arXiv research

The theory of automatic groups is developed, including properties and practical algorithms.

Classifies when homeomorphism groups of stable surfaces have automatic continuity.

Classifies surfaces for pure mapping class groups with automatic continuity.

Procedure groups nonparametric regression curves automatically.

We show that any homomorphism from the homeomorphism group of a compact 2-manifold, with the compact-open topology, or equivalently, with the topology of uniform convergence, into a separable topological group is automatically continuous.

We discuss a problem posed by Gersten: Is every automatic group which does not contain Z+Z subgroup, hyperbolic? To study this question, we define the notion of "n-tracks of length n", which is a structure like Z+Z, and prove its existence in the non-hyperbolic automatic groups with mild conditions. As an application, …

Extends continuity proof to non-compact manifolds and groups.

Let M be a compact manifold, possibly with boundary. We show that the group of homeomorphisms of M has the automatic continuity property: any homomorphism from Homeo(M) to any separable group is necessarily continuous. This answers a question of C. Rosendal. If N is a submanifold of M, the group of homeomorphisms of M …

Groups with specific curvature have a regular language of geodesics.

Exponential growth of stable subgroups in Morse geodesics.

Automatic continuity of polynomial maps and cocycles proved.

Perfect mapping class groups of specific surfaces have no proper subgroups.

Minimal topology on surface homeomorphisms proven.

Study shows compact mapping class groups of infinite type surfaces are never perfect.

In this paper, we consider the formal power series whose n-th coefficient is the number of copies of a given finite graph in the ball of radius n centred at the identity element in the Cayley graph of a finitely generated group and call it the growth function. Epstein, Iano-Fletcher and Uri Zwick proved that the growth…

Autostackability for finitely generated groups is defined via a topological property of the associated Cayley graph which can be encoded in a finite state automaton. Autostackable groups have solvable word problem and an effective inductive procedure for constructing van Kampen diagrams with respect to a canonical fini…

We present a model that can automatically learn alignments between high-dimensional data in an unsupervised manner. Our proposed method casts alignment learning in a framework where both alignment and data are modelled simultaneously. Further, we automatically infer groupings of different types of sequences within the …

We provide an algorithm to solve the word problem in all fundamental groups of closed 3-manifolds; in particular, we show that these groups are autostackable. This provides a common framework for a solution to the word problem in any closed 3-manifold group using finite state automata. We also introduce the notion of a…

Smooth $C^1$ contact maps are always smooth in rigid Carnot groups.

AFR simplifies reducing reliance on spurious features, improving model performance.

We prove that non-elementary hyperbolic groups grow exponentially more quickly than their infinite index quasiconvex subgroups. The proof uses the classical tools of automatic structures and Perron-Frobenius theory. We also extend the main result to relatively hyperbolic groups and cubulated groups. These extensions us…

We propose a novel algorithm for efficiently computing a sparse directed adjacency matrix from a group of time series following a causal graph process. Our solution is scalable for both dense and sparse graphs and automatically selects the LASSO coefficient to obtain an appropriate number of edges in the adjacency matr…

We show that the isomorphism problem is solvable in the class of central extensions of word-hyperbolic groups, and that the isomorphism problem for biautomatic groups reduces to that for biautomatic groups with finite centre. We describe an algorithm that, given an arbitrary finite presentation of an automatic group $Γ…

We show that an isometric action of a compact quantum group on the underlying geodesic metric space of a compact connected Riemannian manifold $(M,g)$ with strictly negative curvature is automatically classical, in the sense that it factors through the action of the isometry group of $(M,g)$. This partially answers a q…

ABM automates feature engineering and variable selection for loss-based models.

