We prove that in each degree divisible by 2 or 3, there are infinitely many totally real number fields that require universal quadratic forms to have arbitrarily large rank.

Following Bridgeman, we demonstrate several families of infinite dilogarithm identities associated with Fibonacci numbers, Lucas numbers, convergents of continued fractions of even periods, and terms arising from various recurrence relations.

As a general framework for the studies of t-structures on triangulated categories and torsion pairs in abelian categories, we introduce the notions of extriangulated categories with negative first extensions and s-torsion pairs. We define a heart of an interval in the poset of s-torsion pairs, which naturally becomes an extriangulated category with a negative first extension. This notion generalises hearts of t-structures on triangulated categories and hearts of twin torsion pairs in abelian categories. In this paper, we show that an interval in the poset of s-torsion pairs is bijectively associated with s-torsion pairs in the corresponding heart. This bijection unifies two well-known bijections: one is the bijection induced by the HRS-tilt of t-structures on triangulated categories. The other is Asai–Pfeifer’s and Tattar’s bijections for torsion pairs in an abelian category, which is related to $\tau$ -tilting reduction and brick labelling.

In this paper we give a complete description of the Bieri–Neumann–Strebel–Renz invariants of the Lodha–Moore groups. The second author previously computed the first two invariants, and here we show that all the higher invariants coincide with the second one, which finishes the complete computation. As a consequence, we present a complete picture of the finiteness properties of normal subgroups of the first Lodha–Moore group. In particular, we show that every finitely presented normal subgroup of the group is of type $\textrm{F}_\infty$ , answering a question posed in Oberwolfach in 2018. The proof involves applying a variation of Bestvina–Brady discrete Morse theory to the so called cluster complex X introduced by the first author. As an application, we also demonstrate that a certain simple group S previously constructed by the first author is of type $\textrm{F}_\infty$ . This provides the first example of a type $\textrm{F}_\infty$ simple group that acts faithfully on the circle by homeomorphisms, but does not admit any nontrivial action by $C^1$ -diffeomorphisms, nor by piecewise linear homeomorphisms, on any 1-manifold.

Conditional on the extended Riemann hypothesis, we show that with high probability, the characteristic polynomial of a random symmetric $\{\pm 1\}$ -matrix is irreducible. This addresses a question raised by Eberhard in recent work. The main innovation in our work is establishing sharp estimates regarding the rank distribution of symmetric random $\{\pm 1\}$ -matrices over $\mathbb{F}_p$ for primes $2 < p \leq \exp(O(n^{1/4}))$ . Previously, such estimates were available only for $p = o(n^{1/8})$ . At the heart of our proof is a way to combine multiple inverse Littlewood–Offord-type results to control the contribution to singularity-type events of vectors in $\mathbb{F}_p^{n}$ with anticoncentration at least $1/p + \Omega(1/p^2)$ . Previously, inverse Littlewood–Offord-type results only allowed control over vectors with anticoncentration at least $C/p$ for some large constant $C > 1$ .

We consider an analogue of Kontsevich’s matrix Airy function where the cubic potential $\textrm{Tr}(\Phi^3)$ is replaced by a quartic term $\textrm{Tr}\!\left(\Phi^4\right)$ . Cumulants of the resulting measure are known to decompose into cycle types for which a recursive system of equations can be established. We develop a new, purely algebraic geometrical solution strategy for the two initial equations of the recursion, based on properties of Cauchy matrices. These structures led in subsequent work to the discovery that the quartic analogue of the Kontsevich model obeys blobbed topological recursion.

