Papers

We study minimal sets of torus homeomorphisms admitting an irrational circle factor whose fibres are thin essential annular continua. For totally irrational pseudo-rotations and homeomorphisms in a nontrivial Dehn-twist class, we prove uniqueness of the minimal set when the fibres are Jordan curves on a residual set of base parameters. The same conclusion holds if the fibre cores are Jordan curves, or if the fibres are locally connected, on a nonmeagre set of parameters. The residual hypothesis cannot be replaced by a full-measure hypothesis, even under area preservation and topological transitivity. For every totally irrational rotation vector, we construct such a map with Jordan-curve fibres almost everywhere and uncountably many pairwise disjoint uniquely ergodic minimal Cantor sets. We also construct examples in every nontrivial Dehn-twist class, with prescribed irrational vertical rotation number and bounded deviations. In both families, the Jordan-curve parameters form a meagre set of full Lebesgue measure.

Adapting a construction of Kurihara in the setting of Drinfeld modular forms, we define Poincaré series on the Drinfeld half-plane $\Omega$. These series are built from products of meromorphic $1$-forms that are naturally associated to geodesics on the Bruhat-Tits tree. We establish convergence under a finiteness condition on the geodesics, verify this condition in several cases, and give sufficient conditions for the resulting cusp forms to be nonzero. For the principal congruence subgroup $\Gamma(\mathfrak{n})$ of $GL_2(\mathbb{F}_q[T])$, we construct explicit linearly independent families of Poincaré series by lifting certain $k$-forms from the components of the analytic reduction of $\Gamma(\mathfrak{n})\backslash\Omega$. We formulate conjectures on the vanishing orders at cusps, and we prove the first of them for an explicit family of Poincaré series by computing the corresponding expansions at the cusps; as an application, we obtain the Drinfeld modular forms $h$ and $\Delta$ as Poincaré series (up to a sign). Finally, for cocompact groups attached to quaternion algebras over $\mathbb{F}_q(T)$ that split at $\infty$, we show that the Poincaré series span the whole space of modular forms of given weight and type.

Let $C$ be a pointed closed convex cone in $\mathbb{R}^n$ with nonempty interior, and let $S^{n-1}$ denote the unit sphere in $\mathbb{R}^n$. The $L_p$ Minkowski problem for $C$-close sets is to determine, for a real number $p$ and a nonzero finite Borel measure $\mu$ defined on $\Omega_{C^\circ}=S^{n-1}\cap \mathrm{int} C^{\circ}$, whether there exists a $C$-close set $\mathds{A}$ such that $\mu$ is the $L_p$ surface area measure of $\mathds{A}$. In this paper, we will solve the problem for $p\in (0,1)$ and for $\mu$ being a nonzero finite Borel measure on $\Omega_{C^\circ}$. Moreover, we establish the continuity of solutions to the $L_p$ Minkowski problem for $p\in [0, 1]$ in several settings.

Preparing interacting quantum systems in low-energy or ground states is a fundamental task in quantum simulation and quantum information processing. In realistic settings, dissipation and active cooling can typically be engineered only on a limited subset of the system. We study the dissipative dynamics of excitation-number conserving quantum systems governed by a GKLS master equation with local jump operators acting on a subset of qubits. We establish sufficient conditions under which such localized dissipation drives the full system to a unique globally attractive zero-excitation state. In particular, we prove that if the Hamiltonian generates excitation transfer described by a graph for which the dissipative subsystem forms a zero forcing set, then the zero-excitation state is the unique globally attractive stationary state. When this state coincides with a ground state of the Hamiltonian, the same mechanism realizes ground-state cooling. Our results provide a graph-theoretic criterion for global state preparation from localized dissipation, which we illustrate using a nearest-neighbor Heisenberg spin chain. For this model, a reduction to the single-excitation sector further yields a scaling estimate with the length of the chain which indicates efficient cooling.

