We introduce a notion of $q$-deformed rational numbers and $q$-deformed continued fractions. A $q$-deformed rational is encoded by a triangulation of a polygon and can be computed recursively. The recursive formula is analogous to the $q$-deformed Pascal identity for the Gaussian binomial coefficients, but the Pascal triangle is replaced by the Farey graph. The coefficients of the polynomials defining the $q$-rational count quiver subrepresentations of the maximal indecomposable representation of the graph dual to the triangulation. Several other properties, such as total positivity properties, $q$-deformation of the Farey graph, matrix presentations and $q$-continuants are given, as well as a relation to the Jones polynomial of rational knots.

Let $M_{n}$ denote a random symmetric $n\times n$ matrix whose upper-diagonal entries are independent and identically distributed Bernoulli random variables (which take values $1$ and $-1$ with probability $1/2$ each). It is widely conjectured that $M_{n}$ is singular with probability at most $(2+o(1))^{-n}$. On the other hand, the best known upper bound on the singularity probability of $M_{n}$, due to Vershynin (2011), is $2^{-n^{c}}$, for some unspecified small constant $c>0$. This improves on a polynomial singularity bound due to Costello, Tao, and Vu (2005), and a bound of Nguyen (2011) showing that the singularity probability decays faster than any polynomial. In this paper, improving on all previous results, we show that the probability of singularity of $M_{n}$ is at most $2^{-n^{1/4}\sqrt{\log n}/1000}$ for all sufficiently large $n$. The proof utilizes and extends a novel combinatorial approach to discrete random matrix theory, which has been recently introduced by the authors together with Luh and Samotij.

Pursuing ideas in [6], we determine the isometry classes of unimodular lattices of rank $28$, as well as the isometry classes of unimodular lattices of rank $29$ without nonzero vectors of norm $\leq 2$. We also provide some invariant that allows to distinguish these lattices and to independently check with a computer that our lists are complete.

We compute the Hodge ideals of $\mathbb{Q}$-divisors in terms of the $V$-filtration induced by a local defining equation, inspired by a result of Saito in the reduced case. We deduce basic properties of Hodge ideals in this generality, and relate them to Bernstein–Sato polynomials. As a consequence of our study we establish general properties of the minimal exponent, a refined version of the log canonical threshold, and bound it in terms of discrepancies on log resolutions, addressing a question of Lichtin and Kollár.

We discuss the $\ell $-adic case of Mazur’s ‘Program B’ over $\mathbb {Q}$: the problem of classifying the possible images of $\ell $-adic Galois representations attached to elliptic curves E over $\mathbb {Q}$, equivalently, classifying the rational points on the corresponding modular curves. The primes $\ell =2$ and $\ell \ge 13$ are addressed by prior work, so we focus on the remaining primes $\ell = 3, 5, 7, 11$. For each of these $\ell $, we compute the directed graph of arithmetically maximal $\ell $-power level modular curves $X_H$, compute explicit equations for all but three of them and classify the rational points on all of them except $X_{\mathrm {ns}}^{+}(N)$, for $N = 27, 25, 49, 121$ and two-level $49$ curves of genus $9$ whose Jacobians have analytic rank $9$.

Aside from the $\ell $-adic images that are known to arise for infinitely many ${\overline {\mathbb {Q}}}$-isomorphism classes of elliptic curves $E/\mathbb {Q}$, we find only 22 exceptional images that arise for any prime $\ell $ and any $E/\mathbb {Q}$ without complex multiplication; these exceptional images are realised by 20 non-CM rational j-invariants. We conjecture that this list of 22 exceptional images is complete and show that any counterexamples must arise from unexpected rational points on $X_{\mathrm {ns}}^+(\ell )$ with $\ell \ge 19$, or one of the six modular curves noted above. This yields a very efficient algorithm to compute the $\ell $-adic images of Galois for any elliptic curve over $\mathbb {Q}$.

In an appendix with John Voight, we generalise Ribet’s observation that simple abelian varieties attached to newforms on $\Gamma _1(N)$ are of $\operatorname {GL}_2$-type; this extends Kolyvagin’s theorem that analytic rank zero implies algebraic rank zero to isogeny factors of the Jacobian of $X_H$.

Let k be an algebraically closed field of prime characteristic p. Let $kGe$ be a block of a group algebra of a finite group G, with normal defect group P and abelian $p'$ inertial quotient L. Then we show that $kGe$ is a matrix algebra over a quantised version of the group algebra of a semidirect product of P with a certain subgroup of L. To do this, we first examine the associated graded algebra, using a Jennings–Quillen style theorem.

