We prove an inequality between the sum of the Betti numbers of a complex projective manifold and its total curvature, and we characterize the complex projective manifolds whose total curvature is minimal. These results extend the classical theorems of Chern and Lashof to complex projective space.

Using the method of Elias and Hogancamp and combinatorics of toric braids from our proof of the rational shuffle conjecture, we give an explicit formula for the triply graded Khovanov–Rozansky homology (superpolynomial) of an arbitrary positive torus knot, thereby proving some of the conjectures of Aganagic and Shakirov, Cherednik, Gorsky and Negut, and Oblomkov, Rasmussen and Shende.

Given a countable group $G$ splitting as a free product $G=G_1\ast \cdots \ast G_k\ast F_N$, we establish classification results for subgroups of the group $\text{Out}(G,\mathcal{ℱ})$ of all outer automorphisms of $G$ that preserve the conjugacy class of each $G_i$. We show that every finitely generated subgroup $H\subseteq \text{Out}(G,\mathcal{ℱ})$ either contains a relatively fully irreducible automorphism, or else it virtually preserves the conjugacy class of a proper free factor relative to the decomposition (the finite generation hypothesis on $H$ can be dropped for $G=F_N$, or more generally when $G$ is toral relatively hyperbolic). In the first case, either $H$ virtually preserves a nonperipheral conjugacy class in $G$, or else $H$ contains an atoroidal automorphism. The key geometric tool to obtain these classification results is a description of the Gromov boundaries of relative versions of the free factor graph $\text{FF}$ and the $\mathcal{𝒵}$–factor graph $\mathcal{𝒵}\text{F}$, as spaces of equivalence classes of arational trees and relatively free arational trees, respectively. We also identify the loxodromic isometries of $\text{FF}$ with the fully irreducible elements of $\text{Out}(G,\mathcal{ℱ})$, and loxodromic isometries of $\mathcal{𝒵}\text{F}$ with the fully irreducible atoroidal outer automorphisms.

We prove that for the harmonic measure associated to a random walk on $\text{Out}(F_r)$ satisfying some mild conditions, a typical tree in the boundary of Outer space is trivalent and nongeometric. This result answers a question of Mladen Bestvina.

Given a complex analytic germ $(X,0)$ in $({ℂ}^n,0)$, the standard Hermitian metric of ${ℂ}^n$ induces a natural arc-length metric on $(X,0)$, called the inner metric. We study the inner metric structure of the germ of an isolated complex surface singularity $(X,0)$ by means of an infinite family of numerical analytic invariants, called inner rates. Our main result is a formula for the Laplacian of the inner rate function on a space of valuations, the nonarchimedean link of $(X,0)$. We deduce in particular that the global data consisting of the topology of $(X,0)$, together with the configuration of a generic hyperplane section and of the polar curve of a generic plane projection of $(X,0)$, completely determine all the inner rates on $(X,0)$, and hence the local metric structure of the germ. Several other applications of our formula are discussed.

We construct a series of homomorphisms from the $Y$–filtration on the monoid of homology cylinders to torsion modules via the mod $\mathbb{Z}$ reduction of the LMO functor. The restrictions of our homomorphisms to the lower central series of the Torelli group do not factor through Morita’s refinement of the Johnson homomorphism. We use it to show that the abelianization of the Johnson kernel of a closed surface has torsion elements. We also determine the third graded quotient $Y_3\mathcal{ℐ}{\mathcal{𝒞}}_{g,1}∕Y_4$ of the $Y$–filtration.

We show the Jordan property for regional fundamental groups of klt singularities of fixed dimension. Furthermore, we prove the existence of effective simultaneous index $1$ covers for $n$–dimensional klt singularities. We give an application to the study of local class groups of klt singularities.

A conjecture of Morel asserts that the sheaf of ${\mathbb{A}}^1$–connected components of a space is ${\mathbb{A}}^1$–invariant. Using purely algebrogeometric methods, we determine the sheaf of ${\mathbb{A}}^1$–connected components of a smooth projective surface, which is birationally ruled over a curve of genus . As a consequence, we show that Morel’s conjecture holds for all smooth projective surfaces over an algebraically closed field of characteristic $0$.

We calculate the homotopy type of $L_1{L}_{K(2)}{S}^0$ and $L_{K(1)}{L}_{K(2)}{S}^0$ at the prime 2, where $L_{K(n)}$ is localization with respect to Morava $K$–theory and $L_1$ localization with respect to $2$–local $K$–theory. In $L_1{L}_{K(2)}{S}^0$ we find all the summands predicted by the Chromatic Splitting Conjecture, but we find some extra summands as well. An essential ingredient in our approach is the analysis of the continuous group cohomology $H^{\ast }({\mathbb{𝔾}}_2,E_0)$, where ${\mathbb{𝔾}}_2$ is the Morava stabilizer group and $E_0=\mathbb{𝕎}[[u_1]]$ is the ring of functions on the height $2$ Lubin–Tate space. We show that the inclusion of the constants $\mathbb{𝕎}\to E_0$ induces an isomorphism on group cohomology, a radical simplification.