In this paper, we study the metric properties of pluriharmonic $\alpha$-Bloch-type mappings $f=h+\overline{g}\in\mathcal{H}(\mathbb{B}^{n},\mathbb{C}^{n})$ on the unit ball $\mathbb{B}^{n}\subset\mathbb{C}^{n}$. By synthesizing local real-part estimates with the subharmonicity of $\log(\vert{} \det Dh(z) \vert{} + \vert{} \det Dg(z) \vert{})$ along hyperbolic geodesics, we establish a globalized non-trivial lower bound across the entire unit ball $\mathbb{B}^{n}$. Using this structural decay estimate, we prove that the invariant weight quantity $D_{f}^{n,\alpha}(z)$ satisfies a global Hölder continuity estimate of order $1/n$ with respect to the Bergman metric $\beta$. Furthermore, we investigate composition operators $C_{\varphi}(f)=f\circ\varphi$ induced by holomorphic self-mappings $\varphi\in\mathcal{A}(\mathbb{B}^{n},\mathbb{B}^{n})$. We establish both sharp sufficient and necessary sampling conditions for $C_{\varphi}$ to be bounded from below on the pluriharmonic $\alpha$-Bloch space $\mathcal{B}_{P(n,\alpha)}$.

Let $\mathbf A$ be a totally nonnegative matrix over a real closed non-Archimedean valued field, and suppose that its valuation $A$ is tropically totally nonnegative. Although the valuations of the classical eigenvalues are weakly majorized by the tropical eigenvalues of $A$, equality can fail through cancellation. We give a necessary and sufficient condition for equality. The only obstruction is dominant cancellation in the principal minors that give the essential coefficients of the tropical characteristic polynomial. In other words, equality holds exactly when the classical and tropical Newton polygons agree at the breakpoints determined by the diagonal entries. We describe the exceptional lifts by explicit polynomial equations in the leading coefficients. We also study these equations on each irreducible component of the admissible parameter space. Finally, we give a complete $3\times 3$ criterion and a family that contains both ordinary lifts with spectral equality and exceptional totally nonnegative lifts without it.

Let $\mathbb H^n$ be the $n$-dimensional hyperbolic space, and $\Delta_{\mathbb H^n}$ be the Laplace–Beltrami operator on $\mathbb H^n$. Let $u=u(t,\cdot)$ be the solution to the wave equation $\partial^2_{t}u-\big(\Delta_{\mathbb H^n}+\frac{1}{4}(n-1)^2\operatorname{Id}\big)u=0$ with initial conditions $u(0,x)=f(x)$ and $\partial_tu(0,x)=0$. In this article we show that for a radial function $f$, $2n/(n-1)\leq p<\infty$, $\gamma<1/p$ and $T\geq 0$, there exists a constant $C_p$ independent of $f$ and $T$ such that $$ \left(\int_T^{T+1}\|u(t,\cdot)\|^p_p\mathrm d t\right)^{\frac1p}\leq C_{p,\gamma}\,\mathrm e^{(n-1)\left(\frac12-\frac1p\right)T}\left\|(-\Delta_{\mathbb H^n})^{\frac{n-1}2\left(\frac12-\frac1p\right)-\frac\gamma2}f\right\|_p. $$ We also show that the exponential part $\mathrm e^{(n-1)(1/2-1/p)T}$ of the bound is optimal.

This paper studies decentralized safe path following for multiple quadrotors on intersecting paths, where safety requires both collision avoidance and strict adherence to pre-assigned routes. The proposed controller reformulates transverse feedback linearization as a constrained quadratic program with four equality constraints: two enforce convergence to and strict adherence to the path, and two prescribe the desired speed and heading. Safety is achieved by relaxing only the along-path speed constraint, while the path and heading constraints remain hard. Under the stated assumptions, and for admissible initial conditions, the controller guarantees that all agents converge to and thereafter follow their assigned paths, avoid collisions, and avoid attitude singularities. We evaluate the controller in the Drake physics engine on non-planar intersecting paths and compare it against two nominal-plus-safety-filter cascades. Code and additional results are available at https://gradslab.github.io/safe_multiquad_pf/.

