Researchers create topologically protected knots in a realizable system.
problem Creating topologically protected vortex knots in experimentally realizable systems.
method Investigated non-Abelian vortices in tetrahedral order in spin-2 Bose--Einstein condensates and bent-core nematic liquid crystals.
result Discovered the first topologically protected knots in an experimentally realizable system.
This paper classifies topological symmetry groups for Petersen family graphs.
problem Understanding symmetries of graphs embedded in 3D space.
method Examined all embeddings of Petersen family graphs in S3 and classified their topological symmetry groups. result Identified all possible groups that can be realized as topological symmetry groups for each graph in the Petersen family.
Introduces sheaf quantization, a topological approach to geometric quantization.
problem Topological realization of WKB-states in geometric quantization.
method Enhancement of constructible sheaves, Betti counterpart of Fukaya--Floer theory.
result Introduction to sheaf quantization as a topological realization of WKB-states.
Extends Nielsen realization to infinite-type surfaces, classifying torsion elements and topological groups.
problem Realizing finite subgroups of mapping class groups on infinite-type surfaces.
method Extending Kerckhoff's result to infinite-type surfaces, using hyperbolic metrics and topological group properties.
result Compact subgroups of mapping class groups are finite, and locally compact subgroups are discrete.
Study shows how to realize Ricci curvature as Reeb vector field for contact 3-manifolds.
problem When can a function be realized as Ricci curvature of a Reeb vector field?
method Topological tools to show realization, resolving singularities depend on contact topology.
result Every admissible function can be realized as Ricci curvature for a singular metric away from a measure zero set.
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 RP1) in the real projective plane. In…
Researchers solved a complex problem for a specific type of 4-manifolds.
problem Realizing mapping classes of finite order on del Pezzo surfaces.
method Synthesized results from reflection group theory and 4-manifold topology.
result Provided both positive and negative examples of realizability.
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.
problem Understanding when pairs of multicurves can be realized as cylinders on translation surfaces.
method Surface topology and flat grafting deformation.
result Pairs of multicurves can be realized as cylinders on some translation surface.
Study describes how to realize periods of meromorphic differentials with specific properties.
problem Realizing meromorphic differentials with given zeros, poles, and topological constraints.
method Complete description of period representations for specified conditions on Riemann surfaces.
result A comprehensive method for realizing meromorphic differentials with prescribed characteristics.
Study detects P-type bifurcations in single system realizations using unreliable kernel density estimates.
problem Detecting P-type bifurcations in signals with unreliable kernel density estimates.
method Create persistence diagrams from single system realization, statistically analyze resulting set, compare point process modeling methods.
result Subsampling outperforms other point process modeling methods in predicting P-type bifurcations.
Solves a problem related to classifying spaces for proper actions and Nielsen Realization.
problem Whether a cocompact proper topological Γ-manifold is equivariantly homotopy equivalent to the classifying space for proper actions.
method Using Poincaré models and assuming a zero-dimensional singular set, the problem is solved in the Poincaré category.
result New results about Brown's problem are obtained under certain conditions on the underlying group.
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.
problem Creating Stein surfaces in complex surfaces.
method Combining Freedman's topology with Eliashberg's holomorphic theory.
result Every tame 2-complex can be isotoped to a Stein surface.
Solves a problem related to Nielsen realization for certain groups.
problem Whether a cocompact proper topological manifold is equivariantly homotopy equivalent to a classifying space.
method Assumes a zero-dimensional singular set and uses properties of hyperbolic groups and aspherical manifolds.
result Solves the problem for specific groups containing a normal torsion-free subgroup.
This paper identifies all topological symmetry groups for Heawood family graphs.
problem Understanding symmetries of spatial graphs in 3D space.
method Analyzing automorphisms of graphs embedded in S3. result All graphs in the Heawood family are intrinsically chiral.
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.
problem Understanding phase transitions in Chern topological insulators.
method Mathematical formulation and explicit families of physical systems.
result Synthetic design of arbitrary Chern jumps in topological phases.
Study conic line arrangements of degree 7, finding their topology and connected components.
problem Understanding the topology of conic line arrangements of degree 7.
method Identifying a π1-equivalent Zariski pair to prove the existence of a conic line arrangement with specific combinatorics. result Determine the number of connected components of conic line arrangements of degree 7.
