arXiv research

Study resolves conjecture linking two algebraic structures on surfaces.

Classifies semi-algebraic surfaces up to bi-Lipschitz homeomorphisms.

Study of cluster and skein algebras for surfaces, showing their connection.

Quantum cluster algebras for surfaces with coefficients defined using skein theory.

Decomposes $SL_3$ skein algebras for surfaces.

It was shown by Fock, Goncharov and Fomin, Shapiro, Thurston that some cluster algebras arise from triangulated orientable suraces. Subsequently Dupont and Palesi generalised this construction to include unpunctured non-orientable surfaces, giving birth to quasi-cluster algebras. Previously we linked this framework to …

Survey of stated skein modules/algebras of 3-manifolds/surfaces.

We define a (co-)Poisson (co)algebra of curves on a bordered surface. A bordered surface is a surface whose boundary have marked points. Curves on the bordered surface are oriented loops and oriented arcs whose endpoints in the set of marked points. We define a (co-)Poisson (co)bracket on the symmetric algebra of a quo…

Method computes centers of Poisson and skein algebras for loops on surfaces.

We generalise surface cluster algebras to the case of infinite surfaces where the surface contains finitely many accumulation points of boundary marked points. To connect different triangulations of an infinite surface, we consider infinite mutation sequences. We show transitivity of infinite mutation sequences on tria…

It was shown by Fomin, Shapiro and Thurston that some cluster algebras arise from orientable surfaces. Subsequently, Dupont and Palesi extended this construction to non-orientable surfaces. We link this framework to Lam and Pylyavskyy's Laurent phenomenon algebras, showing that both orientable and non-orientable unpunc…

We construct examples of complex algebraic surfaces not admitting normal embeddings (in the sense of semialgebraic or subanalytic sets) with image a complex algebraic surface.

Algorithm constructs algebraic curves from translation surfaces.

New method computes automorphisms of surface groups using skein algebras.

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

Minimum algebraic intersection found in hyperbolic surfaces, growing with genus.

Maps Heegaard Floer homology to Hecke algebras for surfaces.

Analogue of classical Hurwitz numbers is defined in the work for regular coverings of surfaces with marked points by seamed surfaces. Class of surfaces includes surfaces of any genus and orientability, with or without boundaries; coverings may have certain singularities over the boundary and marked points. Seamed surfa…

New, algebraic surfaces found in curved spaces.

Proves properties of Torelli Lie algebra for surfaces.

The paper presents a new algebraic structure for planar surfaces.

Analytic curves linked to algebraic ones via Schottky groups.

Classifies area-minimizing surfaces in R^4 as algebraic.

Study on affine surfaces with specific algebraic properties.

An algebraic system is proposed that represent surface cobordisms in thickened surfaces. Module and comodule structures over Frobenius algebras are used for representing essential curves. The proposed structure gives a unified algebraic view of states of categorified Jones polynomials in thickened surfaces and virtual …

We show that the Kauffman bracket skein algebra of any oriented surface F (possibly with marked points in its boundary) has no zero divisors and that its center is generated by knots parallel to the unmarked components of the boundary of F. Furthermore, we show that skein algebras are Noetherian and Ore. Our proofs rel…

Study of algebraic curves and surfaces in flag manifold using twistor geometry.

Paper derives matrix formulae and proves skein relations for non-orientable surfaces in quasi-cluster algebras.

Quantized Coulomb branches linked to skein algebras.

This paper calculates interaction strength for translation surfaces with multiple singularities.

The paper provides presentations for mapping class groups and cluster automorphism groups of surfaces.

Bordered Floer homology associates to a parametrized oriented surface a certain differential graded algebra. We study the properties of this algebra under splittings of the surface. To the circle we associate a differential graded 2-algebra, the nilCoxeter sequential 2-algebra, and to a surface with connected boundary …

Compactifies metrics on K3 surfaces with algebraic description.

Unified study of surfaces using Clifford algebras.

A projective algebraic surface which is homeomorphic to a ruled surface over a curve of genus $g\ge 1$ is itself a ruled surface over a curve of genus $g$. In this note, we prove the analogous result for projective algebraic manifolds of dimension 4 in case $g\ge 2$.

We study on a new kind of surface covered by translation and factorable (TF-type) surfaces in the three dimensional Euclidean space. We consider I and III Laplace-Beltrami operator surfaces of a TF-type surface. Then we obtain degrees and classes of algebraic surfaces of the surfaces using eliminate methods on software…

The study of the topology of real algebraic varieties dates back to the work of Harnack, Klein and Hilbert in the 19th century; in particular, the isotopy type classification of real algebraic curves in real toric surfaces is a classical subject that has undergone considerable evolution. On the other hand, not much is …

Find first (0,2) mirror symmetry examples on Hopf surfaces.

Compute central extension of mapping class group from stated skein algebra

The existence problem for holomorphic structures on vector bundles over non-algebraic surfaces is in general still open. We solve this problem in the case of rank 2 vector bundles over K3 surfaces and in the case of vector bundles of arbitrary rank over all known surfaces of class VII. Our methods, which are based on D…

Researchers compute c-projective symmetry algebras for Kähler surfaces.

In this paper we study the skein algebras of marked surfaces and the skein modules of marked 3-manifolds. Muller showed that skein algebras of totally marked surfaces may be embedded in easy to study algebras known as quantum tori. We first extend Muller's result to permit marked surfaces with unmarked boundary compone…

Proves conjecture linking cluster algebras and skein algebras for surfaces with punctures.

Introduces LRY skein algebras generalizing Kauffman bracket and Roger-Yang skein algebras.

We extend some results of Bonahon, Bullock, Turaev and Wong concerning the skein algebras of closed surfaces to L^e's stated skein algebra associated to open surfaces. We prove that the stated skein algebra with deforming parameter +1 embeds canonically into the centers of the stated skein algebras whose deforming para…

Study identifies specific subvarieties in translation surfaces with quadratic field.

We refine Osserman's argument on the exceptional values of the Gauss map of algebraic minimal surfaces. This gives an effective estimate for the number of exceptional values and the totally ramified value number for a wider class of complete minimal surfaces that includes algebraic minimal surfaces. It also provides a …

We establish basic properties of cluster algebras associated with oriented bordered surfaces with marked points. In particular, we show that the underlying cluster complex of such a cluster algebra does not depend on the choice of coefficients, describe this complex explicitly in terms of "tagged triangulations" of the…