We establish the rationality of simple isolated Cohen–Macaulay codimension $2$ (ICMC2) singularities in all dimensions $\ge 2$ and explicitly compute the vanishing homology of a certain class of threefolds including all the simple ones. ICMC2 singularities are determinantal and can be viewed as a natural generalization of complete intersections. The main tool for our investigations is the so-called Tjurina transformation—a special blowup construction based on the determinantal structure and often compatible with deformations.

We consider noncompact ancient solutions to the mean curvature flow in ${ℝ}^{n+1}$ ($n\ge 3$) which are strictly convex, uniformly two-convex, and noncollapsed. We prove that such an ancient solution is a rotationally symmetric translating soliton.

We show that coinvariants of modules over vertex operator algebras give rise to quasicoherent sheaves on moduli of stable pointed curves. These generalize Verlinde bundles or vector bundles of conformal blocks defined using affine Lie algebras studied first by Tsuchiya, Kanie, Ueno and Yamada, and extend work of others. The sheaves carry a twisted logarithmic $\mathcal{𝒟}$–module structure, and hence support a projectively flat connection. We identify the logarithmic Atiyah algebra acting on them, generalizing work of Tsuchimoto for affine Lie algebras.

We present a cocycle model for elliptic cohomology with complex coefficients in which methods from $2$–dimensional quantum field theory can be used to rigorously construct cocycles. For example, quantizing a theory of vector-bundle-valued fermions yields a cocycle representative of the elliptic Thom class. This constructs the complexified string orientation of elliptic cohomology, which determines a pushforward for families of rational string manifolds. A second pushforward is constructed from quantizing a supersymmetric $\sigma$–model. These two pushforwards agree, giving a precise physical interpretation for the elliptic index theorem with complex coefficients. This both refines and supplies further evidence for the long-conjectured relationship between elliptic cohomology and $2$–dimensional quantum field theory. Analogous methods in supersymmetric mechanics recover path integral constructions of the Mathai–Quillen Thom form in complexified $\text{KO}$–theory and a cocycle representative of the $Â$–class for a family of oriented manifolds.

A Kodaira fibration is a compact, complex surface admitting a holomorphic submersion onto a complex curve such that the fibers have nonconstant moduli. We consider Kodaira fibrations $X$ with nontrivial invariant $\mathbb{Q}$–cohomology in degree $1$, proving that if the dimension of the holomorphic invariants is $1$ or $2$, then $X$ admits a branch covering over a product of curves inducing an isomorphism on rational cohomology in degree $1$. We also study the class of Kodaira fibrations possessing a holomorphic section, and demonstrate that having a section imposes no restriction on possible monodromies.

A subgroup $H\le G$ is said to be almost normal if every conjugate of $H$ is commensurable to $H$. If $H$ is almost normal, there is a well-defined quotient space $G∕H$. We show that if a group $G$ has type $F_{n+1}$ and contains an almost normal coarse ${PD}_n$ subgroup $H$ with $e\left(G∕H\right)=\infty$, then whenever $G^{\prime }$ is quasi-isometric to $G$ it contains an almost normal subgroup $H^{\prime }$ that is quasi-isometric to $H$. Moreover, the quotient spaces $G∕H$ and $G^{\prime }∕H^{\prime }$ are quasi-isometric. This generalises a theorem of Mosher, Sageev and Whyte, who prove the case in which $G∕H$ is quasi-isometric to a finite-valence bushy tree. Using work of Mosher, we generalise a result of Farb and Mosher to show that for many surface group extensions ${\mathrm{\Gamma }}_L$, any group quasi-isometric to ${\mathrm{\Gamma }}_L$ is virtually isomorphic to ${\mathrm{\Gamma }}_L$. We also prove quasi-isometric rigidity for the class of finitely presented $\mathbb{Z}$-by-($\infty$–ended) groups.

We investigate the geometry of asymptotically flat manifolds with controlled holonomy. We show that any end of such manifold admits a torus fibration over an ALE end. In addition, we prove a Hitchin–Thorpe inequality for oriented Ricci-flat $4$–manifolds with curvature decay and controlled holonomy. As an application, we show that any complete, asymptotically flat, Ricci-flat metric on a $4$–manifold which is homeomorphic to ${ℝ}^4$ must be isometric to the Euclidean or the Taub–NUT metric, provided that the tangent cone at infinity is not $ℝ\times {ℝ}_{+}$.

The space $\mathcal{\mathscr{H}}$ of “almost calibrated” $\left(1,1\right)$–forms on a compact Kähler manifold plays an important role in the study of the deformed Hermitian Yang–Mills equation of mirror symmetry, as emphasized by recent work of Collins and Yau (2018), and is related by mirror symmetry to the space of positive Lagrangians studied by Solomon (2013, 2014). This paper initiates the study of the geometry of $\mathcal{\mathscr{H}}$. We show that $\mathcal{\mathscr{H}}$ is an infinite-dimensional Riemannian manifold with nonpositive sectional curvature. In the hypercritical phase case we show that $\mathcal{\mathscr{H}}$ has a well-defined metric structure, and that its completion is a $\text{CAT}\left(0\right)$ geodesic metric space, and hence has an intrinsically defined ideal boundary. Finally, we show that in the hypercritical phase case $\mathcal{\mathscr{H}}$ admits $C^{1,1}$ geodesics, improving a result of Collins and Yau (2018). Using results of Darvas and Lempert (2012) we show that this result is sharp.

Let $V$ be a Banach space all of whose subspaces of a fixed dimension $n$ are isometric, with . In 1932, S Banach asked if under this hypothesis $V$ is necessarily a Hilbert space. In 1967, M Gromov answered it positively for even $n$. We give a positive answer for real $V$ and odd $n$ of the form $n=4k+1$, with the possible exception of $n=133$. Our proof relies on a new characterization of ellipsoids in ${ℝ}^n$ for $n\ge 5$, as the only symmetric convex bodies all of whose linear hyperplane sections are linearly equivalent affine bodies of revolution.

K–polystability is, on the one hand, conjecturally equivalent to the existence of certain canonical Kähler metrics on polarised varieties, and, on the other hand, conjecturally gives the correct notion to form moduli. We introduce a notion of stability for families of K–polystable varieties, extending the classical notion of slope stability of a bundle, viewed as a family of K–polystable varieties via the associated projectivisation. We conjecture that this is the correct condition for forming moduli of fibrations.

Our main result relates this stability condition to Kähler geometry: we prove that the existence of an optimal symplectic connection implies semistability of the fibration. An optimal symplectic connection is a choice of fibrewise constant scalar curvature Kähler metric, satisfying a certain geometric partial differential equation. We conjecture that the existence of such a connection is equivalent to polystability of the fibration. We prove a finite-dimensional analogue of this conjecture, by describing a GIT problem for fibrations embedded in a fixed projective space, and showing that GIT polystability is equivalent to the existence of a zero of a certain moment map.

We correct some oversights in the paper “A spectral sequence for stratified spaces and configuration spaces of points” by the second author. In particular, we explain that an additional hypothesis should be added to Theorem 4.15 in said paper.

This short note corrects an error appearing in the published version of the titular paper and gives a brief discussion of an example illustrating the corrected construction.