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Virtual singular braids are generalizations of singular braids and virtual braids. We define the virtual singular braid monoid via generators and relations, and prove Alexander- and Markov-type theorems for virtual singular links. We also…
Generic relative immersions of compact one-manifolds in the closed unit disk, i.e. divides, provide a powerful combinatorial framework, and allow a topological construction of fibered classical links, for which the monodromy diffeomorphism…
We study real lines on certain Moishezon threefolds which are potentially twistor spaces of 3CP^2. Here, line means a smooth rational curve whose normal bundle is O(1)^2 and the reality implies the invariance under an anti-holomorphic…
We provide a novel framework to compute a discrete vector potential of a given discrete vector field on arbitrary polyhedral meshes. The framework exploits the concept of acyclic matching, a combinatorial tool at the core of discrete Morse…
The Fr\'echet distance is a popular similarity measure that is well-understood for polygonal curves in $\mathbb{R}^d$: near-quadratic time algorithms exist, and conditional lower bounds suggest that these results cannot be improved…
We present a characterisation of blenders based on mapping properties of certain sets of curves that can be rigorously verified by computer-assisted methods. We develop an algorithm to construct these sets of curves that requires only a…
We enumerate complex curves on toric surfaces of any given degree and genus, having a single cusp and nodes as their singularities, and matching appropriately many point constraints. The solution is obtained via tropical enumerative…
A question of Griffiths-Schmid asks when the monodromy group of an algebraic family of complex varieties is arithmetic. We resolve this in the affirmative for the class of algebraic surfaces known as Atiyah-Kodaira manifolds, which have…
The integral variation map and algebraic monodromy of isolated plane curve singularities are important homological invariants of the singularity which are still far from being completely understood. This work provides effective ways of…
We develop a theory of localization for braid group representations associated with objects in braided fusion categories and, more generally, to Yang-Baxter operators in monoidal categories. The essential problem is to determine when a…
Fully braided analog of Faddeev-Reshetikhin-Takhtajan construction of quasitriangular bialgebra $A(X,R)$ is proposed. For given pairing $C$ factor-algebra $A(X,R;C)$ is a dual quantum braided group. Corresponding inhomogeneous quantum group…
We consider the problem of efficiently computing a discrete Morse complex on simplicial complexes of arbitrary dimension and very large size. Based on a common graph-based formalism, we analyze existing data structures for simplicial…
We consider the problem of minimizing a convex objective which is the sum of a smooth part, with Lipschitz continuous gradient, and a nonsmooth part. Inspired by various applications, we focus on the case when the nonsmooth part is a…
Connected components of real algebraic sets are semi-algebraic, i.e. they are described by a boolean formula whose atoms are polynomial constraints with real coefficients. Computing such descriptions finds topical applications in optical…
One approach to studying the Fr\'echet distance is to consider curves that satisfy realistic assumptions. By now, the most popular realistic assumption for curves is $c$-packedness. Existing algorithms for computing the Fr\'echet distance…
A divide is a relative generic immersion of a finite union of copies of the unit interval in the unit disk. A divide defines a classical link in the 3- sphere, which is a fibered link if the image of the immersion is connected. We prove in…
We present a new algorithm for computing integral bases in algebraic function fields of one variable, or equivalently for constructing the normalization of a plane curve. Our basic strategy makes use of the concepts of localization and…
Curve singularities are classical objects of study in algebraic geometry. The key player in their combinatorial structure is the {\it value semigroup}, or its compactification, the {\it value semiring}. One natural problem is to explicitly…
In this paper we give an algorithm that calculates the skeleton of a tame covering of curves over a complete discretely valued field. The algorithm relies on the {{tame simultaneous semistable reduction theorem}}, for which we give a short…
In \cite{Oh22}, the second author defined a complex of groups decomposition of the fundamental group of a finitely generated 2-dimensional special group, called an \emph{intersection complex}, which is a quasi-isometry invariant. In this…