We prove the Turaev–Viro invariants volume conjecture for a “universal” class of cusped hyperbolic $3$-manifolds that produces all $3$-manifolds with empty or toroidal boundary by Dehn filling. This leads to two-sided bounds on the volume of any hyperbolic $3$-manifold with empty or toroidal boundary in terms of the growth rate of the Turaev–Viro invariants of the complement of an appropriate link contained in the manifold. We also provide evidence for a conjecture of Andersen, Masbaum and Ueno (AMU conjecture) about certain quantum representations of surface mapping class groups.

A key step in our proofs is finding a sharp upper bound on the growth rate of the quantum $6j$-symbol evaluated at $q=e\frac{2\pi i}{r}$.

Let $N$ be a smooth manifold that is homeomorphic but not diffeomorphic to a closed hyperbolic manifold $M$. In this paper, we study the extent to which $N$ admits as much symmetry as $M$. Our main results are examples of $N$ that exhibit two extremes of behavior. On the one hand, we find $N$ with maximal symmetry, i.e. $\operatorname{Isom}(M)$ acts on $N$ by isometries with respect to some negatively curved metric on $N$. For these examples, the order of $\operatorname{Isom}(M)$ can be arbitrarily large. On the other hand, we find $N$ with little symmetry, i.e. no subgroup of $\operatorname{Isom}(M)$ of “small” index acts by diffeomorphisms of $N$. The construction of these examples incorporates a variety of techniques including smoothing theory and the Belolipetsky–Lubotzky method for constructing hyperbolic manifolds with a prescribed isometry group.

We analyze the torus fixed loci of Mixed Spin P fields moduli, and deduce its localization formulas with explicit factors. An algorithms toward evaluating quintic’s Gromov–Witten and Fan–Jarvis–Ruan–Witten invariants are derived.

We give a complete classification of the immersed constant mean curvature spheres in a three-sphere with an arbitrary homogeneous metric, by proving that for each $H \in \mathbb{R}$, there exists a constant mean curvature $H$ sphere in the space that is unique up to an ambient isometry.

In this paper we first introduce quermassintegrals for free boundary hypersurfaces in the $(n+1)$-dimensional Euclidean unit ball. Then we solve some related isoperimetric type problems for convex free boundary hypersurfaces, which lead to new Alexandrov–Fenchel inequalities. In particular, for $n = 2$ we obtain a Minkowski-type inequality and for $n = 3$ we obtain an optimal Willmore-type inequality. To prove these estimates, we employ a specifically designed locally constrained inverse harmonic mean curvature flow with free boundary.

With the aim of uniform treatment of multiplicity-free representations of Lie groups, T. Kobayashi introduced the notion of visible actions on complex manifolds. He proved that for a Lie group $U$ the space of global sections of a $U$-equivariant holomorphic vector bundle $\mathcal{W}$ is multiplicity-free if the $U$-action on the base space $X$ is strongly visible and if the isotropy representations on the fibers are multiplicity-free under some compatibility condition on an anti-holomorphic diffeomorphism of $\mathcal{W}$. Especially in the case of the trivial line bundle, his theorem says that if a Lie group acts on the base space strongly visibly then the space of holomorphic functions is multiplicity-free. In short, the visibility is a geometric condition that assures the multiplicity-freeness property. In this article we consider the converse direction when $U$ is a compact real form of a connected complex reductive algebraic group $G$ and $X$ is an irreducible complex algebraic $G$-variety. In this setting the multiplicity-freeness property of the space of algebraic sections of arbitrary $G$-line bundle is equivalent to the sphericity of the $G$-action on $X$. Then our main result says that the multiplicity-freeness property implies the visibility, namely, if $X$ is a $G$-spherical variety then the $U$-action on $X$ is strongly visible.