arXiv research

Study on fundamental groups of framed circle embeddings in 4-manifolds.

For an integral homology 3-sphere embedded asymptotically flatly in an Euclidean space, we find a natural framing extending the standard trivialization on the asymptotically flat part.

Research explores the space-like embeddings in pseudo-hyperbolic space, finding geometric frames and actions.

We establish existence and regularity results for normal Coulomb frames in the normal bundle of two-dimensional surfaces of disc-type embedded in Euclidean spaces of higher dimensions.

Detecting exotic spheres involves analyzing framed configuration spaces.

Improved modeling of chaotic systems using time-delay embeddings and Frenet-Serret frame.

We generalize H. Seifert's algorithm for finding a Seifert surface for a knot or link. The generalization applies to "framed oriented measured lamination links." For knots, a Seifert surface determines a unique framing. In our setting, we analyze the set of framed lamination links which bound Seifert laminations and ar…

Link framings can only change when a 3-manifold has a non-separating sphere.

This paper considers *-graphs in which all vertices have degree 4 or 6, and studies the question of calculating the genus of orientable 2-surfaces into which such graphs may be embedded. A *-graph is a graph endowed with a formal adjacency structure on the half-edges around each vertex, and an embedding of a *-graph is…

We review classical approaches to the problem of isometrically embedding a Riemannian surface into Euclidean 3-space, including coordinate-based approaches exposited by Darboux and Eisenhart, as well as the moving-frames based approaches advocated by Cartan. In particular, the first approach involves reducing the probl…

We construct an infinite family of homology theories of framed links in thickened surfaces, as well as a homology theory whose graded Euler characteristic is exactly the Kauffman bracket of the link in the surface. Both theories are based on ideas coming from Asaeda, Przytycki and Sikora's categorification of the Kauff…

We present two families of exterior differential systems (EDS) for non-isometric embeddings of orthonormal frame bundles over Riemannian spaces of dimension q = 2, 3, 4, 5.... into orthonormal frame bundles over flat spaces of sufficiently higher dimension. We have calculated Cartan characters showing that these EDS sa…

Study of embedding spaces using homotopy theory and operads.

We study the problem of finding the minimal (maximal) genus for a surface where a given four-valent graph with fixed opposite edge structure can be embedded into. We find several partial relations and give new reformulations in combinatorial and knot theoretic languages.

According to Kiyoshi Igusa a generalized Morse function on an n-dimensional manifold M is a smooth function with only Morse and birth-death singularities and a framed function is a generalized Morse function with an additional structure: a framing of the negative eigenspace at each critical point of the function f. In …

Comma.ai's approach to Artificial Intelligence for self-driving cars is based on an agent that learns to clone driver behaviors and plans maneuvers by simulating future events in the road. This paper illustrates one of our research approaches for driving simulation. One where we learn to simulate. Here we investigate v…

We introduce and define "oriented framed measured lamination links" in a 3-manifold $M$. These generalize oriented framed links in 3-manifolds, and are confined to 2-dimensional improperly embedded subsurfaces of the 3-manifold. Just as some framed links bound Seifert surfaces, so also some framed lamination links boun…

The smoothing theory is revised to generalize to different disc embedding spaces.

In this note we describe the recursion relations between two parameter HOMLFY and Kauffman polynomials of framed links These relation correspond to embeddings of quantized universal enveloping algebras. The relation corresponding to embeddings $g_{n}\supset g_{k}\times sl_{n-k}$ where $g_{n}$ is either $so_{2n+1}$, $so…

Predictive Sparse Manifold Transform learns dynamic video sequences.

Invariants for 3-manifolds with embedded links using Hopf algebras.

Voice activity detection (VAD), which classifies frames as speech or non-speech, is an important module in many speech applications including speaker verification. In this paper, we propose a novel method, called self-adaptive soft VAD, to incorporate a deep neural network (DNN)-based VAD into a deep speaker embedding …

We synthesize and extend the previous ideas about appearance of both noncommutative and Finsler geometry in string theory with nonvanishing B--field and/or anholonomic (super) frame structures \cite{vstring,vstr2,vnonc,vncf}. There are investigated the limits to the Einstein gravity and string generalizations containin…

New Alexander polynomial defined for transverse graphs.

