arXiv research

The geography problem is usually stated for simply connected symplectic 4-manifolds. When the first cohomology is nontrivial, however, one can restate the problem taking into account how close the symplectic manifold is to satisfying the conclusion of the Hard Lefschetz Theorem, which is measured by a nonnegative integ…

The geography and botany of smooth/symplectic nonspin 4-manifolds with abelian fundamental group are addressed.

The geography of minimal symplectic 4-manifolds with arbitrary fundamental group and symplectic 6-manifolds with abelian fundamental group of small rank, and with arbitrary fundamental group are addressed.

Study on 4-manifolds with positive scalar curvature.

In this article we consider a version of the geography question for simply-connected symplectic 4-manifolds that takes into account the divisibility of the canonical class as an additional parameter. We also find new examples of 4-manifolds admitting several symplectic structures, inequivalent under deformation and sel…

Questions of geography of various classes of $4$-manifolds have been a central motivating question in $4$-manifold topology. Baykur and Korkmaz asked which small, simply connected, minimal $4$-manifolds admit a genus $2$ Lefschetz fibration. They were able to classify all the possible homeomorphism types and realize al…

The study creates symplectic Lefschetz fibrations and rational blowdowns for new 4-manifolds.

New findings restrict Heegaard Floer homology for certain rational homology spheres.

In this paper, results of J. Park and of B.D Park and Szabo on simply connected symplectic 4-manifolds are re-proven and extended to non-simply connected manifolds using Luttinger surgeries.

The paper characterizes Einstein 4-manifolds with semi-definite curvature and derives inequalities.

In this note, the geography problem in dimension four is reviewed and then its extension to dimension six for the symplectic case is explained. Finally some examples in dimension six are provided.

Geography problem for nonorientable surfaces bounded by knots.

We investigate the tendency for financial instruments to form clusters when there are multiple factors influencing the correlation structure. Specifically, we consider a stock portfolio which contains companies from different industrial sectors, located in several different countries. Both sector membership and geograp…

The paper constructs exotic knotted surfaces and curves in 4-manifolds.

We show that there exist non-formal compact oriented manifolds of dimension $n$ and with first Betti number $b_1=b\geq 0$ if and only if $n\geq 3$ and $b\geq 2$, or $n\geq (7-2b)$ and $0\leq b\leq 2$. Moreover, we present explicit examples for each one of these cases.

Delisle's projection explained by Euler in 18th century.

In \cite{AP3, AHP}, the first author and his collaborators constructed the irreducible symplectic $4$-manifolds that are homeomorphic but not diffeomorphic to $(2n-1){\mathbb{CP}}^{2}\#(2n-1)\overline{\mathbb{CP}}^{2}$ for each integer $n \geq 25$, and the families of simply connected irreducible nonspin symplectic $4$…

We prove that if a contact manifold $(M,ξ)$ is supported by a planar open book, then Euler characteristic and signature of any Stein filling of $(M,ξ)$ is bounded. We also prove a similar finiteness result for contact manifolds supported by spinal open books with planar pages. Moving beyond the geography of Stein filli…

In this paper we construct a family of simply connected, spin, non-complex, symplectic 4-manifolds which cover all but finitely many allowed lattice points $(χ, c)$ lying in $0 \leq c \leq 8.76χ$. Furthermore, as a corollary, we prove that there exist infinitely many exotic smooth structures on $(2n+1)(S^2 \times S^2)$…

New restrictions found on 4-manifolds with pinched curvature.

This work aims to study the Portuguese regional agglomeration process, using the linear form the New Economic Geography models that emphasize the importance of spatial factors (distance, costs of transport and communication) in explaining of the concentration of economic activity in certain locations. In a theoretical …

This note shows every integer can be a signature of a hyperbolic 4-manifold.

Spin Lefschetz fibrations can represent any group and lattice point.

