arXiv research

In this paper we show that there are two symplectic surfaces in the 4-ball which bound the same transverse knot, have the same topology (as abstract surfaces), and are distinguished by the fundamental groups of their complements.

The paper studies Sasaki-Ricci solitons on Sasakian manifolds up to seven dimensions.

Every 4-dimensional infrasolvmanifold $M$ with $β_1(M;\mathbb{Q})>0$ or which is flat or has one of the geometries $\mathbb{N}il^4$, $\mathbb{S}ol_{m,n}^4$, or $\mathbb{S}ol_0^4$ bounds. However there are non-orientable $\mathbb{S}ol_1^4$-manifolds which do not bound. The question remains open for $\mathbb{N}il^3\times…

We show that if $K$ is a fibered ribbon knot in $S^3=\partial B^4$ bounding a ribbon disk $D$, then given an extra transversality condition the fibration on $S^3\setminusν(K)$ extends to a fibration of $B^4\setminusν(D)$. This partially answers a question of Casson and Gordon. In particular, we show the fibration alway…

Researchers compute bounds and formulas for non-orientable 4-genus of torus knots.

The paper explores the genus of surfaces in complex projective spaces and improves minimal genus bounds.

The nonorientable four-ball genus of a knot K is the smallest first Betti number of any smoothly embedded, nonorientable surface F in B^4 bounding K. In contrast to the orientable four-ball genus, which is bounded below by the Murasugi signature, the Ozsvath-Szabo tau-invariant, the Rasmussen s-invariant, the best lowe…

SGD achieves a $O(ε^{-4})$ bound for minimizing gradient norm of smooth functions.

It is known that for coprime integers $p>q\geq 1$, the lens space $L(p^2,pq-1)$ bounds a rational ball, $B_{p,q}$, arising as the 2-fold branched cover of a (smooth) slice disk in $B^4$ bounding the associated 2-bridge knot. Lekilli and Maydanskiy give handle decompositions for each $B_{p,q}$. Whereas, Yamada gives an …

Suppose $k$ centers are fit to $m$ points by heuristically minimizing the $k$-means cost; what is the corresponding fit over the source distribution? This question is resolved here for distributions with $p\geq 4$ bounded moments; in particular, the difference between the sample cost and distribution cost decays with $…

New bounds on nonorientable four-ball genus for torus knots.

The paper shows convergence of Sasaki-Ricci flow on Sasakian 5-manifolds.

Rectified flows achieve optimal sample complexity for generating data.

Combining diffusion models with Langevin dynamics improves posterior sampling efficiency.

Paper formalizes Simon's satisficing through FFSD, proving its equivalence to expected utility theory.

Ribbon tangles are proper embeddings of tori and cylinders in the $4$-ball~$B^4$, "bounding" $3$-manifolds with only ribbon disks as singularities. We construct an Alexander invariant $\mathsf{A}$ of ribbon tangles equipped with a representation of the fundamental group of their exterior in a free abelian group $G$. Th…

The paper proves the existence of Sasaki-Einstein metrics on specific Sasakian manifolds.

Let $(M,g)$ be a two-dimensional compact boundaryless Riemannian manifold with nonpostive curvature, then we shall give improved estimates for the $L^2$-norms of the restrictions of eigenfunctions to unit-length geodesics, compared to the general results of Burq, Gérard and Tzvetkov \cite{burq}. By earlier results of B…

The study bounds the excess of disjoint nonorientable surfaces in a 4-manifold.

Study shows offline RL under $Q^\star$-approximation and partial coverage is harder than previously thought.