We study asymptotic behavior of nonparametric hypersurfaces moving by $\alpha$ powers of Gauss curvature with $\alpha >1/n$. Our work generalizes the results of Oliker [Oli91] for $\alpha =1$.

In the Gromov–Witten theory of a target curve, we consider descendent integrals against the virtual fundamental class relative to the forgetful morphism to the moduli space of curves. We show that cohomology classes obtained in this way lie in the tautological ring.

We study decompositions of complex hyperbolic isometries as products of involutions. We show that $PU(2,1)$ has involution length 4 and commutator length 1 and that, for all $n⩾3$, $PU(n,1)$ has involution length at most 8.

We study the classification problem of holomorphic isometric embeddings of the unit disk into polydisks as in [Ng10, Ch16a]. We give a complete classification of all such holomorphic isometries when the target is the $4$-disk ${\Delta }^4$. Moreover, we classify those holomorphic isometric embeddings with certain prescribed sheeting numbers. In addition, we prove that a known example in the space ${\mathbf{HI}}_k(\Delta ,{\Delta }^{qk};q)$ is globally rigid for any integers $k,q\ge 2$, which generalizes Theorem 1.1 in [Ch16a].

We establish an analytic Hasse principle for linear spaces of affine dimension $m$ on a complete intersection over an algebraic field extension $\mathbb{K}$ of $\mathbb{Q}$. The number of variables required to do this is no larger than what is known for the analogous problem over $\mathbb{Q}$. As an application, we show that any smooth hypersurface over $\mathbb{K}$ whose dimension is large enough in terms of the degree is $\mathbb{K}$-unirational, provided that either the degree is odd or $\mathbb{K}$ is totally imaginary.

We consider the problem of comparing $t$-structures under the derived McKay correspondence and for tilting equivalences. In low-dimensional cases, we relate the $t$-structures via torsion theories arising from additive functions on the triangulated category. As an application, we give a criterion for rationality for surfaces with a tilting bundle. We also show that every smooth projective surface that admits a full, strong, and exceptional collection of line bundles is rational.

Let $N^k\subset S^{k+2}$ be a closed oriented submanifold. Denote its complement by $C\left(N\right)=S^{k+2}\setminus N$. Denote by $\xi \in H^1\left(C\right(N\left)\right)$ the class dual to $N$. The Morse–Novikov number of $C\left(N\right)$ is by definition the minimal possible number of critical points of a regular Morse map $C\left(N\right)\to S^1$ belonging to $\xi$. In the first part of this paper, we study the case where $N$ is the twist frame spun knot associated with an $m$-knot $K$. We obtain a formula that relates the Morse–Novikov numbers of $N$ and $K$ and generalizes the classical results of D. Roseman and E. C. Zeeman about fibrations of spun knots. In the second part, we apply the obtained results to the computation of Morse–Novikov numbers of surface-links in 4-sphere.

We study higher-genus Fan–Jarvis–Ruan–Witten theory of any chain polynomial with any group of symmetries. Precisely, we give an explicit way to compute the cup product of Polishchuk and Vaintrob’s virtual class with the top Chern class of the Hodge bundle. Our formula for this product holds in any genus and without any assumption on the semi-simplicity of the underlying cohomological field theory.

The monster tower is a tower of spaces over a specified base; each space in the tower is a parameter space for curvilinear data up to a specified order. We describe and analyze a natural stratification of these spaces.

In this remark, we show how the monopole Frøyshov invariant, as well as the analogues of the Involutive Heegaard Floer correction terms $\underline{d},\overline{d}$, are related to the $Pin\left(2\right)$-equivariant Floer homology ${\mathit{SWFH}}_{\ast }^G$. We show that the only interesting correction terms of a $Pin\left(2\right)$-space are those coming from the subgroups $\mathbb{Z}/4$, $S^1$, and $Pin\left(2\right)$ itself.

We characterize the cyclic branched covers of the 2-sphere where every homeomorphism of the sphere lifts to a homeomorphism of the covering surface. This answers the question that appeared in an early version of the erratum of Birman and Hilden [2].