Consider, for any integer $n\ge 3$, the set $\operatorname {\mathrm {Pos}}_n$ of all n-periodic tree patterns with positive topological entropy and the set $\operatorname {\mathrm {Irr}}_n\subset \operatorname {\mathrm {Pos}}_n$ of all n-periodic irreducible tree patterns. The aim of this paper is to determine the elements of minimum entropy in the families $\operatorname {\mathrm {Pos}}_n$, $\operatorname {\mathrm {Irr}}_n$ and $\operatorname {\mathrm {Pos}}_n\setminus \operatorname {\mathrm {Irr}}_n$. Let $\unicode{x3bb} _n$ be the unique real root of the polynomial $x^n-2x-1$ in $(1,+\infty )$. We explicitly construct an irreducible n-periodic tree pattern $\mathcal {Q}_n$ whose entropy is $\log (\unicode{x3bb} _n)$. We prove that this entropy is minimum in $\operatorname {\mathrm {Pos}}_n$. Since the pattern $\mathcal {Q}_n$ is irreducible, $\mathcal {Q}_n$ also minimizes the entropy in the family $\operatorname {\mathrm {Irr}}_n$. We also prove that the minimum positive entropy in the set $\operatorname {\mathrm {Pos}}_n\setminus \operatorname {\mathrm {Irr}}_n$ (which is non-empty only for composite integers $n\ge 6$) is $\log (\unicode{x3bb} _{n/p})/p$, where p is the least prime factor of n.

We prove that every locally compact second countable group G arises as the outer automorphism group $\operatorname{Out} M$ of a II1 factor, which was so far only known for totally disconnected groups, compact groups, and a few isolated examples. We obtain this result by proving that every locally compact second countable group is a centralizer group, a class of Polish groups that arise naturally in ergodic theory and that may all be realized as $\operatorname{Out} M$.

Deep neural networks and other modern machine learning models are often susceptible to adversarial attacks. Indeed, an adversary may often be able to change a model’s prediction through a small, directed perturbation of the model’s input – an issue in safety-critical applications. Adversarially robust machine learning is usually based on a minmax optimisation problem that minimises the machine learning loss under maximisation-based adversarial attacks. In this work, we study adversaries that determine their attack using a Bayesian statistical approach rather than maximisation. The resulting Bayesian adversarial robustness problem is a relaxation of the usual minmax problem. To solve this problem, we propose Abram – a continuous-time particle system that shall approximate the gradient flow corresponding to the underlying learning problem. We show that Abram approximates a McKean–Vlasov process and justify the use of Abram by giving assumptions under which the McKean–Vlasov process finds the minimiser of the Bayesian adversarial robustness problem. We discuss two ways to discretise Abram and show its suitability in benchmark adversarial deep learning experiments.

We provide a complete classification of Teichmüller curves occurring in hyperelliptic components of the meromorphic strata of differentials. Using a non-existence criterion based on how Teichmüller curves intersect the boundary of the moduli space we derive a contradiction to the algebraicity of any candidate outside of Hurwitz covers of strata with projective dimension one, and Hurwitz covers of zero residue loci in strata with projective dimension two.

This paper studies twisted signature invariants and twisted linking forms, with a view toward obstructions to knot concordance. Given a knot K and a representation $\rho $ of the knot group, we define a twisted signature function $\sigma _{K,\rho } \colon S^1 \to \mathbb {Z}$. This invariant satisfies many of the same algebraic properties as the classical Levine-Tristram signature $\sigma _K$. When the representation is abelian, $\sigma _{K,\rho }$ recovers $\sigma _K$, while for appropriate metabelian representations, $\sigma _{K,\rho }$ is closely related to the Casson-Gordon invariants. Additionally, we prove satellite formulas for $\sigma _{K,\rho }$ and for twisted Blanchfield forms.

