We prove that every degree-g polynomial in the ψ-classes on ${\bar{\mathcal{M}}}_{\mathit{g},\mathit{n}}$ can be expressed as a sum of tautological classes supported on the boundary with no κ-classes. Such equations, which we refer to as topological recursion relations, can be used to deduce universal equations for the Gromov–Witten invariants of any target.

In 1979, Pisier proved remarkably that a sequence of independent and identically distributed standard Gaussian random variables determines, via random Fourier series, a homogeneous Banach algebra $\mathcal{P}$ strictly contained in $\mathit{C}(\mathbb{T})$, the class of continuous functions on the unit circle $\mathbb{T}$ and strictly containing the classical Wiener algebra $\mathbb{A}(\mathbb{T})$, that is, $\mathbb{A}(\mathbb{T})⫋\mathcal{P}⫋\mathit{C}(\mathbb{T})$. This improved some previous results obtained by Zafran in solving a long-standing problem raised by Katznelson. In this paper, we extend Pisier’s result by showing that any probability measure on the unit circle defines a homogeneous Banach algebra contained in $\mathit{C}(\mathbb{T})$. Thus Pisier algebra is not an isolated object but rather an element in a large class of Pisier-type algebras. We consider the case of spectral measures of stationary sequences of Gaussian random variables and obtain a sufficient condition for the boundedness of the random Fourier series ${\sum }_{\mathit{n}\in \mathbb{Z}}\hat{\mathit{f}}(\mathit{n}){\mathit{\xi }}_{\mathit{n}}exp(2\mathit{\pi }\mathit{i}\mathit{n}\mathit{t})$ in the general setting of dependent random variables $({\mathit{\xi }}_{\mathit{n}})$.

We use bordered Floer theory to study properties of twisted Mazur pattern satellite knots ${\mathit{Q}}_{\mathit{n}}(\mathit{K})$. We prove that ${\mathit{Q}}_{\mathit{n}}(\mathit{K})$ is not Floer homologically thin, with two exceptions. We calculate the 3-genus of ${\mathit{Q}}_{\mathit{n}}(\mathit{K})$ in terms of the twisting parameter n and the 3-genus of the companion K, and we determine when ${\mathit{Q}}_{\mathit{n}}(\mathit{K})$ is fibered. As an application to our results on Floer thickness and 3-genus, we verify the Cosmetic Surgery Conjecture for many of these satellite knots.

Given a natural number k and an orientable surface S of finite type, define the k-curve graph to be the graph with vertices corresponding to isotopy classes of essential simple closed curves on S and with edges corresponding to pairs of such curves admitting representatives that intersect at most k times. We prove that the automorphism group of the k-curve graph of a surface S is isomorphic to the extended mapping class group for all k satisfying $\mathit{k}\le |\mathit{\chi }(\mathit{S})|-512$. We prove the same result for the so-called systolic complex, a variant of the curve graph with many complete subgraphs coming from interesting collections of systoles with respect to a hyperbolic metric. This resolves a conjecture of Schmutz Schaller.

We study the local equivalence problem for real-analytic (${\mathcal{C}}^{\mathit{\omega }}$) hypersurfaces ${\mathit{M}}^5\subset {\mathbb{C}}^3$ that, in some holomorphic coordinates $({\mathit{z}}_1,{\mathit{z}}_2,\mathit{w})\in {\mathbb{C}}^3$ with $\mathit{w}=\mathit{u}+\sqrt{-1}\mathit{v}$, are rigid in the sense that their graphing functions

$$\mathit{u}=\mathit{F}({\mathit{z}}_1,{\mathit{z}}_2,{\bar{\mathit{z}}}_1,{\bar{\mathit{z}}}_2)$$

are independent of v. Specifically, we study the group ${\mathsf{Hol}}_{\mathsf{rigid}}(\mathit{M})$ of rigid local biholomorphic transformations of the form

$$({\mathit{z}}_1,{\mathit{z}}_2,\mathit{w})⟼({\mathit{f}}_1({\mathit{z}}_1,{\mathit{z}}_2),{\mathit{f}}_2({\mathit{z}}_1,{\mathit{z}}_2),\mathit{a}\mathit{w}+\mathit{g}({\mathit{z}}_1,{\mathit{z}}_2)),$$

where $\mathit{a}\in \mathbb{R}\setminus \{0\}$ and $\mathit{D}({\mathit{f}}_1,{\mathit{f}}_2)/\mathit{D}({\mathit{z}}_1,{\mathit{z}}_2)\ne 0$, which preserve the rigidity of hypersurfaces.

