Paper solves a long-standing problem with curvature estimates.
problem Long-standing problem in curvature equation.
method Global curvature estimate for the Hessian equation.
result Solves a long-standing problem in curvature equation.
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We consider the corresponding Christoffel-Minkowski problem for curvature measures. The existence of star-shaped -convex bodies with prescribed -th curvature measures () has been a longstanding problem. This is settled in this paper through the establishment of a crucial a priori estimate for the c…
In this paper we present the solution to a longstanding problem of differential geometry: Lie's third theorem for Lie algebroids. We show that the integrability problem is controlled by two computable obstructions. As applications we derive, explain and improve the known integrability results, we establish integrabilit…
We found unique tori with same curvatures using isometric transformations.
problem Determining if metric and mean curvature uniquely define a torus.
method Constructed Bonnet pairs of tori using isothermic surfaces and conformal transformations.
result Explicit construction of compact Bonnet pairs of tori.
We present the solution of a longstanding internal problem of noncommutative geometry, namely the computation of the index of a transversally elliptic operator on an arbitrary foliation. The new and crucial ingredient is a certain Hopf algebra associated to the transverse frame bundle. Its cyclic cohomology is defined …
We show that Kervaire invariant one elements in the homotopy groups of spheres exist only in dimensions at most 126. By Browder's Theorem, this means that smooth framed manifolds of Kervaire invariant one exist only in dimensions 2, 6, 14, 30, 62, and possibly 126. With the exception of dimension 126 this resolves a lo…
We give a quick tour through many of the classical results in the field of minimal submanifolds, starting at the definition. The field of minimal submanifolds remains extremely active and has very recently seen major developments that have solved many longstanding open problems and conjectures; for more on this, see th…
Toric Quasifoldsmath.SG
Toric quasifolds extend toric geometry to non-rational polytopes.
problem Extending toric geometry to non-rational polytopes.
method Introduced toric quasifolds to solve symplectic extension problem.
result Illustrated toric quasifolds and their atlases.
Symplectic realization is a longstanding problem which can be traced back to Sophus Lie. In this paper, we present an explicit solution to this problem for an arbitrary holomorphic Poisson manifold. More precisely, for any holomorphic Poisson manifold , we prove that there exists a holomorphic symplectic struct…
Proves common stellar subdivisions for all PL homeomorphic polyhedra.
problem Proving common stellar subdivisions for all PL homeomorphic polyhedra.
method Proved weighted strong factorization conjecture and iterated blowups.
result Every two PL homeomorphic polyhedra have a common stellar subdivision.
The VC-dimension of a set system is a way to capture its complexity and has been a key parameter studied extensively in machine learning and geometry communities. In this paper, we resolve two longstanding open problems on bounding the VC-dimension of two fundamental set systems: -fold unions/intersections of half-s…
It is a longstanding problem to determine the precise relationship between the geodesic length spectrum of a hyperbolic manifold and its commensurability class. A well known result of Reid, for instance, shows that the geodesic length spectrum of an arithmetic hyperbolic surface determines the surface's commensurabilit…
One of the longstanding open problems in spectral graph clustering (SGC) is the so-called model order selection problem: automated selection of the correct number of clusters. This is equivalent to the problem of finding the number of connected components or communities in an undirected graph. We propose automated mode…
New findings on knot operations challenge a long-standing conjecture.
problem Understanding equivariant unknotting numbers of strongly invertible knots.
method Study of symmetric crossing change operations for strongly invertible knots.
result The equivariant unknotting number is not additive under connected sum.
