Embeddings of mapping tori for end-periodic graph maps are proven.
problem Embedding mapping tori of end-periodic graph maps into finite complexes.
method Flowline-preserving homotopy equivalence and π1-injective map. result Every mapping class of Γ arising from an end-periodic homotopy equivalence contains a representative whose mapping torus realizes such an embedding.
The paper defines and proves the existence of train track maps on graphs of groups.
problem Understanding homotopy equivalences in graphs of groups.
method Developed the theory of train track maps on graphs of groups, defining maps and homotopy equivalences.
result Any homotopy equivalence of a graph of groups may be represented by a relative train track map under certain conditions.
Maps preserving edges on curve graphs and Hatcher-Thurston graphs are induced by homeomorphisms of surfaces.
problem Characterizing maps preserving edges on curve graphs and Hatcher-Thurston graphs.
method Proving edge-preserving maps on curve graphs and Hatcher-Thurston graphs are induced by homeomorphisms of surfaces.
result Edge-preserving maps on curve graphs and Hatcher-Thurston graphs are unique up to isotopy when (g,n)eq(2,0). The paper studies higher order Dirichlet-to-Neumann maps on graphs and their eigenvalues.
problem Analyzing eigenvalues of higher order Dirichlet-to-Neumann maps on graphs.
method Introducing and studying higher order Dirichlet-to-Neumann maps on graphs, deriving estimates on eigenvalues.
result Raulot-Savo-type estimates on the eigenvalues of the DtN maps.
The paper explores non-amenability in infinite-type surfaces and graphs.
problem Determining non-amenability in mapping class groups of infinite-type surfaces and graphs.
method Analyzes mapping class groups of infinite-type surfaces and graphs, provides examples and exhibits classes of groups.
result Completely determines non-amenability of mapping class groups of infinite-type surfaces and graphs.
Anosov maps on torus curve graphs have positive integer translation lengths.
problem Understanding the translation lengths of Anosov maps on curve graphs of tori.
method Constructive proof and algorithm for calculating exact translation lengths.
result The stable translation length of an Anosov map on the curve graph is always a positive integer.
The paper defines when surfaces are homotopy equivalent to graphs and explores their mapping class groups.
problem Understanding when surfaces are homotopy equivalent to graphs.
method Analyzes second-countable orientable surfaces with noncompact boundary.
result Defines a necessary and sufficient condition for surfaces to be homotopy equivalent to graphs.
Shifts are not type-preserving on surface graphs.
problem Understanding the type-preserving property of shift maps on surface graphs.
method Analyzing Dehn twists and shift maps on arc, curve, and relative arc graphs of surfaces.
result Shift maps are not type-preserving on surfaces with isolated punctures.
The curve graph and related graphs are hyperbolic and have quasi-tree fibers.
problem Understanding the structure of the curve graph and related graphs.
method Analyzing a sequence of graphs with Lipschitz maps and proving hyperbolicity and quasi-tree properties.
result The graphs in the sequence are hyperbolic and have quasi-tree fibers, leading to bounds on asymptotic dimension and acylindrical actions.
The paper studies maps on graphs and their fibers, focusing on hyperelliptic graphs.
problem Analyzing maps on graphs and their fibers.
method Analyzes maps Phi and Phi* on Culler--Vogtmann's outer space CV_n and its quotient T_n, focusing on hyperelliptic graphs.
result Fibers of Phi in T_n are aspherical and pi_1-injective subspaces.
Study of mapping class groups on infinite graphs, focusing on their large-scale geometry.
problem Understanding the large-scale geometry of mapping class groups on infinite graphs.
method Using coarse geometry techniques, classify coarsely bounded groups and compute asymptotic dimension.
result Identify conditions for global and local coarsely bounded pure mapping class groups of infinite rank graphs.
Study graph manifolds, knots, and mapping classes via profinite groups.
problem Classifying graph manifolds by their profinite fundamental groups.
method Analysis of commensurability, knot behavior, and mapping classes.
result Exteriors of graph knots are distinguished by profinite fundamental groups.
Study on harmonic maps between cones, linking degrees to graph Laplacian eigenvalues.
problem Understanding harmonic maps between singular spaces.
method Analyzing homogeneous harmonic maps between simplicial cones and their degrees.
result Degrees of homogeneous harmonic maps are related to eigenvalues of discrete graph Laplacians.
Enhances graph classification models on small datasets.
problem Over-fitting and undergeneralization on small-scale benchmark datasets.
method Data augmentation via graph structure transformation and model evolution framework.
result Average improvement of 3 - 13% accuracy on graph classification tasks.