As an example, we calculate the associated graded of the basic algebra of the nonprincipal block in the case of a semidirect product of an extraspecial p-group P of exponent p and order $p^3$ with a quaternion group of order eight with the centre acting trivially. In the case of $p=3$, we give explicit generators and relations for the basic algebra as a quantised version of $kP$. As a second example, we give explicit generators and relations in the case of a group of shape $2^{1+4}:3^{1+2}$ in characteristic two.

Tao (2018) showed that in order to prove the Lonely Runner Conjecture (LRC) up to $n+1$ runners it suffices to consider positive integer velocities in the order of $n^{O(n^2)}$. Using the zonotopal reinterpretation of the conjecture due to the first and third authors (2017) we here drastically improve this result, showing that velocities up to $\binom {n+1}{2}^{n-1} \le n^{2n}$ are enough.

We prove the same finite-checking result, with the same bound, for the more general shifted Lonely Runner Conjecture (sLRC), except in this case our result depends on the solution of a question, that we dub the Lonely Vector Problem (LVP), about sumsets of n rational vectors in dimension two. We also prove the same finite-checking bound for a further generalization of sLRC that concerns cosimple zonotopes with n generators, a class of lattice zonotopes that we introduce.

In the last sections we look at dimensions two and three. In dimension two we prove our generalized version of sLRC (hence we reprove the sLRC for four runners), and in dimension three we show that to prove sLRC for five runners it suffices to look at velocities adding up to $195$.

We give new descriptions of the Bruhat order and Demazure products of affine Weyl groups in terms of the weight function of the quantum Bruhat graph. These results can be understood to describe certain closure relations concerning the Iwahori–Bruhat decomposition of an algebraic group. As an application towards affine Deligne–Lusztig varieties, we present a new formula for generic Newton points.

The limited dependence between the additive and the multiplicative structure of fields is in the background of a number of explicit constructions of various types of pseudorandom objects. In this direction we study the size of the intersection of (the additive) translates of fibers of the (multiplicative) norm function over finite fields. Besides extending earlier upper bounds, our main focus here is on obtaining lower bounds.

From our results we conclude several consequences in extremal combinatorics. Our motivation is the projective norm graph $\text {NG}(q,t)$ and its small subgraph statistics. $\text {NG}(q,t)$ provides a tight construction for the Turán number of complete bipartite graphs $K_{t,s}$ with $s>(t-1)!$; in particular, it does not contain $K_{t, (t-1)!+1}$. Yet, for $t\geq 4$ it is not even known whether $\text {NG}(q,t)$ contains $K_{t,t}$. The determination of the largest integer $s(t)$, such that $\text {NG}(q,t)$ contains $K_{t,s(t)}$ for all large enough prime powers q is an important open question with far-reaching consequences, and the best known bounds, $t-1\leq s(t) \leq (t-1)!$, are very far apart. In this paper we settle the first open case and establish $s(4)=6$. Along the way we also count subgraphs of $\text {NG}(q,t)$ isomorphic to H, for any fixed $3$-degenerate graph H, and find that projective norm graphs are quasirandom with respect to these parameters. These results go beyond the consequences of the Expander Mixing Lemma and also imply extensions of the work of Alon and Shikhelman on generalized Turán numbers.

Finally, we also give an elementary proof of the $K_{4,s}$-freeness of $\text {NG}(q,t)$ for $s= 6(\sum _{i=0}^{t-4} q^i) +1$. This was known before for $t=4$ only, via a less direct algebro-geometric argument.

Let $\pi :X\rightarrow Z$ be a Fano type fibration with $\dim X-\dim Z=d$ and let $(X,B)$ be an $\epsilon $-lc pair with $K_X+B\sim _{\mathbb {R}} 0/Z$. The canonical bundle formula gives $(Z,B_Z+M_Z)$ where $B_Z$ is the discriminant divisor and $M_Z$ is the moduli divisor which is determined up to $\mathbb {R}$-linear equivalence. Shokurov conjectured that one can choose $M_Z\geqslant 0$ such that $(Z,B_Z+M_Z)$ is $\delta $-lc where $\delta $ only depends on $d,\epsilon $. Very recently, this conjecture was proved by Birkar [8]. For $d=1$ and $\epsilon =1$, Han, Jiang, and Luo [13] gave the optimal value of $\delta =1/2$. In this paper, we give the optimal value of $\delta $ for $d=1$ and arbitrary $0<\epsilon \leqslant 1$.