arXiv research

Study conic line arrangements of degree 7, finding their topology and connected components.

The paper finds free semigroups in dense subgroups of Lie groups with critical exponents arbitrarily close to the subgroup's.

Two unique conic-line arrangements with degree 9 are found.

Paper finds infinite pairs of fiber-type curves with same topology but different embeddings.

The invariant $\mathcal{I}(\mathcal{A},ξ,γ)$ was first introduced by E. Artal, V. Florens and the author. Inspired by the idea of G. Rybnikov, we obtain a multiplicativity theorem of this invariant under the gluing of two arrangements along a triangle. An application of this theorem is to prove that the extended Rybnik…

The paper explores conic-line arrangements via Poncelet's theorem and finds families of reducible curves.

Proves a representation that separates a subgroup in RAAGs.

The main result of this article is a refinement of the well-known subgroup separability results of Hall and Scott for free and surface groups. We show that for any finitely generated subgroup, there is a finite dimensional representation of the free or surface group that separates the subgroup in the induced Zariski to…

Let ${\mathcal {B}}$ be a reducible reduced plane curve. We introduce a new point of view to study the topology of $(\PP^2, {\mathcal {B}})$ via Galois covers and Alexander polynomials. We show its effectiveness through examples of Zariski $N$-plets for conic and conic-quartic configurations.

We construct a topological invariant of algebraic plane curves, which is in some sense an adaptation of the linking number of knot theory. This invariant is shown to be a generalization of the I-invariant of line arrangements developed by the first author with Artal and Florens. We give two practical tools for computin…

A central question in the study of line arrangements in the complex projective plane $\mathbb{CP}^2$ is: when does the combinatorial data of the arrangement determine its topological properties? In the present work, we introduce a topological invariant of complexified real line arrangements, the chamber weight. This in…

Study conic-line arrangements of degree 7, finding their topology and components.

New invariant detects non-homeomorphic arrangements with similar coefficients.

New invariant identifies complex line arrangements with same combinatorics but different embeddings.

We study 3 basic questions about fundamental groups of algebraic varieties. For a morphism, is being surjective on $π_1$ preserved by base change? What is the connection between openness in the Zariski and in the Euclidean topologies? Which morphisms have the path lifting property?

A $k$-Artal arrangement is a reducible algebraic curve composed of a smooth cubic and $k$ inflectional tangents. By studying the topological properties of their subarrangements, we prove that for $k=3,4,5,6$, there exist Zariski pairs of $k$-Artal arrangements. These Zariki pairs can be distinguished in a geometric way…

Floating geodesic planes in Hitchin manifolds have fractal closures with non-integer dimensions.

Classifies Zariski closures of positive representations in Lie groups.

Study trisections on rational elliptic surfaces to find new Zariski pairs.

The study finds conditions for certain groups to be dense in a specific mathematical space.

We present a new proof of the following theorem of Benoist-Quint: Let $G:=SO^\circ(d,1)$, $d\ge 2$ and $Δ<G$ a cocompact lattice. Any orbit of a Zariski dense subgroup $Γ$ of $G$ is either finite or dense in $Δ\backslash G$. While Benoist and Quint's proof is based on the classification of stationary measures, our proo…

New lattices in higher dimensions have dense surface subgroups.

We show that a surface group contained in a reductive real algebraic group can be deformed to become Zariski dense, unless its Zariski closure acts transitively on a Hermitian symmetric space of tube type. This is a kind of converse to a rigidity result of Burger, Iozzi and Wienhard.

Deforms surface groups to be Zariski dense in SL(n,R)

We prove that every Bers slice of quasi-Fuchsian space is Zariski dense in the character variety.

The paper finds dense subgroups in certain Lie groups.

In this paper, complement-equivalent arithmetic Zariski pairs will be exhibited answering in the negative a question by Eyral-Oka on these curves and their groups. A complement-equivalent arithmetic Zariski pair is a pair of complex projective plane curves having Galois-conjugate equations in some number field whose co…

Paper finds surface groups can deform in reductive symmetric spaces.

We show that in any $\mathbb{Q}$-Gorenstein flat family of klt singularities, normalized volumes are lower semicontinuous with respect to the Zariski topology. A quick consequence is that smooth points have the largest normalized volume among all klt singularities. Using an alternative characterization of K-semistabili…

We begin by showing that commensurators of Zariski dense subgroups of isometry groups of symmetric spaces of non-compact type are discrete provided that the limit set on the Furstenberg boundary is not invariant under the action of a (virtual) simple factor. In particular for rank one or simple Lie groups, Zariski dens…

Bi-Lipschitz rigidity theorem for dense subgroups of algebraic groups.

Generic Hitchin representations generate dense subgroups.

Maximal representations in symplectic lattices proven for most cases.

We study the Abel-Jacobi map for bisections of a certain rational elliptic surface. As an application, we construct examples of Zariski $N$-plets for conic arrangements.

Proves unique maps from certain spaces to others.

Let $S$ be an elliptic surface over a smooth curve $C$ with a section $O$. We denote its generic fiber by $E_S$. For a divisor $D$ on $S$, we canonically associate a $C(C)$-rational point $P_D$. In this note, we give a description of $P_D$ of $E_S$, when the rank of the group of $C(C)$-rational points is one. We apply …

Character variety of Whitehead link described in detail.

The study explores deformations of discrete subgroups in non-compact homogeneous spaces.

We prove that any flat family $(\mathcal{ F}_u)_{u\in U}$ of rank 2 torsion-free sheaves on a Gauduchon surface defines a continuous map on the semi-stable locus $U^{\mathrm {ss}}:=\{u\in U \ |\ \mathcal{ F}_u\hbox{ is slope semi-stable}\}$ with values in the Donaldson-Uhlenbeck compactification of the corresponding in…

Hyperbolic groups' infinite orbits spread evenly in spaces.

Study of translation covers of platonic solids reveals monodromy group structures.

Complex projective manifolds without rational curves are quotients of Abelian varieties.

Study Cremona transformations in weighted projective planes to find rational cuspidal curves and Zariski pairs.

We define the higher-order Alexander modules $A_{n,i}(\mathcal{U})$ and higher-order degrees $δ_{n,i}(\mathcal{U})$ which are invariants of a complex hypersurface complement $\mathcal{U}$. These invariants come from the module structure of the homology of certain solvable covers of the hypersurface complement. Such inv…

We introduce and study a new class of representations of surface groups into Lie groups of Hermitian type, called weakly maximal representations. They are defined in terms of invariants in bounded cohomology and extend considerably the scope of maximal representations. We prove that weakly maximal representations are d…

Odd-dimensional SL(n,Q) contains dense surface subgroups.

Using the invariant developed in [6], we differentiate four arrangements with the same combinatorial information but in different deformation classes. From these arrangements, we construct four other arrangements such that there is no orientation-preserving homeomorphism between them. Furthermore, some couples of arran…

The study explores deformations of standard locally homogeneous spaces.