New problems defined for maps onto graphs related to Reeb spaces.
problem Constructing explicit functions that yield specific graphs as Reeb spaces.
method Defining new classes of maps onto graphs and considering problems for these classes.
result New insights and challenges in constructing functions for specific Reeb spaces.
Graph neural network predicts optimal coarse-grained mapping operators.
problem Optimal coarse-grained mapping operators selection for molecular dynamics simulations.
method Graph Neural Network (DSGPM) trained on expert-annotated data.
result DSGPM outperforms state-of-the-art methods in graph segmentation.
Study proves hyperfiniteness of mapping class group actions on surface graphs.
problem Hyperfiniteness of mapping class group actions on surface graphs.
method Infinite unicorn paths and Gromov boundaries of arc and curve graphs.
result Proves hyperfiniteness of orbit equivalence relations induced by mapping class group actions.
Study of mapping class groups of infinite graphs, focusing on their finiteness and commensurability.
problem Understanding the finiteness properties and commensurability of mapping class groups of infinite graphs.
method Investigation of asymptotically rigid mapping class groups, construction of explicit presentations, and analysis of algebraic and geometric properties.
result Graph Houghton groups are not commensurable with other known Houghton-type groups, defining a new class of groups.
Study shows pants graph automorphisms match mapping class groups of nonorientable surfaces.
problem Understanding automorphisms of pants graphs on nonorientable surfaces.
method Analyzing mapping class groups and proving isomorphism.
result Automorphism group of pants graphs isomorphic to mapping class groups.
The study examines translation lengths of pseudo-Anosov maps on curve graphs.
problem Estimating translation lengths of pseudo-Anosov maps on curve graphs.
method Analyzing geodesic axes and powers of Dehn twists.
result Determining minimal translation lengths and optimizing map ratios.
The paper classifies dense conjugacy classes in mapping class groups of locally finite graphs.
problem Identifying which mapping class groups have dense conjugacy classes.
method Developed flux homomorphisms and combinatorial criteria for stability.
result A complete classification for self-similar locally finite graphs and a criterion for stability.
New graphs model periodic mapping classes and Seifert manifolds.
problem Modeling periodic mapping classes and Seifert manifolds.
method Introduced general tête-à-tête graphs and algorithms to model and reverse model Seifert manifolds and periodic mapping classes.
result General tête-à-tête graphs model all periodic mapping classes.
A map from a circle to a graph splits if pre-image diameters are small.
problem Can a map from a circle to a graph split if pre-image diameters are small?
method Examines maps from a circle to a graph with small pre-image diameters.
result A map splits if pre-image diameters are small enough.
Parabolic mapping class acts on curve graphs of infinite type surfaces.
problem Understanding parabolic isometries on curve graphs of infinite type surfaces.
method Fine curve graph tools to prove existence of parabolic isometries.
result Existence of parabolic isometries on graphs of curves of infinite type surfaces.
The paper characterizes graph manifolds using fold maps and embeddability of polyhedra.
problem Understanding the global topologies of graph manifolds.
method Using fold maps into the plane and embeddability of polyhedra in 3-manifolds.
result Characterizes graph manifolds via fold maps and polyhedra embeddability.
Study of flip graphs and their automorphism groups for infinite-type surfaces.
problem Understanding automorphism groups of flip graphs for infinite-type surfaces.
method Examined the relationship between mapping class groups and flip graphs for infinite-type surfaces.
result Extended mapping class groups are isomorphic to proper subgroups of automorphism groups of flip graphs.
Study of pure mapping class groups on infinite graphs.
problem Classifying graphs with specific mapping class groups.
method Completely classified graphs with pure mapping class groups.
result Established semidirect product decomposition and computed first integral cohomology.
Study of infinite-type surfaces' automorphisms and graph structures.
problem Understanding automorphisms of infinite-type surfaces.
method Isomorphic mappings between extended mapping class groups and graph automorphism groups.
result Extended mapping class groups are isomorphic to graph automorphism groups.
Study of graphs from hexagon decompositions of surfaces.
problem Understanding geometric properties of hexagon decompositions.
method Define and analyze graphs associated with hexagon decompositions of surfaces.
result Quasi-isometric relationships between studied graphs and known groups.
Unified feature maps for graph kernels improve efficiency without sacrificing accuracy.
problem Efficiently applying non-linear kernel methods to large-scale graph data.
method Constructing feature maps for graph kernels, analyzing feasibility, and proposing algorithms.
result Explicit feature maps can achieve similar accuracy to kernel trick methods but with significantly reduced computation time.
