Papers

Let $G$ be a complex reductive affine algebraic group with its symmetric tensor category $\mathcal{C}_G$ of finite-dimensional rational representations. We prove that every simple commutative algebra object with at most countable dimension in the direct limit completion $\operatorname{Ind}(\mathcal{C}_G)$ of $\mathcal{C}_G$ is isomorphic to the algebra $\mathcal{O}(G/H)$ of regular functions on the homogeneous space $G/H$, for some reductive algebraic subgroup $H$ of $G$. Then we apply this result to the theory of vertex operator algebra extensions. In particular, assuming the strong rationality of the $ \operatorname{A}_5$-orbifold of the vertex operator algebra $V_{\mathcal{L}_2}$ associated with the rank-one root lattice $\mathcal{L}_2:= \sqrt{2}\mathbb{Z}$, we classify all the not necessarily rational simple CFT type preunitary vertex operator algebra extensions of the simple unitary Virasoro vertex operator algebra $L(1,0)$ with central charge $c=1$ satisfying a certain spectrum condition. This result is the vertex operator algebra analogue of a conformal net result by Feng Xu. Every strongly rational preunitary vertex operator algebra extension of $L(1,0)$ satisfies the above spectrum condition because of the congruence subgroup modularity property of its characters. As a consequence, we get a complete classification result for strongly rational $c=1$ vertex operator algebras, up to the strong rationality of $V_{\mathcal{L}_2}^{\operatorname{A}_5}$.

We study the Blaschke–Santaló diagram associated with the volume, the diameter, and the Cheeger constant. In the class of bounded open subsets of $\mathbb{R}^m$, $m\ge 2$, we give a complete description of the diagram. In the class of convex bodies of $\mathbb R^m$, we prove that the diagram is closed and simply connected as it is given by the region between two continuous functions for which qualitative properties such as monotonicity, local behavior, and growth estimates are studied.

We prove Conjecture 1.9 of Gabrielov-Novikov-Shapiro. More precisely, for every $p>0$, every nonconstant restriction to a line of a potential generated by $n$ point charges has at most $2n-1$ critical points. The bound is sharp, even for positive charges. The proof uses a duality with an auxiliary planar potential generated by collinear charges and Morse theory.

Generative models for implied volatility surfaces must produce outputs that satisfy static no-arbitrage constraints. We study these constraints in latent space. For a fixed generator, we assign each latent code a scalar margin determined by the no-arbitrage conditions of the generated surface. The codes with nonnegative margin form the admissible latent set. We establish conditions under which strictly admissible codes remain admissible under small perturbations and the boundary of the admissible set is characterized by zero margin. For regular boundary components, we formulate a level-set equation whose local dynamics are directed toward the zero-margin set. The analysis treats the generator as a map from latent variables to surfaces and is therefore not restricted to a particular architecture. It applies to variational autoencoders, generative adversarial networks, and other generative models with a deterministic realization map. Numerical tests recover known boundaries in analytic examples. Experiments with a variational autoencoder trained on Heston surfaces show that similar reconstruction errors can correspond to different admissible regions and that the latent prior may be concentrated inside such a region. The computed boundary can also be used to modify latent codes that generate violating surfaces.

Metropolitan regions are simultaneously pursuing several interventions to improve mobility, accessibility, and energy efficiency, necessitating integrated tools to understand how these policies interact to affect travel behavior, energy use, and infrastructure needs. This paper evaluates the combined impacts of electrification, freight demand management, road pricing, parking reform, and transit expansion on the Chicago metropolitan transportation system in 2050, using a business-as-usual (BAU) scenario as the baseline. We employ POLARIS, a large-scale agent-based modeling framework calibrated to 2019 conditions, to simulate nine policy scenarios for the seven-county northeastern Illinois region. The framework co-simulates activity-based passenger demand, endogenous freight generation, multimodal traffic assignment, and transit operations, with charging infrastructure and freight operations optimized for each case. Our findings reveal that under the high electrification scenario, total fuel mass declines by 68% while total charging energy increases by approximately 4-8x from BAU, resulting in a peak power demand near 4 GW concentrated in the urban core. Furthermore, freight management policies reduce freight VMT by increasing trip frequency but shortening distances, smart road pricing most effectively reduces auto VMT, and transit expansion boosts ridership by 18% relative to BAU. By presenting the first integrated, agent-based scenario framework for Chicago that jointly evaluates these interventions, this study provides actionable insights for regional transportation planning, grid infrastructure investment, and emissions reduction, highlighting the value of targeted charger upgrades and coordinated policy bundles.

