Sharp stability result for maps near infinitely concentrated minimisers.
problem Stability of maps near minimisers with infinite concentration.
method Dynamic approach to deform maps into harmonic maps, controlling topology changes.
result Sharp quantitative estimates on map distance to infinitely concentrated minimisers.
New results on hypersurfaces show no branch points, improving smoothness.
problem Analyzing area minimising hypersurfaces mod p without branch points.
method General analysis of immersed stable minimal hypersurfaces with alternating orientation.
result Area minimising hypersurfaces mod p do not admit immersed branch points.
Harmonic maps stability under small perturbations of boundary data.
problem Stability of minimising harmonic maps under small perturbations of boundary data.
method Analysis of W1,p perturbations of boundary data and energy minimisers. result Energy minimisers close to the original map in Hölder norm.
We study the problem of finding strain-minimising stream surfaces in a divergence-free vector field. These surfaces are generated by motions of seed curves that propagate through the field in a strain minimising manner, i.e., they move without stretching or shrinking, preserving the length of their arbitrary arc. In ge…
New frame method simplifies solving variational problems with Euclidean symmetry.
problem Solving variational problems with Euclidean symmetry.
method Rotation Minimising frame and symbolic invariant calculus.
result Noether's conservation laws and Euler-Lagrange equations derived directly.
Loss minimisation fails to capture epistemic uncertainty in second-order predictors.
problem Capturing epistemic uncertainty in machine learning models.
method Analysis of a second-order learner approach using loss minimisation.
result Loss minimisation does not faithfully represent epistemic uncertainty in second-order predictors.
Study characterizes hulls and capacities on Riemannian manifolds, proving isoperimetric inequalities.
problem Characterizing hulls and capacities on Riemannian manifolds.
method Investigates strictly outward minimising hulls and uses p-capacities to recover their areas.
result Sharp isoperimetric inequality on complete noncompact manifolds with nonnegative Ricci curvature.
Proves existence and regularity of spherical minimizers for lipid membrane energy.
problem Existence and regularity of minimizers for the Canham-Helfrich energy.
method Establishes lower semicontinuity and proves existence through weak convergence of immersions.
result Proves existence and regularity of minimizers for the Canham-Helfrich energy on spheres.
Characterizes harmonic morphisms preserving minimal submanifolds and finds novel area-minimising hypercones.
problem Understanding harmonic morphisms and their relationship to minimal submanifolds.
method Characterization of harmonic morphisms as weakly horizontally conformal maps preserving minimal submanifold equations, derivation of reduction properties for other co-dimensions, application to find novel area-minimising hypercones.
result Novel family of degree 4 area-minimising hypercones in R^m, m≥32.
Proves convexity of minimizers in energy functions with convex potentials.
problem Connectedness and convexity of minimizers in energy functions involving surface tensions and convex potentials.
method Introduces a 'two-point function' to measure lack of convexity and prove negative second variation of the energy.
result Positively answers an old question of Almgren about connectedness and convexity of minimizers.
We study the Calabi functional on a ruled surface over a genus two curve. For polarisations which do not admit an extremal metric we describe the behaviour of a minimising sequence splitting the manifold into pieces. We also show that the Calabi flow starting from a metric with suitable symmetry gives such a minimising…
Proves strict inequality for minimizers of Willmore energy under isoperimetric constraints.
problem Minimizing the Willmore energy under isoperimetric constraints.
method Connected sum approach, building on previous work by Keller-Mondino-Rivière.
result Existence of minimizers for the isoperimetric constrained Willmore problem in every genus.
In this paper, we formulate a method for minimising the expectation value of the procurement cost of electricity in two popular spot markets: {\it day-ahead} and {\it intra-day}, under the assumption that expectation value of unit prices and the distributions of prediction errors for the electricity demand traded in tw…
The Duffing oscillator's parameters are identified online using variational message passing.
problem Estimating parameters of a nonlinear Duffing oscillator in real-time.
method Variational message passing on a factor graph of the Duffing oscillator's generative model.
result The online inference procedure performs as well as offline methods.
The paper analyzes greedy algorithms for MMD minimization, showing their efficiency and approximation error.
problem Minimizing Maximum Mean Discrepancy (MMD) for probability measure quantization.
method Iterative algorithms including kernel herding, greedy MMD minimization, and Sequential Bayesian Quadrature (SBQ).
result The greedy algorithms have a lower approximation error than SBQ, but are significantly faster.
