The study proves lower semi-continuity of the index for minimal surfaces using viscosity methods.
problem Proving lower semi-continuity of the index for minimal surfaces.
method Using Hilbert manifold structure and viscous approximation of the area.
result Proves lower semi-continuity of the index for minimal surfaces.
Study on volume continuity of Lagrangian submanifolds.
problem Lower semi-continuity of Lagrangian volume.
method Analysis of volume properties with respect to Hofer- and γ-distances.
result Volume is γ-lower semi-continuous in two specific cases.
Study continuity of minimal surface areas in hyperbolic 3-manifolds near short geodesics.
problem Understanding the interaction between minimal surfaces and short geodesics in hyperbolic 3-manifolds.
method Geometric convergence of hyperbolic manifolds, lower semi-continuity, continuity of A1. result Lower semi-continuity and continuity of A1 if a minimal surface satisfies certain hypotheses. The paper proves continuity of Morse index for Ricci shrinkers.
problem Lower and upper semi-continuity of the Morse index for gradient Ricci shrinkers.
method Adapting and refining recent arguments on CMC hypersurfaces and polynomially weighted Sobolev spaces, with techniques for non-compact shrinkers.
result Identifies a condition ensuring the Morse index of asymptotically conical shrinkers is bounded below by the f-index of their asymptotic cone.
In this note we continue the analysis of metric measure space with variable ricci curvature bounds. First, we study (κ,N)-convex functions on metric spaces where κ is a lower semi-continuous function, and gradient flow curves in the sense of a new evolution variational inequality that captures the information that …
The study optimizes cell membranes' shapes based on curvature and proves existence of minimizers.
problem Optimizing cell membranes' shapes with respect to curvature.
method Modeling cell membranes as optimal shapes with L2-deficit of mean curvature to spontaneous curvature, and proving lower semi-continuity and existence of minimizers. result Smoothly embedded minimizers and diameter bounds are obtained.
New algorithm estimates semi-continuous data density using entropy maximization.
problem Estimating density functions for semi-continuous data.
method Maximum entropy principle, requiring only constraint function samples.
result Estimate has significantly less bias compared to existing methods.
Characterizes conditions for quotient spaces of decompositions to be manifolds.
problem Conditions for quotient spaces of decompositions to be manifolds.
method Generalized characterizations of upper semi-continuity for decomposition into one for a class decomposition.
result Characterizations of necessary and sufficient conditions for quotient spaces of decompositions to be k-manifolds (k=1,2). We present a viscosity approach to the min-max construction of closed geodesics on compact Riemannian manifolds of arbitrary dimension. We also construct counter-examples in dimension 1 and 2 to the ε-regularity in the convergence procedure. Furthermore, we prove the lower semi-continuity of the index o…
Study Galois groupoids of discret Painlevé equations.
problem Computing Galois groupoids for discret Painlevé equations.
method Using semi-continuity theorem for Galois groupoid in confluence of difference to differential equations.
result Computed Galois groupoids for discret Painlevé equations.
New method shows stability of Willmore immersions' Morse index and nullity.
problem Stability of Morse index and nullity of Willmore immersions.
method Upper semi-continuity method applied to Willmore immersions.
result Sum of Morse index and nullity is upper semi-continuous.
The paper proves the existence of singular cscK metrics on smoothable varieties.
problem Existence of singular cscK metrics on smoothable varieties.
method Developing a strong topology of pluripotential theory in families and uniform estimates for cscK metrics.
result Existence of singular cscK metrics on Q-Gorenstein smoothable klt varieties when the Mabuchi functional is coercive. We investigate a semi-continuity property for stability conditions for sheaves that is important for the problem of variation of the moduli spaces as the stability condition changes. We place this in the context of a notion of stability previously considered by the authors, called multi-Gieseker-stability, that general…
We prove a Gamma-convergence result for a family of bending energies defined on smooth surfaces in R3 equipped with a director field. The energies strongly penalize the deviation of the director from the surface unit normal and control the derivatives of the director. Such type of energies for example arise…
Stability of branched immersions with energy constraints.
problem Stability of branched Willmore immersions with bounded energy.
method Refined analysis of fourth-order differential operators with regular singularities.
result Sum of Morse index and nullity is lower semi-continuous.
Investigates stability properties of Haezendonck-Goovaerts premium principles in Orlicz spaces.
problem Stability properties of Haezendonck-Goovaerts premium principles in various Orlicz spaces.
method Analysis of stability properties including Fatou and Lebesgue properties, and continuity with respect to Φ-weak convergence. result Haezendonck-Goovaerts principles satisfy the Fatou property and Lebesgue property under certain conditions.
