A pattern knot in a solid torus defines a self-map of the smooth knot concordance group. We prove that if the winding number of a pattern is even but not divisible by 8, then the corresponding map is not a homomorphism, thus partially establishing a conjecture of Hedden.

We extend the work of N. Zubrilina on murmuration of modular forms to the case when prime-indexed coefficients are replaced by squares of primes. Our key observation is that the shape of the murmuration density is the same.

Let G be a locally compact, Hausdorff, second countable groupoid and A be a separable, $C_0(G^{(0)})$-nuclear, G-$C^*$-algebra. We prove the existence of quasi-invariant, completely positive and contractive lifts for equivariant, completely positive and contractive maps from A into a separable, quotient $C^*$-algebra. Along the way, we construct the Busby invariant for G-actions.

Granville–Soundararajan, Harper–Nikeghbali–Radziwiłł and Heap–Lindqvist independently established an asymptotic for the even natural moments of partial sums of random multiplicative functions defined over integers. Building on these works, we study the even natural moments of partial sums of Steinhaus random multiplicative functions defined over function fields. Using a combination of analytic arguments and combinatorial arguments, we obtain asymptotic expressions for all the even natural moments in the large field limit and large degree limit, as well as an exact expression for the fourth moment.

We provide new upper bounds for sums of certain arithmetic functions in many variables at polynomial arguments and, exploiting recent progress on the mean-value of the Erdős—Hooley $\Delta$-function, we derive lower bounds for the cardinality of those integers not exceeding a given limit that are expressible as certain sums of powers.

We study the freeness problem for multiplicative subgroups of $\operatorname{SL}_2(\mathbb{Q})$. For $q = r/p$ in $\mathbb{Q} \cap (0,4)$, where p is prime and $\gcd(r,p)=1$, we initiate the study of the algebraic structure of the group $\Delta_q$ generated by

\[A = \begin{pmatrix}1 & 0 \\ 1 & 1\end{pmatrix} \text{ and } Q_q = \begin{pmatrix}1 & q \\ 0 & 1\end{pmatrix}.\]

$\Delta_{r/p} = \overline{\Gamma}_1^{(p)}(r)$

$\operatorname{SL}_2(\mathbb{Z}[{1}/{p}])$

$r \leq 4$

$r = 5$

$r \in \{ p-1, p+1, (p+1)/2 \}$

$\Delta_{r/p}$

$J_2(r)$

$\operatorname{SL}_2(\mathbb{Z}[{1}/{p}])$

$J_2(r)$

It is conjectured that for any fixed relatively prime positive integers a,b and c all greater than 1 there is at most one solution to the equation $a^x+b^y=c^z$ in positive integers x, y and z, except for specific cases. We develop the methods in our previous work which rely on a variety from Baker’s theory and thoroughly study the conjecture for cases where c is small relative to a or b. Using restrictions derived from the hypothesis that there is more than one solution to the equation, we obtain a number of finiteness results on the conjecture. In particular, we find some, presumably infinitely many, new values of c with the property that for each such c the conjecture holds true except for only finitely many pairs of a and b. Most importantly we prove that if $c=13$ then the equation has at most one solution, except for $(a,b)=(3,10)$ or (10,3) each of which gives exactly two solutions. Further, our study with the help of the Schmidt Subspace Theorem among others more, brings strong contributions to the study of Pillai’s type Diophantine equations, notably a general and satisfactory result on a well-known conjecture of M. Bennett on the equation $a^x-b^y=c$ for any fixed positive integers a,b and c with both a and b greater than 1. Some conditional results are presented under the abc-conjecture as well.

In this short paper, we prove that the restriction conjecture for the (hyperbolic) paraboloid in $\mathbb{R}^{d}$ implies the $l^p$-decoupling theorem for the (hyperbolic) paraboloid in $\mathbb{R}^{2d-1}$. In particular, this gives a simple proof of the $l^p$ decoupling theorem for the (hyperbolic) paraboloid in $\mathbb{R}^3$.

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

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

$a_p(E)$

$E/\mathbb{Q}$