Multisections generalize Heegaard splittings and trisections to higher dimensions.
problem Decomposing higher-dimensional manifolds into 1-handlebodies with specific intersection properties.
method Considering multisections of higher-dimensional smooth or PL closed orientable manifolds, and associating diagrams to them.
result Multisection diagrams uniquely determine manifolds in all dimensions up to 6.
New method to encode Weinstein 4-manifolds using multisections with divides.
problem Encoding and manipulating Weinstein 4-manifolds.
method Using multisection diagrams with divides and algorithms based on Kirby-Weinstein handle decompositions and PALFs.
result Definition and symplectic implementation of monodromy on multisections with divides.
Multisections exist in all dimensions and can be put into bridge positions.
problem Existence and bridge positions of submanifolds in multisected manifolds.
method Definition and existence of multisections in arbitrary dimensions; generalization of bridge trisections.
result Existence of multisections in all dimensions and their bridge positions.
The paper constructs multisections for m-spun 3-manifolds in higher dimensions.
problem Finding multisections for higher-dimensional manifolds.
method Extending the concept of spun 3-manifolds to higher dimensions, constructing multisections and diagrams.
result Infinitely many examples of non-diffeomorphic multisected manifolds in all dimensions.
Efficient multisections found for odd-dimensional tori.
problem Finding optimal multisections of odd-dimensional tori.
method Constructing multisections with specific properties (genus n, symmetry) for odd-dimensional tori. result Optimal multisections of genus n found for odd-dimensional tori. New method for decomposing manifolds into 1-handlebodies.
problem Decomposing manifolds into 1-handlebodies.
method Multisections of surface bundles and fiber bundles over the circle.
result Multisections exist for surface bundles and fiber bundles over the circle.
Paper proves uniqueness of bridge multisections for surfaces in 4-space.
problem Tackles the combinatorial description of surface links in 4-space.
method Develops a surgery operation called band surgery to prove uniqueness.
result Proves any n-valent graph is the spine of a bridge multisection for an unknotted surface. New method to compute homology and intersection form of 4-manifolds.
problem Computing homology and intersection form of 4-manifolds.
method Using multisections to define complexes and retraction onto CW-complex.
result Efficient proofs of homology and intersection form from multisection diagrams.
Study of curves in rational surfaces using multisections and torus actions.
problem Understanding curves in rational surfaces using multisections and torus actions.
method Analysis of multisections of embedded surfaces in rational 4-manifolds with torus actions.
result Every smooth, complex curve in CP^1 × CP^1 can be put in efficient bridge position with respect to a genus one 4-section.
The study shows manifolds with special generic maps also have nice multisections.
problem Characterizing manifolds with special generic maps and multisections.
method Analyzing manifolds with special generic maps and their properties, and showing how these maps restrict the differentiable structures of spheres and manifolds.
result Manifolds admitting special generic maps also admit nice generalized multisections.
Characterizes Lefschetz fibrations and multisections via monodromy factorizations.
problem Classifying Lefschetz fibrations and multisections.
method Using monodromy factorizations and Hurwitz equivalence.
result Two Lefschetz fibrations with multisections are isomorphic if and only if their monodromy factorizations are related by a finite collection of modifications.
Multisections generalize trisections for 4-manifolds, allowing complex operations and explicit diagrams.
problem Decomposing arbitrary smooth 4-manifolds into simpler pieces.
method Introducing multisections as a sequence of cut systems on a surface, enabling smooth operations and explicit diagrams.
result Elliptic fibrations admit genus 3 multisections, with explicit diagrams.
We initiate a study of positive multisections of Lefschetz fibrations via positive factorizations in framed mapping class groups of surfaces. Using our methods, one can effectively capture various interesting symplectic surfaces in symplectic 4-manifolds as multisections, such as Seiberg-Witten basic classes and except…
This paper generalizes Heegaard splittings to all dimensions using triangulations.
problem Decomposing piecewise linear manifolds into simpler components.
method Using triangulations, the paper proves every closed piecewise linear manifold has a multisection.
result Every closed piecewise linear n-manifold has a multisection.
We give examples of foliations that answer two questions posed by Mitsumatsu and Vogt about the genus minimising properties of closed leaves of 2-dimensional foliations on 4-manifolds. By studying stable commutator lengths in certain stable mapping class groups, we also answer an asymptotic version of another question …
Defines pants distance for knotted surfaces in 4-manifolds.
problem Complexity of knotted surfaces in 4-manifolds.
method Defining pants distance and proving standard form conditions.
result Determines conditions for standard form of knotted surfaces.