The study connects specific circle embeddings to 4-manifold diffeomorphisms.

Identifies a mod-$p$ triple cup product for rational homology 3-spheres with specific first homology.

Unified geometric framework for Brownian motion on various manifolds.

One of the key challenges of visual perception is to extract abstract models of 3D objects and object categories from visual measurements, which are affected by complex nuisance factors such as viewpoint, occlusion, motion, and deformations. Starting from the recent idea of viewpoint factorization, we propose a new app…

We prove that no $14$-connected (resp. $30$-connected) stably parallelizable manifold $N^{30}$ (resp. $N^{62}$) of dimension $30$ (resp. $62$) with the Arf-Kervaire invariant 1 can be smoothly embedded into $\mathbb{R}^{36}$ (resp. $\mathbb{R}^{83}$).

We show that generalised geometry gives a unified description of maximally supersymmetric consistent truncations of ten- and eleven-dimensional supergravity. In all cases the reduction manifold admits a "generalised parallelisation" with a frame algebra with constant coefficients. The consistent truncation then arises …

Researchers found 5 local fields to uniquely describe 3D director fields, related through 6 differential relations.

New method for classifying disk embeddings in 4-manifolds.

Study of embedding calculus using infinite operads.

These are lecture notes on the rigidity of submanifolds of projective space "resembling" compact Hermitian symmetric spaces in their homogeneous embeddings. Recent results are surveyed, along with their classical predecessors. The notes include an introduction to moving frames in projective geometry, an exposition of t…

E. Cartan's method of moving frames is applied to 3-dimensional manifolds $M$ which are CR-embedded in 5-dimensional real hyperquadrics $Q$ in order to classify $M$ up to CR symmetries of $Q$ given by the action of one of the Lie groups $SU(3,1)$ or $SU(2,2)$. In the latter case, the CR structure of $M$ derives from a …

Maximal representations are studied using tree embeddings and geodesic currents.

The paper characterizes when a 2-sphere can be embedded in a knot trace.

In this paper we show that if one writes down the structure equations for the evolution of a curve embedded in an (n)-dimensional Riemannian manifold with constant curvature this leads to a symplectic, a Hamiltonian and an hereditary operator. This gives us a natural connection between finite dimensional geometry, infi…

Consider the nonstandard embedding of SO(3) into SO(5) given by the 5-dimensional irreducible representation of SO(3), henceforth called SO(3)_\ir. In this note, we study the topology and the differential geometry of 5-dimensional Riemannian manifolds carrying such an SO(3)_\ir structure, i.\,e. with a reduction of the…

The homotopy fiber of the inclusion from the long embedding space to the long immersion space is known to be an iterated based loop space (if the codimension is greater than two). In this paper we deloop the homotopy fiber to obtain the topological Stiefel manifold, combining results of Lashof and of Lees. We also give…

In this paper, we generalize the famous Hasimoto's transformation by showing that the dynamics of a closed unidimensional vortex filament embedded in a three-dimensional manifold of constant curvature gives rise under Hasimoto's transformation to the non-linear Schrodinger equation. We also give a natural interpretatio…

The paper simplifies embedding spaces in manifolds by attaching handles.

We investigate deep neural network performance in the textindependent speaker recognition task. We demonstrate that using angular softmax activation at the last classification layer of a classification neural network instead of a simple softmax activation allows to train a more generalized discriminative speaker embedd…

Acoustic Neighbor Embeddings map speech and text to fixed dimensions for phonetic confusability.

Speaker embeddings are continuous-value vector representations that allow easy comparison between voices of speakers with simple geometric operations. Among others, i-vector and x-vector have emerged as the mainstream methods for speaker embedding. In this paper, we illustrate the use of modern computation platform to …

The paper proves a concordance theorem for surfaces with a specific invariant.

In this paper, we propose "personal VAD", a system to detect the voice activity of a target speaker at the frame level. This system is useful for gating the inputs to a streaming on-device speech recognition system, such that it only triggers for the target user, which helps reduce the computational cost and battery co…

Study cobordisms of nested manifolds and their invariants.