The braided Ptolemy-Thompson group $T^*$ is an extension of the Thompson group $T$ by the full braid group $B_{\infty}$ on infinitely many strands. This group is a simplified version of the acyclic extension considered by Greenberg and Sergiescu, and can be viewed as a mapping class group of a certain infinite planar s…

We introduce the notion of an EZ-structure on a group. Delta-hyperbolic groups and CAT(0)-groups have EZ-structures. We show torsion-free groups having an EZ-structure automatically have an action by homeomorphisms on a closed (high-dimensional) ball, which is well-behaved away from a "bad limit set" in the boundary of…

This document is a practical guide to computations using an automatic structure for the mapping class group of a once-punctured, oriented surface $S$. We describe a quadratic time algorithm for the word problem in this group, which can be implemented efficiently with pencil and paper. The input of the algorithm is a wo…

Simple construction of Lie 2-groups from loop group extensions.

The study explores homeomorphism groups of self-similar 2-manifolds, including the 2-sphere and Cantor set.

We propose a privacy-enhanced matrix factorization recommender that exploits the fact that users can often be grouped together by interest. This allows a form of "hiding in the crowd" privacy. We introduce a novel matrix factorization approach suited to making recommendations in a shared group (or nym) setting and the …

New groups prevent certain geometric actions on spaces.

Exponential proportion of pseudo-Anosovs in mapping class groups.

Fed-FEARE model extracts rules from multiple agencies' data securely.

We introduce a financial portfolio optimization framework that allows us to automatically select the relevant assets and estimate their weights by relying on a sorted $\ell_1$-Norm penalization, henceforth SLOPE. Our approach is able to group constituents with similar correlation properties, and with the same underlyin…

The exponential growth of complex data demands fully automatic clustering. Gaussian mixture models (GMMs) provide uncertainty-aware grouping but often require expertise to specify hyperparameters, e.g., component count and covariance structure. While mclust (R) automates this via Bayesian Information Criterion (BIC), P…

Functional groups (FGs) are molecular substructures that are served as a foundation for analyzing and predicting chemical properties of molecules. Automatic discovery of FGs will impact various fields of research, including medicinal chemistry and material sciences, by reducing the amount of lab experiments required fo…

Automatically identifies geometric flat outputs for robotic systems.

This paper considers extractive summarisation in a comparative setting: given two or more document groups (e.g., separated by publication time), the goal is to select a small number of documents that are representative of each group, and also maximally distinguishable from other groups. We formulate a set of new object…

The famous Švarc-Milnor Lemma says that a group $G$ acting properly and cocompactly via isometries on a length space $X$ is finitely generated and induces a quasi-isometry equivalence $g\to g\cdot x_0$ for any $x_0\in X$. We redefine the concept of coarseness so that the proof of the Lemma is automatic.

The compact-open topology is minimal on diffeomorphism and homeomorphism groups of most smooth manifolds.

We have developed an automatic sleep stage classification algorithm based on deep residual neural networks and raw polysomnogram signals. Briefly, the raw data is passed through 50 convolutional layers before subsequent classification into one of five sleep stages. Three model configurations were trained on 1850 polyso…

Following \cite{citeSavelyevVirtualMorsetheoryon$Omega$Ham$(Momega)$.}, we develop here a connection between Morse theory for the (positive) Hofer length functional $L: Ω\text {Ham}(M, ω) \to \mathbb{R}$, with Gromov-Witten/Floer theory, for monotone symplectic manifolds $(M, ω)$. This gives some immediate restrictio…

We consider a problem of grouping multiple graphs into several clusters using singular value thesholding and non-negative factorization. We derive a model selection information criterion to estimate the number of clusters. We demonstrate our approach using "Swimmer data set" as well as simulated data set, and compare i…

Develops platforms to analyze social media data for human behavior and emotions.

In previous work, joint with Bux, Fluch, Marschler and Witzel, we proved that the braided Thompson groups are of type $\textrm{F}_\infty$. The proof utilized certain contractible cube complexes, which in this paper we prove are CAT(0). We then use this fact to compute the geometric invariants $Σ^m(F_{\textrm{br}})$ of …

Study on self-similar surfaces and their mapping class groups generated by involutions.

Study conformal Killing forms on specific nilpotent Lie groups.