Paper finds optimal shapes for minimizing average lengths of billiard trajectories in specific polygons.
problem Finding optimal shapes to minimize the average length of billiard trajectories.
method Used techniques from Teichmüller theory.
result Optimal shapes minimize average lengths of billiard trajectories in specific polygons.
A neural flow method minimizes Willmore energy for 2-surfaces in 3D space.
problem Minimizing Willmore energy for closed oriented 2-surfaces in 3D space.
method Introducing neural Willmore flow to model and minimize the Willmore energy using neural architectures.
result The neural flow reproduces expected round sphere and Clifford torus for genus 0 and 1 surfaces, respectively, and finds minimal Willmore surfaces for genus 2.
Choose two points in the tangent bundle of the Euclidean plane (x,X),(y,Y)∈TR2. In this work we characterise the immersed length minimising paths with a prescribed bound on the curvature starting at x, tangent to X; finishing at y, tangent to Y, in each connected component of the space of paths…
Surveying mass in 2D hyperbolic geometry, overcoming challenges via minimisation.
problem Defining mass in 2D hyperbolic geometry.
method Minimisation using positive energy theorem and gluing theorems.
result Construction of novel initial data sets with controlled mass.
Study proves properties of constant mean curvature hypersurfaces in high-dimensional spaces.
problem Properties of constant mean curvature hypersurfaces in high-dimensional spaces.
method Proves properties of constant mean curvature hypersurfaces using min-max procedure and surgery.
result Every tangent cone at each isolated singularity is area-minimising.
A method improves Cryo-EM 3D map refinement by regularizing rotation estimation.
problem Noise-robustness vs. data-consistency in Cryo-EM 3D map reconstruction.
method Ellipsoidal support lifting (ESL) for regularizing and approximating the global minimizer over Riemannian manifolds.
result The induced bias due to regularizing effect of ESL estimates better rotations than global optimisation.
In this paper we formulate in general terms an approach to prove strong consistency of the Empirical Risk Minimisation inductive principle applied to the prototype or distance based clustering. This approach was motivated by the Divisive Information-Theoretic Feature Clustering model in probabilistic space with Kullbac…
Study minimizes crossing points of up to 12 curves on a genus 2 surface.
problem Minimizing intersection points of curves on a surface.
method Analyzes systems of up to 12 simple closed curves on a genus 2 surface to find the minimum crossing number.
result Determines the minimal crossing number of up to 12 curves on a genus 2 surface and proves the minimization systems are unique.
The study finds solutions to a curvature minimisation problem in fixed-length curves.
problem Minimizing the L∞-norm of curvature among curves of fixed length. method Characterized solutions by a system of differential equations and classified the structure of solutions.
result Characterized solutions to the L∞-norm of curvature problem. Study on earthquake metric on Teichmüller space, proving properties and new completions.
problem Understanding the earthquake metric on Teichmüller space.
method Proofs of properties, new completions, and interpretation of the metric.
result Coincidence of various completions for the earthquake metric.
For a bounded smooth domain in the plane and smooth boundary data we consider the minimisation of the Willmore functional for graphs subject to Dirichlet or Navier boundary conditions. For H2-regular graphs we show that bounds for the Willmore energy imply area and diameter bounds. We then consider the L1-lower s…
New method finds points for approximating distributions faster.
problem Approximating target probability distributions using finite points.
method Stationary MMD points computed via MMD gradient flows.
result Stationary MMD points converge faster than global minimizers.
PINNs solve differential geometry problems in complex shapes.
problem Solving differential geometry problems in complex shapes.
method Training neural networks with loss functions inspired by differential conditions.
result PINNs are effective for differential geometry problems.
This paper introduces a new method to train normalizing flows using precision-recall divergences.
problem Training generative models with mode dropping and low-quality samples.
method Introduces PR-divergences and proposes a novel generative model to minimize precision-recall trade-offs.
result Normalizing flows can be trained to achieve specific precision-recall trade-offs using PR-divergences.
Scheme minimizes p-elastic energy of curves over time.
problem Minimizing p-elastic energy of curves over time. method Minimizing movement scheme with approximate normal graphs.
result Short-time existence and lower bound on solution's lifetime.
