arXiv research

When a solenoid is embedded in three space, its complement is an open three manifold. We discuss the geometry and fundamental groups of such manifolds, and show that the complements of different solenoids (arising from different inverse limits) have different fundamental groups. Embeddings of the same solenoid can give…

Study shows mapping class groups differ for $h$-cobordant manifolds.

New 2-knots found with same knot group but different quandles.

New method identifies differences between groups in low-dimensional data representations.

New filling functions for groups with coefficients show different asymptotic behavior.

Feature noise causes loss discrepancies across groups even with equal data.

Clustering groups similar data points into clusters.

The homological and homotopical Dehn functions are different ways of measuring the difficulty of filling a closed curve inside a group or a space. The homological Dehn function measures fillings of cycles by chains, while the homotopical Dehn function measures fillings of curves by disks. Since the two definitions invo…

The difference between slice and doubly-slice knots is reflected in algebra by the difference between metabolic and hyperbolic Blanchfield linking forms. We exploit this algebraic distinction to refine the classical Witt group of linking forms by defining a `double Witt group' of linking forms. We calculate the double …

The paper classifies Ricci solitons on specific Lorentzian Lie groups.

Given a symmetry $τ$ of a closed Riemann surface $S$, there exists an extended Kleinian group $K$, whose orientation-preserving half is a Schottky group $Γ$ uniformizing $S$, such that $K/Γ$ induces $\langle τ\rangle$; the group $K$ is called an extended Schottky group. A geometrical structural description, in terms of…

Anosov subgroups generalize convex-cocompact groups in hyperbolic geometry.

Choosing a reference group in Oaxaca-Blinder decomposition can reverse conclusions.

Study geodesic Lie groups' convergence to limits with quantitative estimates.

Study shows different fundamental groups for manifolds with same limit.

We propose a unified framework in which the different constructions of cohomology groups for topological and Lie groups can all be treated on equal footings. In particular, we show that the cohomology of "locally continuous" cochains (respectively "locally smooth" in the case of Lie groups) fits into this framework, wh…

Unified study of homological representations of mapping class groups.

Study develops a new method for creating fair models.

There are classification tasks that take as inputs groups of images rather than single images. In order to address such situations, we introduce a nested multi-instance deep network. The approach is generic in that it is applicable to general data instances, not just images. The network has several convolutional neural…

Hecke's theorem on different generalized to 3-manifolds.

An extended Kleinian group whose orientation-preserving half is a Schottky group is called an extended Schottky group. These groups correspond to the real points in the Schottky space. Their geometric structures is well known and it permits to provide information on the locus of fixed points of symmetries of handlebodi…

Doubly fair dynamic pricing ensures equal prices for different groups over time.

We construct a map from fundamental groups of complements to some plane configurations to the groups $G_{n}^{k}$ for large $k$. We discuss connection between the groups $G_{n}^{k}$ for different $G_{n}^{k}$ and their geometric realization.

We generalize the asymptotic faithfulness of the skein quantum $SU(2)$ representations of mapping class groups of orientable closed surfaces to skein $SU(3)$. Skein quantum representations of mapping class groups are different from the Reshetikin-Turaev ones from quantum groups or geometric quantization because they ar…

Topological surgery in dimension $3$ is intrinsically connected with the classification of $3$-manifolds and with patterns of natural phenomena. In this expository paper, we present two different approaches for understanding and visualizing the process of $3$-dimensional surgery. In the first approach, we view the proc…

A new approach to group fairness treats it as a bargaining problem.

MultiDendrograms is a Java-written application that computes agglomerative hierarchical clusterings of data. Starting from a distances (or weights) matrix, MultiDendrograms is able to calculate its dendrograms using the most common agglomerative hierarchical clustering methods. The application implements a variable-gro…

GMLP learns feature groups for tabular data without known structure.

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

CST detects discrimination by comparing protected and non-protected individuals with a counterfactual.

New method measures treatment effects across different groups.

Study of Sasaki groups vs Kähler groups, showing distinct behaviors.

Classifies Morse boundaries of 3-manifold groups.

In this paper we introduce the framed pure braid group on $n$ strands of an oriented surface, a topological generalisation of the pure braid group $P_n$. We give different equivalents definitions for framed pure braid groups and we study exact sequences relating these groups with other generalisations of $P_n$, usually…

New groups with distinct Dehn functions and properties.

We propose a novel end-to-end neural network architecture that, once trained, directly outputs a probabilistic clustering of a batch of input examples in one pass. It estimates a distribution over the number of clusters $k$, and for each $1 \leq k \leq k_\mathrm{max}$, a distribution over the individual cluster assignm…

AgABC improves ABC algorithm by balancing exploration and exploitation.

The paper introduces group-representative clustering to ensure fair representation of different groups in clusters.

Recent results in coupled or temporal graphical models offer schemes for estimating the relationship structure between features when the data come from related (but distinct) longitudinal sources. A novel application of these ideas is for analyzing group-level differences, i.e., in identifying if trends of estimated ob…

Clustering partitions a dataset such that observations placed together in a group are similar but different from those in other groups. Hierarchical and $K$-means clustering are two approaches but have different strengths and weaknesses. For instance, hierarchical clustering identifies groups in a tree-like structure b…

New method finds sparse groups of input variables for neural networks.

There are many statistical tests that verify the null hypothesis: the variable of interest has the same distribution among k-groups. But once the null hypothesis is rejected, how to present the structure of dissimilarity between groups? In this article, we introduce The Merging Path Plot - a methodology, and factorMerg…

Datasets containing large samples of time-to-event data arising from several small heterogeneous groups are commonly encountered in statistics. This presents problems as they cannot be pooled directly due to their heterogeneity or analyzed individually because of their small sample size. Bayesian nonparametric modellin…

Study virtual braid groups, proving a key subgroup result.

We construct a smooth Lie group structure on the group of real analytic diffeomorphisms of a compact analytic manifold with corners. This generalises the known analogous results in the situation where the real analytic manifold has no corners. Additionally our approach uses a different construction.

Using the Geometrization Theorem we prove a result on centralizers in fundamental groups of 3-manifolds. This result had been obtained by Jaco and Shalen and by Johannson using different techniques.

Automorphisms of Riemann surfaces with specific properties are studied.

Study reveals uniform difference in stretch factors between genus two handlebody group and outer automorphism group.