Groups of homotopy equivalences of graphs help realize compact subgroups.
problem Realizing compact subgroups of homotopy equivalences of graphs.
method Introduced a Polish group topology on the group of proper homotopy equivalences and proved the Nielsen Realization theorem.
result Compact subgroups of homotopy equivalences can be realized by simplicial isomorphisms of graphs.
New experiments show Gauss diagrams not all as simple as previously thought.
problem Determining which Gauss diagrams correspond to knots.
method Experimental mathematics using Prolog and constraint satisfaction.
result Found counterexamples showing existing criteria are incomplete.
Study smooth embeddings of line configurations in complex projective plane.
problem Realizing line configurations as smooth 2-spheres.
method Lattice-theoretic arguments based on Donaldson's diagonalization theorem.
result Established a stronger obstruction in the smooth category.
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.
problem Proving non-realizability of specific mapping class groups.
method Analyzing compactly supported and full mapping class groups of surfaces with genus 3 or order 6 symmetries.
result Proven non-realizability of mapping class groups for surfaces with genus 3 or order 6 symmetries.
Research explores real algebraic realization of round fold maps of codimension -1.
problem Real algebraic realization of round fold maps of codimension -1.
method Generalizes canonical projections of unit spheres to round fold maps and discusses their real algebraic realization.
result Developed new studies in real algebraic geometry focusing on round fold maps of codimension -1.
TKFT models computation via smooth vector fields, simulating functions in a single dynamical step.
problem Modeling computation in a single step.
method Established Topological Kleene Field Theory (TKFT) as a new model of computation.
result Any computable function can be simulated in a single go of a dynamical system.
The paper examines circle graphs of Gauss diagrams and finds counterexamples to previous descriptions.
problem Problems with previous descriptions of realizable Gauss diagrams.
method Experimental checking and formulation of new descriptions of realizable circle graphs.
result New descriptions of realizable circle graphs and an algorithm for checking realizability.
Study of projective Fraïssé limits of trees with confluent epimorphisms.
problem Finite trees with monotone epimorphisms do not amalgamate.
method Developed new mappings and properties of continua to apply to topological graphs.
result Topological realization of the Fraïssé limit of finite trees with ramification vertices of order at most 3 is the Wa\. zewski dendrite D3. Analytic realization of Thom-Smale complex for G-manifolds.
problem Realizing Thom-Smale complex for G-manifolds with Lie group action.
method Using G-invariant Witten instanton complex associated with a Morse-Bott function.
result Generalized Thom-Smale complex for G-manifolds including horizontal direction influence.
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.
problem Understanding which topological surfaces can be leaves in minimal hyperbolic surface laminations.
method Analyzes topological obstructions and embeddings of Cantor tree-like surfaces.
result All possible topological types of leaves can be simultaneously embedded in a lamination.
New topological realization of Kontsevich graph complex for large dimensions.
problem Understanding the rational homotopy groups of Diff partial(D2k).
method Construction of a chain map from Kontsevich graph complex to rational singular chain complex.
result New elements in rational homotopy groups of BDiff partial(D2k) determined by cycles in graph complex.
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.
problem Realization of Dehn twists as finite order diffeomorphisms on spin 4-manifolds.
method Use Y. Kato's 10/8-type inequality for involutions and its refinement.
result Dehn twists about specific spheres in certain 4-manifolds are not homotopic to finite order diffeomorphisms.
Study conic-line arrangements of degree 7, finding their topology and components.
problem Understanding the topology and components of conic-line arrangements of degree 7.
method Examined the embedded topology using SageMath and Sirocco package.
result Determined the number of connected components and fundamental groups.
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.
problem Calculating Wilson surfaces in higher gauge theory.
method Derived geometric framework, topological coadjoint orbit model, functional integral framework.
result Strong evidence that model underlies Wilson surfaces partition function.
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.
problem Understanding braid groups and their actions.
method Interactive art piece illustrating two braid group definitions and their action on free group.
result Demonstrates all plane motions in a single subspace.
New theorem links tropical phased matroids to higher-dimensional spheres.
problem Understanding topological properties of tropical phased matroids.
method Proving homeomorphism between topological order complex and a sphere.
result Topological order complex of tropical phased matroids is a (2n−3)-sphere. 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 D arc algebra using this singular cobordism construction.
Develops Floer cohomology for 4-manifolds with involutions and links.
problem Floer cohomology for 4-manifolds with involutions and links.
method Floer cohomology framework for spin^c 4-manifolds with involution and oriented links.
result Proves Frøyshov-type inequalities relating 4-manifold topological quantities and homology cobordism invariants.