The ${\tilde{\mathit{A}}}_{\mathit{n}}$ Coxeter groups are known to not be systolic or cocompactly cubulated for $\mathit{n}\ge 3$. We prove that these groups act geometrically on weakly modular graphs, a weak notion of nonpositive curvature generalizing the 1-skeleta of $\mathrm{CAT}(0)$ cube complexes and systolic complexes. To prove weak modularity, we describe the canonical embeddings of the 1-skeleta of ${\tilde{\mathit{A}}}_{\mathit{n}}$ Coxeter complexes into the Euclidean spaces ${\mathbb{R}}^{\mathit{n}+1}$. We also prove that the 1-skeleta of buildings of type ${\tilde{\mathit{A}}}_3$ are weakly modular.

We explore the structure of invariant measures on compact Kähler manifolds with Hamiltonian torus actions. We derive a formula for conditional measures on the orbits of the complex torus and use it to prove a conditional statement about uniqueness of solutions to the g-Monge–Ampère equation.

Let K be a compact convex domain in the Euclidean plane. The mixed area $\mathit{A}(\mathit{K},-\mathit{K})$ of K and $-\mathit{K}$ can be bounded from above by $1/(6\sqrt{3})\mathit{L}{(\mathit{K})}^2$, where $\mathit{L}(\mathit{K})$ is the perimeter of K. This was proved by Ulrich Betke and Wolfgang Weil [5]. They also showed that if K is a polygon, then equality holds if and only if K is a regular triangle. We prove that among all convex domains, equality holds only in this case, as conjectured by Betke and Weil. This is achieved by establishing a stronger stability result for the geometric inequality $6\sqrt{3}\mathit{A}(\mathit{K},-\mathit{K})\le \mathit{L}{(\mathit{K})}^2$.

We show that the difference between the topological 4-genus of a knot and the minimal genus of a surface bounded by that knot that can be decomposed into a smooth concordance followed by an algebraically simple locally flat surface can be arbitrarily large. This extends work of Hedden, Livingston, and Ruberman showing that there are topologically slice knots which are not smoothly concordant to any knot with trivial Alexander polynomial.

In this paper, we compare the notions of double sliceness for links. The main result is showing that a large family of 2-component Montesinos links are not strongly doubly slice despite being weakly doubly slice and having doubly slice components. Our principal obstruction to strong double slicing comes by considering branched double covers. To this end, we prove a result classifying Seifert fibered spaces that admit a smooth embedding into integer homology ${\mathit{S}}^1\times {\mathit{S}}^3$s by maps inducing surjections on the first homology group. A number of other results and examples pertaining to doubly slice links are also given.

Following Haken [Ha] and Casson and Gordon [CG], it was shown in [Sc] that given a reducing sphere or ∂-reducing disk E in a Heegaard split manifold M, the Heegaard surface T can be isotoped so that it intersects E in a single circle. Here we show that when this is achieved by two different positionings of T, one can be moved to the other by a sequence of

∙ isotopies of T rel E,

∙ pushing a stabilizing pair of T through E, and

∙ eyegelass twists of T.

This last move is inspired by one of Powell’s proposed generators for the Goeritz group [Po].

We consider the distribution in residue classes modulo primes p of Euler’s totient function $\mathit{\varphi }(\mathit{n})$ and the sum-of-proper-divisors function $\mathit{s}(\mathit{n}):=\mathit{\sigma }(\mathit{n})-\mathit{n}$. We prove that the values of $\mathit{\varphi }(\mathit{n})$ for $\mathit{n}\le \mathit{x}$ that are coprime to p are asymptotically uniformly distributed among the $\mathit{p}-1$ coprime residue classes modulo p, uniformly for $5\le \mathit{p}\le {(log\mathit{x})}^{\mathit{A}}$ (with A fixed but arbitrary). We also show that the values of $\mathit{s}(\mathit{n})$ for n composite are uniformly distributed among all p residue classes modulo every $\mathit{p}\le {(log\mathit{x})}^{\mathit{A}}$. These appear to be the first results of their kind where the modulus is allowed to grow substantially with x.

In this paper, we provide several results regarding the structure of derived categories of (nested) Hilbert schemes of points. We show that the criteria of Krug–Sosna and Addington for the universal ideal sheaf functor to be fully faithful resp. a $\mathbb{P}$-functor are sharp. Then we show how to embed multiple copies of the derived category of the surface using these fully faithful functors. We also give a semiorthogonal decomposition for the nested Hilbert scheme of points on a surface, and finally we give an alternative proof of a semiorthogonal decomposition due to Toda for the symmetric product of a curve.

We estimate Weyl sums over the integers with sum of binary digits either fixed or restricted by some congruence condition. In our proofs, we use ideas that go back to a paper by Banks, Conflitti, and the first author (2002). Moreover, we apply the “main conjecture” on the Vinogradov mean value theorem established by Bourgain, Demeter, and Guth (2016) and Wooley (2016, 2019). We use our result to give an estimate of the discrepancy of point sets defined by the values of polynomials at arguments having the sum of binary digits restricted in different ways.

We show that for any manifold M, compact or homeomorphic to the interior of a compact manifold, the homeomorphism group of M has automatic continuity. The same is true for the relative homeomorphism group $Homeo(\mathit{M},\mathit{X})$ where X is homeomorphic to the union of a Cantor set and a (possibly, empty) finite set, and the big mapping class group $Homeo(\mathit{M},\mathit{X})/{Homeo}_0(\mathit{M},\mathit{X})$. In other words, the algebraic structure of these groups is extremely rigid and determines their topology in a very strong way.