arXiv research

The minimal stratum in Prym loci have been the first source of infinitely many primitive, but not algebraically primitive Teichmueller curves. We show that the stratum Prym(2,1,1) contains no such Teichmueller curve and the stratum Prym(2,2) at most 92 such Teichmueller curves. This complements the recent progress esta…

This paper is devoted to the classification of connected components of Prym eigenform loci in the strata H(2,2)^odd and H(1,1,2) in the Abelian differentials bundle in genus 3. These loci, discovered by McMullen are GL^+(2,R)-invariant submanifolds (of complex dimension 3) that project to the locus of Riemann surfaces …

We prove that for each discriminant $D \equiv 0,1 \mod 4, D \not\in\{4,9\}$, the corresponding Prym eigenform locus discovered by McMullen in the stratum $\mathcal{H}(6)$ is connected. Thus, the projection of any of those loci in the moduli space is a single Teichmüller curve. Along the way, we obtain a classification …

Computes constants for specific geometric structures.

This paper is devoted to the classification of GL^+(2,R)-orbit closures of surfaces in the intersection of the Prym eigenform locus with various strata of quadratic differentials. We show that the following dichotomy holds: an orbit is either closed or dense in a connected component of the Prym eigenform locus. The pro…

The paper calculates the size of origami orbit graphs in complex surfaces.

This paper deals with Prym eigenforms which are introduced previously by McMullen. We prove several results on the directional flow on those surfaces, related to complete periodicity (introduced by Calta). More precisely we show that any homological direction is algebraically periodic, and any direction of a regular cl…

We study the rational Picard group of the projectivized moduli space of holomorphic n-differentials on complex genus g stable curves. We define (n - 1) natural classes in this Picard group that we call Prym-Tyurin classes. We express these classes as linear combinations of boundary divisors and the divisor of n-differe…

Origami graphs' Euler characteristics grow as origami complexity increases.

We calculate the Masur-Veech volume of the gothic locus $\mathcal{G}$ in the stratum $\mathcal{H}(2^{3})$ of genus four. Our method is based on the use of the formulae for the Euler characteristics of gothic Teichmüller curves to determine the number of lattice points of given area. We also use this method to recalcula…

No primitive Teichmüller curves found in Prym(2,2).

Study shows periodic points of Prym eigenforms in specific genera.

Researchers compute cohomology of mapping class groups with Prym representations, showing instability for large genus.

Unique map connects curve pairs to abelian varieties.

We survey Lagrangian fibrations of holomorphic symplectic varieties, both compact and non-compact, whose fibres are Jacobians and Prym varieties.

Researchers study Prym representations for handlebody groups, focusing on cyclic cases.

Prym-Teichmüller curves $W_D(4)$ constitute the main examples of known primitive Teichmüller curves in the moduli space $\mathcal{M}_3$. We determine, for each non-square discriminant $D>1$, the number and type of orbifold points in $W_D(4)$. These results, together with the formulas of Lanneau-Nguyen and Möller for th…

Given a Prym-Teichmüller curve in $\mathcal{M}_3$, this note provides an invariant that sorts the cusp prototypes of Lanneau and Nguyen by component. This can be seen as an analogue of McMullen's genus $2$ spin invariant, although the source of this invariant is different. Moreover, we describe the Galois action on the…

Algorithm finds periodic points on Veech surfaces.

For each discriminant $D>1$, McMullen constructed the Prym-Teichmüller curves $W_D(4)$ and $W_D(6)$ in $\mathcal{M}_{3}$ and $\mathcal{M}_{4}$, which constitute one of the few known infinite families of geometrically primitive Teichmüller curves. In the present paper, we determine for each $D$ the number and type of or…

Proves effective Chen ranks conjecture for Koszul modules.

We determine the action of the Torelli group on the equivariant cohomology of the space of flat SL(2,C) connections on a closed Riemann surface. We show that the trivial part of the action contains the equivariant cohomology of the even component of the space of flat PSL(2,C) connections. The non-trivial part consists …

Describes spectral data for singular fibres of a specific Hitchin system.