Given a general polarized $K3$ surface $S\subset \mathbb P^g$ of genus $g\le 14$, we study projections of minimal degree and their variational structure. In particular, we prove that the degree of irrationality of all such surfaces is at most $4$, and that for $g=7,8,9,11$ there are no rational maps of degree $3$ induced by the primitive linear system. Our methods combine vector bundle techniques à la Lazarsfeld with derived category tools and also make use of the rich theory of singular curves on $K3$ surfaces.

We conduct a theoretical analysis of the performance of $\beta $-encoders. The $\beta $-encoders are A/D (analogue-to-digital) encoders, the design of which is based on the expansion of real numbers with noninteger radix. For the practical use of such encoders, it is important to have theoretical upper bounds of their errors. We investigate the generating function of the Perron–Frobenius operator of the corresponding one-dimensional map and deduce the invariant measure of it. Using this, we derive an approximate value of the upper bound of the mean squared error of the quantization process of such encoders. We also discuss the results from a numerical viewpoint.

We study the behaviour of Kauffman bracket skein modules of 3-manifolds under gluing along surfaces. For this we extend this notion to $3$-manifolds with marking consisting of open intervals and circles in the boundary. The new module is called the stated skein module.

The first results concern non-injectivity of certain natural maps defined when forming connected sums along spheres or disks. These maps are injective for surfaces or for generic quantum parameter, but we show that in general they are not when the quantum parameter is a root of 1. We show that when the quantum parameter is a root of 1, the empty skein is zero in a connected sum where each constituent manifold has non-empty marking. We also prove various non-injectivity results for the Chebyshev-Frobenius map and the map induced by deleting marked balls.

We then interpret stated skein modules as a monoidal symmetric functor from a category of “decorated cobordisms” to a category of algebras and their bimodules. We apply this to deduce properties of stated skein modules as a Van-Kampen like theorem, a computation through Heegaard decompositions and a relation to Hochshild homology for trivial circle bundles over surfaces.

We show that Calabi–Yau fibrations over curves are uniformly K-stable in an adiabatic sense if and only if the base curves are K-stable in the log-twisted sense. Moreover, we prove that there are cscK metrics for such fibrations when the total spaces are smooth.

This article is dedicated to investigating limit behaviours of invariant measures with respect to delay and system parameters of 3D Navier–Stokes–Voigt equations. Firstly, the well-posedness of such a system is obtained on arbitrary open sets that satisfy the Poincaré inequality, and then a unique minimal pullback attractor is attained by using the energy equation method and asymptotic compactness property. Furthermore, we construct a family of invariant Borel probability measures, which are supported on the pullback attractors. Specifically, when the external forcing terms are periodic in time, the periodic invariant measure can be obtained. Finally, as the delay approaches zero and system parameters tend to some numbers, the limit of the invariant measure sequences for this class of equations must be the invariant measure of the corresponding limit equations.

What proportion of integers $n \leq N$ may be expressed as $x^2 + dy^2$ for some $d \leq \Delta $, with $x,y$ integers? Writing $\Delta = (\log N)^{\log 2} 2^{\alpha \sqrt {\log \log N}}$ for some $\alpha \in (-\infty , \infty )$, we show that the answer is $\Phi (\alpha ) + o(1)$, where $\Phi $ is the Gaussian distribution function $\Phi (\alpha ) = \frac {1}{\sqrt {2\pi }} \int ^{\alpha }_{-\infty } e^{-x^2/2} dx$.

A consequence of this is a phase transition: Almost none of the integers $n \leq N$ can be represented by $x^2 + dy^2$ with $d \leq (\log N)^{\log 2 - \varepsilon }$, but almost all of them can be represented by $x^2 + dy^2$ with $d \leq (\log N)^{\log 2 + \varepsilon}\kern-1.5pt$.

We describe the structure of regular codimension $1$ foliations with numerically projectively flat tangent bundle on complex projective manifolds of dimension at least $4$. Along the way, we prove that either the normal bundle of a regular codimension $1$ foliation is pseudo-effective, or its conormal bundle is nef.