In this paper, we study locally strongly convex Tchebychev hypersurfaces, namely the {\it centroaffine totally umbilical hypersurfaces}, in the -dimensional affine space . We first make an ordinary-looking observation that such hypersurfaces are characterized by having a Riemannian structure ad…
We introduce novel communication strategies in synchronous distributed Deep Learning consisting of decentralized gradient reduction orchestration and computational graph-aware grouping of gradient tensors. These new techniques produce an optimal overlap between computation and communication and result in near-linear sc…
Predicting flood for any location at times of extreme storms is a longstanding problem that has utmost importance in emergency management. Conventional methods that aim to predict water levels in streams use advanced hydrological models still lack of giving accurate forecasts everywhere. This study aims to explore arti…
We reconsider the training objective of Generative Adversarial Networks (GANs) from the mixed Nash Equilibria (NE) perspective. Inspired by the classical prox methods, we develop a novel algorithmic framework for GANs via an infinite-dimensional two-player game and prove rigorous convergence rates to the mixed NE, reso…
Given a singular Schubert variety Z in a compact Hermitian symmetric space it is a longstanding question to determine when Z is homologous to a smooth variety Y. We identify those Schubert varieties for which there exist first-order obstructions to the existence of Y. This extends (independent) work of M. Walters, R. B…
One of the longstanding problems in spectral graph clustering (SGC) is the so-called model order selection problem: automated selection of the correct number of clusters. This is equivalent to the problem of finding the number of connected components or communities in an undirected graph. In this paper, we propose AMOS…
Study shows not all smooth paths are optimal in certain geometric structures.
problem Existence of non-smooth sub-Riemannian minimizing geodesics.
method Constructed a but not length-minimizer example.
result Found a real-analytic sub-Riemannian structure with non-smooth minimizers.
Some of recent developments, including recent results, ideas, techniques, and approaches, in the study of degenerate partial differential equations are surveyed and analyzed. Several examples of nonlinear degenerate, even mixed, partial differential equations, are presented, which arise naturally in some longstanding, …
A longstanding question in superstring/ theory is does it predict supersymmetry below the string scale? We formulate and discuss a necessary condition for this to be true; this is the mathematical conjecture that all stable, compact Ricci flat manifolds have special holonomy in dimensions below eleven. Almost equiva…
Symplectormophism groups of non-compact manifolds, orbifold balls, and a space of Lagrangiansmath.SG
We establish connections between contact isometry groups of certain contact manifolds and compactly supported symplectomorphism groups of their symplectizations. We apply these results to investigate the space of symplectic embeddings of balls with a single conical singularity at the origin. Using similar ideas, we als…
The Burau representation is a natural action of the braid group B_n on the free Z[t,t^{-1}]-module of rank n-1. It is a longstanding open problem to determine for which values of n this representation is faithful. It is known to be faithful for n=3. Moody has shown that it is not faithful for n>8 and Long and Paton imp…
Uniform convergence of metrics on vortex moduli space in Bradlow limit.
problem Understanding the geometry of vortex moduli spaces.
method Proof of uniform convergence of metrics using normalized metric and Fubini-Study metric.
result Establishes the Fubini-Study metric as the limit of the normalized metric in the Bradlow limit.
The cosmetic surgery conjecture is a longstanding conjecture in 3-manifold theory. We present a theorem about exceptional cosmetic surgery for homology spheres. Along the way we prove that if the surgery is not a small seifert -homology sphere or a toroidal irreducible non-Seifert surgery then t…
A longstanding problem for Deep Neural Networks (DNNs) is understanding their puzzling ability to generalize well. We approach this problem through the unconventional angle of \textit{cognitive abstraction mechanisms}, drawing inspiration from recent neuroscience work, allowing us to define the Cognitive Neural Activat…
New neural network solves Nirenberg problem for curvature on sphere.
problem Prescribing Gaussian curvature on for metrics conformal to the round metric.
method Mesh-free physics-informed neural network (PINN) that directly parametrises the conformal factor.
result Neural network achieves very low losses for realisable curvatures, distinguishing them from non-realisable ones.
Many problems in sequential decision making and stochastic control often have natural multiscale structure: sub-tasks are assembled together to accomplish complex goals. Systematically inferring and leveraging hierarchical structure, particularly beyond a single level of abstraction, has remained a longstanding challen…
New algorithm detects anomalies by forcing samples to displace mass in low-density regions.
problem Detecting anomalies in datasets.
method Mass Repulsing Optimal Transport (MROT) approach.
result Our algorithm improves anomaly detection over existing methods.
Paper solves long-standing Gaussian curvature conjecture for minimal graphs.
problem Gaussian curvature of minimal graphs over the unit disk.
method Complex-analytic methods, conformal harmonic parameterization.
result Sharp estimate for Gaussian curvature at the origin of minimal graphs.