Stability of Morse index for Yang-Mills connections in 4D.
problem Stability of critical points in Yang-Mills energy relaxation.
method Establishing lower semi-continuity of Morse index and upper continuity of Morse index plus nullity.
result Yang-Mills fields are more stable than harmonic maps in 4D.
Generates critical points in manifolds for higher index min-max problems.
problem Finding critical points in Banach manifolds with Finsler structures.
method Minmax hierarchy and viscosity approach to minimal surfaces.
result Characterization of minimal surfaces in S3 and conjecture for next surfaces. Continuous-time pricing-hedging duality for European options.
problem Finding the minimal superhedging price of path-dependent European options.
method Formulates a duality between analytic and probabilistic problems, using simple trading strategies and semi-continuous claims.
result The minimal superhedging price equals the supremum of expectations over all martingale measures.
Study minimizers for isoperimetric problem with log-convex density.
problem Minimizing weighted perimeter for sets with fixed weighted volume.
method Analyzing minimizers for the weighted isoperimetric problem with a specific density form.
result Centred balls are minimizers for the weighted isoperimetric problem with log-convex density.
New solver SR2 tackles deep neural network training with nonsmooth regularization.
problem Training deep neural networks with nonsmooth regularization to achieve sparsity and efficiency.
method Combines adaptive quadratic regularization with proximal stochastic gradient principles.
result Established worst-case iteration complexity of O(ε^−2) for SR2.
We define a norm on the homology of a foliated manifold, which refines and majorizes the usual Gromov norm on homology. This norm depends in an upper semi-continuous way on the underlying foliation, in the geometric topology, and can therefore be used to study the question of which foliations arise as geometric limits …
Study the rank of Nijenhuis tensor on parallelizable almost complex manifolds.
problem Understanding the rank of Nijenhuis tensor on parallelizable almost complex manifolds.
method Reduction of computations to solving PDEs, explicit solutions on specific manifolds, analysis of curve of almost complex structures, classification of Lie algebras.
result Classification of Lie algebras admitting almost complex structures with specific Nijenhuis tensor ranks.
Study of critical points in Ginzburg-Landau approximation with stability results.
problem Stability of critical points in Ginzburg-Landau approximation.
method Application of previous joint method with T. Rivière for upper semi-continuity of extended Morse index.
result Upper semi-continuity of extended Morse index for sequences of critical points.
What length of rope (of given diameter) is required to tie a particular knot? To answer this question, we define some new notions of thickness for a space curve, one based on Gromov's distortion, and another generalizing the thickness of Litherland, Simon et al. We prove a basic inequality between these thickness measu…
For a compact almost complex 4-manifold (M,J), we study the subgroups HJ± of H2(M,R) consisting of cohomology classes representable by J-invariant, respectively, J-anti-invariant 2-forms. If b+=1, we show that for generic almost complex structures on M, the subgroup HJ− is trivial. …
New invariants detect Fano varieties' K-stability.
problem Detecting K-stability in Fano varieties.
method Introducing valuative invariants based on higher moments.
result Can detect K-stability of Fano varieties.
We consider some natural infinitesimal Einstein deformations on Sasakian and 3-Sasakian manifolds. Some of these are infinitesimal deformations of Killing spinors and further some integrate to actual Killing spinor deformations. In particular, on 3-Sasakian 7 manifolds these yield infinitesimal Einstein deformations pr…
Unified framework for training neural networks with non-smooth, non-convex regularizers.
problem Training neural networks with non-smooth, non-convex regularizers.
method ProxGen framework for stochastic proximal gradient descent.
result ProxGen framework achieves the same convergence rate as standard methods and outperforms subgradient-based approaches.
Paper finds unique viscosity solution to complex control problems.
problem Complex stochastic control problems with singular terminal state constraints.
method Establishes existence of unique nonnegative continuous viscosity solution using novel comparison principle.
result Unique viscosity solution to HJB equation for linear-quadratic control problems.
Study counts rational curves on hyperKähler ALE 4-manifolds.
problem Counting rational curves on hyperKähler ALE 4-manifolds.
method Using the period map of the twistor space, count rational curves and specify complex structures.
result The integer function on the parameter space is lower semi-continuous.
For a smooth, closed n-manifold M, we define an upper semi-continuous integer-valued complexity function on H1(M;R) using Morse theory. This measures how far an integral class is from being a fiber of a fibration. The fact complexity minimisers are open generalises Tischler's result on the openness of …
New algorithm minimizes sum of three functions with linear operator.
problem Minimizing the sum of three convex functions with a linear operator.
method Proposes a new primal-dual algorithm for the problem.
result Proves convergence and provides convergence rates.