With any holomorphic vector bundle $E$ over a compact base one can associate a line bundle, denoted $O_{PE}(1)$. According to a conjecture of Griffiths, if $O_{PE}(1)$ admits a positively curved hermitian metric $k$, then $E$ also admits a positively curved hermitian metric, $h$. In this paper we show that, while the conjecture may be correct, it is not possible to obtain $h$ out of $k$ by a fiberwise construction that is functorial.

We study quantitative stability for the fractional Yamabe functional introduced by González and Qing. Its minimizers correspond to conformal metrics with constant fractional curvature. We prove that the deficit of the fractional Yamabe functional controls, with a suitable power, the distance to the set of minimizers. Our approach is based on a Lyapunov–Schmidt reduction for the nonlocal functional, together with the extension characterization of the fractional conformal Laplacian, which provides the regularity theory needed for the local analysis. As a consequence, we also obtain an analogous quantitative stability estimate for the associated extension functional.

Can an artificial intelligence (AI) generate a scientific hypothesis outside a human collaborator's active hypothesis space (AHS), and can human-AI research be organized to make such breakthroughs more likely? We document such a case while proving a theorem that connects two basic organizing mechanisms of statistical physics: collective behavior arising in zero field from competing interactions and that induced or controlled by an external field. A zero-field $O(n)$-vector open chain with arbitrary inhomogeneous nearest- and next-nearest-neighbor interaction functions $U_i(S_i\cdot{S}_{i+1})$ and $V_i(S_i\cdot{S}_{i+2})$ is microscopically, via a temperature-independent mapping at the Hamiltonian level, equivalent to a simpler $O(n)$ open chain with nearest-neighbor interaction $V_i(\boldsymbol\sigma_i\cdot\boldsymbol\sigma_{i+1})$ and axial single-spin potential $U_i(\sigma_i^z)$ for every integer $n\ge1$ and every system size $L\ge1$. The homogeneous linear specialization maps the foundational frustrated $J_1$-$J_2$ model onto the canonical $J$-$h$ field model—with $n=1,2,3$ being the Ising, XY, and Heisenberg classical spin models, respectively; the theorem resolved a longstanding challenge for $n=3$ published in 1990. Its proof was done with an AI-synthesized recursive Householder moving frame and understood via a human-recognized hidden reciprocity. An analogous theorem holds when the continuous $O(n)$ spins are replaced by the $q$-state Potts spins, implying a closed-form exact solution of the $J_1$-$J_2$ standard Potts open chain for every $q\ge2$ and every $L\ge1$. The emergence of these theorems from a human-AI co-development framework suggests that sustained AI involvement throughout a systematic research program may incubate autonomous scientific breakthroughs and make aspects of the discovery process experimentally testable.

A social network perspective on the human microbiome is crucial for understanding how the interplay between within-host microbial dynamics and between-host transmission influences microbial community stability, host metabolism, and population health and nutrition. Here, we provide a mathematical and numerical analysis of human-associated microbial communities in dyadic social ties and within a small network. By utilizing a generalized Lotka-Volterra model, we simulate within-host dynamics while incorporating inter-host microbial sharing driven by social interactions. Furthermore, the study accounts for individual variability in microbial behavior across different hosts. Our work identifies the conditions under which microbial sharing can have long-lasting effects, offering insights into how social networks shape the human microbiome.