Study of 3-manifolds in 5-sphere using bridge decompositions.
problem Understanding embeddings of 3-manifolds in 5-sphere.
method Introduce and study bridge decompositions, use multisections of 5-manifolds.
result Every embedded 3-manifold admits a bridge decomposition.
New invariant defined for Weinstein domains, related to Kirby-Thompson's invariant.
problem Defining a symplectic invariant for Weinstein domains.
method Contact cut graph, Lefschetz fibrations, multisections with divides.
result Definition of Weinstein L-invariant and its relation to Kirby-Thompson's invariant. The paper finds infinitely many 4-manifolds with non-isotopic sections.
problem Finding non-isotopic sections in 4-manifolds.
method Using Nielsen equivalence, the paper constructs infinitely many 4-manifolds with non-isotopic sections.
result Infinitely many 4-manifolds have non-isotopic sections.
In this paper we use continuous family of multisections of the moduli space of pseudo holomorphic discs to partially improve, in the case of real coefficient, the construction of Lagrangian Floer cohomology of which the author developed jointly with Oh-Ohta-Ono. Namely we associate cyclically symmetric filtered A infin…
Geometric models create fractional quantum anyons.
problem Constructing fractional quantum anyons.
method 4D edge-cone orbifold geometries with embedded 2D surfaces.
result Anyon states arise from braid representations of surface braids.
Smooth 5-manifolds can be decomposed into three pieces.
problem Decomposing smooth 5-manifolds into three pieces.
method Introduced a trisection of a 5-manifold, compatible with any desired trisection of its boundary.
result Every smooth, oriented, compact 5-manifold admits a smooth trisection.
4-manifolds can be uniquely described as loops of Morse functions.
problem Describing 4-manifolds as loops of Morse functions.
method Proving uniqueness for each of four descriptions of 4-manifolds.
result Two loops of Morse functions yielding diffeomorphic 4-manifolds are related by specific moves.
Revisits and proves a reparametrization theorem for multi-valued graphs in higher codimension.
problem Analyzing multi-valued sections of vector bundles and proving a reparametrization theorem.
method Develops properties of Q-multisections and provides a geometric proof. result Elementary and purely geometric proof of a reparametrization theorem for multi-valued graphs.
SDP algorithm for community detection in unbalanced networks.
problem Community detection in networks with latent community structure.
method Semidefinite programming (SDP) algorithm for exact recovery of communities.
result Achieves exact recovery of latent communities up to information-theoretic limits.
This paper constructs a symplectic manifold from a Riemannian one.
problem Creating a symplectic manifold from a given Riemannian manifold.
method Using Ontaneda's theorem and Donaldson's results, the paper constructs a symplectic manifold S from a Riemannian manifold N.
result The construction of a symplectic manifold S from a Riemannian manifold N.
Cieliebak, Mundet i Riera and Salamon recently formulated a definition of branched submanifold of Euclidean space in connection with their discussion of multivalued sections and the Euler class. This note proposes an intrinsic definition of a weighted branched manifold Z that is obtained from the usual definition of or…
Closed Riemannian 4 or 5-manifolds contain branched immersed closed minimal surfaces.
problem Existence of classical minimal surfaces in 4 and 5-manifolds
method Harmonic replacement method
result Proves the existence of branched immersed closed minimal surfaces in 4 and 5-manifolds
New methods decompose manifolds into submanifolds via fold maps.
problem Understanding the topologies and differentiable structures of manifolds globally.
method Explicit decompositions of manifolds via fold maps, generalizing Morse functions.
result Decompositions of manifolds into lower dimensional spaces via fold maps.
New tools classify symplectic fillings of contact 3-manifolds.
problem Classifying symplectic fillings of contact 3-manifolds.
method Spinal open book decompositions and bordered Lefschetz fibrations.
result Symplectic fillings of contact 3-manifolds are deformation equivalent to complements of positive multisections in bordered Lefschetz fibrations.
Semidefinite programming solves community detection in stochastic block models.
problem Identifying community-like structures in graphs with varying probabilities of connections.
method Semidefinite programming (SDP) algorithms for exact recovery in the Stochastic Block Model (SBM).
result SDP-based algorithms achieve exact recovery in the SBM with high probability under certain conditions.
Study symplectic fillings of sandwiched singularities.
problem Contrast deformation theory and symplectic topology of Milnor fibers.
method Develop an analog of de Jong--van Straten's theory in the symplectic setting using spinal open books and nearly Lefschetz fibrations.
result Minimal symplectic fillings of links are generated by certain immersed disk arrangements.