After performing a Cartan-type reduction to an appropriate $\{\mathit{e}\}$-structure, we find exactly two primary invariants ${\mathit{I}}_0$ and ${\mathit{V}}_0$, which we express explicitly in terms of the 5-jet of the graphing function F of M. The identical vanishing $0\equiv {\mathit{I}}_0({\mathit{J}}^5\mathit{F})\equiv {\mathit{V}}_0({\mathit{J}}^5\mathit{F})$ then provides a necessary and sufficient condition for M to be locally rigidly biholomorphic to the known model hypersurface

$${\mathit{M}}_{\mathsf{LC}}:\mathit{u}=\frac{{\mathit{z}}_1{\bar{\mathit{z}}}_1+1/2{\mathit{z}}_1^2{\bar{\mathit{z}}}_2+1/2{\bar{\mathit{z}}}_1^2{\mathit{z}}_2}{1-{\mathit{z}}_2{\bar{\mathit{z}}}_2}.$$

We establish that always $\mathsf{dim}{\mathsf{Hol}}_{\mathsf{rigid}}(\mathit{M})\le 7=\mathsf{dim}{\mathsf{Hol}}_{\mathsf{rigid}}({\mathit{M}}_{\mathsf{LC}})$.

If one of these two primary invariants ${\mathit{I}}_0\not\equiv 0$ or ${\mathit{V}}_0\not\equiv 0$ does not vanish identically, then on either of the two Zariski-open sets $\{\mathit{p}\in \mathit{M}:{\mathit{I}}_0(\mathit{p})\ne 0\}$ or $\{\mathit{p}\in \mathit{M}:{\mathit{V}}_0(\mathit{p})\ne 0\}$, we show that this rigid equivalence problem between rigid hypersurfaces reduces to an equivalence problem for a certain five-dimensional $\{\mathit{e}\}$-structure on M, that is, we get an invariant absolute parallelism on ${\mathit{M}}^5$. Hence $\mathsf{dim}{\mathsf{Hol}}_{\mathsf{rigid}}(\mathit{M})$ drops from 7 to 5, illustrating the gap phenomenon.

We show that noetherian splinters ascend under essentially étale homomorphisms. Along the way, we also prove that the henselization of a noetherian local splinter is always a splinter and that the completion of a local splinter with geometrically regular formal fibers is a splinter. Finally, we give an example of a (nonexcellent) Gorenstein local splinter with mild singularities whose completion is not a splinter. Our results provide evidence for a strengthening of the direct summand theorem, namely that regular maps preserve the splinter property.

The main goal of this paper is to study the different definitions of generating sequences appearing in the literature. We present these definitions and show that under certain situations they are equivalent. We also present an example that shows that they are not, in general, equivalent. We also present the relation of generating sequences and key polynomials.

A ring is said to be rigid if it admits no nonzero locally nilpotent derivations, and an affine variety is rigid if its coordinate ring is rigid. In this paper, we improve some techniques for determining the rigidity of k-domains (affine varieties) over a field k of characteristic zero. First, we generalize the ABC theorem. Then we study locally nilpotent derivations of a simple algebraic extension $\mathit{R}[\mathit{z}]$ of a k-domain R, where $\mathit{r}{\mathit{z}}^{\mathit{n}}\in \mathit{R}$ for some nonzero $\mathit{r}\in \mathit{R}$ and some positive integer n. Subsequently, we study locally nilpotent derivations and rigidity of an extension $\mathit{R}[\mathit{x},\mathit{y}]$ of R such that ${\mathit{r}}_1{\mathit{x}}^{\mathit{m}}{\mathit{y}}^{\mathit{n}}\in \mathit{R}$ or ${\mathit{r}}_1{\mathit{x}}^{\mathit{m}}+{\mathit{r}}_2{\mathit{y}}^{\mathit{n}}\in \mathit{R}$ for some nonzero ${\mathit{r}}_1,{\mathit{r}}_2\in \mathit{R}$ and some positive integers $\mathit{m},\mathit{n}$. Finally, as applications of these general results, we prove the rigidity of some quadrinomial varieties and Pham–Brieskorn hypersurfaces.

Given a dominant rational self-map on a projective variety over a number field, we can define the arithmetic degree at a rational point. It is known that the arithmetic degree at any point is less than or equal to the first dynamical degree. In this paper, we show that there are densely many $\bar{\mathbb{Q}}$-rational points with maximal arithmetic degree (i.e., whose arithmetic degree is equal to the first dynamical degree) for self-morphisms on projective varieties. For unirational varieties and Abelian varieties, we show that there are densely many rational points with maximal arithmetic degree over a sufficiently large number field. We also give a generalization of a result of Kawaguchi and Silverman in the Appendix.