The set of points that escape to infinity under iteration of a cosine map, that is, of the form $C_{a,\,b} \colon z \mapsto ae^z+be^{-z}$ for $a,\,b\in \mathbb{C}^\ast$ , consists of a collection of injective curves, called dynamic rays. If a critical value of $C_{a,\,b}$ escapes to infinity, then some of its dynamic rays overlap pairwise and split at critical points. We consider a large subclass of cosine maps with escaping critical values, including the map $z\mapsto \cosh(z)$ . We provide an explicit topological model for their dynamics on their Julia sets. We do so by first providing a model for the dynamics near infinity of any cosine map, and then modifying it to reflect the splitting of rays for functions of the subclass we study. As an application, we give an explicit combinatorial description of the overlap occurring between the dynamic rays of $z\mapsto \cosh(z)$ , and conclude that no two of its dynamic rays land together.

Let K be any field of characteristic two and let $U_1$ and $W_1$ be the Lie algebras of the derivations of the algebra of Laurent polynomials $K[t,t^{-1}]$ and of the polynomial ring K[t], respectively. The algebras $U_1$ and $W_1$ are equipped with natural $\mathbb{Z}$ -gradings. In this paper, we provide bases for the graded identities of $U_1$ and $W_1$ , and we prove that they do not admit any finite basis.

Hilbert–Kunz multiplicity and F-signature are numerical invariants of commutative rings in positive characteristic that measure severity of singularities: for a regular ring both invariants are equal to one and the converse holds under mild assumptions. A natural question is for what singular rings these invariants are closest to one. For Hilbert–Kunz multiplicity this question was first considered by the last two authors and attracted significant attention. In this paper, we study this question, i.e., an upper bound, for F-signature and revisit lower bounds on Hilbert–Kunz multiplicity.

We show that the only finite quasi-simple non-abelian groups that can faithfully act on rationally connected threefolds are the following groups: ${\mathfrak{A}}_5$ , ${\text{PSL}}_2(\textbf{F}_7)$ , ${\mathfrak{A}}_6$ , ${\text{SL}}_2(\textbf{F}_8)$ , ${\mathfrak{A}}_7$ , ${\text{PSp}}_4(\textbf{F}_3)$ , ${\text{SL}}_2(\textbf{F}_{7})$ , $2.{\mathfrak{A}}_5$ , $2.{\mathfrak{A}}_6$ , $3.{\mathfrak{A}}_6$ or $6.{\mathfrak{A}}_6$ . All of these groups with a possible exception of $2.{\mathfrak{A}}_6$ and $6.{\mathfrak{A}}_6$ indeed act on some rationally connected threefolds.

In a 1916 paper, Ramanujan studied the additive convolution $S_{a, b}(n)$ of sum-of-divisors functions $\sigma_a(n)$ and $\sigma_b(n)$ , and proved an asymptotic formula for it when a and b are positive odd integers. He also conjectured that his asymptotic formula should hold for all positive real a and b. Ramanujan’s conjecture was subsequently proved by Ingham, and then by Halberstam with a power saving error term.

In this paper, we give a new proof of Ramanujan’s conjecture that obtains lower order terms in the asymptotics for most ranges of the parameters. We also describe a connection to a counting problem in geometric topology that was studied in the second author’s thesis and which served as our initial motivation in studying this sum.

For a domain G in the one-point compactification $\overline{\mathbb{R}}^n = {\mathbb{R}}^n \cup \{ \infty\}$ of ${\mathbb{R}}^n, n \geqslant 2$ , we characterise the completeness of the modulus metric $\mu_G$ in terms of a potential-theoretic thickness condition of $\partial G\,,$ Martio’s M-condition [ 35 ]. Next, we prove that $\partial G$ is uniformly perfect if and only if $\mu_G$ admits a minorant in terms of a Möbius invariant metric. Several applications to quasiconformal maps are given.

In 1974, Erdős posed the following problem. Given an oriented graph H, determine or estimate the maximum possible number of H-free orientations of an n-vertex graph. When H is a tournament, the answer was determined precisely for sufficiently large n by Alon and Yuster. In general, when the underlying undirected graph of H contains a cycle, one can obtain accurate bounds by combining an observation of Kozma and Moran with celebrated results on the number of F-free graphs. As the main contribution of the paper, we resolve all remaining cases in an asymptotic sense, thereby giving a rather complete answer to Erdős’s question. Moreover, we determine the answer exactly when H is an odd cycle and n is sufficiently large, answering a question of Araújo, Botler and Mota.

