arXiv research

X-ray transform on H-type groups solved, revealing function injectivity.

Injective X-ray transform on Heisenberg group for regular functions.

A new method slices and sums radial kernels faster.

Scientific imaging techniques such as optical and electron microscopy and computed tomography (CT) scanning are used to study the 3D structure of an object through 2D observations. These observations are related to the original 3D object through orthogonal integral projections. For common 3D reconstruction algorithms, …

We establish a general slice theorem for the action of a locally convex Lie group on a locally convex manifold, which generalizes the classical slice theorem of Palais to infinite dimensions. We discuss two important settings under which the assumptions of this theorem are fulfilled. First, using Glöckner's inverse fun…

Unified method for deriving ridgelet transforms for various neural network architectures.

The paper introduces new estimators for multivariate functions using Fourier methods.

From Furuta's $\frac{10}{8}$ theorem, we derive a smooth slicing obstruction for knots in $S^3$ using a spin $4$-manifold whose boundary is $0$-surgery on a knot. We show that this obstruction is able to detect torsion elements in the smooth concordance group and find topologically slice knots which are not smoothly sl…

A quaternionic version of Picard's theorem limits how many values a slice regular function can avoid.

In this paper, we proved a rigidity theorem of the Hodge metric for concave horizontal slices and a local rigidity theorem for the monodromy representation.

Study improves understanding of Ricci curvature in manifolds.

The paper uses Fourier integral theorem for estimating multivariate distributions.

Study proves obstructions to equivariantly slice strongly negative amphichiral knots.

Here we explore a variety of properties of intrinsic flat convergence. We introduce the sliced filling volume and interval sliced filling volume and explore the relationship between these notions, the tetrahedral property and the disappearance of points under intrinsic flat convergence. We prove two new Gromov-Hausdorf…

The study establishes uncertainty principles on harmonic manifolds of rank one.

Establish a unified framework for negative results in Fourier analysis.

In this paper, we prove the existence of maximal slices in anti-de Sitter spaces (ADS spaces) with small boundary data at spatial infinity. The main arguments is implicit function theorem. We also get a necessary and sufficient condition for boundary behavior of totally geodesic slice in ADS space. Moreover, we show th…

A general slice theorem for the action of a Fréchet Lie group on a Fréchet manifolds is established. The Nash-Moser theorem provides the fundamental tool to generalize the result of Palais to this infinite-dimensional setting. The presented slice theorem is illustrated by its application to gauge theories: the action o…

We prove that the Fourier--Laplace--Nahm transform for connections on the projective line is a hyper-Kähler isometry.

In this paper we prove a new inversion theorem and a refinement of an old support theorem for two Radon transforms on a symmetric space. Included are some new identities for the Abel transform and some results about the Fourier transform from a joint work with Rawat, Sengupta and Sitaram.

Algorithm finds ribbon disks for alternating knots, resolving sliceness for most prime knots.

Study spherical Fourier transform on hypergeometric type harmonic manifolds.

The paper derives statistics of multi-factor functions from their Fourier transforms.

We establish the slice-ribbon conjecture for a large family of Montesinos' knots by means of Donaldson's theorem on the intersection forms of definite 4-manifolds.

We review the well-known slice theorem of Ebin for the action of the diffeomorphism group on the space of Riemannian metrics of a closed manifold. We present advances in the study of the spaces of Riemannian metrics, and produce a more concise proof for the existence of slices.

The paper proves the consistency and efficiency of a volatility estimator in noisy data.

New theorem connects handle-ribbon knots to slice derivatives.

We prove a Slice Theorem around closed leaves in a singular Riemannian foliation, and we use it to study the $C^\infty$-algebra of smooth basic functions, generalizing to the inhomogeneous setting a number of results by G.~Schwarz. In particular, in the infinitesimal case we show that this algebra is generated by a fin…

We investigate the homogeneity of certain kind of slices of the complete complexification of a proper complex equifocal submanifold in a symmetric space of non-compact type.

This article concerns cotangent-lifted Lie group actions; our goal is to find local and ``semi-global'' normal forms for these and associated structures. Our main result is a constructive cotangent bundle slice theorem that extends the Hamiltonian slice theorem of Marle, Guillemin and Sternberg. The result applies to a…

New integral theorems improve density function estimations.

The Fourier transform of Heegaard Floer d-invariants helps classify 3-manifolds.

The slicing number of a knot, $u_s(K)$, is the minimum number of crossing changes required to convert $K$ to a slice knot. This invariant is bounded above by the unknotting number and below by the slice genus $g_s(K)$. We show that for many knots, previous bounds on unknotting number obtained by Ozsvath and Szabo and b…

The paper decomposes metrics on manifolds with boundaries.

Let p and q be distinct integers greater than one. We show that the 2-component pretzel link P(p,q,-p,-q) is not slice, even though it has a ribbon mutant, by using 3-fold branched covers and an obstruction based on Donaldson's diagonalization theorem. As a consequence, we prove the slice-ribbon conjecture for 4-strand…

This paper studies equivariant cohomology of slice groupoids using linearization theorems.

We introduce Tristram-Levine signatures of virtual knots and use them to investigate virtual knot concordance. The signatures are defined first for almost classical knots, which are virtual knots admitting homologically trivial representations. The signatures and $ω$-signatures are shown to give bounds on the topologic…

We use the famous knot-theoretic consequence of Freedman's disc theorem---knots with trivial Alexander polynomial bound a locally-flat disc in the 4-ball---to prove the following generalization. The degree of the Alexander polynomial of a knot is an upper bound for twice its topological slice genus. We provide examples…

Lecture notes on kernel functions and Random Fourier Features.

The algebraic genus of a knot is an invariant that arises when one considers upper bounds for the topological slice genus coming from Freedman's theorem that Alexander polynomial one knots are topologically slice. This paper develops null-homologous twisting operations as a tool for studying the algebraic genus and, co…

Extend classical theory of affine processes to path-dependent setting

In the early 1980's Mike Freedman showed that all knots with trivial Alexander polynomial are topologically slice (with fundamental group Z). This paper contains the first new examples of topologically slice knots. In fact, we give a sufficient homological condition under which a knot is slice with fundamental group Z …

Study cohomogeneity one RCD-spaces, proving structural results and constructing new examples.

This paper contains the results of efforts to determine values of the smooth and the topological slice genus of 11- and 12-crossing knots. Upper bounds for these genera were produced by using a computer to search for genus one concordances between knots. For the topological slice genus further upper bounds were produce…

Lower bounds on rational slice genus using Heegaard Floer invariants.

The paper proves that rational concordance of double twist knots is reciprocal.

The study explores deep and shallow slice knots in 4-manifolds, linking them to conjectures and proving existence and nonexistence results.

Extends De Leeuw theorems to noncommutative groups and multipliers.