Study examines mean curvature flow on graphs of maps between manifolds.
problem Investigating mean curvature flow on graphs of maps between manifolds with bounded geometry.
method Investigates the mean curvature flow of graphs of smooth length-decreasing maps f:RmoN. result Uniform decay estimates for all derivatives of order ≥2 of ft along the flow. Optimal Euclidean structure minimizes energy in weighted toroidal graphs.
problem Finding the optimal Euclidean structure for weighted toroidal graphs.
method Minimizing Dirichlet energy over all possible Euclidean structures and realizations within a fixed homotopy class.
result The optimal Euclidean structure induces a weighted Delaunay decomposition.
Study uniformly differentiable graphs in Carnot groups, proving area formulas.
problem Characterize uniformly differentiable intrinsic graphs in Carnot groups.
method Characterize uniform intrinsic differentiability via Hölder properties of projections of vector fields.
result Explicit area formula for uniformly intrinsically differentiable maps in Carnot groups.
The study examines when mapping class groups are quasi-isometric to graphs of curves.
problem When is the mapping class group of an infinite-type surface quasi-isometric to a graph of curves?
method Using the work of Rosendal, Mann, and Rafi, the study defines a necessary and sufficient condition called translatability for a mapping class group to be quasi-isometric to a graph of curves.
result The mapping class group of the plane minus a Cantor set is quasi-isometric to the loop graph defined by Bavard.
Study of endperiodic maps on infinite graphs, proving homotopy and eigenvalue properties.
problem Understanding endperiodic maps on infinite graphs with finitely many ends.
method Adapting relative train track maps and combinatorial techniques to infinite type setting.
result Any generalized endperiodic map is homotopic to a relative train track map.
3D manifolds can map to a plane with specific curve patterns.
problem Characterizing 3D manifolds that can map to the plane with certain curve patterns.
method Analyzing fold maps and their critical value sets.
result Closed orientable 3-manifolds admit round fold maps into the plane if and only if they are graph manifolds.
Edge-preserving maps on curve graphs of low genus surfaces are induced by unique homeomorphisms.
problem Characterizing edge-preserving maps on curve graphs of low genus surfaces.
method Proving edge-preserving maps on curve graphs of genus 0 or 1 with at least 5 or 3 boundary components are induced by homeomorphisms.
result Edge-preserving maps on curve graphs of low genus surfaces are induced by unique homeomorphisms.
Characterizes Lipschitz continuity using microlocal sheaf theory.
problem Understanding Lipschitz continuity on differential manifolds.
method Microlocal sheaf theory applied to the graph of continuous maps.
result Characterization of Lipschitz continuity in microlocal terms.
New theory classifies knotted spheres in 4D space.
problem Classifying knotted punctured spheres in 4D space.
method Diagrammatic theory of welded graphs, Tube map extension, Milnor invariants.
result Complete link-homotopy classification of knotted punctured spheres.
Study shows surfaces without certain curves have infinite orbit graph.
problem Characterizing surfaces with specific curve properties.
method Utilized tools from mapping class group geometry.
result Infinite-invariance index 1 surfaces lack good curve graphs.
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.
Godin introduced the categories of open closed fat graphs Fatoc and admissible fat graphs Fatad as models of the mapping class group of open closed cobordism. We use the contractibility of the arc complex to give a new proof of Godin's result that Fatad is a model of the mapping class group of open-close…
We give a necessary and sufficient condition for the mapping class group of the pair of the 3-sphere and a graph embedded in it to be isomorphic to the topological symmetry group of the embedded graph.
We consider the mean curvature flow of the graph of a smooth map f:R2→R2 between two-dimensional Euclidean spaces. If f satisfies an area-decreasing property, the solution exists for all times and the evolving submanifold stays the graph of an area-decreasing map ft. Further, we prove unifo…
We construct maps on hat Heegaard Floer homology for cobordisms decorated with graphs. The graph TQFT allows for cobordisms with disconnected ends. Our construction uses Juhász's sutured Floer TQFT. We compute the maps for several elementary graph cobordisms. As an application, we compute the action of the fundamental …
Characterizes lamination spaces of graphs on a pair of pants.
problem Understanding lamination spaces of graphs on a pair of pants.
method Identifying lamination spaces as lattice polytopes and using graph exploration technique.
result Characterizes the polytopes that arise as lamination spaces of graphs on a pair of pants.
WEGL embeds graphs in a vector space for faster machine learning.
problem Efficiently embedding graphs for machine learning tasks.
method Wasserstein distance for node embedding similarity, Monge maps for graph representation.
result State-of-the-art classification performance with superior computational efficiency.
For given closed orientable 3-manifolds M and N let cD(M,N) be the set of mapping degrees from M to N. We address the problem: For which N, cD(M,N) is finite for all M? The answer is known in Thurston's picture of closed orientable irreducible 3-manifolds unless the target is a non-trivial graph manifol…