Over perfect fields of characteristic $p\ge5$, the unipotent radicals of the height-one special, Witt-Jacobson, and Hamiltonian automorphism groups in at least three, two, and four variables, respectively, attain the largest derived length allowed by their nilpotency classes. More strongly, every term of each derived series equals the corresponding power-of-two term of the lower central series. These equalities hold on finite-field points, with explicit Frattini quotients and lower $p$-central series. For the special family, a Cartier quotient measures the difference between the congruence and lower central filtrations. The Hamiltonian correction comes from flux and the socle line; the Witt-Jacobson family has none. We also identify the contact automorphism group scheme and, at arbitrary divided-power height, compute the Cartier-Verschiebung maps on restricted derivations and the Verschiebung filtration and distribution algebra of the Frobenius thickening of the special radical.

The shifted congruence series of finite special Nottingham jet groups is realized by an integral augmentation filtration. For volume-preserving formal substitutions in characteristic $p\ge5$ and $n\ge3$ variables, the filtration induced by the action on truncated power series has the congruence subgroups as its dimension subgroups. The canonical weighted augmentation filtration has the same dimension subgroups. The group symbols form a degree-truncated Lie algebra of divergence-free polynomial fields. Since brackets with quadratic fields give the subspace of exact fields in every coefficient degree at least three, the Lie indecomposables occur only in degrees $(p-1)(n-1)+pr$, with $r\ge0$, where they are determinant-weighted Frobenius twists of tensor products of symmetric powers with the defining module.

We describe the Lean 4 formalization of Langlands's Second Main Lemma for local epsilon factors over nonarchimedean local fields, in mixed and equal characteristic. The lemma compares the induction expressions attached to compatible characters of the intermediate fields of a bicyclic extension of prime-square degree. We explain the mathematical constructions and comparisons that constitute the proof, and identify their Lean declarations.

We conjecture a positive four-shift identity for rank-two Nahm sums with denominator steps $(1,31)$, giving a mixed-base realization of the Monster McKay–Thompson series $T_{31A}$ through the classical Rogers–Ramanujan function $U(1,31)$. A theta dissection directly derives the modular target. For the underlying quadratic exponent $3r^2+31rs+93s^2$, we prove nonmodularity of every individual sum with rational linear terms and normalization, together with an exclusion from finite Fricke systems with meromorphic finite-width expansions at infinity. Exact agreement through $q^{2000}$ and six vanishing radial corrections support the proposed combination; a five-term dilogarithm certificate and an exact amplitude calculation prove its leading asymptotic behavior. We further propose a three-component Fricke law for a dual combination, with an explicit icosahedral matrix involving fifth-root constants, and a quadratic identity linking its components to $T_{31A}$. The dual translation law and matrix algebra are proved; its Fricke transformation and moonshine identity remain conjectural, supported respectively by numerical evaluations and exact finite coefficient checks.

We develop a structural theory of the Gaussian mixed-norm Fock spaces $\mathcal F_{\alpha}^{p,q}$, where $\alpha>0$ and $0<p,q\leq\infty$, which separate angular $L^p$-means from radial Gaussian $L^q$-summability. We first prove the sharp circle-mean estimate $M_p(f,r)\leq C_{\alpha,q}(1+r)^{-\frac{1}{q}} e^{\frac{\alpha}{2}r^2}\lVert f\rVert_{\mathcal F_{\alpha}^{p,q}}$, $r\geq0$, where $M_p(f,r)$ denotes the angular $L^p$-mean of $f$ on the circle $\lvert z\rvert=r$ with the convention $\frac{1}{\infty}=0$, and show that the exponent $-\frac{1}{q}$ is optimal. The proof is based on a local mass principle for convex Laplace-type exponents, which also yields an annular discretization of the mixed Gaussian norm. Using this discretization, we characterize the continuous embeddings between mixed-norm Fock spaces and obtain an atomic decomposition in terms of normalized Fock kernels, with coefficients belonging to a corresponding annular mixed sequence space. We further characterize Carleson and vanishing Carleson measures and identify the continuous duals in the Banach, quasi-Banach, and endpoint regimes. For $p<1$, the atomic decomposition relies on a quantitative re-centering theorem that controls expansions of normalized Fock kernels under perturbations of their centers. This theorem is of independent interest even for the Hilbert Fock space $\mathcal F_{\alpha}^{2}$.

