arXiv research

New minimal surfaces found using Toda lattice and integrable systems.

New lattices in higher dimensions have dense surface subgroups.

The study finds surface subgroups in cocompact lattices of $H^{2n}$ for $n\geq2$.

Proves effective slope gaps for lattice surfaces.

Algorithm constructs surfaces with specific Veech groups in lattice strata.

The paper studies alternating links in thickened surfaces using flow lattices and disc mutations.

The paper proves residual finiteness of certain lattices and constructs surfaces with specific fundamental groups.

My main results are simple formulas for the surface area of d-dimensional lattice polytopes using Ehrhart theory.

We study the translation surfaces obtained by considering the unfoldings of the surfaces of Platonic solids. We show that they are all lattice surfaces and we compute the topology of the associated Teichmüller curves. Using an algorithm that can be used generally to compute Teichmüller curves of translation covers of p…

The study finds that certain hyperbolic manifolds contain subgroups isomorphic to surface groups.

Generalizes Kauffman's clock theorem to surfaces.

The objective of this paper is to present some geometric aspects of surfaces associated with theta function solutions of the periodic 2D-Toda lattice. For this purpose we identify the $(N^2-1)$-dimensional Euclidean space with the ${\frak su}(N)$ algebra which allows us to construct the generalized Weierstrass formula …

Fix a K3 lattice $Λ$ of rank two and $L\inΛ$ a big and nef divisor that is positive enough. We prove that the generic $Λ$-polarised K3 surface has an integral nodal rational curve in the linear system $|L|$, in particular strengthening previous work of the first named author. The technique is by degeneration, and also …

Principal binets generalize curvature line surfaces to square lattices and are a discrete integrable system.

We derive results on the distribution of directions of saddle connections on translation surfaces using only the Birkhoff ergodic theorem applied to the geodesic flow on the moduli space of translation surfaces. Our techniques, together with an approximation argument, also give an alternative proof of a weak version of…

The paper finds dense subgroups in certain Lie groups.

Researchers calculate complexity of billiard paths in regular polygons.

Tropical geometry and weighted lattices improve curve and surface fitting.

Periodic surfaces have a limited number of bending modes, equal to their membrane modes.

Method computes harmonic and conformal maps from point clouds.

Counting lattice points in moduli space of Klein surfaces.

Building on work of Kapouleas and Yang, we construct sequences of minimal surfaces embedded in the round 3-sphere which converge to the Clifford torus counted with multiplicity two and have second fundamental form blowing up at every point of the torus and genus tending to infinity. Each surface in a given sequence res…

Minimal crossing number found in arithmetic curve systems.

We give a systematic method to calculate some homological data from the global monodromy of a topological elliptic surface. We apply this method to the cases 1) the transcendental lattice of an extremal elliptic K3 surface, 2) the torsion part of Mordell-Weil group of a general elliptic surface, and 3) the Mordell-Weil…

Study connects K3 surfaces to holomorphic metrics, solving complex structure variation.

Study flat metrics from right prisms, finding non-lattice surfaces with translation coverings.

This paper classifies commensurability of Deligne-Mostow lattices.

We construct some non-arithmetic ball quotients as branched covers of a quotient of an Abelian surface by a finite group, and compare them with lattices that previously appear in the literature. This gives an alternative construction, which is independent of the computer, of some lattices constructed by the author with…

An integral hyperbolic lattice is called reflective if its automorphism group is generated by reflections, up to finite index. Since 1981, it is known that their number is essentially finite. We show that K3 surfaces over C with reflective Picard lattices can be characterized in terms of compositions of their self-corr…

Maximal representations in symplectic lattices proven for most cases.

We construct two infinite families of ball quotient compactifications birational to bielliptic surfaces. For each family, the volume spectrum of the associated noncompact finite volume ball quotient surfaces is the set of all positive integral multiples of $\frac{8}{3}π^{2}$, i.e., they attain all possible volumes of c…

Analytic lattice cohomology defined for isolated singularities, linking to Heegaard Floer cohomology.

We classify the minimum volume smooth complex hyperbolic surfaces that admit smooth toroidal compactifications, and we explicitly construct their compactifications. There are five such surfaces and they are all arithmetic, i.e., they are associated with quotients of the ball by an arithmetic lattice. Moreover, the asso…

We find explicit bases for naturally defined lattices over a ring of algebraic integers in the SO(3) TQFT-modules of surfaces at roots of unity of odd prime order. Some applications relating quantum invariants to classical 3-manifold topology are given.

We consider Zimmer's program of lattice actions on surfaces by PL homomorphisms. It is proved that when the surface is not the torus or Klein bottle the action of any finite-index subgroup of SL(n,Z), n>4, (more generally for any 2-big lattice), factors through a finite group action. The proof is based on an establishm…

Researchers compute gap distributions for saddle connection directions on specific translation surfaces.

New method approximates hyperbolic lattices using cube complexes.

We study representations of lattices of PU(m,1) into PU(n,1). We show that if a representation is reductive and if m is at least 2, then there exists a finite energy harmonic equivariant map from complex hyperbolic m-space to complex hyperbolic n-space. This allows us to give a differential geometric proof of rigidity …

This note presents a formula for the enumerative invariants of arbitrary genus in toric surfaces. The formula computes the number of curves of a given genus through a collection of generic points in the surface. The answer is given in terms of certain lattice paths in the relevant Newton polygon. If the toric surface i…

Study decomposability of Lagrangian classes on K3 surfaces.

Study on geodesics and dihedral groups in lattices.

Study orbits of discrete lattice actions on the plane, derive new results for Veech surfaces.

In the present article, we provide examples of fake quadrics, that is, minimal complex surfaces of general type with the same numerical invariants as the smooth quadric in $\PP ^3$ which are quotients of the bidisc by an irreducible lattice of automorphisms. Moreover, we list classes of arithmetic lattices over a real …

The study finds bounds on homologically independent loops on hyperelliptic hyperbolic surfaces.

We show that cocompact lattices in rank one simple Lie groups of non-compact type distinct from SO(2m,1) (m>0) contain surface subgroups.

The study finds non-uniform lattices with thin Hitchin representations in specific Lie groups.

Criterion for congruence RFRS towers in hyperbolic lattices.

Researchers reinterpret complex hyperbolic orbifolds using line arrangements.