Given a polarised abelian variety over a number field, we provide totally explicit upper bounds for the cardinality of the rational points whose Néron-Tate height is less than a small threshold. These imply new estimates for the number of torsion points as well as the minimal height of a non-torsion point. Our bounds involve the Faltings height and dimension of the abelian variety together with the degrees of the polarisation and the number field but we also get a stronger statement where we use certain successive minima associated to the period lattice at a fixed archimedean place, in the spirit of a result of David for elliptic curves.

Let F be a non-archimedean locally compact field of residual characteristic p, let $G=\operatorname {GL}_{r}(F)$ and let $\widetilde {G}$ be an n-fold metaplectic cover of G with $\operatorname {gcd}(n,p)=1$. We study the category $\operatorname {Rep}_{\mathfrak {s}}(\widetilde {G})$ of complex smooth representations of $\widetilde {G}$ having inertial equivalence class $\mathfrak {s}=(\widetilde {M},\mathcal {O})$, which is a block of the category $\operatorname {Rep}(\widetilde {G})$, following the ‘type theoretical’ strategy of Bushnell-Kutzko.

Precisely, first we construct a ‘maximal simple type’ $(\widetilde {J_{M}},\widetilde {\lambda }_{M})$ of $\widetilde {M}$ as an $\mathfrak {s}_{M}$-type, where $\mathfrak {s}_{M}=(\widetilde {M},\mathcal {O})$ is the related cuspidal inertial equivalence class of $\widetilde {M}$. Along the way, we prove the folklore conjecture that every cuspidal representation of $\widetilde {M}$ could be constructed explicitly by a compact induction. Secondly, we construct ‘simple types’ $(\widetilde {J},\widetilde {\lambda })$ of $\widetilde {G}$ and prove that each of them is an $\mathfrak {s}$-type of a certain block $\operatorname {Rep}_{\mathfrak {s}}(\widetilde {G})$. When $\widetilde {G}$ is either a Kazhdan-Patterson cover or Savin’s cover, the corresponding blocks turn out to be those containing discrete series representations of $\widetilde {G}$. Finally, for a simple type $(\widetilde {J},\widetilde {\lambda })$ of $\widetilde {G}$, we describe the related Hecke algebra $\mathcal {H}(\widetilde {G},\widetilde {\lambda })$, which turns out to be not far from an affine Hecke algebra of type A, and is exactly so if $\widetilde {G}$ is one of the two special covers mentioned above.

We leave the construction of a ‘semi-simple type’ related to a general block $\operatorname {Rep}_{\mathfrak {s}}(\widetilde {G})$ to a future phase of the work.

We show that the sets of $d$-dimensional Latin hypercubes over a non-empty set $X$, with $d$ running over the positive integers, determine an operad which is isomorphic to a sub-operad of the endomorphism operad of $X$. We generalise this to categories with finite products, and then further to internal versions for certain Cartesian closed monoidal categories with pullbacks.

Mukai’s program in [16] seeks to recover a K3 surface X from any curve C on it by exhibiting it as a Fourier–Mukai partner to a Brill–Noether locus of vector bundles on the curve. In the case X has Picard number one and the curve $C\in |H|$ is primitive, this was confirmed by Feyzbakhsh in [11, 13] for $g\geq 11$ and $g\neq 12$. More recently, Feyzbakhsh has shown in [12] that certain moduli spaces of stable bundles on X are isomorphic to the Brill–Noether locus of curves in $|H|$ if g is sufficiently large. In this paper, we work with irreducible curves in a nonprimitive ample linear system $|mH|$ and prove that Mukai’s program is valid for any irreducible curve when $g\neq 2$, $mg\geq 11$ and $mg\neq 12$. Furthermore, we introduce the destabilising regions to improve Feyzbakhsh’s analysis in [12]. We show that there are hyper-Kähler varieties as Brill–Noether loci of curves in every dimension.