For a class of $\mathbb{R}^d$-ations and $\mathbb{Z}^d$-actions on the $n$-dimensional torus $\mathbb{T}^n$, we characterize their unique ergodicity and establish a theorem of Weyl type. This result allows us to establish an isomorphism between the Banach algebra of quasi-periodic functions with spectrum in a given $\mathbb{Z}$-module and the Banach algebra of periodic functions on a torus. This, in return, allows us to give a very simple proof of Hausdorff-Young inequalities for Besicovitch almost periodic functions. The regularity of the parent function of a quasi-periodic function is also studied.

Building on the work of Miller et al. [Fibonacci Quarterly, 2022], we show that it is impossible to "walk to infinity" along the Fibonacci sequence in any integer base $b\geq 2$ when at most $N$ digits are appended per step. Our proof method is base-independent, yielding the bound $$L \;\leq\; 2N\log_\varphi b \,+\, O(1),$$ uniformly in the starting term, without relying on base-specific periodicity computations (here, $\varphi=\frac{1+\sqrt{5}}{2}$). Our approach extends to certain Lucas sequences.

We adapt Caro's notion of overholonomicity to give a definition of holonomic D-cap-modules on rigid analytic spaces. We prove stability under five of the six operations (both inverse image functors, duality, and both direct image functors for projective morphisms), as well as base change results. Up to the open problem of stability under tensor products, we obtain an analogue of the usual six-functor formalism for holonomic D-modules.

Using the stable twisted trace formula for the triality automorphism, we show the adjoint lifting (to GL(8)) of cuspidal representations of GL(3) with a discrete series local component. We also describe the possible isobaric decompositions of the resulting automorphic representations on GL(8).

For $C/\mathbb{Q}_p$ a complete algebraically closed field, we construct a collection of non-isomorphic rank two $\mathbb{Q}_p$-local systems on $\mathbb{P}^1_C$ indexed by $C$. This implies that the de Jong fundamental group $\pi_{1,\mathrm{dJ}}(\mathbb{P}^1_C)$ depends on $C$ and, if $C$ has cardinality $>2^{\mathbb{N}}$, that $\pi_{1,\mathrm{dJ}}(\mathbb{P}^1_C)$ is not topologically countably generated. The argument in fact applies to any connected rigid analytic variety over $C$ with a non-constant function to $\mathbb{P}^1_C$.

We show the de Jong fundamental group of any non-trivial abelian variety over a complete algebraically closed extension of $\mathbb{Q}_p$ is non-abelian.

In our paper from 1985 we have constructed two integrals of the Riemann's function $Z^2(t)$ over two disconnected sets with asymptotically equal measures such that these two integrals differ by considerably big excess. In the present paper we use the formula for that excess to construct a new $\zeta$-equivalent of the Fermat-Wiles theorem on a two-parametric set of lemniscates of Bernoulli.

We propose and investigate a strategy toward a proof of the Riemann Hypothesis based on a spectral realization of its non-trivial zeros. Our approach constructs self-adjoint operators obtained as rank-one perturbations of the spectral triple associated with the scaling operator on the interval $[\lambda^{-1}, \lambda]$. The construction only involves the Euler products over the primes $p \leq x = \lambda^2$ and produces self-adjoint operators whose spectra coincide, with striking numerical accuracy, with the lowest non-trivial zeros of $\zeta(1/2 + i s)$, even for small values of $x$. The theoretical foundation rests on the framework introduced in "Spectral triples and zeta-cycles" (Enseign. Math. 69 (2023), no. 1-2, 93-148), together with the extension in "Quadratic Forms, Real Zeros and Echoes of the Spectral Action" (Commun. Math. Phys. (2025)) of the classical Caratheodory-Fejer theorem for Toeplitz matrices, which guarantees the necessary self-adjointness. Numerical experiments show that the spectra of the operators converge towards the zeros of $\zeta(1/2 + i s)$ as the parameters $N, \lambda \to \infty$. A rigorous proof of this convergence would establish the Riemann Hypothesis. We further compute the regularized determinants of these operators and discuss the analytic role they play in controlling and potentially proving the above result by showing that, suitably normalized, they converge towards the Riemann $\Xi$ function.

