential of AI technologies in quantitative investment.
New uncertainty principle for Schrödinger equations on hyperbolic manifolds.
problem Uncertainty principle for Schrödinger equations on hyperbolic manifolds.
method General strategy of Escauriaza-Kenig-Ponce-Vega, new Carleman estimates, logarithmic convexity, new mollifier and weight function.
result Similar rigidity phenomenon as in Euclidean space persists in hyperbolic geometry.
Unique continuation property for measures in high dimensions.
problem Understanding the structure of measures in high-dimensional spaces.
method Analyzing locally uniformly distributed measures and their supports.
result Locally uniformly distributed measures satisfy a unique continuation property.
This is the content of the lectures given by the author at the winter school KAWA3 held at the University of Barcelona in 2012 from January 30 to February 3. The main goal was to give an account of viscosity techniques and to apply them to degenerate Complex Monge-Ampère equations following recent works of P. Eyssidieu…
ARTEMIS combines deep learning and symbolic reasoning for financial predictions.
problem Lack of interpretability and economic principles in deep learning models in finance.
method Neuro-symbolic framework combining neural operators, stochastic differential equations, and symbolic distillation.
result ARTEMIS achieves state-of-the-art directional accuracy, outperforming all baselines on synthetic crash regime.
We consider a stochastic optimal control problem in a market model with temporary and permanent price impact, which is related to an expected utility maximization problem under finite fuel constraint. We establish the initial condition fulfilled by the corresponding value function and show its first regularity property…
We prove several unique continuation results for biharmonic maps between Riemannian manifolds.
The sum-product or belief propagation (BP) algorithm is a widely used message-passing technique for computing approximate marginals in graphical models. We introduce a new technique, called stochastic orthogonal series message-passing (SOSMP), for computing the BP fixed point in models with continuous random variables.…
Unique solutions found for Plateau problems in smooth and continuous calibrations.
problem Finding unique solutions to the Plateau problem for specific types of currents.
method Boundary regularity theory for area-minimizing currents and unique continuation argument.
result Every compactly supported smoothly or continuously calibrated integral current is the unique solution to the Plateau problem for its boundary data.
We investigate a class of quadratic-exponential growth BSDEs with jumps. The quadratic structure introduced by Barrieu & El Karoui (2013) yields the universal bounds on the possible solutions. With local Lipschitz continuity and the so-called A_gamma-condition for the comparison principle to hold, we prove the existenc…
Study shows how optimal transport behaves in higher dimensions.
problem Characterizing optimal transport in higher dimensions with Euclidean distance.
method Investigates the small regularization limit of entropic optimal transport.
result The limiting transport plan is supported on transport rays and uniquely minimizes a relative entropy functional.
Minimal surfaces reflect across spheres, proving annulus uniqueness.
problem Uniqueness of free boundary minimal annuli in balls.
method Reflection principle applied to minimal surfaces meeting spheres at 90 degrees.
result Every embedded free boundary minimal annulus in a ball is the critical catenoid.
The paper proves the law of one price in a continuous-time setting without friction.
problem Identifying conditions under which the law of one price holds in a continuous-time setting without frictions.
method Formulating a new mechanism for LOP failure and proving a novel variant of the uniform boundedness principle.
result Establishes the equivalence of the economic concept of LOP with the probabilistic property of the existence of a local $\scr{E}$-martingale state price density.
Curvature measures uniquely determined by invariance under embeddings.
problem Characterizing curvature measures uniquely.
method Applied Weyl principle and Künneth-type formula.
result Curvature measures uniquely characterized by invariance under isometric embeddings.
AlphaForgeBench evaluates LLMs as quantitative researchers, not trading agents, to address instability in financial decision-making.
problem Behavioral instability of LLMs in sequential decision-making under financial uncertainty.
method Proposes AlphaForgeBench, a framework that requires LLMs to generate executable alpha factors and compose factor-based trading strategies.
result Eliminates execution-induced instability and provides a rigorous benchmark for evaluating financial reasoning.