arXiv research

Summary of pure cactus groups and circle points.

Corrects earlier work on surface orbifold pure braid groups.

Explains the pure cactus group of degree three and its relation to four points on a circle.

We show that the fundamental group of the space of ordered affine-equivalent configurations of at least five points in the real plane is isomorphic to the pure braid group modulo its centre. In the case of four points this fundamental group is free with eleven generators.

New matrices link point motions to braid groups.

Planar pure braids form a group that acts on a CAT(0) cubical complex.

Global fixed points in low-dimensional surface group space correspond to trivial representations.

The pure braid group cannot be realized as area-preserving homeomorphisms.

Study Farrell cohomology for non-orientable surfaces, classifying subgroup conjugacy.

Proves transitivity of a specific class of quadratic polynomials.

Finite type invariants (also known as Vassiliev invariants) of pure braids are considered from a group-theoretic point of view. New results include a construction of a universal invariant with integer coefficients based on the Magnus expansion of a free group and a calculation of numbers of independent invariants of ea…

Geometric study of linear neural networks identifies pure and spurious critical points.

This paper gives a new interpretation of the virtual braid group in terms of a strict monoidal category SC that is freely generated by one object and three morphisms, two of the morphisms corresponding to basic pure virtual braids and one morphism corresponding to a transposition in the symmetric group. The key to this…

We prove that the "minus" version of Lipshitz's double-point enhanced grid homology is a knot invariant through purely combinatorial means.

We construct a group $Γ_{n}^{4}$ corresponding to the motion of points in $\mathbb{R}^{3}$ from the point of view of Delaunay triangulations. We study homomorphisms from pure braids on $n$ strands to the product of copies of $Γ_{n}^{4}$. We will also study the group of pure braids in $\mathbb{R}^{3}$, which is describe…

We give several new positive finite presentations for the pure braid group that are easy to remember and simple in form. All of our presentations involve a metric on the punctured disc so that the punctures are arranged "convexly", which is why we describe them as geometric presentaitons. Motivated by a presentation fo…

The study examines singularities and geometric properties of surfaces derived from frontals with specific singular points.

New curves share invariant up to any fixed order.

Study of spaces of pure braids and string links using diagrams and integrals.

Combinatorial proof of grid homology properties.

Paper shows how Non-Abelian T-duality solves pure spinor equations in supersymmetric vacua.

RSHT algorithm simplifies complex shapes to points.

We present an intriguing question about lattice points in triangles where Pick's formula is "almost correct". The question has its origin in knot theory, but its statement is purely combinatorial. After more than 30 years the topological question was recently solved, but the lattice point problem is still open.

We explore martingale and convex duality techniques to study optimal investment strategies that maximize expected risk-averse utility from consumption and terminal wealth. We consider a market model with jumps driven by (multivariate) marked point processes and so-called non-linear wealth dynamics which allows to take …

This paper is a comment on the survey paper by Biau and Scornet (2016) about random forests. We focus on the problem of quantifying the impact of each ingredient of random forests on their performance. We show that such a quantification is possible for a simple pure forest , leading to conclusions that could apply more…

We present a generally covariant approach to quantum mechanics in which generalized positions, momenta and time variables are treated as coordinates on a fundamental "phase-spacetime." We show that this covariant starting point makes quantization into a purely geometric flatness condition. This makes quantum mechanics …

We define a family of representations $\{ρ_n\}_{n\geq 0}$ of a pure braid group $P_{2k}$. These representations are obtained from an action of $P_{2k}$ on a certain type of $A_2$ web space with color $n$. The $A_2$ web space is a generalization of the Kauffman bracket skein module of a disk with marked points on its bo…

Study bounds and asymptotic behavior of largest purely non-free actions on compact Riemann surfaces.

We consider a purely algebraic result. Then given a circle or cyclic group of prime order action on a manifold, we will use it to estimate the lower bound of the number of fixed points. We also give an obstruction to the existence of $\mathbb{Z}_p$ action on manifolds with isolated fixed points when $p$ is a prime.

A new method joins two arcs with a degree of freedom.