We study the one-parameter family of Hessian metrics $g_\alpha=\alpha h_++(1-\alpha)h_-$, $\alpha\in[0,1]$, formed by convexly combining the log-partition functions (not the densities) of the multinomial and negative multinomial Fisher information Hessians. Four results are proved in full. (1) $g_\alpha\succ0$ for all $\alpha\in[0,1]$; its Gaussian curvature ($n=2$) is $-\tfrac14$ at $\alpha=0$ and $+\tfrac14$ at $\alpha=1$, but non-constant for intermediate $\alpha$, changing sign at a closed-form critical value $\alpha^\ast(w)$, with an analytic uniqueness proof. (2) Viewing $g_\alpha$ as the spatial metric of a Kasner-type volume equation, positive-definiteness forces one conserved quantity $\beta\le0$, while a second, independent quantity $C=4\det K$ governs the equation's branch (bounded oscillatory, parabolic, or unbounded growth) and is not fixed by $\beta$ or positive-definiteness alone; we show this discriminant dictionary is a theorem about the abstract matrix equation, to which $g_\alpha$ supplies initial data of either sign of $C$ without the trajectories remaining on $\{g_\alpha(\theta)\}$ itself. (3) $\Psi_\alpha$ is identified as the log-partition function of a compound distribution: a negative-binomial latent count $M$ (parameter $r=1-\alpha$), augmented by a Bernoulli parity bit and split multinomially, with non-negative base measure for every $\alpha$. (4) The associated Bregman divergence obeys a generalised Pythagorean theorem whose projection onto the symmetric locus is the arithmetic mean, independent of $\alpha$; the endpoints' curvature sign matches the Gauss–Bonnet excess of genuine geodesic triangles. Numerical checks are kept separate from analytic proofs throughout.

This paper presents a probabilistic perspective on large language models (LLMs), developed with the aim of bringing together, in a single self-contained account, tools that are usually treated separately across the literature. LLMs are described through probability measures on the set of sequences of tokens, specified via their autoregressive conditional distributions. Training is formulated as a maximum-likelihood estimation problem, addressed by stochastic gradient methods, while text generation is viewed as the sequential simulation of the resulting stochastic process. The role of the asymmetry of the Kullback–Leibler divergence in text generation is examined in relation with characteristic phenomena such as hallucination and the distinction between statistical plausibility and truth. As a complementary illustration of the same viewpoint, we also discuss diffusion models, built around the score function, which cast generation not as sequential token prediction but as the simulation of a reverse-time stochastic process transforming noise into data both in discrete and continuous time.

We report certified computations on four no-three-in-line questions, two in the cube and two in the plane, by one method: exact decision procedures inside symmetry strata, every witness re-verified by a second program sharing no code, every number traced to a public journal. (I) Let a(n) be the largest number of points of {0,...,n-1}^3 with no three collinear (A399138). We determine a(1),...,a(6) = 1, 8, 16, 28, 40, 64 with DRAT-certified unsatisfiability proofs, give certified lower bounds a(7) >= 73, a(8) >= 94, a(9) >= 116, a(10) >= 138, a(11) >= 164, and prove a(p) >= p^2 for every prime p. The optima share a layer structure 2n^2 - 2n + 4 that provably fails at n = 5 and n = 7. (II) For b(n), the largest number with no four coplanar (A280537), nineteen public configurations give lower bounds for 9 <= n <= 29, four of which improve the known bounds and their monotone closure (b(12) >= 31, b(21) >= 47, b(22) >= 49, b(27) >= 56); the cyclically invariant subspace has maximum 23 at n = 9 and 26 at n = 10; four kinds of symmetry are incompatible with the problem; no upper bound beyond 3n is known. (III) The Guy-Kelly first-moment heuristic for the plane is audited against exact counts (A000755 to n = 20): its corrected constant comes out in closed form, pi/sqrt(3), the threshold crossing n = 493 is reproduced, and its error is shown to depend on the shape of the question rather than on n alone, with an unbounded multiplier; whether the residual error is Theta(n) or Theta(n ln n), which decides whether the constant survives, cannot be told by counting, and we measure how far from telling we are. (IV) The direction spectrum of the 2n-point solutions is measured, and a line model with no fitted parameter reproduces its shape (fifteen constants within 12%, seven predicted blind) but not its scale. Withdrawn claims are kept in the text; each part states what it does not establish.