The study shows that normalized volumes of singularities can only jump down at countably many subvarieties.
problem Understanding the semi-continuity of normalized volumes in singularities.
method Using a flat family of klt singularities and an alternative characterization of K-semistability.
result K-semistability is a very generic or empty condition in Q-Gorenstein flat families of log Fano pairs. If (X,J) is an almost complex manifold, then a function u is said to be plurisubharmonic on X if it is upper semi-continuous and its restriction to every local pseudo-holomorphic curve is subharmonic. As in the complex case, it is conjectured that plurisubharmonicity is equivalent to the fact that the (1,1)-cur…
We propose a theory "a la Conley" for cone fields using a notion of relaxed orbits based on cone enlargements, in the spirit of space time geometry. We work in the setting of closed (or equivalently semi-continuous) cone fields with singularities. This setting contains (for questions which are parametrization independe…
We address the restriction problem for viscosity subsolutions of a fully nonlinear PDE on a manifold Z. The constraints on the restrictions of smooth subsolutions to a submanifold X in Z determine a restricted subequation on X. The problem is to show that general (upper semi-continuous) subsolutions restrict to satisfy…
Hodge numbers of Sasakian manifolds remain unchanged under deformations.
problem Invariance of Hodge numbers under deformations of Sasakian manifolds.
method Analysis of deformations of Sasakian structures and use of transversely elliptic operators.
result Hodge numbers are invariant under arbitrary deformations of the Sasakian structure.
This is an essay on potential theory for geometric plurisubharmonic functions. It begins with a given closed subset G of the Grassmann bundle G(p,TX) of tangent p-planes to a riemannian manifold X. This determines a nonlinear partial differential equation which is convex but never uniformly elliptic (p < dim X). …
This paper consists of two parts. In the first one we study the behaviour of medial axes (skeletons) of closed sets in a connected complete Riemannian manifold M under deformations. The second one is devoted to a similar study of conflict sets. We apply a new approach to the deformation process. Instead of …
The study examines properties of projection onto differentiable manifolds.
problem Investigating the maximal domain for orthogonal projections onto submanifolds.
method Analyzes properties of the projection map on C1 and Ck submanifolds with Lipschitz conditions. result Characterizes the frontier function describing the maximal domain of the projection map.
This note establishes smooth approximation from above for J-plurisubharmonic functions on an almost complex manifold (X,J). The following theorem is proved. Suppose X is J-pseudoconvex, i.e., X admits a smooth strictly J-plurisubharmonic exhaustion function. Let u be an (upper semi-continuous) J-plurisubharmonic functi…
The paper studies properties of image Milnor number and stability of germs.
problem Understanding the stability and properties of isolated instability germs.
method Analyzes three properties of image Milnor number for germs with isolated instability.
result Establishes the conservation of image Milnor number and proves weak Mond's conjecture.
Analyzes Kähler-Einstein metrics on families of Fano varieties.
problem Establishing Kähler-Einstein metrics on Fano varieties in families.
method Analytic method to show unique Kähler-Einstein metrics on neighboring fibers.
result Uniform a priori estimates and continuous variation of Kähler-Einstein potentials.
This paper tackles co-design of neural hardware and software to improve efficiency.
problem Designing efficient deep learning systems that consider both hardware and software optimizations together.
method Developed a constrained Bayesian optimization framework to automatically identify profitable design points in the joint hardware/software design space.
result Improved energy-delay product by 18% (ResNet) and 40% (DQN) over hand-tuned systems.
This paper is intended as the first step of a programme aiming to prove in the long run the long-conjectured closedness under holomorphic deformations of compact complex manifolds that are bimeromorphically equivalent to compact Kähler manifolds, known as Fujiki {\it class} C manifolds. Our main idea is to exp…
The paper proves stability of critical points for conformally invariant Lagrangians.
problem Stability of critical points for conformally invariant Lagrangians under weak convergence.
method Upper-semi-continuity of Morse index plus nullity established for critical points.
result The sum of Morse indices and nullity is bounded from above by the sum of the Morse indices plus the nullity of the weak limit and bubbles.
Bayesian hierarchical tensor factorization model for international trade flows
problem Sparse semi-continuous tensor data modeling
method Bayesian hierarchical tensor factorization with Poisson and Gamma models
result Identifies multiway dependence in trade flows