The moduli space of genus 3 translation surfaces with a single zero has two connected components. We show that in the odd connected component H^{odd}(4) the only GL^+(2,R) orbit closures are closed orbits, the Prym locus Q(3,-1^3), and H^{odd}(4). Together with work of Matheus-Wright, this implies that there are only f…

A correspondence between 1) rank 2 completely integrable systems of Jacobians of algebraic curves and 2) (holomorphically) symplectic surfaces was established in a previous paper by the first author. A more general abelian variety that occurs as a Liouville torus of integrable systems is a prym variety associated to a …

This paper is devoted to the classification of the infinite families of Teichmuller curves generated by Prym eigenforms of genus 3 having a single zero. These curves were discovered by McMullen. The main invariants of our classification is the discriminant D of the corresponding quadratic order, and the generators of t…

A well-known conjecture asserts that the mapping class group of a surface (possibly with punctures/boundary) does not virtually surject onto $\Z$ if the genus of the surface is large. We prove that if this conjecture holds for some genus, then it also holds for all larger genera. We also prove that if there is a counte…

Study shows how $\operatorname{SL}_2$ Hitchin connection at level four behaves.

New groups found that don't virtually algebraically fiber, related to mapping class group orbits.

Researchers create explicit representations for skein algebras of small surfaces, revealing their Azumaya loci.

In this note, we describe a procedure to construct generalized complex structures with an arbitrarily large number of type change loci on products of the circle with a connected sum of closed 3-manifolds. The loci need not be isotopic.

Study geodesics and shortest arcs on Lie groups with specific metrics.

In this paper, we study Fuchsian loci of ${\rm PSL}_n(\mathbb{R})$-Hitchin components. In particular, using the Bonahon-Dreyer parametrization of ${\rm PSL}_n(\mathbb{R})$-Hitchin components, we give an explicit parametrization of Fuchsian loci of a pair of pants.

We survey the cohomology jumping loci and the Alexander-type invariants associated to a space, or to its fundamental group. Though most of the material is expository, we provide new examples and applications, which in turn raise several questions and conjectures. The jump loci of a space X come in two basic flavors: th…

As an increasing number of genome-wide association studies reveal the limitations of attempting to explain phenotypic heritability by single genetic loci, there is growing interest for associating complex phenotypes with sets of genetic loci. While several methods for multi-locus mapping have been proposed, it is often…

Study Riemannian metrics on lens spaces, find cut loci and diameters.

Study Riemannian metrics on lens spaces, find cut loci and diameters.

We introduce the natural and fairly general notion of a subanalytic bundle (with a finite dimensional vector space $P$ of sections) on a subanalytic subset $X$ of a real analytic manifold $M$, and prove that when $M$ is compact, there is a Baire subset $U$ of sections in $P$ whose zero-loci in $X$ have tubular neighbou…

New method studies discriminantal loci of algebraic varieties.

New upper bound for geodesic complexity derived from cut locus decompositions.

Study curvature loci of 3-manifolds in R^6 and R^5.

Study geodesics and shortest arcs on Lie groups with specific metrics.

Generalizes results on Bieri-Neumann-Strebel-Renz invariants and tropical varieties.

Study geodesics on a special cylinder with arbitrary wind.

New potentials found for sheaves on Calabi-Yau 4-folds.

Let $X\subset \mathbb P^N$ be a scroll over a smooth curve $C$ and let $Ł=\mathcal O_{\mathbb P^N}(1)|_X$ denote the hyperplane bundle. The special geometry of $X$ implies that some sheaves related to the principal part bundles of $Ł$ are locally free. The inflectional loci of $X$ can be expressed in terms of these she…

Study Alexander invariants and cohomology jump loci in group extensions with trivial monodromy.

We show that some riemannian manifolds diffeomorphic to the sphere have the property that the cut loci of general points are smoothly embedded closed disks of codimension one. Ellipsoids with distinct axes are typical examples of such manifolds.