We introduce several Ramsey-theoretic properties of 0-dimensional ambits and obtain their dynamical characterizations. In consequence, we obtain Ramsey-theoretic criteria for triviality and for profiniteness of some important invariants (in particular, of the Ellis groups) of the ambits in question. This yields a criterion for the structural property that each distal minimal factor of the given ambit is a profinite flow. Another result is a criterion for metrizability of the minimal left ideals in the Ellis semigroup. Then we study three specializations of the above abstract context: to first order theories, to definable groups, and to the classical Kechris-Pestov-Todorčević theory, recovering known and obtaining new results in each of these contexts.

We obtain a strong decomposition theorem for invariant global types in a monadically NIP theory. From this, we prove that if an $n$-type $tp(\bar{a}/MC)$ does not fork over $M$ then $\bar{a}=(\bar{f},\bar{d})$ where $tp(\bar{d}/MC)$ is $M$-definable and $tp(\bar{f}/MC\bar{d})$ is finitely satisfied in $M$. Over arbitrary base sets $B$, we prove that an $n$-type $tp(\bar{a}/BC)$ does not fork over $B$ if and only if $tp(a_i/BC)$ does not fork over $B$ for each singleton $a_i\in\bar{a}$. We show that monadically NIP theories satisfy density of definable types among non-forking extensions.

A sub-quorum coloring is a partial vertex coloring in which every colored vertex sees at least half of its colored closed neighborhood in its own color. We study the maximum number of colors $\psq(G)$ from structural, algorithmic and extremal viewpoints. Every $2$-independent set yields a sub-quorum coloring, so $\psq(G)\ge\beta_2(G)$. We prove the degree bound $\psq(G)\le \Delta(G)|V(G)|/(\Delta(G)+\delta(G)-1)$ for $\delta(G)\ge2$, and obtain an exact matching characterization for graphs of maximum degree at most three. Consequently every cubic graph of order $n$ satisfies $n/2\le\psq(G)\le3n/5$, with a stability identity at the upper endpoint. For trees we give an edge representation and a linear-time dynamic program that computes both $\psq(T)$ and an optimal coloring; we also show on the double star $S_{2,2}$ that $\psq$ can exceed the dissociation number even on trees. We prove NP-completeness even for bipartite graphs and determine exact values for several classical families. For rectangular grids, we formulate the conjecture $\psq(G_{m,n})=\beta_2(G_{m,n})=F(m,n)$, recover the dissociation-number formula from known $3$-path vertex-cover results, prove the ladder case directly, and verify the conjecture for every fixed width $2\le m\le11$; widths $3$ through $11$ use exact transfer certificates. For hypercubes we prove $\psq(Q_n)=\beta_2(Q_n)=2^{n-1}$ for every $n\ge2$. Finally, a Nordhaus–Gaddum argument gives $\psq(G)+\psq(\overline G)>\log_2 |V(G)|$ and leads to a complementary upper-bound problem.

In this paper, we determine the precise asymptotics of the number of parts of stable regularity equipartitions in terms of the Littlestone dimension: every graph $G$ of Littlestone dimension $\operatorname{Lit}(G)\leq\ell$ has a regular equipartition into excellent sets with $(1 + o_{\epsilon\to 0,\ell}(1))\cdot\epsilon^{-\ell-1}$ parts and in the other direction, for every $\ell\in\mathbb{N}_+$, there is an infinite family of graphs, all of Littlestone dimension $\ell$, whose equipartitions into good sets must have size at least $(1 + o_{\epsilon\to 0,\ell}(1))\cdot\epsilon^{-\ell-1}$. Dropping the equitability condition, we determine the asymptotics of non-equitable partitions up to a multiplicative $\log(1/\epsilon)$: every graph $G$ with $\operatorname{Lit}(G)\leq\ell$ has a regular partition into excellent sets with $(1 + o_{\epsilon\to 0,\ell}(1))\cdot\epsilon^{-\ell}\cdot\ln(1/\epsilon)$ parts and in the other direction, for every $\ell\in\mathbb{N}_+$, there is an infinite family of graphs, all of Littlestone dimension $\ell$, whose partitions into good sets must have size at least $(1 + o_{\epsilon\to 0,\ell}(1))\cdot\epsilon^{-\ell}$. We also show that such partition can be obtained algorithmically efficiently in an approximation scheme fashion: replacing the $o_{\epsilon\to 0,\ell}(1)$ term above by a constant $c > 0$, we obtain randomized $O_{c,\epsilon,\ell}(n\cdot\log(n))$-time algorithms for partitions/equipartitions into good sets, a deterministic $O_{c,\epsilon,\ell}(n^2)$-time algorithm for partitions into good sets, a deterministic $O_{c,\epsilon,\ell}(n^6)$-time algorithm for equipartitions into good sets, a deterministic $O_{c,\ell,\epsilon}(1)\cdot n^{O(\ell\cdot 2^{2\cdot\ell+4})}$-time algorithm for partitions/equipartitions into excellent sets, and $O_{c,\epsilon,\ell}(\log(n+1))$-space algorithms for partitions/equipartitions into good/excellent sets.