We propose a unified framework to address a family of classical mixed-integer optimization problems with logically constrained decision variables, including network design, facility location, unit commitment, sparse portfolio selection, binary quadratic optimization, sparse principal analysis and sparse learning proble…
Study geodesics in sub-Riemannian manifolds, resolving open questions.
problem Understanding geodesics in sub-Riemannian geometry, especially those that lose regularity.
method Constructing examples and using a lifting procedure.
result Existence of non-smooth and branching minimizing geodesics in real-analytic sub-Riemannian manifolds and Carnot groups.
Revisits PCA with new formulations and insights.
problem Improving PCA formulations and understanding.
method Difference-of-convex (DC) framework, kernelizability, out-of-sample applicability, simultaneous iteration, DCA perspective.
result PCA-like problems are kernelizable and have new optimization perspectives.
A topological version of a longstanding conjecture of H. Hopf, originally proposed by W. Thurston, states that the sign of the Euler characteristic of a closed aspherical manifold of dimension depends only on the parity of . Gromov defined several hyperbolization functors which produce an aspherical manifold …
Optimal strategy found for identifying best arm in bandits with small gap.
problem Best arm identification in two-armed bandits with a fixed budget and small gap.
method Neyman allocation rule augmented with inverse probability weighting.
result Proposed strategy is asymptotically optimal when gap is small.
This survey explores compact geodesic orbit manifolds and their properties.
problem Classifying compact geodesic orbit manifolds.
method Review and analysis of existing results.
result Study of geodesic orbit condition for $\SU(5)/\s(\U(2) imes \U(2))$.
Improving sample efficiency has been a longstanding goal in reinforcement learning. This paper proposes algorithm: a sample efficient policy gradient method with stochastic mirror descent. In , a novel variance-reduced policy gradient estimator is presented to improve sample efficiency.…
New PDEs of mixed type emerge in fluid mechanics and geometry.
problem Analysis of nonlinear PDEs of mixed type.
method Through historical problems and recent trends.
result Many PDEs are of mixed type, requiring new analysis.
The paper proves the existence of a folded annulus with multiple creases.
problem Existence of a folded annulus with multiple creases.
method Analyze developable surfaces, use normal curvature and relative torsion, compute geometric descriptors, prove propagation of folds.
result Proves the existence of a folded annulus with multiple creases.
In mixtures-of-experts (ME) model, where a number of submodels (experts) are combined, there have been two longstanding problems: (i) how many experts should be chosen, given the size of the training data? (ii) given the total number of parameters, is it better to use a few very complex experts, or is it better to comb…
A new method for Bayesian inference in high dimensions using projected Stein variational gradient descent.
problem Bayesian inference challenges in high-dimensional data.
method Adapting Stein variational gradient descent to exploit intrinsic low dimensionality of data.
result pSVGD is more accurate and efficient than SVGD, especially in high-dimensional settings.
Although optimization is the longstanding algorithmic backbone of machine learning, new models still require the time-consuming implementation of new solvers. As a result, there are thousands of implementations of optimization algorithms for machine learning problems. A natural question is, if it is always necessary to…
High Quality Prediction of Protein Q8 Secondary Structure by Diverse Neural Network Architecturescs.LG
We tackle the problem of protein secondary structure prediction using a common task framework. This lead to the introduction of multiple ideas for neural architectures based on state of the art building blocks, used in this task for the first time. We take a principled machine learning approach, which provides genuine,…
Study curves in non-orientable surfaces with specific intersection properties.
problem Enumerate and understand curves in non-orientable surfaces with intersection constraints.
method Generalized construction of Malestein-Rivin-Theran to non-orientable surfaces.
result Lower bound for maximum number of curves in generic non-orientable surface.
Off-policy reinforcement learning has many applications including: learning from demonstration, learning multiple goal seeking policies in parallel, and representing predictive knowledge. Recently there has been an proliferation of new policy-evaluation algorithms that fill a longstanding algorithmic void in reinforcem…
In this paper we provide a systematic discussion of how to incorporate orientation preserving symmetries into the treatment of Willmore surfaces via the loop group method. In this context we first develop a general treatment of Willmore surfaces admitting orientation preserving symmetries, and then show how to induce f…