arXiv research

Classifies and constructs all extendable automorphisms of closed surfaces over the 3-sphere.

Study shows automorphisms of Markov surfaces share periodic points if they share a common iterate.

Classifies periodic diffeomorphisms on surfaces commuting with specific involutions.

Paper disproves Wright's periodic map conjecture.

Finite group actions on surfaces extend to 3-manifolds.

Algorithm finds periodic points on Veech surfaces.

Study shows periodic points of Prym eigenforms in specific genera.

We show that every automorphism $α$ of a free group $F_k$ of finite rank $k$ has {\it asymptotically periodic} dynamics on $F_k$ and its boundary $\partial F_k$: there exists a positive power $α^q$ such that every element of the compactum $F_k \cup \partial F_k$ converges to a fixed point under iteration of $α^q$.

A graph $G$ is said to be $p$-periodic, if the automorphism group $Aut(G)$ contains an element of order $p$ which preserves no edges. In this paper, we investigate the behavior of graph polynomials (Negmai and Tutte) with respect to graph periodicity. In particular, we prove that if $p$ is a prime, then the coefficient…

Automorphisms of surfaces extendable over 4-sphere with invariant spin structures.

The mapping class group of a Heegaard splitting is the group of automorphisms of the ambient 3-manifold that take the surface onto itself, modulo isotopies that keep the surface on itself. We characterize the mapping classes that restrict to periodic and reducible automorphisms of the surface.

We provide an effective algorithm for determining whether an element of the outer automorphism group of a free group is fully irreducible. Our method produces a finite list which can be checked for periodic proper free factors.

Decomposes axis bundles into cubist structures for fully irreducible outer automorphisms.

Various Seiberg-Witten Floer cohomologies are defined for a closed, oriented 3-manifold; and if it is the mapping torus of an area-preserving surface automorphism, it has an associated periodic Floer homology as defined by Michael Hutchings. We construct an isomorphism between a certain version of Seiberg-Witten Floer …

Tête-à-tête graphs were introduced by N. A'Campo in 2010 with the goal of modeling the monodromy of isolated plane curves. Mixed tête-à-tête graphs provide a generalization which define mixed tête-à-tête twists, which are pseudo-periodic automorphisms on surfaces. We characterize the mixed tête-à-tête twists as those p…

We classify isotopy classes of automorphisms (self-homeomorphisms) of 3-manifolds satisfying the Thurston Geometrization Conjecture. The classification is similar to the classification of automorphisms of surfaces developed by Nielsen and Thurston, except an automorphism of a reducible manifold must first be written as…

The thesis shows how automorphisms of hyperbolic groups can be represented by train track maps.

Embeddings of mapping tori for end-periodic graph maps are proven.

The study examines the stretch factors of outer automorphisms and their latent symmetry.

New periodic solutions found in 2n-body problem, braids of pseudo-Anosov type with stretch factors as metallic ratios.

We define the Kobayashi quotient of a complex variety by identifying points with vanishing Kobayashi pseudodistance between them and show that if a compact complex manifold has an automorphism whose order is infinite, then the fibers of this quotient map are nontrivial. We prove that the Kobayashi quotients associated …

Two flows are topologically almost commensurable if, up to removing finitely many periodic orbits and taking finite coverings, they are topologically equivalent. We prove that all suspensions of automorphisms of the 2-dimensional torus and all geodesic flows on unit tangent bundles to hyperbolic 2-orbifolds are pairwis…

In this note, we establish a relationship between fractional Dehn twist coefficients of Riemann surface automorphisms and modular invariants of holomorphic families of algebraic curves. Specially, we give a characterization of pseudo-periodic maps with nontrivial fractional Dehn twist coefficients. We also obtain some …

The study explores automorphisms and centralizers in free group outer automorphisms.

The (torsion) complexity of a finite edge-weighted graph is defined to be the order of the torsion subgroup of the abelian group presented by its Laplacian matrix. When G is d-periodic (i.e., G has a free action of the rank-d free abelian group by graph automorphisms, with finite quotient) the Mahler measure of its Lap…

Complexity of signed graphs linked to Alexander polynomials and Lehmer's question.

In this paper, we will construct an example of a closed Riemann surface $X$ that can be realized as a quotient of a triply periodic polyhedral surface $Π\subset \mathbb{R}^3$ where the Weierstrass points of $X$ coincide with the vertices of $Π.$ First we construct $Π$ by attaching Platonic solids in a periodic manner a…

The study examines aperiodicity properties of automorphism groups of free products of groups.

Study of quadratic form associated with surface automorphisms and its applications to singularity theory.

A natural family of affine cubic surfaces arises from SL(2)-characters of the 4-holed sphere and the 1-holed torus. The ideal locus is a tritangent plane which is generic in the sense that the cubic curve at infinity consists of three lines pairwise intersecting in three double points. We show that every affine cubic s…

Classifies charge-3 monopoles with symmetry, identifying new spectral curves.

We study the one parameter family of genus 2 Riemann surfaces defined by the orbit of the L-shaped translation surface tiled by three squares under the Teichmüller geodesic flow. These surfaces are real algebraic curves with three real components. We are interested in describing these surfaces by their period matrices.…

Geometric quantization extended to arbitrary connected spaces using path integration.

We develop a global Poincaré residue formula to study period integrals of families of complex manifolds. For any compact complex manifold $X$ equipped with a linear system $V^*$ of generically smooth CY hypersurfaces, the formula expresses period integrals in terms of a canonical global meromorphic top form on $X$. Two…

Study Swan modules and homotopy types, resolving Wall and Dyer questions.

We present a technique for the enumeration of all isotopically distinct ways of tiling a hyperbolic surface of finite genus, possibly nonorientable and with punctures and boundary. This provides a generalization of the enumeration of Delaney-Dress combinatorial tiling theory on the basis of isotopic tiling theory. To a…

Two markets should be considered isomorphic if they are financially indistinguishable. We define a notion of isomorphism for financial markets in both discrete and continuous time. We then seek to identify the distinct isomorphism classes, that is to classify markets. We classify complete one-period markets. We define …

Automorphisms of handlebodies arise naturally in the a classification of automorphisms of three-manifolds. Among automorphisms of handlebodies, there are certain automorphisms called irreducible (or generic), which are analogues of pseudo-Anosov automorphisms of surfaces. We show that irreducible automorphisms of handl…

Study shows RAAG automorphisms and outer automorphisms are not relatively hyperbolic.

Study knot invariants using automorphism groups of free nilpotent groups.

Constructs Cartan geometries from automorphism behaviors.

In this paper, we investigate the structure of the automorphism groups of pure braid groups. We prove that, for $n>3$, $\Aut(P_n)$ is generated by the subgroup $\Aut_c(P_n)$ of central automorphisms of $P_n$, the subgroup $\Aut(B_n)$ of restrictions of automorphisms of $B_n$ on $P_n$ and one extra automorphism $w_n$. W…

Automorphism groups of Hopf manifolds are finite and have a bounded order.

Automorphisms of pants complex are shown to be inner.

Study of modular representations in homology of congruence subgroups.

Finite groups can be automorphism groups of translation surfaces with poles.

Automorphisms of Lie algebras and their root systems are fully lifted.

Parabolic automorphisms on hyperkahler manifolds act ergodically on fibers.