We construct a measurable aggregator for the value processes associated with stochastic control problems, where the criterion to optimise is given by solutions to semi-martingale backward stochastic differential equations (BSDEs) with jumps. The results can be applied to control problems where the triplet of semi-martingale characteristics is controlled in a possibly non-dominated case or where uncertainty about the characteristics is present in the optimisation. The construction also provides a time-consistent system of fully nonlinear conditional expectations on the Skorokhod space. We also construct an appropriate path-regularisation of the value function and prove a corresponding dynamic programming principle. The generality we seek allows for the treatment of controlled diffusions, pure-jump processes, and discrete-time processes in a unified setting.

We establish the Margulis-Soifer dichotomy for one-relator groups: every one-relator group is either virtually solvable or has a maximal subgroup of infinite index. We also present examples of one-relator groups with and without free maximal subgroups of infinite index, as well as examples that possess both free and non-free infinite index maximal subgroups. Triviality of the Frattini subgroup is also shown for all non-solvable one-relator groups. We close the paper with a short list of questions.

We show that the $ \mathrm{K}_1 $-functor modeled on simple reductive groups over semilocal rings is solvable if the isotropic rank is at least $ 2 $ and that the Tits index is neither $ {}^{2} \mathsf{E}_{6, 2}^{16'} $ nor $ \mathsf{E}_{8, 2}^{78} $. For these two Tits indices the result is already known, but assuming that the base ring contains a field. Our result implies that the elementary subgroup (or its derived subgroup) is the maximal perfect subgroup of the reductive group.

We prove that every Voronoi cell of a unit-ball packing of $\mathbb{R}^4$ has volume at least $8$, with equality only at the $D_4$ configuration, so that the twenty-four-cell conjecture holds and the density of a sphere packing in four dimensions is at most $\pi^2/16$. The proof runs through the number of contacts of a cell. Up to twenty-two a covering estimate suffices; at twenty-three the cell is bounded through an exact volume identity inside a ball and a semidefinite certificate for one inequality between pair angles; at twenty-four the configuration is the root system, which we prove from the second level of the semidefinite hierarchy with its equality case, the positive kernel verified in exact arithmetic.

We consider the three-dimensional incompressible Navier–Stokes–Cahn–Hilliard (NSCH) system in domains perforated by no-slip obstacles of diameter of order \(\varepsilon^3\) separated by distances of order \(\varepsilon\). Both viscosity and mobility may depend on the phase variable. At this critical Stokes-capacity scale, the limiting velocity \(\boldsymbol u\) and phase field \(\phi\) satisfy an NSCH system with the additional Brinkman resistance \(\nu(\phi)\mathbf B\boldsymbol u\), where \(\mathbf B\) is determined by the exterior Stokes capacity of the reference obstacle. The principal analytical difficulty is the interaction between the phase-dependent viscosity and the order-one energy concentration of the critical Stokes correctors. The variational chemical-potential identity yields, after scalar extension, strong convergence of the phase in \(L^2(0,T;H^1(\Omega))\), and hence strong convergence of the phase-dependent viscosity in the same topology. Combined with a cellwise Hardy multiplier estimate, this compactness can be transferred through the concentrated corrector layer. The resulting weighted-capacity statement applies to uniformly positive and bounded coefficient sequences converging strongly in \(L^2(0,T;H^1(\Omega))\) and simultaneously identifies the effective Brinkman force and the corresponding viscous dissipation lower bound. We also investigate a vanishing capillary coefficient \(\lambda_\varepsilon\to0\). After the natural velocity scaling, the limit is an unsteady Stokes–Brinkman equation coupled to an unadvected Cahn–Hilliard equation.

