arXiv research

Maps preserving mass and injective on boundary are isometries.

Study on ALH manifolds with boundary, showing surjectivity of scalar curvature map and mass rigidity.

New mass and staticity concepts derived from weighted curvature maps.

Deep learning reduces noise in weak lensing mass maps using GANs.

The paper analyzes harmonic maps to flat model spaces to prove geometric stability of the positive mass theorem.

Establishes inequality for multiple black holes, proving mass lower bound.

Study on fractional mass for codimension-two currents, proving equi-coercivity and Γ-convergence.

We propose a definition of quasi-local mass based on the Penrose Inequality. Two further definitions are given by measuring distortions of the exponential map.

New optimal transport method handles mass creation and destruction.

New cones in 4D space found with minimal mass.

Given a transportation cost $c: M \times\bar M \to\mathbf{R}$, optimal maps minimize the total cost of moving masses from $M$ to $\bar M$. We find a pseudo-metric and a calibration form on $M\times\bar M$ such that the graph of an optimal map is a calibrated maximal submanifold. We define the mass of space-like current…

Improved Sobolev mappings in Carnot groups with weaker assumptions.

Solves a problem posed by Brezis and Mironescu about least mass of area-minimizing currents.

We define barycentric coordinates on a Riemannian manifold using Karcher's center of mass technique applied to point masses for n+1 sufficiently close points, determining an n-dimensional Riemannian simplex defined as a "Karcher simplex." Specifically, a set of weights is mapped to the Riemannian center of mass for the…

Deep Relevance Regularization improves neural network performance in tumor typing.

We demonstrate the potential of Deep Learning methods for measurements of cosmological parameters from density fields, focusing on the extraction of non-Gaussian information. We consider weak lensing mass maps as our dataset. We aim for our method to be able to distinguish between five models, which were chosen to lie …

Using McCann's transportation map, we establish a transport inequality on compact manifolds with positive Ricci curvature. This inequality contains the sharp spectral comparison estimates.

Let M be a compact Riemannian manifold and let $μ$,d be the associated measure and distance on M. Robert McCann obtained, generalizing results for the Euclidean case by Yann Brenier, the polar factorization of Borel maps S : M -> M pushing forward $μ$ to a measure $ν$: each S factors uniquely a.e. into the composition …

In this paper we study the ergodic theory of the geodesic flow on negatively curved geometrically finite manifolds. We prove that the measure theoretic entropy is upper semicontinuous when there is no loss of mass. In case we are losing mass, the critical exponents of parabolic subgroups of the fundamental group have a…

NN-EVCLUS uses neural networks to cluster data with uncertainty.

We explore geometric aspects of bubble convergence for harmonic maps. More precisely, we show that the formation of bubbles is characterised by the local excess of curvature on the target manifold. We give a universal estimate for curvature concentration masses at each bubble point and show that there is no curvature l…

We prove that tangent cones to 2-dimensional calibrated cycles are unique. Using this result we prove a rate of convergence for the mass of the blow-up of a calibrated integral 2-cycle towards the limiting density. With the same techniques, we can also prove such a rate for J-holomorphic maps between almost complex man…

Suppose M is a noncompact connected n-manifold and m is a good Radon measure of M with m(bdry M) = 0. Let H(M; m) denote the group of m-preserving homeomorphisms of M equipped with the compact-open topology and H_E(M; m) denote the subgroup consisting of all h in H(M; m) which fix the ends of M. Each h in H_E(M; m) mov…

Study gauge freedoms in elastic wave equations and Dirichlet-to-Neumann map.

We consider 3+1 rotationally symmetric Lorentzian Einstein spacetime manifolds with $Λ>0$ and reduce the equations to 2+1 Einstein equations coupled to `shifted' wave maps. Subsequently, we prove various (explicit) positive mass-energy theorems. No smallness is assumed.

We present effective methods to compute equivariant harmonic maps from the universal cover of a surface into a nonpositively curved space. By discretizing the theory appropriately, we show that the energy functional is strongly convex and derive convergence of the discrete heat flow to the energy minimizer, with explic…

Paper presents a method for identifying isotope envelopes in MALDI-ToF data.

Local mappings relate dual and primal factor graphs for efficient marginal probability estimation.

These lectures were a part of the geometry course held during the Fall 2011 Mathematics Advanced Study Semesters (MASS) Program at Penn State (\url{http://www.math.psu.edu/mass/}). The lectures are meant to be accessible to advanced undergraduate and early graduate students in mathematics. We have placed a great emphas…

We explain how to derive largeness constraints in scalar curvature geometry using some basic splitting results and the potential theory on singular area minimizing hypersurfaces. This includes a variety of results like the non-existence of positive scalar curvature metrics on enlargeable manifolds or simplified proofs …

Study explores warped geometries of tensor manifolds, finding non-geodesic connections for some parameters.

Consider transportation of one distribution of mass onto another, chosen to optimize the total expected cost, where cost per unit mass transported from x to y is given by a smooth function c(x,y). If the source density f^+(x) is bounded away from zero and infinity in an open region U' \subset R^n, and the target densit…

Generative adversarial networks (GANs) are the state of the art in generative modeling. Unfortunately, most GAN methods are susceptible to mode collapse, meaning that they tend to capture only a subset of the modes of the true distribution. A possible way of dealing with this problem is to use an ensemble of GANs, wher…

The paper reviews advances in estimating and understanding optimal transport maps.

The paper studies how adding a 'Gauge Mass' term breaks gauge symmetry in Yang-Mills-Higgs systems and analyzes the resulting behavior.

The X-ADM mass is shown to be equivalent to the ADM mass, proving the X-positive mass theorem in all dimensions.

Study on sphere-valued maps, proving energy convergence and current limits.

A new geometric cocycle measures mass of hyperbolic manifolds.

The paper studies nonlinear mass concepts in 3-manifolds with nonnegative scalar curvature.

Study on Hausdorff dimension of singular CR Yamabe problem.

Equivalence proven for isocapacitary mass notions.

For a given $1$-Lipschitz map $u\colon\mathbb{R}^n\to\mathbb{R}^m$ we define a partition, up to a set of Lebesgue measure zero, of $\mathbb{R}^n$ into maximal closed convex sets such that restriction of $u$ is an isometry on these sets. We consider a disintegration, with respect to this partition, of a log-concave meas…

The paper defines a new mass quantity for 3-manifolds and proves a positive mass theorem.

New algorithms solve partial optimal transport problems for applications like PU learning.

Characterizes infinite harmonic maps using 1-currents.

Introduce new boundary mass for asymptotically flat half-manifolds

The paper establishes geometric inequalities for quasi-local masses.

We prove directly without using a density theorem that (i) the ADM mass defined in the usual way on an asymptotically flat manifold is equal to the mass defined intrinsically using Ricci tensor; (ii) the Hamiltonian formulation of center of mass and the center of mass defined intrinsically using Ricci tensor are the sa…