Given a permutation group G, the derangement graph of G is defined with vertex set G, where two elements x and y are adjacent if and only if $xy^{-1}$ is a derangement. We establish that if G is transitive with degree exceeding 30, then the derangement graph of G contains a complete subgraph with four vertices. In the process, we determine all transitive groups whose derangement graph does not contain a complete subgraph on four vertices. As a consequence, if G is a normal subgroup of A such that $|A : G| = 3$ and U is a subgroup of G satisfying $G = \bigcup _{a \in A} U^a$, then $|G : U| \leq 10$. This provides support for a conjecture by Neumann and Praeger concerning Kronecker classes.

We continue our investigation of the fractal uncertainty principle (FUP) for random fractal sets. In the prequel Eswarathasan and Han [‘Fractal uncertainty principle for discrete Cantor sets with random alphabet’, Math. Res. Lett. 30(6) (2023), 1657–1679], we considered the Cantor sets in the discrete setting with alphabets randomly chosen from a base of digits so the dimension $\delta \in (0,\frac 23)$. We proved that, with overwhelming probability, the FUP with an exponent $\ge \frac 12-\frac 34\delta -\varepsilon $ holds for these discrete Cantor sets with random alphabets. In this paper, we construct random Cantor sets with dimension $\delta \in (0,\frac 23)$ in $\mathbb {R}$ via a different random procedure from the previous one used in Eswarathasan and Han [‘Fractal uncertainty principle for discrete Cantor sets with random alphabet’, Math. Res. Lett. 30(6) (2023), 1657–1679]. We prove that, with overwhelming probability, the FUP with an exponent $\ge \frac 12-\frac 34\delta -\varepsilon $ holds. The proof follows from establishing a Fourier decay estimate of the corresponding random Cantor measures, which is in turn based on a concentration of measure phenomenon in an appropriate probability space for the random Cantor sets.

We study the boundedness of the Mordell–Weil rank and the growth of the v-primary part of the Tate–Shafarevich group of p-supersingular abelian varieties of $\mathrm {GL}_2$-type with real multiplication over $\mathbb Z_p$-extensions of number fields, where v is a prime lying above p. Building on the work of Iovita and Pollack in the case of elliptic curves, under precise ramification and splitting conditions on p, we construct explicit systems of local points using the theory of Lubin–Tate formal groups. We then define signed Coleman maps, which in turn allow us to formulate and analyse signed Selmer groups. Assuming these Selmer groups are cotorsion, we prove that the Mordell–Weil groups are bounded over any subextensions of the ${\mathbb Z}_p$-extension and provide an asymptotic formula for the growth of the v-primary part of the Tate–Shafarevich groups. Our results extend those of Kobayashi, Pollack, and Sprung on p-supersingular elliptic curves.

Let M be a compact three-dimensional Riemannian manifold with non-negative Ricci curvature and a non-empty boundary $\partial M$. Fraser and Li [2] established a compactness theorem for the space of compact, properly embedded minimal surfaces of fixed topological type in M with a free boundary on $\partial M$, assuming that $\partial M$ is strictly convex with respect to the inward unit normal. In this paper, we show that the strict convexity condition on $\partial M$ cannot be relaxed.

We define the tropical Tevelev degrees, ${\mathsf{Tev}}_g^{\mathtt{trop}}$, as the degree of a natural finite morphism between certain tropical moduli spaces, in analogy to the algebraic case. We develop an explicit combinatorial construction that computes ${\mathsf{Tev}}_g^{\mathtt{trop}} = 2^g$. We prove that these tropical enumerative invariants agree with their algebraic counterparts, giving an independent tropical computation of the algebraic degrees ${\mathsf{Tev}}_g$.

This paper addresses the prediction of positive rank for elliptic curves without the need to find a point of infinite order or compute L-functions. While the most common method relies on parity conjectures, a recent technique introduced by Dokchitser, Wiersema and Evans predicts positive rank based on the value of a certain product of Tamagawa numbers, raising questions about its relationship to parity. We show that their method is a subset of the parity conjectures approach: whenever their method predicts positive rank, so does the use of parity conjectures. To establish this, we extend previous work on Brauer relations and regulator constants to a broader setting involving combinations of permutation modules known as K-relations. A central ingredient in our argument is demonstrating a compatibility between Tamagawa numbers and local root numbers.

For a finite extension F of ${\mathbb Q}_p$ and $n \geq 1$, we show that the category of Lubin-Tate bundles on the $(n-1)$-dimensional Drinfeld symmetric space is equivalent to the category of finite-dimensional smooth representations of the group of units of the division algebra of invariant $1/n$ over F.

For a uniformly locally finite metric space $(X, d)$, we investigate coarse flows on its uniform Roe algebra $\mathrm {C}^*_u(X)$, defined as one-parameter groups of automorphisms whose differentiable elements include all partial isometries arising from partial translations on X. We first show that any flow $\sigma $ on $\mathrm {C}^*_u(X)$ corresponds to a (possibly unbounded) self-adjoint operator h on $\ell _2(X)$ such that $\sigma _t(a) = e^{ith} a e^{-ith}$ for all $t \in \mathbb {R}$, allowing us to focus on operators h that generate flows on $\mathrm {C}^*_u(X)$.

Assuming Yu’s property A, we prove that a self-adjoint operator h on $\ell _2(X)$ induces a coarse flow on $\mathrm {C}^*_u(X)$ if and only if h can be expressed as $h = a + d$, where $a \in \mathrm {C}^*_u(X)$ and d is a diagonal operator with entries forming a coarse function on X. We further study cocycle equivalence and cocycle perturbations of coarse flows, showing that, under property A, any coarse flow is a cocycle perturbation of a diagonal flow. Finally, for self-adjoint operators h and k that induce coarse flows on $\mathrm {C}^*_u(X)$, we characterize conditions under which the associated flows are either cocycle perturbations of each other or cocycle conjugate to each other. In particular, if $h - k$ is bounded, then the flow induced by h is a cocycle perturbation of the flow induced by k.