arXiv research

Extends tangle theory to include undetermined crossings in periodic structures.

Algorithm decides if pseudo-Anosov flows have perfect fits.

We apply Cartan's method of equivalence to find a covering for the modified Khokhlov - Zabolotskaya equation.

Studied $L^2-$invariants on stratified spaces, proving their stability.

We study conformal actions of connected nilpotent Lie groups on compact pseudo-Riemannian manifolds. We prove that if a type-(p,q) compact manifold M supports a conformal action of a connected nilpotent group H, then the degree of nilpotence of H is at most 2p+1, assuming p <= q; further, if this maximal degree is atta…

We first prove rigidity results for pseudo-Anosov flows in prototypes of toroidal 3-manifolds: we show that a pseudo-Anosov flow in a Seifert fibered manifold is up to finite covers topologically equivalent to a geodesic flow and we show that a pseudo-Anosov flow in a solv manifold is topologically equivalent to a susp…

For each surface $S$ of genus $g>2$ we construct pairs of conjugate pseudo-Anosov maps, $\varphi_1$ and $\varphi_2$, and two non-equivalent covers $p_i: \tilde S \longrightarrow S$, $i=1,2$, so that the lift of $\varphi_1$ to $\tilde S$ with respect to $p_1$ coincides with that of $\varphi_2$ with respect to $p_2$. The…

In hyperbolic L-spaces, we find multiple pseudo-Anosov flows with unique properties.

We apply the technique of integrable extensions to the symmetry pseudo-group of the r-th mdKP equation. This gives another look on deriving known coverings and allows us to find new coverings for this equation.

We discuss a relation between deformed cohomologies of symmetry pseudo-groups and coverings of differential equations. Examples include the potential Khokhlov--Zabolotskaya equation and the Boyer--Finley equation.

A foliation is R-covered if the leaf space in the universal cover is homeomorphic to the real numbers. We show that, up to topological conjugacy, there are at most two pseudo-Anosov flows transverse to such a foliation. If there are two, then the foliation is weakly conjugate to the the stable foliation of an R-covered…

We derive Wahlquist - Estabrook forms of the covering of Plebanski's second heavenly equation from Maurer - Cartan forms of its symmetry pseudo-group.

Tiny complexes share 3-5 triangles in common coverings.

Horizontal surgery on pseudo-Anosov flows yields almost equivalent flows.

By combining the ideas of Cartan's equivalence method and the method of the equivariant moving frame for pseudo-groups, we develop an efficient method for solving equivalence problems arising from horizontal Lie pseudo-group actions. The key is a pseudo-group analog of the classic result that characterizes congruence o…

Two pseudo-Riemannian metrics $g$ and $\bar g$ are geodesically equivalent, if they share the same (unparameterized) geodesics. We give a complete local description of such metrics which solves the natural generalisation of Beltrami problem for pseudo-Riemannian metrics.

In this note, we show that, if a pseudo-Anosov map $φ:S\to S$ admits a finite cover whose action on the first homology has spectral radius greater than $1$, then the monodromy of any fibered structure of any finite cover of the mapping torus $M_φ$ has the same property.

The paper studies global invertibility of maps on Finsler manifolds.

Solve Beltrami problem in dimension two

Paper defines and analyzes mathematical equivalence of periodic tangles.

The paper studies entropy in branched covers of 3-manifolds.

Han discusses variants of digital covering maps and their equivalences.

New method shows pseudo-Anosov flows on graph manifolds can be simplified.

Study of flows on 7D manifolds with holomorphic properties.

New insights into pseudo-Anosov flows with special periodic orbits.

New bounds on homological eigenvalues relate to Weil-Petersson length.

Odd covers have one Anosov flow, even covers have two.

We find contact integrable extensions and coverings for the r-th double modified dispersionless Kadomtsev--Petviashvili equation.

The paper finds pseudo-Anosov-like maps on an infinite ladder surface.

We construct local normal forms of pseudo-Riemannian projectively equivalent 2-dimensional metrics.

Minimal constructions of meanders and hyperelliptic pillowcase covers help in understanding ratio-optimizing pseudo-Anosovs.

The paper introduces colorings and invariants for twisted links and shows how double coverings can be equivalent.

We construct the first known examples of compact pseudo-Riemannian manifolds having an essential group of conformal transformations, and which are not conformally flat. Our examples cover all types $(p,q)$, with $2 \leq p \leq q$.

In this article, we summarize the results on symmetric conformal geometries. We review the results following from the general theory of symmetric parabolic geometries and prove several new results for symmetric conformal geometries. In particular, we show that each symmetric conformal geometry is either locally flat or…

Study Agol cycles for pseudo-Anosov 3-braids.

Let $Σ$ be a compact orientable surface of finite type with at least one boundary component. Let $f \in \textup{Mod}(Σ)$ be a pseudo Anosov mapping class. We prove a conjecture of McMullen by showing that there exists a finite cover $\widetildeΣ \to Σ$ and a lift $\widetilde{f}$ of $f$ such that $\wt{f}_*: H_1(\wtΣ; \m…

We prove global results about actions of cocompact lattices in higher-rank simple Lie groups on closed manifolds endowed with either a projective class of connections or a conformal class of pseudo-Riemannian metrics of signature $(p,q)$, with $\min(p,q) \geq 2$. In the continuity of a recent article, provided that suc…

In this paper we provide a negative answer to a question of Farb about the relation between the algebraic degree of the stretch factor of a pseudo-Anosov homeomorphism and the genus of the surface on which it is defined.

Study the topology of Ricci limit spaces using Gromov-Hausdorff limits.

Generalizes covering lemmas in metric spaces, finding equivalent properties.

Floer cohomology is computed for certain elements of the mapping class group of a surface $Σ$ of genus $g>1$ which are compositions of positive and negative dehn twists along some loops in $Σ$. The computations cover a certain class of pseudo-Anasov maps.

Characterizes pseudo-Anosov orbit spaces via bifoliated planes

The paper proves left-orderability for certain Dehn fillings of pseudo-Anosov mapping tori.

Characterizes covers using simple closed curves on surfaces.

We prove an important partial case of the pseudo-Riemannian version of the projective Lichnerowicz conjecture stating that a complete manifold admitting an essential group of projective transformations is the round sphere (up to a finite cover).

Study multiplicity-free covering of graded manifolds, proving equivalence of categories.

Proves harmonicity equivalence on manifold metrics.

We apply the technique of integrable extensions to the symmetry pseudo-group of the dKP-hyper CR interpolating equation. This allows us to find a covering for this equation and to construct multi-valued Einstein-Weyl structures.