Pure exploration (aka active testing) is the fundamental task of sequentially gathering information to answer a query about a stochastic environment. Good algorithms make few mistakes and take few samples. Lower bounds (for multi-armed bandit models with arms in an exponential family) reveal that the sample complexity …

We compare the sample complexity of private learning [Kasiviswanathan et al. 2008] and sanitization~[Blum et al. 2008] under pure $ε$-differential privacy [Dwork et al. TCC 2006] and approximate $(ε,δ)$-differential privacy [Dwork et al. Eurocrypt 2006]. We show that the sample complexity of these tasks under approxima…

PureTS uses simple linear models to improve long-term time series forecasting.

A purely combinatorial compactification of the configuration space of n (>4) distinct points with equal weights in the real projective line was introduced by M. Yoshida. We geometrize it so that it will be a real hyperbolic cone-manifold of finite volume with dimension n-3. Then, we vary weights for points. The geometr…

We give a metric characterization of the scalar curvature of a smooth Riemannian manifold, analyzing the maximal distance between $(n+1)$ points in infinitesimally small neighborhoods of a point. Since this characterization is purely in terms of the distance function, it could be used to approach the problem of definin…

In the present paper, we define an invariant of free links valued in a free product of some copies of $\mathbb{Z}_{2}$. In \cite{Ma2} the second named author constructed a connection between classical braid group and group presentation generated by elements corresponding to horizontal trisecants. This approach does not…

Study kernels of mapping class group representations on surface configuration spaces.

The triple linking number of an oriented surface link was defined as an analogical notion of the linking number of a classical link. We consider a certain $m$-component $T^2$-link ($m \geq 3$) determined from two commutative pure $m$-braids $a$ and $b$. We present the triple linking number of such a $T^2$-link, by usin…

The article constructs stochastic integration in Riemannian manifolds.

We construct an explicit bundle with flat connection on the configuration space of n points of a complex curve. This enables one to recover the `formality' isomorphism between the Lie algebra of the prounipotent completion of the pure braid group of n points on a surface and an explicitly presented Lie algebra t_{g,n} …

In this paper we apply the theory of finitely generated FI-modules developed by Church, Ellenberg and Farb to certain sequences of rational cohomology groups. Our main examples are the cohomology of the moduli space of n-pointed curves, the cohomology of the pure mapping class group of surfaces and some manifolds of hi…

Let $ξ$ and $η$ be two non--commuting isometries of the hyperbolic $3$--space $\mathbb{H}^3$ so that $Γ=\langleξ,η\rangle$ is a purely loxodromic free Kleinian group. For $γ\inΓ$ and $z\in\mathbb{H}^3$, let $d_γz$ denote the distance between $z$ and $γ\cdot z$. Let $z_1$ and $z_2$ be the mid-points of the shortest geod…

We study the problem of estimating the mean of a random vector $X$ given a sample of $N$ independent, identically distributed points. We introduce a new estimator that achieves a purely sub-Gaussian performance under the only condition that the second moment of $X$ exists. The estimator is based on a novel concept of a…

We study the geometric properties of a $2m$-dimensional complex manifold $\mathcal{M}$ admitting a holomorphic reduction of the frame bundle to the structure group $P \subset \mathrm{Spin}(2m,\mathbb{C})$, the stabiliser of the line spanned by a pure spinor at a point. Geometrically, $\mathcal{M}$ is endowed with a hol…

A novel one-class classifier fusion method for robust anomaly detection.

We consider several classes of knotted objects, namely usual, virtual and welded pure braids and string links, and two equivalence relations on those objects, induced by either self-crossing changes or self-virtualizations. We provide a number of results which point out the differences between these various notions. Th…

A new invariant for pure braids is defined and shown not to be trivial.

We use the energy gap result of pure Yang-Mills equation [Feehan P.M.N., Adv. Math. 312 (2017), 547-587, arXiv:1502.00668] to prove another energy gap result of complex Yang-Mills equations [Gagliardo M., Uhlenbeck K., J. Fixed Point Theory Appl. 11 (2012), 185-198, arXiv:1401.7366], when Riemannian manifold $X$ of dim…