We show that exact self-testing of a quantum correlation does not imply robust self-testing. Mančinska and Schmidt previously established such a separation for non-local games, but their obstruction to robustness arises from two distinct optimal correlations, with the self-tested one remaining robust. We establish the separation for a single correlation. Our proof is operator-algebraic. The main technical contribution is a general construction that associates a synchronous binary correlation to any finitely presented C*-algebra with unitary generators and real-coefficient relations. The implementing states of the correlation correspond exactly to the tracial states on the algebra. We apply this construction to an algebra with a unique finite-dimensional tracial state and a distinct amenable tracial state. By a characterization established by Zhao and independently by Kar, the resulting correlation is a self-test but not a robust self-test.

We prove a PL bordism version of Gromov's linearity conjecture over a broad class of asphericalization groups for manifolds endowed with faithful representations in all dimensions. Since every group embeds into an asphericalization group, this bordism version of the conjecture holds after allowing the target group to be enlarged. Our results apply in both the oriented and unoriented settings. In the PL category, no bounded local geometry is required. As an application, we obtain improved linear bounds for the Cheeger–Gromov $L^2$ $\rho$-invariants of PL $(4k-1)$-manifolds associated with their universal covers, replacing the previous superfactorial growth in dimension by factorial growth. The proofs are based on a quantitative asphericalization method that provides explicit control of the complexity of the resulting bordisms.

Generalized covering radii measure how many columns of a parity-check matrix are needed to generate several syndromes simultaneously. Their finite-geometric counterparts are $(\rho,t)$-saturating sets, for which every $t$-dimensional subspace is contained in a subspace generated by at most $\rho$ prescribed vectors. We investigate this covering problem for the norm-one configurations associated with generalized Zetterberg codes. We establish the upper bound $2t+1$ over every nonbinary finite field and in an explicit binary range, together with complementary lower bounds obtained by counting subspaces and constructing subfield obstructions. For an explicit range of large $t$, these configurations are $t$-strong blocking sets, and the $t^{\rm th}$ generalized covering radius attains its minimum possible value $t$. For binary Zetterberg codes, we determine the second generalized covering radius in every extension degree and prove that the third radius is seven for an infinite subfamily.

This article is the second part of a review devoted to spacetime singularities in classical General Relativity. While Primer I developed the mathematical framework for defining, classifying, and proving the existence of singularities, the present work examines their concrete realization in spatially homogeneous cosmological models. After reviewing the geometrical foundations of homogeneity and the Bianchi classification, we derive the field equations for orthogonal and tilted perfect fluid cosmologies and use the Kasner and Bianchi I solutions as elementary prototypes of anisotropic singular behaviour. Raychaudhuri focusing provides general conditions under which the homogeneous Cauchy development reaches a finite proper time boundary. We then distinguish matter singularities, characterized by divergent density and Ricci curvature, from more subtle non-scalar or whimper singularities associated with Cauchy horizons and extreme fluid tilt. The classical Ellis-King analysis is reconsidered in the light of later developments, including rigorous curvature-blow-up results for Bianchi VIII and IX models, kinematic singularities in geodesically complete tilted spacetimes, and the recent reappearance of whimper behaviour in dipole cosmology. The resulting picture shows that even spatially homogeneous cosmologies exhibit a remarkably rich hierarchy of singular boundaries. The detailed dynamical approach to these boundaries - Hamiltonian cosmology, Kasner transitions, Mixmaster dynamics and the BKL conjecture - is reserved for Primer III.

We establish graph-theoretic analogues of the Brauer-Kuroda relations for ramified Galois covers of finite graphs with arbitrary finite Galois groups. These formulas relate the numbers of spanning trees of intermediate quotient graphs and recover the corresponding formulas for unramified covers. Our proofs use $h$-functions defined on multiplicity spaces of complex representations of the Galois group.