arXiv research

An algorithm calculates Gabai width for thousands of knots.

Paper proves vanishing homology groups for certain hyperbolic groups.

A generalization of the affine-geometric Wirtinger inequality for curves to hypersurfaces is given.

The Wirtinger number of a virtual link is the minimum number of generators of the link group over all meridional presentations in which every relation is an iterated Wirtinger relation arising in a diagram. We prove that the Wirtinger number of a virtual link equals its virtual bridge number. Since the Wirtinger number…

We define the {\it Wirtinger number} of a link, an invariant closely related to the meridional rank. The Wirtinger number is the minimum number of generators of the fundamental group of the link complement over all meridional presentations in which every relation is an iterated Wirtinger relation arising in a diagram. …

The computation of the fundamental group of the complement of an algebraic plane curve has been theoretically solved since Zariski-van Kampen, but actual computations are usually cumbersome. In this work, we describe the notion of Wirtinger presentation of such a group relying on the real picture of the curve and with …

Sharp bounds for curve isoperimetric deficit derived.

Considering Wirtinger's inequality for piece-wise equipartite functions we find a discrete version of this classical inequality. The main tool we use is the theorem of classification of isometries. Our approach provides a new elementary proof of Wirtinger's inequality that also allows to study the case of equality. Mor…

New method calculates bridge indices of spatial graphs using diagram colorings and Wirtinger number.

Unified approach to discrete and smooth isoperimetric inequalities of arbitrary order.

We study the phase retrieval problem, which solves quadratic system of equations, i.e., recovers a vector $\boldsymbol{x}\in \mathbb{R}^n$ from its magnitude measurements $y_i=|\langle \boldsymbol{a}_i, \boldsymbol{x}\rangle|, i=1,..., m$. We develop a gradient-like algorithm (referred to as RWF representing reshaped W…

New formula for knot group representations and hyperbolic structures.

We study optimal curvature-free inequalities of the type discovered by C. Loewner and M. Gromov, using a generalisation of the Wirtinger inequality for the comass. Using a model for the classifying space BS^3 built inductively out of BS^1, we prove that the symmetric metrics of certain two-point homogeneous manifolds t…

Study on weakly G-slim complexes and non-positive immersions for group presentations.

For any subvariety of a compact holomorphic symplectic Kaehler manifold, we define the number W(X), which we call Wirtinger number. We show that $W(X)\leq 1$, and the equality is reached if and only if the subvariety $X\subset M$ is trianalytic, i. e. compactible with the hyperkaehler structure on M. For a sequence $X_…

New theory classifies knotted spheres in 4D space.

Paper introduces danceability index as a new bridge index definition.

The paper proves existence and uniqueness of slant immersions in complex space forms.

Hadamard Wirtinger Flow recovers sparse signals from fewer measurements.

Over-the-air computation (AirComp) shows great promise to support fast data fusion in Internet-of-Things (IoT) networks. AirComp typically computes desired functions of distributed sensing data by exploiting superposed data transmission in multiple access channels. To overcome its reliance on channel station informatio…

We prove an inequality that generalizes the Fan-Taussky-Todd discrete analog of the Wirtinger inequality. It is equivalent to an estimate on the spectral gap of a weighted discrete Laplacian on the circle. The proof uses a geometric construction related to the discrete isoperimetric problem on the surface of a cone. In…

Link's sphere number equals its bridge number.

We consider the robust phase retrieval problem of recovering the unknown signal from the magnitude-only measurements, where the measurements can be contaminated by both sparse arbitrary corruption and bounded random noise. We propose a new nonconvex algorithm for robust phase retrieval, namely Robust Wirtinger Flow to …

This research explores complex-valued neural networks and their implementation.

This paper considers the noisy sparse phase retrieval problem: recovering a sparse signal $x \in \mathbb{R}^p$ from noisy quadratic measurements $y_j = (a_j' x )^2 + ε_j$, $j=1, \ldots, m$, with independent sub-exponential noise $ε_j$. The goals are to understand the effect of the sparsity of $x$ on the estimation prec…

New method for quandle presentations of surface knots in 4-manifolds.

The paper presents fundamental groups of complements of shadows in 4-balls.

Free surface-links are shown to be ribbon links.

We introduce \textit{dual graph diagrams} representing oriented knots and links. We use these combinatorial structures to define corresponding algebraic structures we call \textit{biquasiles} whose axioms are motivated by dual graph Reidemeister moves, generalizing the Dehn presentation of the knot group analogously to…

A classical link in 3-space can be represented by a Gauss paragraph encoding a link diagram in a combinatorial way. A Gauss paragraph may code not a classical link diagram, but a diagram with virtual crossings. We present a criterion and a linear algorithm detecting whether a Gauss paragraph encodes a classical link. W…

We prove the meridional rank conjecture for twisted links and arborescent links associated to bipartite trees with even weights. These links are substantial generalizations of pretzels and two-bridge links, respectively. Lower bounds on meridional rank are obtained via Coxeter quotients of the groups of link complement…

Paper tackles phase retrieval with robust gradient descent for noisy data.

In this paper, a regional knot invariant is constructed. Like the Wirtinger presentation of a knot group, each planar region contributes a generator, and each crossing contributes a relation. The invariant is call a tridle of the link. As in the quandle theory, one can define Alexander quandle and get Alexander polynom…

A new deep learning model improves phase retrieval performance.

Empirical study compares finite- and infinite-width BNNs, revealing performance differences under model mismatch.

The isospectral problem for p-widths is solved using Zoll metrics on S^2.

Width trees link link invariants and bridge number.

Lectures on deep learning properties in infinite and large-width networks.

Computed p-widths for hemisphere, first for manifolds with boundary.

Polygon $p$-widths are found via billiard trajectories.

A number of results for C$^2$-smooth surfaces of constant width in Euclidean 3-space ${\mathbb{E}}^3$ are obtained. In particular, an integral inequality for constant width surfaces is established. This is used to prove that the ratio of volume to cubed width of a constant width surface is reduced by shrinking it along…

Computed p-widths for real projective plane.

Study bounds Urysohn width of manifolds under surgeries.

Residual networks with block width max(d_x, d_y) approximate all functions.

The paper connects complex normalizing flows to Kähler-Ricci flows using geometric and statistical perspectives.

Study calculates fundamental groups of torus knots using algebraic topology.

Existing nonconvex statistical optimization theory and methods crucially rely on the correct specification of the underlying "true" statistical models. To address this issue, we take a first step towards taming model misspecification by studying the high-dimensional sparse phase retrieval problem with misspecified link…

While studying the existence of closed geodesics and minimal hypersurfaces in compact manifolds, the concept of width was introduced in different contexts. Generally, the width is realized by the energy of the closed geodesics or the volume of minimal hypersurfaces, which are found by the Minimax argument. Recently, Ma…