arXiv research

Algorithm classifies saddle-focus singularities in Hamiltonian systems.

We summarize properties of 3-manifold groups, with a particular focus on the consequences of the recent results of Ian Agol, Jeremy Kahn, Vladimir Markovic and Dani Wise.

The paper explores mapping class groups and their quantum field theory representations.

Survey of gauge theory for families of 4-manifolds.

We focus our attention on the notion of intrinsic Lipschitz graphs, inside a special class of metric spaces i.e. the Carnot groups. More precisely, we provide a characterization of locally intrinsic Lipschitz functions in Carnot groups of step 2 in terms of their intrinsic distributional gradients.

Survey of combination theorems in geometry and dynamics.

We study properties of a generalization of the Mahler measure to elements in group rings, in terms of the Lueck-Fuglede-Kadison determinant. Our main focus is the variation of the Mahler measure when the base group is changed. In particular, we study how to obtain the Mahler measure over an infinite group as limit of M…

We study a normalized version of the second order renormalization group flow on closed Riemannian surfaces. We discuss some general properties of this flow and establish several basic formulas. In particular, we focus on surfaces with zero and positive Euler characteristic.

Survey on handlebody groups and their properties.

We focus on the cohomology of the $k$-th nilpotent quotient of the free group, $F/F_k$. This paper describes all the group 2-, 3-cocycles in terms of Massey products, and gives expressions for some of the 3-cocycles. We also give simple proofs of some of the results on Milnor invariants and the Johnson-Morita homomorph…

We study a family of spheres with constant mean curvature (CMC) in the Riemannian Heisenberg group $H^1$. These spheres are conjectured to be the isoperimetric sets of $H^1$. We prove several results supporting this conjecture. We also focus our attention on the sub-Riemannian limit.

Study the monodromy and center-focus problems for rational maps defined by products of generic lines.

The study explores normal generators for mapping class groups and their properties.

In this article, we study connections between representation theory and efficient solutions to the conjugacy problem on finitely generated groups. The main focus is on the conjugacy problem in conjugacy separable groups, where we measure efficiency in terms of the size of the quotients required to distinguish a distinc…

Following previous work of the second author, we establish more properties of groups of circle homeomorphisms which admit invariant laminations. In this paper, we focus on a certain type of such groups-so-called pseudo-fibered groups, and show that many 3-manifold groups are examples of pseudo-fibered groups. We then p…

Maps in Carnot groups are equivalent to solutions of a PDE system.

Universal inequalities for Laplacian eigenvalues on discrete groups.

Study deformation spaces of irregular isomonodromy systems on Riemann surfaces.

These lecture notes concern the algebraic geometry of the character variety of a finitely generated group in SL(2,C) from the point of view of skein modules. We focus on the case of surface and 3-manifolds groups and construct the Reidemeister torsion as a rational volume form on the character variety.

Braid groups help create complex surfaces in algebraic geometry.

For a Coxeter group $W$ we have an associating bi-linear form $B$ on a real vector space. We assume that $B$ has the signature $(n-1,1)$. In this case we have the Cannon-Thurston map for $W$, that is, a $W$-equivariant continuous surjection from the Gromov boundary of $W$ to the limit set of $W$. We focus on the case w…

We present an algebraic method to study four-dimensional toric varieties by lifting matrix equations from the special linear group ${\rm SL}_2({\mathbb Z})$ to its preimage in the universal cover of ${\rm SL}_2({\mathbb R})$. With this method we recover the classification of two-dimensional toric fans, and obtain a des…

Study on curvatures of surfaces in specific Lie groups.

Theory of relatively Anosov representations using flow methods.

Study on mappings between nonrigid Carnot groups, proving quasisymmetric rigidity.

Study on spheres in simply-connected 4-manifolds with abelian complements.

Researchers describe and compare decompositions of Poincaré duality pairs.

Quantum theory uses modular group representations to assign invariants to 3-manifolds.

We study junctions of Wilson lines in refined SU(N) Chern-Simons theory and their local relations. We focus on junctions of Wilson lines in antisymmetric and symmetric powers of the fundamental representation and propose a set of local relations which realize one-parameter deformations of quantum groups $\dot{U}_{q}(\m…

A smooth five-dimensional s-cobordism becomes a smooth product if stabilized by a finite number n of $S^2xS^2x[0,1]$'s. We show that for amenable fundamental groups, the minimal n is subextensive in covers, i.e., n(cover)/index(cover) has limit 0. We focus on the notion of sweepout width, which is a bridge between 4-di…

Torelli group cannot be realized as area-preserving homeomorphisms.

We focus on two kinds of infinite index subgroups of the mapping class group of a surface associated with a Lagrangian submodule of the first homology of a surface. These subgroups, called Lagrangian mapping class groups, are known to play important roles in the interaction between the mapping class group and finite-ty…

Study on Seifert 4-manifolds to classify profinite rigidity.

We propose a program to study groups acting faithfully on S^1 in terms of number of pairwise transverse dense invariant laminations. We give some examples of groups which admit a small number of invariant laminations as an introduction to such groups. Main focus of the present paper is to characterize Fuchsian groups i…

Study presentations of groups that can be generalised over continuous open group monomorphisms.

Overview of dynamics in algebraic correspondences and their connections.

Study on moduli space of bi-invariant metrics in Lie groups.

The paper classifies homogeneous hypersurfaces in three 4D Thurston geometries.

The paper explores invariant vs non-invariant complex structures on Lie groups.

Abstract: Extends Drinfeld correspondence to infinite-dimensional Lie groups.

Study on subgroups' evolution in 3D Lie groups using mean curvature flow.

The book explores Lie groups and Carnot-Carathéodory spaces, highlighting their applications in metric geometry and geometric group theory.

We study singular hyperkahler quotients of the cotangent bundle of a complex semisimple Lie group as stratified spaces whose strata are hyperkahler. We focus on one particular case where the stratification satisfies the frontier condition and the partial order on the set of strata can be described explicitly by Lie the…

In this paper we study the groups of contactomorphisms of a closed contact manifold from a topological viewpoint. First we construct examples of contact forms on spheres whose Reeb flow has a dense orbit. Then we show that the unitary group U(n+1) is homotopically essential in the group of contactomorphisms of the stan…

Adds examples to Goeritz groups for a specific type of 3-manifold splitting.

This article is about Artin's braid group and its role in knot theory. We set ourselves two goals: (i) to provide enough of the essential background so that our review would be accessible to graduate students, and (ii) to focus on those parts of the subject in which major progress was made, or interesting new proofs of…

Study bi-graded Lie algebras and their applications.

Researchers derive $q$-series for $SO(3)$ and $OSp(1|2)$ groups.