We define the symmetric braid index $b_s(K)$ of a ribbon knot K to be the smallest index of a braid whose closure yields a symmetric union diagram of K, and derive a Khovanov-homological characterisation of knots with $b_s(K)$ at most three. As applications, we show that there exist knots whose symmetric braid index is strictly greater than the braid index, and deduce that every chiral slice knot with determinant one has braid index at least four. We also calculate bounds for $b_s(K)$ for prime ribbon knots with at most 11 crossings.

Let S be a fine and saturated (fs) log scheme, and let F be a group scheme over the underlying scheme of S which is étale locally representable by (1) a finite dimensional $\mathbb{Q}$-vector space, or (2) a finite rank free abelian group, or (3) a finite abelian group. We give a full description of all the higher direct images of F from the Kummer log flat site to the classical flat site. In particular, we show that: in case (1) the higher direct images of F vanish; and in case (2) the first higher direct image of F vanishes and the nth ($n\gt 1$) higher direct image of F is isomorphic to the $(n-1)$-th higher direct image of $F\otimes_{{\mathbb Z}}{\mathbb Q}/{\mathbb Z}$. In the end, we make some computations when the base is a standard henselian log trait or a Dedekind scheme endowed with the log structure associated to a finite set of closed points.

We use a graph to define a new stability condition for algebraic moduli spaces of rational curves. We characterise when the tropical compactification of the moduli space agrees with the theory of geometric tropicalisation. The characterisation statement occurs only when the graph is complete multipartite.

An action of a group G on a set X is said to be quasi-n-transitive if the diagonal action of G on $X^n$ has only finitely many orbits. We show that branch groups, a special class of groups of automorphisms of rooted trees, cannot act quasi-2-transitively on infinite sets.

Non-autonomous self-similar sets are a family of compact sets which are, in some sense, highly homogeneous in space but highly inhomogeneous in scale. The main purpose of this paper is to clarify various regularity properties and separation conditions relevant for the fine local scaling properties of these sets. A simple application of our results is a precise formula for the Assouad dimension of non-autonomous self-similar sets in $\mathbb{R}^d$ satisfying a certain “bounded neighbourhood” condition, which generalises earlier work of Li–Li–Miao–Xi and Olson–Robinson–Sharples. We also see that the bounded neighbourhood assumption is, in few different senses, as general as possible.

For an ideal I in a Noetherian ring R, the Fitting ideals $\mathrm{Fitt}_j(I)$ are studied. We discuss the question of when $\mathrm{Fitt}_j(I)=I$ or $\sqrt{\mathrm{Fitt}_j(I)}=\sqrt{I}$ for some j. A classical case is the Hilbert–Burch theorem when $j=1$ and I is a perfect ideal of grade 2 in a local ring.

The 3-dimensional Heisenberg group can be equipped with three different types of left-invariant Lorentzian metric, according to whether the center of the Lie algebra is spacelike, timelike or null. Using the second of these types, we study spacelike surfaces of mean curvature zero. These surfaces with singularities are associated with harmonic maps into the 2-sphere. We show that the generic singularities are cuspidal edge, swallowtail and cuspidal cross-cap. We also give the loop group construction for these surfaces, and the criteria on the loop group potentials for the different generic singularities. Lastly, we solve the Cauchy problem for harmonic maps into the 2-sphere using loop groups, and use this to give a geometric characterisation of the singularities. We use these results to prove that a regular spacelike maximal disc with null boundary must have at least two cuspidal cross-cap singularities on the boundary.

In this paper, we prove that the hitting probability of the Minkowski sum of fractal percolations can be characterised by capacity. Then we extend this result to Minkowski sums of general random sets in $\mathbb Z^d$, including ranges of random walks and critical branching random walks, whose hitting probabilities are described by Newtonian capacity individually.

We discuss the Singer conjecture and Gromov–Lück inequality $\chi\geq |\sigma|$ for aspherical complex surfaces. We give a proof of the Singer conjecture for aspherical complex surfaces with residually finite fundamental group that does not rely on Gromov’s Kähler groups theory. Without the residually finiteness assumption, we observe that this conjecture can be proven for all aspherical complex surfaces except possibly those in Class $\mathrm{VII}_0^+$ (a positive answer to the global spherical shell conjecture would rule out the existence of aspherical surfaces in this class). We also sharpen the Gromov-Lück inequality for aspherical complex surfaces that are not in Class $\mathrm{VII}_0^+$. This is achieved by connecting the circle of ideas of the Singer conjecture with the study of Reid’s conjecture.

In the early 2000s, Ramakrishna asked the question: for the elliptic curve

\[E\;:\; y^2 = x^3 - x,\]

$a_p(E)$

$E/\mathbb{Q}$

For every positive integer d, we show that there must exist an absolute constant $c \gt 0$ such that the following holds: for any integer $n \geqslant cd^{7}$ and any red-blue colouring of the one-dimensional subspaces of $\mathbb{F}_{2}^{n}$, there must exist either a d-dimensional subspace for which all of its one-dimensional subspaces get coloured red or a 2-dimensional subspace for which all of its one-dimensional subspaces get coloured blue. This answers recent questions of Nelson and Nomoto, and confirms that for any even plane binary matroid N, the class of N-free, claw-free binary matroids is polynomially $\chi$-bounded.

Our argument will proceed via a reduction to a well-studied additive combinatorics problem, originally posed by Green: given a set $A \subset \mathbb{F}_{2}^{n}$ with density $\alpha \in [0,1]$, what is the largest subspace that we can find in $A+A$? Our main contribution to the story is a new result for this problem in the regime where $1/\alpha$ is large with respect to n, which utilises ideas from the recent breakthrough paper of Kelley and Meka on sets of integers without three-term arithmetic progressions.

Let E be an elliptic curve defined over ${{\mathbb{Q}}}$ which has good ordinary reduction at the prime p. Let K be a number field with at least one complex prime which we assume to be totally imaginary if $p=2$. We prove several equivalent criteria for the validity of the $\mathfrak{M}_H(G)$-property for ${{\mathbb{Z}}}_p$-extensions other than the cyclotomic extension inside a fixed ${{\mathbb{Z}}}_p^2$-extension $K_\infty/K$. The equivalent conditions involve the growth of $\mu$-invariants of the Selmer groups over intermediate shifted ${{\mathbb{Z}}}_p$-extensions in $K_\infty$, and the boundedness of $\lambda$-invariants as one runs over ${{\mathbb{Z}}}_p$-extensions of K inside of $K_\infty$.

Using these criteria we also derive several applications. For example, we can bound the number of ${{\mathbb{Z}}}_p$-extensions of K inside $K_\infty$ over which the Mordell–Weil rank of E is not bounded, thereby proving special cases of a conjecture of Mazur. Moreover, we show that the validity of the $\mathfrak{M}_H(G)$-property sometimes can be shifted to a larger base field K′.