Closed geodesics associated with indefinite binary quadratic forms, or equivalently with real quadratic irrationals, have long been studied as geometric $\mathrm{SL}_2(\mathbb{Z})$-invariants. Building on the Birman-Williams approach to Lorenz knots and following the notion of modular knots introduced by Ghys, this article investigates the topological $\mathrm{SL}_2(\mathbb{Z})$-invariants arising from modular knots. Our main focus is the Alexander polynomial of modular knots. Using the Burau representation, we highlight two contrasting features of this family. On the one hand, for each fixed degree, only finitely many Alexander polynomials of modular knots occur. On the other hand, any integer appears as a coefficient of the Alexander polynomial of some modular knot, and coefficients of the same sign can occur in runs of arbitrarily long length.

The $k$-Markov numbers, introduced by Gyoda and Matsushita, are those which appear in positive integral solutions to $x^2 + y^2 + z^2 + k(xy + xz + yz) = (3+3k)xyz$. When $k =0$, this recovers the ordinary Markov numbers. A long-standing question in the theory of Markov numbers is Frobenius's unicity conjecture, concerning whether every Markov number is the maximum in a unique solution triple. Aigner gave a series of weaker, related conjectures which were confirmed to be true by Lee, Li, Rabideau, and Schiffler using techniques from the theory of cluster algebras. We show here that $k$-Markov numbers also satisfy Aigner's conjectures.

We develop an explicit $p$-adic integration theory for Igusa towers of modular Siegel manifolds, which finds applications to explicit reciprocity laws.

We present a simple $q$-ary family of single-error-correcting, double-error-detecting (SEC--DED) linear codes whose parity checks are tied directly to the base-$p$ ($q=p$ prime) digits of the coordinate index. For blocklength $n=p^r$ the construction uses only $r+1$ parity checks -- \emph{near-Hamming} overhead -- and admits an index-based decoder that runs in a single pass with constant-time location and magnitude recovery from the syndromes. Based on the prototype, we develop two extensions: Code A1, which removes specific redundant trits to achieve higher information rate and support variable-length encoding; and Code A2, which incorporates two group-sum checks together with a 3-wise XOR linear independence condition on index subsets, yielding a ternary distance-4 (SEC--TED) variant. Furthermore, we demonstrate how the framework generalizes via $n$-wise XOR linearly independent sets to construct codes with distance $d = n + 1$, notably recovering the ternary Golay code for $n = 5$ -- showing both structural generality and a serendipitous link to optimal classical codes. Our contribution is not optimality but \emph{implementational simplicity} and an \emph{array-friendly} structure: the checks are digitwise and global sums, the mapping from syndromes to error location is explicit, and the SEC--TED upgrade is modular. We position the scheme against classical $q$-ary Hamming and SPC/product-code baselines and provide a small comparison of parity overhead, decoding work, and two-error behavior.

We study fixed points of a function arising in a representation theory of the Drinfeld modules by the bounded linear operators on a Hilbert space. We prove that such points correspond to number fields of the class number one. As an application, one gets a solution to the Gauss conjecture for the real quadratic fields of class number one.

We consider the $q$-deformation of rational numbers introduced recently by Morier-Genoud and Ovsienko. We propose three enumerative interpretations of these $q$-rationals: in terms of a new version of Ostrowski's numeration system for integers, in terms of order ideals of fence posets and in terms of perfect matchings of snake graphs. Contrary to previous results which are restricted to rational numbers greater than one, our interpretations work for all positive rational numbers and are based on a single combinatorial object for defining both the numerator and denominator. The proofs rest on order-preserving bijections between posets over these objects. We recover a formula for a $q$-analog of Markoff numbers. We also deduce a fourth interpretation given in terms of the integer points inside a polytope in $\mathbb{R}^k$ on both sides of a hyperplane where $k$ is the length of the continued fraction expansion.

These are lecture notes for a course in Winter 2022/23, updated and completed in October 2025. The goal of the lectures is to present some recent developments around six-functor formalisms, in particular: the abstract theory of 6-functor formalisms; the 2-category of cohomological correspondences, and resulting simplifications in the proofs of Poincaré--Verdier duality results; the relation between 6-functor formalisms and ``geometric rings''; many examples of 6-functor formalisms, both old and new.