We investigate the discrete spectrum of finitely many point interactions for Neumann and Robin Laplacians on special unbounded and exterior $C^{1,1}$ domains in dimensions two and three. The operators are realized as self-adjoint extensions through an ordinary boundary triple whose gamma field and Weyl matrix are constructed from the Robin Green kernel. Eigenvalues below the background spectrum are characterized by the Weyl matrix, yielding an exact finite-dimensional counting formula. If the background operator is non-negative, this also gives the number of negative eigenvalues without assuming a finite zero-energy limit of the Weyl matrix. In the one-centre case we identify the critical coupling and prove that the unique eigenvalue branch is real analytic, strictly increasing, and strictly concave. For scalar multicentre couplings $\Theta=\alpha I_N$, sufficiently strong attraction produces exactly $N$ eigenvalues below the background spectrum, and all of them have universal leading asymptotics coinciding with the whole-space laws. If the bottom of the background spectrum is an isolated eigenvalue, the branch exists for every finite coupling and we determine its leading decoupling asymptotics as the coupling tends to $+\infty$. Explicit exterior-sphere and exterior-disk models illustrate the critical couplings and their threshold behaviour.

We strengthen a recent selection game equivalence proved by L. Chiozini involving spaces and their associated spaces of continuous real-valued functions. This strengthening establishes that the Rothberger game is perfect- and Markov-information dual to the strong closed discrete game on the space of real-valued continuous functions. In the process, we extend the theory of strategic equivalences for the second player in variations of the Menger and Rothberger games involving large covers while noting explicitly how large covers present serious obstacles to applying strategy translation results used by the authors in similar scenarios in previous papers. We also briefly comment on subbasic selection games introduced by D. Guerrero Sánchez and V.V. Tkachuk.

For a point $P$ on a smooth projective curve $C$ of genus $g$, the section ring $R_d = R(C,\mathcal{O}_C(dP))$ can be minimally presented as a quotient $S_d/I_d$ where $S_d$ is a $\mathbb{Z}$-graded polynomial ring. Motivated by Green's $N_p$ properties for projective embeddings, we investigate the syzygies of $R_d$ over $S_d$ in low degrees $d$ when $R_d$ is not generated in degree 1. We bound the degrees of the generators of $R_d$ and prove uniform column-by-column bounds on the support of the Betti table of $R_d$ over $S_d$. We compute the weighted regularity of $R_d$ and show that if $d$ is larger than the Frobenius number of $P$ then $R_d$ satisfies the weighted $N_p$ condition, where $p=g-1-\binom{d-g}{2}$. Finally, we give sufficient criteria for the Betti numbers to be determined explicitly and show that for ordinary points, the resolution of $R_{g+1}$ is pure.

We continue the study of the basic theory of object-unital groupoid graded rings. In this work, we are especially interested in nilpotency conditions on the graded Jacobson radical. We introduce the concept of left/right objectwise nilpotency of graded ideals, and prove that this condition is appropriate for obtaining graded generalizations of the Hopkins–Levitzki theorem. Although this condition is not symmetric, we show that its two-sided version is suitable for defining gr-semiprimary rings. It is known that one-sided $\Gamma_0$-artinian rings need not be $\Gamma_0$-noetherian, but using our tools we prove that one-sided gr-hereditary $\Gamma_0$-artinian rings, two-sided $\Gamma_0$-artinian rings, and $d$-finitely generated one-sided $\Gamma_0$-artinian rings are $\Gamma_0$-noetherian. However, the first class need not be gr-semiprimary, whereas the other two always are. We illustrate our results with several (counter)examples, especially involving graded upper triangular matrices.

Generic categories arising in representation stability and equivariant commutative algebra are difficult to analyze in positive characteristic, as these categories rarely admit enough (or even any) finitely generated injectives. Our main observation is that they are nonetheless an increasing union of Serre subcategories, each with finitely many simples and enough finite length injectives. We package this into an axiomatic framework which is widely applicable. In particular, we use this to analyze (1) VI-modules in non-describing characteristic, recovering Nagpal's classification of simple generic VI-modules, and (2) GL-equivariant modules over truncated polynomial rings and exterior algebras in infinitely many variables.

