arXiv research

Computes the component group of real reductive groups.

Identifies Anosov representations of hyperbolic triangle groups in SL(3,R).

Study topological components of surface group representations into SL(2,R) and PSL(2,R).

Computes the component group of arbitrary real algebraic groups.

Survey on Higgs bundle moduli spaces and their connected components.

We consider the "intrinsic" symmetry group of a two-component link $L$, defined to be the image $Σ(L)$ of the natural homomorphism from the standard symmetry group $\MCG(S^3,L)$ to the product $\MCG(S^3) \cross \MCG(L)$. This group, first defined by Whitten in 1969, records directly whether $L$ is isotopic to a link $L…

Research examines the distribution of curve components in random multicurves.

Lifts isometries in orbit spaces for compact groups.

Connected components of Morse boundaries are studied in graph of groups.

Surveying Hitchin representations of Fuchsian groups.

In this paper we describe the space of maximal components of the character variety of surface group representations into PSp(4,R) and Sp(4,R). For every rank 2 real Lie group of Hermitian type, we construct a mapping class group invariant complex structure on the maximal components. For the groups PSp(4,R) and Sp(4,R),…

We prove that the identity component of the holomorphic isometry group of a Sasaki-Einstein metric is the identity component of a maximal compact subgroup of its automorphism group.

Algorithm finds connected components on Lie groups for multi-orientation image analysis.

Explicit computation of symplectic form for $\mathrm{PGL}_n(\mathbb{R})$-Hitchin component.

We extend the notion of Hitchin component from surface groups to orbifold groups and prove that this gives new examples of higher Teichmüller spaces. We show that the Hitchin component of an orbifold group is homeomorphic to an open ball and we compute its dimension explicitly. We then give applications to the study of…

We propose a new method for supervised learning, especially suited to wide data where the number of features is much greater than the number of observations. The method combines the lasso ($\ell_1$) sparsity penalty with a quadratic penalty that shrinks the coefficient vector toward the leading principal components of …

Study the first homology group for a specific mapping class group.

We discuss a general procedure for using characteristic classes to study the components of the gauge group for a principal G-bundle. To illustrate this, we work out the case where G is the projective unitary group.

Study of intrinsic symmetry groups of links, finding counterexamples.

Compactifies maximal component of surface group representations into a closed ball.

Study links using quandles and groups, proving key properties.

Paper proves Torelli group's finiteness for surfaces with 2 boundaries.

This note proves properties of surface-links with trivial components.

We prove that each Torelli group of an orientable surface with any number of boundary components is at least exponentially distorted in the mapping class group by using Broaddus-Farb-Putman's techniques. Further we show that the distortion of each Torelli group in the level $d$ mapping class group is the same as that o…

New bounds on sample size for identifying mixture models with grouped samples.

The automorphisms group of the 3-dimensional Reeb component with complex leaves is computed in the case where the component is obtained by the Hopf construction and the holonomy of the boundary leaf is not tangent to the identity to the infinite order. Combined with a previous work, for 3-dimensional Reeb components ob…

Let H be a closed, noncompact subgroup of a simple Lie group G, such that G/H admits an invariant Lorentz metric. We show that if G = SO(2,n), with n > 2, then the identity component of H is conjugate to the identity component of SO(1,n). Also, if G = SO(1,n), with n > 2, then the identity component of H is conjugate t…

In this paper we compute a presentation for the group of ring motions of the split union of a Hopf link with Euclidean components and a Euclidean circle. A key part of this work is the study of a short exact sequence of groups of ring motions of general ring links in $\mathbb{R}^3$. This sequence allowed us to build th…

Algorithm estimates nonparametric mixtures from grouped data.

Goldman symplectic form and complex structure compatible on $\mathrm{SL}(3,\mathbb R)$ Hitchin component.

We show that the last few components in principal component analysis of the correlation matrix of a group of stocks may contain useful financial information by identifying highly correlated pairs or larger groups of stocks. The results of this type of analysis can easily be included in the information an investor uses …

Study pressure metrics for cusped Hitchin representations.

A new method for fair PCA ensures balanced error across groups.

We review the standard Hopf construction of Reeb components with leafwise complex structure and determine the group of leafwise holomorphic smooth automorphisms for tame Reeb components in the case of complex leaf dimension one. For this, we solve the Schröder type functional equation on the half line for expanding dif…

Proposes a boosting framework for sparsity in grouped covariates.

The group of volume preserving diffeomorphisms, the group of symplectomorphisms and the group of contactomorphisms constitute the classical groups of diffeomorphisms. The first homology groups of the compactly supported identity components of the first two groups have been computed by Thurston and Banyaga, respectively…

When estimating finite mixture models, it is common to make assumptions on the mixture components, such as parametric assumptions. In this work, we make no distributional assumptions on the mixture components and instead assume that observations from the mixture model are grouped, such that observations in the same gro…

Finite mixture models are statistical models which appear in many problems in statistics and machine learning. In such models it is assumed that data are drawn from random probability measures, called mixture components, which are themselves drawn from a probability measure P over probability measures. When estimating …

This paper describes connected components of the strata of holomorphic abelian differentials on marked Riemann surfaces with prescribed degrees of zeros. Unlike the case for unmarked Riemann surfaces, we find there can be many connected components, distinguished by roots of the cotangent bundle of the surface. In the c…

The paper identifies a component of representations mapping modular group elements to isometries with unique fixed points.

Given a closed, oriented surface X of genus g>1, and a semisimple Lie group G, let R_G be the moduli space of reductive representations of the fundamental group of X in G. We determine the number of connected components of R_PGL(n,R), for n>=4 even. In order to have a first division of connected components, we first cl…

Unified proof of Teichmüller components using robust submanifolds.

In the present paper, we define an invariant of free links valued in a free product of some copies of $\mathbb{Z}_{2}$. In \cite{Ma2} the second named author constructed a connection between classical braid group and group presentation generated by elements corresponding to horizontal trisecants. This approach does not…

The linear slice of quasi-Fuchsian once-punctured torus groups is defined by fixing the complex length of some simple closed curve to be a fixed positive real number. It is known that the linear slice is a union of disks, and it always has one standard component containing Fuchsian groups. Komori and Yamashita proved t…

New hyperbolic 3-pseudomanifolds with unique properties.

Study on string links invariant under associator choice and Grothendieck--Teichmüller group action.

New method counts link components from Thompson group elements.

Many data-driven approaches exist to extract neural representations of functional magnetic resonance imaging (fMRI) data, but most of them lack a proper probabilistic formulation. We propose a group level scalable probabilistic sparse factor analysis (psFA) allowing spatially sparse maps, component pruning using automa…