arXiv research

New 4-manifolds with distinct smoothings via group actions.

Study shows infinitely many ways to compactify surfaces as ball quotients.

We prove that for every natural number k there are simply connected topological four-manifolds which have at leat k distinct smooth structures supporting Einstein metrics, and also have infinitely many distinct smooth structures not supporting Einstein metrics. Moreover, all these smooth structures become diffeomorphic…

The study classifies smooth structures on specific complex projective spaces.

New lattices from 4-manifolds show manifold properties.

The study examines smooth structures and inertia groups of specific manifolds.

The abstract discusses smooth structures on a fake real projective space.

We provide examples of families of (log) smooth canonically polarized varieties, including smooth weighted pointed curves and smooth hypersurfaces in $P^3$ with large degree such that the Chow semistable limits under distinct pluricanonical embeddings do not stabilize.

Study of choosing distinct points on cubic curves, proving impossibility.

We introduce a new technique that is used to show that the complex projective plane blown up at 6, 7, or 8 points has infinitely many distinct smooth structures. None of these smooth structures admit smoothly embedded spheres with self-intersection -1, i.e. they are minimal. In addition, none these smooth structures ad…

Motivated by a construction of Fintushel and Stern, we show that the topological 4--manifold $CP^2#5{\bar CP^2}$ supports infinitely many distinct smooth structures.

Let P be a knot in an unknotted solid torus (i.e. a satellite operator or pattern), K a knot in S^3 and P(K) the satellite of K with pattern P. For any satellite operator P, this correspondence gives a function P : C -> C on the set of smooth concordance classes of knots. We give examples of winding number one satellit…

The study identifies criteria for 3-manifolds to be boundaries of exotic 4-manifolds.

We show examples of pairs of smooth, compact, homeomorphic 4-manifolds, whose diffeomorphism types are distinguished by the topology of the singular sets of smooth stable maps defined on them. In this distinction we rely on results from Seiberg-Witten theory.

Study on knots in 3-manifolds using classical techniques.

Paper finds infinitely many smooth solutions to complex geometry problem.

Motivated by Stipsicz and Szabó's exotic 4-manifolds with b_2^+=3 and b_2^-=8, we construct a family of simply connected smooth 4-manifolds with b_2^+=3 and b_2^-=8. As a corollary, we conclude that the topological 4-manifold 3{CP}^2#8{-CP}^2 admits infinitely many distinct smooth structures.

We review the basic definition of a stack and apply it to the topological and smooth settings. We then address two subtleties of the theory: the correct definition of a ``stack over a stack'' and the distinction between small stacks (which are algebraic objects) and large stacks (which are generalized spaces).

The study explores smooth structures on specific four-manifolds with cyclic groups, finding many admit infinitely many smooth structures.

We say that a topologically embedded 3-sphere in a smoothing of Euclidean 4-space is a barrier provided, roughly, no diffeomorphism of the 4-manifold moves the 3-sphere off itself. In this paper we construct infinitely many one parameter families of distinct smoothings of 4-space with barrier 3-spheres. \par The existe…

We consider billiard trajectories in a smooth convex body in $\mathbb R^d$ and estimate the number of distinct periodic trajectories that make exactly $p$ reflections per period at the boundary of the body. In the case of prime $p$ we obtain the lower bound $(d-2)(p-1)+2$, which is much better than the previous estimat…

A new loss function improves deep learning performance without class separation constraints.

The study finds smooth structures on specific 4-manifolds with even fundamental groups.

Paper constructs infinitely many tangent functors on diffeological spaces.

In high dimensions, there are many distinct manifolds with nonnegative curvature.

Consider a domain D in R^3 which is convex (possibly all R^3) or which is smooth and bounded. Given any open surface M, we prove that there exists a complete, proper minimal immersion f : M --> D. Moreover, if D is smooth and bounded, then we prove that the immersion f : M --> D can be chosen so that the limit sets of …

Smooth and symplectic symmetries of an infinite family of distinct exotic $K3$ surfaces are studied, and comparison with the corresponding symmetries of the standard $K3$ is made. The action on the $K3$ lattice induced by a smooth finite group action is shown to be strongly restricted, and as a result, nonsmoothability…

The paper proves that any smooth curve can have two similar inscribed rectangles.

The paper characterizes global hyperbolicity in Lorentzian manifolds without relying on manifold topology.

New disks fill Legendrian knots without being smoothly isotopic.

RAMBO optimizes multi-regime problems by discovering and modeling distinct energy basins.

We show that the minimal volume entropy of closed manifolds remains unaffected when nonessential manifolds are added in a connected sum. We combine this result with the stable cohomotopy invariant of Bauer-Furuta in order to present an infinite family of four-manifolds with the following properties: 1) They have positi…

Introduces a new triple coproduct for knots on surfaces, preserving local crossing patterns.

The study finds stably free modules and distinct 2-complexes for large ranks.

This paper investigates the question of which smooth compact 4-manifolds admit Riemannian metrics that minimize the L2-norm of the curvature tensor. Metrics with this property are called OPTIMAL; Einstein metrics and scalar-flat anti-self-dual metrics provide us with two interesting classes of examples. Using twistor m…

We show that, for any two orientable smooth open 4-manifolds $X_0,X_1$ which are homeomorphic, their cotangent bundles $T^*X_0,T^*X_1$ are symplectomorphic with their canonical symplectic structure. In particular, for any smooth manifold $R$ homeomorphic to $\mathbb{R}^4$, the standard Stein structure on $T^*R$ is Stei…

In this article, we shall investigate the relationship between the existence or non-existence of non-singular solutions to the normalized Ricci flow and smooth structures on closed 4-manifolds, where non-singular solutions to the normalized Ricci flow are solutions which exist for all time $t \in [0, \infty)$ with unif…

It is known that an arbitrary smooth, oriented 4-manifold admits the structure of what is called a broken Lefschetz fibration. Given a broken fibration, there are certain modifications, realized as homotopies of the fibration map, that enable one to construct infinitely many distinct fibrations of the same manifold. Th…

Existence of minimizers for Klein bottles in 4D space.

We construct an infinite order cork (W,f), which means that W is a smooth compact contractible 4-manifold with Stein structure, and f is a self diffeomorphism of the boundary of W, such that the n-fold composition maps f^{n}=f o f o... o f give rise to smoothly distinct corks (W, f^{n}) for sufficiently large values of…

Study on choosing points on cubic curves, answering some questions about their flexibility.

We show that non-degenerate hyperquadrics in R^{n+2} admit no skew branes. Stated more traditionally, a compact codimension-one immersed submanifold of a non-degenerate hyperquadric of euclidean space must have parallel tangent spaces at two distinct points. Similar results have been proven by others, but (except for e…

Paper refines RNN training by analyzing smoothness and attractors.

The paper constructs infinitely many surfaces with specific mean curvature.

Improved learning rates with new smoothness measure.

Theory of 2-vector bundles for smooth manifolds developed.

Study finds many nonplanar minimal spheres in elongated ellipsoids.

Study on diffeotopy groups of non-compact 4-manifolds, extending previous results.