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.
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. 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.
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.
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 …
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.
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.
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.
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.
In this article, we characterize isomorphism classes of Lefschetz fibrations with multisections via their monodromy factorizations. We prove that two Lefschetz fibrations with multisections are isomorphic if and only if their monodromy factorizations in the relevant mapping class groups are related to each other by a f…
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…
Recently Gay and Kirby described a new decomposition of smooth closed 4-manifolds called a trisection. This paper generalises Heegaard splittings of 3-manifolds and trisections of 4-manifolds to all dimensions, using triangulations as a key tool. In particular, we prove that every closed piecewise linear n-mani…
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.
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 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. We show that the "geometric models of matter" approach proposed by the first author can be used to construct models of anyon quasiparticles with fractional quantum numbers, using 4-dimensional edge-cone orbifold geometries with orbifold singularities along embedded 2-dimensional surfaces. The anyon states arise through…
Gay and Kirby introduced the notion of a trisection of a smooth 4-manifold, which is a decomposition of the 4-manifold into three elementary pieces. Rubinstein and Tillmann later extended this idea to construct multisections of piecewise-linear (PL) manifolds in all dimensions. Given a PL manifold Y of dimension n,…
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…
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.
We analyze a notion of multiple valued sections of a vector bundle over an abstract smooth Riemannian manifold, which was suggested by W. Allard in the unpublished note "Some useful techniques for dealing with multiple valued functions" and generalizes Almgren's Q-valued functions. We study some relevant properties o…
Let M be a compact oriented even-dimensional manifold. This note constructs a compact symplectic manifold S of the same dimension and a map f from S to M of strictly positive degree. The construction relies on two deep results: the first is a theorem of Ontaneda that gives a Riemannian manifold N of tightly pinched neg…
We propose a semidefinite programming (SDP) algorithm for community detection in the stochastic block model, a popular model for networks with latent community structure. We prove that our algorithm achieves exact recovery of the latent communities, up to the information-theoretic limits determined by Abbe and Sandon (…
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…
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.
We consider the problem of identifying underlying community-like structures in graphs. Towards this end we study the Stochastic Block Model (SBM) on k-clusters: a random model on n=km vertices, partitioned in k equal sized clusters, with edges sampled independently across clusters with probability q and within …
New knots found with Seifert genus not matching minimal genus Seifert surfaces.
problem Discrepancy between Seifert genus and minimal genus Seifert surfaces.
method Constructed knots with specific genus and handle numbers to demonstrate the discrepancy.
result Found knots where Seifert genus is not realized by minimal genus Seifert surfaces.
Construct divide knots with specific genus properties.
problem Understanding the difference between smooth and topological four-genus for knots.
method Construct divide knots with controlled smooth and topological four-genus ratios.
result For strongly quasipositive fibred knots, the ratio between smooth and topological four-genus can be made arbitrarily close to zero.
The concordance genus of a knot K is the minimum Seifert genus of all knots smoothly concordant to K. Concordance genus is bounded below by the 4-ball genus and above by the Seifert genus. We give a lower bound for the concordance genus of K coming from the knot Floer complex of K. As an application, we prove that ther…
The paper generalizes the T-genus to characterize slice knots and slice genus.
problem Characterizing slice knots and slice genus using the T-genus. method Generalizing the T-genus to provide a 3-dimensional characterization of the slice genus. result The difference between the T-genus and the slice genus can be arbitrarily large. Characterizes the Z-genus of boundary links using Blanchfield forms.
problem Determining the Z-genus of boundary links.
method Characterization using Blanchfield forms.
result Variant of shake genus equals Z-genus of knot.
The concordance genus of a knot is the least genus of any knot in its concordance class. Although difficult to compute, it is a useful invariant that highlights the distinction between the three-genus and four-genus. In this paper we define and discuss the stable concordance genus of a knot, which describes the behavio…
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.
Classifies 3-braid knots with maximal 4-genus using McCoy's method.
problem Classifying 3-braid knots with maximal 4-genus.
method McCoy's twisting method and Xu normal form.
result Upper bounds for the topological 4-genus of 3-braid knots.
Study shows non-equivariant and equivariant non-orientable 4-genus of periodic knots can differ.
problem Exploring differences in non-orientable 4-genus for periodic knots.
method Analyzed p-periodic knots, showing differences in equivariant and non-equivariant non-orientable 4-genus.
result Differences exist in non-equivariant and equivariant non-orientable 4-genus for periodic knots.
Study Khovanov homology of Turaev genus one links, finding a trivial summand.
problem Analyzing Khovanov homology of Turaev genus one links.
method Examined Khovanov homology in specific polynomial gradings.
result Found a trivial summand in Khovanov homology of Turaev genus one links.
Contact connected sums do not increase support genus.
problem Understanding how support genus changes under contact connected sums.
method Analyzing the support genus of contact connected sums of 3-manifolds.
result The support genus of contact connected sums is at most the maximum of the summands' support genera.
The concordance genus of a knot is the least genus of any knot in its concordance class. It is bounded above by the genus of the knot, and bounded below by the slice genus, two well-studied invariants. In this paper we consider the concordance genus of 11--crossing prime knots. This analysis resolves the concordance ge…
This article is about the graph genus of certain well studied graphs in surface theory: the curve, pants and flip graphs. We study both the genus of these graphs and the genus of their quotients by the mapping class group. The full graphs, except for in some low complexity cases, all have infinite genus. The curve grap…
Classifies knot traces with specific trisection genus limits.
problem Classifying knot traces with specific trisection genus limits.
method Classifying knot traces with specific trisection genus limits.
result Infinitely many knots have traces with trisection genus 3 and 4, and arbitrarily large trisection genus.
Classifies Teichmüller curves in genus three from genus two.
problem Classifying Teichmüller curves in genus three from genus two.
method Action on homology modulo two of affine groups of translation surfaces.
result Classification of orbits of square-tiled surfaces.
The paper proves trisection genus of Akbulut cork and constructs many corks with trisection genus 3.
problem Determining the trisection genus of contractible 4-manifolds and exotic pairs.
method Proved trisection genus of Akbulut cork and constructed corks with trisection genus 3.
result First examples of contractible 4-manifolds with determined trisection genus except for 4-ball.
New expanders for mean curvature flow contradict genus-reduction conjecture.
problem Contradicting Ilmanen's genus-reduction conjecture for mean curvature flow.
method Construct new expanders asymptotic to cones arising from shrinkers.
result Existence of expanders of arbitrarily large genus.
Study shows systole behavior changes significantly for large genus hyperbolic surfaces.
problem Understanding systole behavior in large genus hyperbolic surfaces.
method Analysis of random surfaces with respect to Weil-Petersson volume.
result Expected value of separating systole behaves like 2logg for large genus.