A compact oriented 4-manifold is defined to be of ``superconformal simple type'' if certain polynomials in the basic classes (constructed using the Seiberg-Witten invariants) vanish identically. We show that all known 4-manifolds of $b_2^+>1$ are of superconformal simple type, and that the numerical invariants of 4-man…

Khovanov homology invariant under Conway mutation.

Scoping review of EO-ML methods for causal inference in poverty geography.

We prove that for any contact 3-manifold supported by a spinal open book decomposition with planar pages, there is a universal bound on the Euler characteristic and signature of its minimal symplectic fillings. The proof is an application of the spine removal surgery operation recently introduced in joint work of the a…

The correlation functions of supersymmetric gauge theories on a four-manifold X can sometimes be expressed in terms of topological invariants of X. We show how the existence of superconformal fixed points in the gauge theory can provide nontrivial information about four-manifold topology. In particular, in the example …

Khovanov multicurves are restricted to linear components.

Lagrange's map construction ideas influenced later mathematicians.

Describes the Teichmüller stack and its variants, answering questions about orbifold points and local models.

I review some recent results on four-manifold invariants which have been obtained in the context of topological quantum field theory. I focus on three different aspects: (a) the computation of correlation functions, which give explicit results for the Donaldson invariants of non-simply connected manifolds, and for gene…

For any given finitely presented group G there exists a closed oriented 4-manifold with fundamental group G. What pairs of integers can occur as the signature and Euler characteristic of such a 4-manifold? The first part of this paper develops general theory related to this question and applies this theory to resolve t…

New techniques create irreducible 4-manifolds with specific properties.

The study explores relations between various knot invariants.

Constructs 4-manifolds with positive Euler characteristic proving a conjecture.

Clarifies construction of K-contact non-Sasakian Smale-Barden manifolds.

Geography effect is investigated for the Chinese stock market including the Shanghai and Shenzhen stock markets, based on the daily data of individual stocks. The Shanghai city and the Guangdong province can be identified in the stock geographical sector. By investigating a geographical correlation on a geographical pa…

We consider the connected-sum method of constructing compact Riemannian 7-manifolds with holonomy G_2 developed in math.DG/0012189. The method requires pairs of projective complex threefolds endowed with anticanonical K3 divisors, the latter `matching' via a certain non-holomorphic map. Suitable examples of threefolds …

Constructs symplectic surface bundles with positive signatures.

We exhibit many examples of closed symplectic manifolds on which there is an autonomous Hamiltonian whose associated flow has no nonconstant periodic orbits (the only previous explicit example in the literature was the torus T^2n (n\geq 2) with an irrational symplectic structure). The underlying smooth manifolds of our…

Let X_1, X_2 be symplectic 4-manifolds containing symplectic surfaces F_1,F_2 of identical positive genus and opposite squares. Let Z denote the symplectic sum of X_1 and X_2 along the F_k. Using relative Gromov--Witten theory, we determine precisely when the symplectic 4-manifold Z is minimal (i.e., cannot be blown do…

In studying the "11/8-Conjecture" on the Geography Problem in 4-dimensional topology, Furuta proposed a question on the existence of Pin(2)-equivariant stable maps between certain representation spheres. In this paper, we present a complete solution to this problem by analyzing the Pin(2)-equivariant Mahowald invariant…

The aim of this paper is to use mapping class group relations to approach the `geography' problem for Stein fillings of a contact 3-manifold. In particular, we adapt a formula of Endo and Nagami so as to calculate the signature of such fillings as a sum of the signatures of basic relations in the monodromy of a related…

The paper constructs metrics with negative curvature on complex manifolds.

The research community has increasing interest in autonomous driving research, despite the resource intensity of obtaining representative real world data. Existing self-driving datasets are limited in the scale and variation of the environments they capture, even though generalization within and between operating regio…

We investigate symmetry properties of peculiar modules, a Heegaard Floer invariant of 4-ended tangles which the author introduced in [arXiv:1712.05050]. In particular, we give an almost complete answer to the geography problem for components of peculiar modules of tangles. As a main application, we show that Conway mut…

Researchers propose a new approach to the Hopf problem for aspherical varieties.