arXiv research

Researchers create topologically protected knots in a realizable system.

This paper classifies topological symmetry groups for Petersen family graphs.

Introduces sheaf quantization, a topological approach to geometric quantization.

Extends Nielsen realization to infinite-type surfaces, classifying torsion elements and topological groups.

A venerable problem in combinatorics and geometry asks whether a given incidence relation may be realized by a configuration of points and lines. The classic version of this would ask for algebraic lines over some field or possibly real pseudolines: embedded circles (isotopic to $RP^1$) in the real projective plane. In…

Researchers solved a complex problem for a specific type of 4-manifolds.

This is a survey on two closely related subjects. First, we review the study of topological structure of `finite type' components of spaces of Bridgeland's stability conditions on triangulated categories. The key is to understand Happel-Reiten-Smalo tilting as tiling of cells. Second, we review topological realizations…

This paper explores how pairs of multicurves can be realized as cylinders on translation surfaces.

Study describes how to realize periods of meromorphic differentials with specific properties.

Study detects P-type bifurcations in single system realizations using unreliable kernel density estimates.

Solves a problem related to classifying spaces for proper actions and Nielsen Realization.

The symmetries of complex molecular structures can be modeled by the {\em topological symmetry group} of the underlying embedded graph. It is therefore important to understand which topological symmetry groups can be realized by particular abstract graphs. This question has been answered for complete graphs; it is natu…

New method constructs Stein surfaces using topological isotopy.

Solves a problem related to Nielsen realization for certain groups.

This paper identifies all topological symmetry groups for Heawood family graphs.

Friedl and Kim show any taut sutured manifold can be realized as a twisted homology product, but their proof gives no practical description of how complicated the realizing representation needs to be. We give a number of results illustrating the relationship between the topology of a taut sutured handlebody and the com…

Quantum phase diagrams for Chern topological insulators show jumps at critical loci.

Study conic line arrangements of degree 7, finding their topology and connected components.

Groups of homotopy equivalences of graphs help realize compact subgroups.

Given a contact 3-manifold we consider the problem of when a given function can be realized as the Ricci curvature of a Reeb vector field for the contact structure. We will use topological tools to show that every admissible function can be realized as such Ricci curvature for a singular metric which is an honest compa…

New experiments show Gauss diagrams not all as simple as previously thought.

Study smooth embeddings of line configurations in complex projective plane.

The Hessian Topology is a subject having interesting relations with several areas, for instance, differential geometry, implicit differential equations, analysis and singularity theory. In this article we study the problem of realization of a real plane curve as the Hessian curve of a smooth function. The plane curves …

Study shows certain mapping class groups cannot be realized as subgroup of homeomorphisms.

Research explores real algebraic realization of round fold maps of codimension -1.

TKFT models computation via smooth vector fields, simulating functions in a single dynamical step.

The paper examines circle graphs of Gauss diagrams and finds counterexamples to previous descriptions.

Study of projective Fraïssé limits of trees with confluent epimorphisms.

Analytic realization of Thom-Smale complex for G-manifolds.

In this paper, we study face vectors of simplicial posets that are the face posets of cell decompositions of topological manifolds without boundary. We characterize all possible face vectors of simplicial posets whose geometric realizations are homeomorphic to the product of spheres. As a corollary, we obtain the chara…

Study topological constraints for minimal hyperbolic surface laminations.

New topological realization of Kontsevich graph complex for large dimensions.

A unified summary is given of the existence theory of Stein manifolds in all dimensions, based on published and pending literature. Eliashberg's characterization of manifolds admitting Stein structures requires an extra delicate hypothesis in complex dimension 2, which can be eliminated by passing to the topological se…

Study shows Dehn twists on certain 4-manifolds cannot be realized by finite order diffeomorphisms.

Study conic-line arrangements of degree 7, finding their topology and components.

Let $M$ be a closed, oriented manifold of dimension $d$. Let $LM$ be the space of smooth loops in $M$. Chas and Sullivan recently defined a product on the homology $H_*(LM)$ of degree $-d$. They then investigated other structure that this product induces, including a Batalin -Vilkovisky structure, and a Lie algebra str…

We investigate several topological and combinatorial properties of line arrangements. We associate to a line arrangement a link obtained by intersecting the arrangement with some sphere. Several topics are discussed: (a) some link configurations can be realized by complex line arrangements but not by real line arrangem…

We formalize the arithmetic topology, i.e. a relationship between knots and primes. Namely, using the notion of a cluster C*-algebra we construct a functor from the category of 3-dimensional manifolds M to a category of algebraic number fields K, such that the prime ideals (ideals, resp.) in the ring of integers of K c…

We explain how, starting with a stack of D4-branes ending on an NS5-brane in type IIA string theory, one can, via T-duality and the topological-holomorphic nature of the relevant worldvolume theories, relate (i) the lattice models realized by Costello's 4d Chern-Simons theory, (ii) links in 3d analytically-continued Ch…

We define 2-dimensional topological substitutions. A tiling of the Euclidean plane, or of the hyperbolic plane, is substitutive if the underlying 2-complex can be obtained by iteration of a 2-dimensional topological substitution. We prove that there is no primitive substitutive tiling of the hyperbolic plane $\mathbb{H…

Given a Delaunay decomposition of a compact hyperbolic surface, one may record the topological data of the decomposition, together with the intersection angles between the `empty disks' circumscribing the regions of the decomposition. The main result of this paper is a characterization of when a given topological decom…

New model calculates Wilson surfaces in higher gauge theory.

We explore the topology of real Lagrangian submanifolds in a toric symplectic manifold which come from involutive symmetries on its moment polytope. We establish a real analog of the Delzant construction for those real Lagrangians, which says that their diffeomorphism type is determined by combinatorial data. As an app…

Topological data analysis (TDA) has emerged as one of the most promising techniques to reconstruct the unknown shapes of high-dimensional spaces from observed data samples. TDA, thus, yields key shape descriptors in the form of persistent topological features that can be used for any supervised or unsupervised learning…

Interactive art piece shows braid groups and plane motions.

New theorem links tropical phased matroids to higher-dimensional spheres.

We classify topologically trivial Legendrian $Θ$-graphs and identify the complete family of nondestabilizeable Legendrian realizations in this topological class. In contrast to all known results for Legendrian knots, this is an infinite family of graphs. We also show that any planar graph that contains a subdivision of…

We combinatorially describe the $2$-category of singular cobordisms, called (rank one) foams, which governs the functorial version of Khovanov homology. As an application we topologically realize the type $\mathrm{D}$ arc algebra using this singular cobordism construction.