CRUMB: Efficient Prior Fitted Network Inference via Distributionally Matched Context Batching
problem Inference of tabular foundation models with large training datasets
method CRUMB (Clustered Retrieval Using Minimised-MMD Batching)
result CRUMB outperforms state-of-the-art context selection strategies on the TabArena benchmark
On the one hand, we prove that the Clifford torus in C2 is unstable for Lagrangian mean curvature flow under arbitrarily small Hamiltonian perturbations, even though it is Hamiltonian F-stable and locally area minimising under Hamiltonian variations. On the other hand, we show that the Clifford torus is r…
Develops strategies to minimize trading costs in volatile markets.
problem Minimizing trading costs in volatile markets with uncertain asset price paths.
method Constructs dynamic, pathwise optimal trade execution strategies using random Young differential equations.
result Good trade execution strategies minimize trading costs in a pathwise sense, not just expected costs.
MIME uses mutual information minimization for better exploration in environments with abrupt transitions.
problem Agents struggle at abrupt environmental transitions.
method MIME learns a latent representation without predicting future states.
result MIME outperforms surprisal-driven agents at transition boundaries.
Generalizes skyrmion theory to gauged maps with G-action.
problem Finding non-trivial minimizers of gauged skyrmion energy.
method Introduces a gauged energy functional and studies BPS equations.
result Classifies solutions for G=mU(1) and G=mSU(2). Solves the Poisson problem for elastic plates with specific boundary conditions.
problem Finding an immersed surface minimizing Germain's elastic energy.
method Minimizes total curvature energy E(Σ) variationally. result The minimum is an immersed disk with branch points, extending to a C0,α Gauss map. The paper uses the technique of finite-dimensional approximation to show that a constant scalr curvature Kahler metric (on a polarised algebraic variety without holomorphic vector fields) minimises the Mabuchi functional.
In this article we discuss the problem of calculating optimal model-independent (robust) bounds for the price of Asian options with discrete and continuous averaging. We will give geometric characterisations of the maximising and the minimising pricing model for certain types of Asian options in discrete and continuous…
Study optimal consumption and investment for investors with Epstein-Zin preferences.
problem Optimal consumption and investment for investors with Epstein-Zin preferences in an incomplete market.
method Variational characterisation and direct method to prove existence of optimal policies.
result Existence and uniqueness of optimal consumption and investment policies.
Study minimum ribbonlength of immersed flat knots and links.
problem Finding the minimum ribbonlength for immersed planar knots and links.
method Embedding into disk diagram space to find length minimizers.
result Computed minimal ribbonlength for some knot and link diagrams.
Study examines insider trading in short-selling restricted markets.
problem Analyzing insider trading opportunities in short-selling prohibited markets.
method Introducing minimal supermartingale measure and analyzing its properties in relation to minimal martingale measure.
result Conditions under which both measures fail to exist, indicating insider information affecting market perception.
New algorithm minimizes FE objectives for synthetic AIF agents.
problem Minimizing Free Energy for synthetic active inference agents.
method Variational message passing on Constrained FFGs.
result Message passing algorithms induce epistemic behavior in AIF agents.
This work proposes a model for geodesic distances and flows on manifolds.
problem Geodesic distances and flows on differentiable manifolds.
method Manifold-augmented Eikonal equation solutions.
result Geodesic flow provides globally length-minimizing curves.
Smooth approximations near singularities of constant mean curvature surfaces are found.
problem Finding smooth approximations for constant mean curvature surfaces near singular points.
method Proving the existence of sequences of smooth CMC hypersurfaces converging to a given one in a ball centered at the singularity.
result Smooth approximations exist in a ball centered at the singularity of a CMC hypersurface.
The question of optimal portfolio is addressed. The conventional Markowitz portfolio optimisation is discussed and the shortcomings due to non-Gaussian security returns are outlined. A method is proposed to minimise the likelihood of extreme non-Gaussian drawdowns of the portfolio value. The theory is called Leptokurti…
New method uses kernel Stein discrepancy for measure transport without strict continuity constraints.
problem Minimizing Kullback-Leibler divergence for posterior approximation.
method Proposes minimizing kernel Stein discrepancy instead of Kullback-Leibler divergence.
result Demonstrates consistency and competitiveness of the new method.
The paper studies properties of Sliced Wasserstein energy for discrete measures.
problem Optimizing discrete probability measures using Sliced Wasserstein loss.
method Investigates the regularity and optimisation properties of the Sliced Wasserstein energy and its Monte-Carlo approximation.
result Convergence results on the critical points of Monte-Carlo approximations to the Sliced Wasserstein energy.
Denoising autoencoders (DAEs) are powerful deep learning models used for feature extraction, data generation and network pre-training. DAEs consist of an encoder and decoder which may be trained simultaneously to minimise a loss (function) between an input and the reconstruction of a corrupted version of the input. The…