We prove that the $\frac{1}{2}$–snowflake of any finite-dimensional normed space $X$ embeds into a Hilbert space with quadratic average distortion

$$O\left(\sqrt{logdim\left(X\right)}\right).$$

We deduce from this (optimal) statement that if an $n$–vertex expander embeds with average distortion $D⩾1$ into $X$, then necessarily $dim\left(X\right)⩾n^{\mathrm{\Omega }\left(1∕D\right)}$, which is sharp by the work of Johnson, Lindenstrauss and Schechtman (1987). This improves over the previously best-known bound $dim\left(X\right)\gtrsim {\left(logn\right)}^2∕D^2$ of Linial, London and Rabinovich (1995), strengthens a theorem of Matoušek (1996) which resolved questions of Johnson and Lindenstrauss (1982), Bourgain (1985) and Arias-de-Reyna and Rodríguez-Piazza (1992), and answers negatively a question that was posed (for algorithmic purposes) by Andoni, Nguyen, Nikolov, Razenshteyn and Waingarten (2016).

We study the relationship between three compactifications of the moduli space of gauge equivalence classes of Hermitian Yang–Mills connections on a fixed Hermitian vector bundle over a projective algebraic manifold of arbitrary dimension. Via the Donaldson–Uhlenbeck–Yau theorem, this space is analytically isomorphic to the moduli space of stable holomorphic vector bundles, and as such it admits an algebraic compactification by Gieseker–Maruyama semistable torsion-free sheaves. A recent construction due to the first and third authors gives another compactification as a moduli space of slope semistable sheaves. Following fundamental work of Tian generalising the analysis of Uhlenbeck and Donaldson in complex dimension two, we define a gauge-theoretic compactification by adding certain gauge equivalence classes of ideal connections at the boundary. Extending work of Jun Li in the case of bundles on algebraic surfaces, we exhibit comparison maps from the sheaf-theoretic compactifications and prove their continuity. The continuity, together with a delicate analysis of the fibres of the map from the moduli space of slope semistable sheaves, allows us to endow the gauge-theoretic compactification with the structure of a complex analytic space.

We show that the commensurator of any quasiconvex abelian subgroup in a biautomatic group is small, in the sense that it has finite image in the abstract commensurator of the subgroup. Using this criterion we exhibit groups that are $CAT\left(0\right)$ but not biautomatic. These groups also resolve a number of other questions concerning $CAT\left(0\right)$ groups.

We refine Khovanov homology in the presence of an involution on the link. This refinement takes the form of a triply graded theory, arising from a pair of filtrations. We focus primarily on strongly invertible knots and show, for instance, that this refinement is able to detect mutation.

We give two constructions of functorial topological realizations for schemes of finite type over the field $ℂ\left(\left(t\right)\right)$ of formal Laurent series with complex coefficients, with values in the homotopy category of spaces over the circle. The problem of constructing such a realization was stated by D Treumann, motivated by certain questions in mirror symmetry. The first construction uses spreading out and the usual Betti realization over $ℂ$. The second uses generalized semistable models and the log Betti realization defined by Kato and Nakayama, and applies to smooth rigid analytic spaces as well. We provide comparison theorems between the two constructions and relate them to the étale homotopy type and de Rham cohomology. As an illustration of the second construction, we treat two examples, the Tate curve and the nonarchimedean Hopf surface.

We construct local and global solutions to the Kähler–Ricci flow from a noncollapsed Kähler manifold with curvature bounded from below. Combining with the mollification technique of McLeod–Simon–Topping, we show that the Gromov–Hausdorff limit of a sequence of complete noncompact noncollapsed Kähler manifolds with orthogonal bisectional curvature and Ricci curvature bounded from below is homeomorphic to a complex manifold. We also use it to study the complex structure of complete Kähler manifolds with nonnegative orthogonal bisectional curvature, nonnegative Ricci curvature and maximal volume growth.

We prove a mixed curvature analogue of Gromov’s Almost Flat Manifolds Theorem for upper sectional and lower Bakry–Emery Ricci curvature bounds.

We study the analytic tangent cones of admissible Hermitian Yang–Mills connections near a homogeneous singularity of a reflexive sheaf, and relate it to the Harder–Narasimhan–Seshadri filtration. We also give an algebrogeometric characterization of the bubbling set.

Partial quotients of moment-angle complexes are topological analogues of smooth, not necessarily compact toric varieties. In 1998, Buchstaber and Panov proposed a formula for the cohomology ring of such a partial quotient in terms of a torsion product involving the corresponding Stanley–Reisner ring. We show that their formula gives the correct cup product if $2$ is invertible in the chosen coefficient ring, but not in general. We rectify this by defining an explicit deformation of the canonical multiplication on the torsion product.

We show that the minimum weight of a weighted blowup of ${\mathbb{𝔸}}^d$ with $𝜀$–log canonical singularities is bounded by a constant depending only on $𝜀$ and $d$. This was conjectured by Birkar.

Using the recent classification of $4$–dimensional empty simplices by Iglesias-Valiño and Santos, we work out an explicit bound for blowups of ${\mathbb{𝔸}}^4$ with terminal singularities: the smallest weight is always at most $32$, and at most $6$ in all but finitely many cases.