arXiv research

In this article, we prove a generalization of a theorem of Lisca-Matic to Stein cobordisms and develop a method for distinguishing certain Stein cobordisms using rotation numbers. Using these results along with standard techniques from convex surface theory and classifications of tight contact structures on certain 3-m…

The computation of the cobordism group of Morse functions on unoriented surfaces using Stein factorizations.

The paper classifies symplectic fillings of lens spaces and constructs cobordisms.

Clarifies properties of Ozsvath-Szabo contact invariant.

An isolated complex surface singularity induces a canonical contact structure on its link. In this paper, we initiate the study of the existence problem of Stein cobordisms between these contact structures depending on the properties of singularities. As a first step we construct an explicit Stein cobordism from any co…

Suppose S is a compact surface with boundary, and let g be a diffeomorphism of S which fixes the boundary pointwise. We denote by (M_{S,g},ξ_{S,g})$ the contact 3-manifold compatible with the open book (S,g). In this article, we construct a Stein cobordism from the contact connected sum (M_{S,h},ξ_{S,h}) # (M_{S,g},ξ_{…

We continue our study of contact structures on manifolds of dimension at least five using complex surgery theory. We show that in each dimension 2q+1 > 3 there are 'maximal' almost contact manifolds to which there is a Stein cobordism from any other (2q+1)-dimensional contact manifold. We show that the product M x S^2 …

We show that the pre-order defined on the category of contact manifolds by arbitrary symplectic cobordisms is considerably less rigid than its counterparts for exact or Stein cobordisms: in particular, we exhibit large new classes of contact 3-manifolds which are symplectically cobordant to something overtwisted, or to…

We study the topology of exact and Stein fillings of the canonical contact structure on the unit cotangent bundle of a closed surface $Σ_g$, where $g$ is at least 2. In particular, we prove a uniqueness theorem asserting that any Stein filling must be s-cobordant rel boundary to the disk cotangent bundle of $Σ_g$. For …

Embeds 3-manifolds in symplectic 4-manifolds with constraints.

In this paper, we show that the Ozsváth-Szabó contact invariant $c^+(ξ)\in HF^+(-Y)$ of a contact 3-manifold $(Y,ξ)$ can be calculated combinatorially if $Y$ is the boundary of a certain type of plumbing $X$, and $ξ$ is induced by a Stein structure on $X$. Our technique uses an algorithm of Ozsváth and Szabó to determi…

A spinal open book decomposition on a contact manifold is a generalization of a supporting open book which exists naturally e.g. on the boundary of a symplectic filling with a Lefschetz fibration over any compact oriented surface with boundary. In this first paper of a two-part series, we introduce the basic notions re…

In this paper we prove a vanishing theorem for the contact Ozsvath--Szabo invariants of certain contact 3--manifolds having positive Giroux torsion. We use this result to establish similar vanishing results for contact structures with underlying 3--manifolds admitting either a torus fibration over the circle or a Seife…

Stein's method improves probabilistic inference and learning.

GC Stein manifolds characterized with embeddings and functions.

Stochastic Stein Discrepancies improve inference efficiency.

We give a bordism-theoretic characterisation of those closed almost contact (2q+1)-manifolds (with q > 2) which admit a Stein fillable contact structure. Our method is to apply Eliashberg's h-principle for Stein manifolds in the setting of Kreck's modified surgery. As an application, we show that any simply connected a…

Improved sampling method using regularized Stein Variational Gradient Flow.

Develops Stein's method for Riemannian manifolds using diffusion.

The study extends cobordism theory to complex sections, defining and calculating cobordism groups.

It is shown that every subcritical Stein manifold is deformation equivalent to the product of a Stein manifold with $\C$.

Regularized Stein thinning improves MCMC output approximations.

Study Stein and Milnor fillings of links from surface singularities.

Stein's method (Stein, 1973; 1981) is a powerful tool for statistical applications and has significantly impacted machine learning. Stein's lemma plays an essential role in Stein's method. Previous applications of Stein's lemma either required strong technical assumptions or were limited to Gaussian distributions with …

Study cobordisms of nested manifolds and their invariants.

The paper introduces a new invariant for cobordism classes of manifolds and extends cobordism groups.

Stein transport improves Bayesian inference with faster convergence and reduced variance.

Defines cobordism maps connecting Khovanov and instanton homologies.

The study explores conditions for $h$-cobordisms between smooth 4-manifolds.

In this paper we survey results on the existence of holomorphic embeddings and immersions of Stein manifolds into complex manifolds. Most results pertain to proper maps into Stein manifolds. We include a new result saying that every continuous map $X\to Y$ between Stein manifolds is homotopic to a proper holomorphic em…

The paper improves Stein importance sampling for Markov chain samples.

Counterexample found for Stein property of certain solvable Lie groups.

Study shows Khovanov homology's relation to decomposable Lagrangian cobordisms.

A new framework improves kernel Stein discrepancy tests for validating distributions.

Stein variational gradient descent (SVGD) is a deterministic sampling algorithm that iteratively transports a set of particles to approximate given distributions, based on an efficient gradient-based update that guarantees to optimally decrease the KL divergence within a function space. This paper develops the first th…

2-stein submanifolds in space forms have constant curvature if normal connection is flat or codimension is 2.

We show that, under a certain condition, contact 5-manifolds can `coarsely' distinguish smooth structures on compact Stein 4-manifolds via contact open books. We also give a simple sufficient condition for an infinite family of Stein 4-manifolds to have an infinite subfamily of pairwise non-diffeomorphic Stein 4-manifo…

The article examines how many stabilizations are needed to transform 5D s-cobordisms into product cobordisms.

We give an algorithm which produces infinitely many pairwise exotic Stein fillings of the same contact 3-manifolds, applying positive allowable Lefschetz fibrations over the disk. As a corollary, for a large class of Stein fillings, we realize the topological invariants (i.e. fundamental group, homology group, homology…

Sliced kernelized Stein discrepancy improves goodness-of-fit tests and model learning in high dimensions.

Extends cobordism groups of immersions to projections with new results.

For any integer $n\geq 2$, we construct an infinite family of Stein fillable contact $(4n-1)$-manifolds each of which admits infinitely many pairwise homotopy inequivalent Stein fillings.

The square root of Fredholm determinants causes numerical instabilities in option pricing models.

Study Stein fillings of planar contact 3-manifolds with relative trisection genus 2.

Study continuity and Hölder estimates for solutions on Stein spaces.

Improving scalability and stability of Stein discrepancies for scalable goodness-of-fit testing

We give complete geometric invariants of cobordisms of fold maps with oriented singular set and cobordisms of even codimensional fold maps. These invariants are given in terms of cobordisms of stably framed manifolds and cobordisms of immersions with prescribed normal bundles defined by the author in his earlier works.

Much of machine learning relies on comparing distributions with discrepancy measures. Stein's method creates discrepancy measures between two distributions that require only the unnormalized density of one and samples from the other. Stein discrepancies can be combined with kernels to define kernelized Stein discrepanc…