We prove that analogues of the Hardy–Littlewood generalised twin prime conjecture for almost primes hold on average. Our main theorem establishes an asymptotic formula for the number of integers $n=p_1p_2 \leq X$ such that $n+h$ is a product of exactly two primes which holds for almost all $|h|\leq H$ with $\log^{19+\varepsilon}X\leq H\leq X^{1-\varepsilon}$ , under a restriction on the size of one of the prime factors of n and $n+h$ . Additionally, we consider correlations $n,n+h$ where n is a prime and $n+h$ has exactly two prime factors, establishing an asymptotic formula which holds for almost all $|h| \leq H$ with $X^{1/6+\varepsilon}\leq H\leq X^{1-\varepsilon}$ .

We show that any set of distinct homotopy classes of simple closed curves on the torus that pairwise intersect at most k times has size $k+O(\sqrt k \log k)$ . Prior to this work, a lemma of Agol, together with the state of the art bounds for the size of prime gaps, implied the error term $O(k^{21/40})$ , and in fact the assumption of the Riemann hypothesis improved this error term to the one we obtain $O(\sqrt k\log k)$ . By contrast, our methods are elementary, combinatorial, and geometric.

We compute, via motivic wall-crossing, the generating function of virtual motives of the Quot scheme of points on ${\mathbb{A}}^3$ , generalising to higher rank a result of Behrend–Bryan–Szendrői. We show that this motivic partition function converges to a Gaussian distribution, extending a result of Morrison.

On small Seifert fibered spaces $M(e_0;\,r_1,r_2,r_3)$ with $e_0\neq-1,-2,$ all tight contact structures are Stein fillable. This is not the case for $e_0=-1$ or $-2$ . However, for negative twisting structures it is expected that they are all symplectically fillable. Here, we characterise fillable structures among zero-twisting contact structures on small Seifert fibered spaces of the form $M\left({-}1;\,r_1,r_2,r_3\right)$ . The result is obtained by analysing monodromy factorizations of associated planar open books.

For G a profinite group, we construct an equivalence between rational G-Mackey functors and a certain full subcategory of G-sheaves over the space of closed subgroups of G called Weyl-G-sheaves. This subcategory consists of those sheaves whose stalk over a subgroup K is K-fixed.

This extends the classification of rational G-Mackey functors for finite G of Thévenaz and Webb, and Greenlees and May to a new class of examples. Moreover, this equivalence is instrumental in the classification of rational G-spectra for profinite G, as given in the second author’s thesis.

Since 1984, many authors have studied the dynamics of maps of the form $\mathcal{E}_a(z) = e^z - a$ , with $a > 1$ . It is now well-known that the Julia set of such a map has an intricate topological structure known as a Cantor bouquet, and much is known about the dynamical properties of these functions.

It is rather surprising that many of the interesting dynamical properties of the maps $\mathcal{E}_a$ actually arise from their elementary function theoretic structure, rather than as a result of analyticity. We show this by studying a large class of continuous $\mathbb{R}^2$ maps, which, in general, are not even quasiregular, but are somehow analogous to $\mathcal{E}_a$ . We define analogues of the Fatou and the Julia set and we prove that this class has very similar dynamical properties to those of $\mathcal{E}_a$ , including the Cantor bouquet structure, which is closely related to several topological properties of the endpoints of the Julia set.

We investigate which higher rank simple Lie groups admit profinitely but not abstractly commensurable lattices. We show that no such examples exist for the complex forms of type $E_8$ , $F_4$ , and $G_2$ . In contrast, there are arbitrarily many such examples in all other higher rank Lie groups, except possibly $\textrm{SL}_{2n+1}(\mathbb{R})$ , $\textrm{SL}_{2n+1}(\mathbb{C})$ , $\textrm{SL}_n(\mathbb{H})$ , or groups of type $E_6$ .