arXiv research

The paper classifies orbit closures of symplectic Lie algebras.

Classifies Zariski closures of positive representations in Lie groups.

We show that if a braid $B$ can be parametrised in a certain way, then previous work can be extended to a construction of a polynomial $f:\mathbb{R}^4\to\mathbb{R}^2$ with the closure of $B$ as the link of an isolated singularity of $f$, showing that the closure of $B$ is real algebraic. In particular, we prove that cl…

Characterizes closures of test configurations and algebraic singularity types.

Proof confirms conjecture for certain braids and their closures.

We study a regular closure operator in the category of quandles. We show that the regular closure operator and the pullback closure operator corresponding to the reflector from the category of quandles to its full subcategory of trivial quandles coincide, we give a simple description of this closure operator, and analy…

Study of $g$-vector cones in cluster algebras from weighted orbifolds.

We prove that every algebraic curve X defined over the algebraic closure of the rationals is birational over the complex numbers to a Teichmuller curve.

Paper constructs connections on curves with specific Galois groups.

Let K be an algebraically closed field of characteristic zero, endowed with a complete nonarchimedean norm. Let X be a K-rigid analytic variety and Σa semianalytic subset of X. Then the closure of Σin X with respect to the canonical topology is again semianalytic. The proof uses Embedded Resolution of Singularities.

Recently, Cochran and Harvey defined torsion-free derived series of groups and proved an injectivity theorem on the associated torsion-free quotients. We show that there is a universal construction which extends such an injectivity theorem to an isomorphism theorem. Our result relates injectivity theorems to a certain …

The paper connects arithmetic invariants of hyperbolic 3-manifolds.

Our paper is devoted to the study of the holonomy groups of Finsler surfaces using the methods of infinite dimensional Lie theory. The notion of infinitesimal holonomy algebra will be introduced, by the smallest Lie algebra of vector fields on an indicatrix, containing the curvature vector fields and their horizontal c…

Minimal rational curves on compactified symmetric spaces are orbit-closures of 1-parameter subgroups.

We study the Morton-Franks-Williams inequality for closures of simple braids (also known as positive permutation braids). This allows to prove, in a simple way, that the set of simple braids is a orthonormal basis for the inner product of the Hecke algebra of the braid group defined by Kálmán, who first obtained this r…

The object of this paper is to study GL(2,R) orbit closures in hyperelliptic components of strata of abelian differentials. The main result is that all higher rank affine invariant submanifolds in hyperelliptic components are branched covering constructions, i.e. every translation surface in the affine invariant subman…

Proves knots in handlebodies can be represented as plats of braids.

In this paper we show that the normal closure of the mth power of a half-twist has infinite index in the mapping class group of a punctured sphere. Furthermore, in some cases we prove that the quotient of the mapping class group of the punctured sphere by the normal closure of a power of a half-twist contains a free ab…

We study a subset of square free positive braids and we give a few algebraic characterizations of them and one geometric characterization: the set of positive braids whose closures are unlinks. We describe canonical forms of these braids and of their conjugacy classes.

We construct a representation of the braid groups in a cluster C*-algebra coming from a triangulation of the Riemann surface S with one or two cusps. It is shown that the Laurent polynomials attached to the K-theory of such an algebra are topological invariants of the closure of braids. In particular, the Jones and HOM…

We connect the algebraic geometry and representation theory associated to Freudenthal's magic square. We give unified geometric descriptions of several classes of orbit closures, describing their hyperplane sections and desingularizations, and interpreting them in terms of composition algebras. In particular, we show h…

We prove that a Bers slice is never algebraic, meaning that its Zariski closure in the character variety has strictly larger dimension. A corollary is that skinning maps are never constant. The proof uses grafting and the theory of complex projective structures.

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

Let A denote the algebraic closure of the rationals Q in the complex numbers C. Suppose G is a torsion-free group which contains a congruence subgroup as a normal subgroup of finite index and denote by U(G) the C-algebra of closed densely defined unbounded operators affiliated to the group von Neumann algebra. We prove…

To a braid $β\in Br_n$ we associate a complex of sheaves $S_β$ on $Hilb_n(C^2)$ such that the previously defined triply graded link homology of the closure $L(β)$ is isomorphic to the homology of $S_β$. The construction of $S_β$ relies on the Chern functor $CH: MF_n^{st}\to D^{per}_{C^*\times C^*}(Hilb_n(C^2))$ defined…

We show that a germ of a real analytic Lorentz metric on ${\bf R}^3$ which is locally homogeneous on an open set containing the origin in its closure is necessarily locally homogeneous. We classifiy Lie algebras that can act quasihomogeneously---meaning they act transitively on an open set admitting the origin in its c…

New algebraic structure helps distinguish braids.

Solves double coset problem for braid group H_n.

Holomorphic tensors on algebraic cones are invariant under certain group actions.

We develop a diagrammatic formalism for calculating the Alexander polynomial of the closure of a braid as a state-sum. Our main tools are the Markov trace formulas for the HOMFLY-PT polynomial and Young's semi-normal representations of the Iwahori-Hecke algebras of type A.

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.

Translation surfaces can be defined in an elementary way via polygons, and arise naturally in in the study of various basic dynamical systems. They can also be defined as Abelian differentials on Riemann surfaces, and have moduli spaces called strata that are related to the moduli space of Riemann surfaces. There is a …

The theorems of M. Ratner, describing the finite ergodic invariant measures and the orbit closures for unipotent flows on homogeneous spaces of Lie groups, are extended for actions of subgroups generated by unipotent elements. More precisely: Let G be a Lie group (not necessarily connected) and Gamma a closed subgroup …

It is known that the hyperbolic plane admits a countable infinity of compactifications into a closed disk such that the isometric action of SL(2;R) acts analytically on the compactified space. We prove that among those compactifications, only the two most classical ones (namely the closures of Poincaré's disk and Klein…

Study of bonded knots and braids with new algebraic models.

Classifies components of k-differentials and their orbit closures.

We generalize the notion of a Lie algebroid over infinite jet bundle by replacing the variational anchor with an N-tuple of differential operators whose images in the Lie algebra of evolutionary vector fields of the jet space are subject to collective commutation closure. The linear space of such operators becomes an a…

In the moduli space M_g of genus g Riemann surfaces, consider the locus RM_O of Riemann surfaces whose Jacobians have real multiplication by the order O in a totally real number field F of degree g. If g = 2 or 3, we compute the closure of RM_O in the Deligne-Mumford compactification of M_g and the closure of the locus…

We finish the proof of the conjecture of F. Bogomolov and F. Pop: Let $F_{1}$ and $F_{2}$ be fields finitely-generated and of transcendence degree $\geq 2$ over $k_{1}$ and $k_{2}$, respectively, where $k_{1}$ is either $\bar{\mathbb{Q}}$ or $\bar{\mathbb{F}}_{p}$, and $k_{2}$ is algebraically closed. We denote by $G_{…

In 2001, Khovanov and Seidel constructed a faithful action of the (m+1)-strand braid group on the derived category of left modules over a quiver algebra, A_m. We interpret the Hochschild homology of the Khovanov-Seidel braid invariant as a direct summand of the sutured Khovanov homology of the annular braid closure.

The Hecke algebra H_n contains well known idempotents E_λ which are indexed by Young diagrams with n cells. They were originally described by Gyoja. A skein theoretical description of E_λ was given by Aiston and Morton. The closure of E_λ becomes an element Q_λ of the skein of the annulus. In this skein, they are known…

For a stratified symplectic space, a suitable concept of stratified Kaehler polarization, defined in terms of an appropriate Lie-Rinehart algebra, encapsulates Kaehler polarizations on the strata and the behaviour of the polarizations across the strata and leads to the notion of stratified Kaehler space. This notion es…

In this paper we construct a homomorphism of the affine braid group $Br_n^{aff}$ in the convolution algebra of the equivariant matrix factorizations on the space $\overline{\mathcal{X}}_2=\mathfrak{b}_n\times GL_n\times\mathfrak{n}_n$ considered in the earlier paper of the authors. We explain that the pull-back on the …

A notion known as smooth envelope, or superposition closure, appears naturally in several approaches to generalized smooth manifolds which were proposed in the last decades. Such an operation is indispensable in order to perform differential calculus. A derivation of the enveloping algebra can be restricted to the orig…

Study left-invariant pseudo-Riemannian metrics on Lie groups focusing on null cone Lie algebras.

We give a new definition of the knot invariant associated to the Lie algebra su_{N+1}. The knot or link must be presented as the plat closure of a braid. The invariant is then a homological intersection pairing between two submanifolds of a configuration space of points in a disk. This generalizes previous work on the …

The paper studies representations of braid groups via curves and finds conditions for their Zariski closure and arithmeticity.

The study shows knots from 3-braids cannot be concordant to a specific Legendrian unknot.