We consider, for any irrational $\alpha$ and interval $I \subset \mathbb{T}$, the 2-interval coding subshift $X^{(I, \alpha)}$ induced by coding orbits under repeated rotation by $\alpha$ via membership in $I$ or $I^c$. Each sequence $c \in X^{(I, \alpha)}$ has an associated Schrödinger operator $H_c$, and in \cite{kaminaga} it was proved that if the continued fraction of $\alpha$ has digits with limsup at least $4$, then almost every $c \in X^{(I, \alpha)}$ has so-called $3$-block Gordon structure, which implies that the operator $H_c$ has no eigenvalues. We significantly improve this result by completely characterizing almost-sure $3$-block Gordon structure, proving that in fact it holds for all $(\alpha,I)$ except for a countable set of pairs $(\alpha, |I|)$ where $\alpha$ is Möbius equivalent to the silver mean and $|I| \in \mathbb{Z}\alpha + f(\alpha)$ where $f(\alpha)$ is a specific infinite series taking value either $\frac{1}{2}, \frac{\alpha}{2}$, or $\frac{\alpha+1}{2}$. We also show that for a set of $\alpha$ of full measure, and for every $I$, the set of points whose orbit codings do not have $3$-block Gordon structure has Hausdorff dimension bounded away from $1$.

We construct the absolute algebraic geometry of the compactification $\overline{\text{Spec}\, \mathbf Z}$ by amalgamating the affine absolute curve $(\text{Spec}\, \mathbf Z)_{\mathbf{F}_{1}}$ with an archimedean component defined over the signed extension $\mathbf{F}_{1^2}$ of $\mathbf{F}_1$. By adjoining a formal imaginary unit to the absolute projective line, we obtain an equivariant topos endowed with a canonical geometric inversion symmetry, which induces the twistor real structure on its complex points. This archimedean geometry is incorporated into a global absolute curve defined as an internal object of the odd arithmetic topos, dual to the multiplicative monoid of odd positive integers, and governed by the intrinsic Hopf structure of spherical $\mathbf{F}_{1^2}$-algebras. The restriction of the absolute Frobenius action to odd integers is forced arithmetically by the extension of scalars to $\mathbf{F}_{1^2}$. On complex points, the resulting dynamics simultaneously generates the Adams operations and complex conjugation on real Hodge structures. At the categorical level, the odd arithmetic topos originates in the pericyclic category, whose $\lambda$-operations provide a conceptual interpretation of the local factors of geometric L-functions.

Let $G$ be a finite group, let $N$ be a normal subgroup of $G$, and let $\theta$ be an irreducible $p$-Brauer character of $N$, for a prime $p$. It is well known that the number of irreducible Brauer characters of $G$ lying over $\theta$ equals the number of $p$-regular $\theta$-good conjugacy classes of $G_{\theta}/N$. In this note we express the number of real-valued irreducible Brauer characters of $G$ lying over $\theta$ in terms of the $p$-regular $\theta$-good conjugacy classes of $G_{\theta}/N$. This is a modular analogue of one of the main results in a recent paper (arXiv:2301.01177) by John Murray.

We study the spatial Maki–Thompson rumor model on infinite, connected graphs of bounded degree. Spreaders transmit the rumor to ignorant neighbors but become stiflers upon contacting non-ignorant neighbors. We prove extinction on Cayley graphs of linear growth, for every \(\lambda,\alpha>0\) and every initial configuration with finitely many non-ignorant vertices, and establish an explicit extinction criterion on arbitrary bounded-degree graphs for processes started from finitely many spreaders. On Cayley graphs of superlinear growth, we prove survival from a single spreader whenever the ratio of the stifling to the transmission rate lies below an explicit threshold depending only on the maximum degree. On Cayley graphs of polynomial growth of degree \(D\ge2\), we further show that the range has positive lower density with positive probability. Under a stronger condition, macroscopic annuli contain a surface-order number of simultaneously active spreaders for a total duration bounded uniformly away from zero. When \(\alpha>0\), every finite region eventually contains no spreaders, so global survival forces the rumor to move continually into new regions. Under either subcriticality or a sufficiently small stifling rate, we prove central limit theorems for the final stifler density and the total spreader occupation time, together with a functional central limit theorem for the empirical survival function. These results follow from a central limit theorem for stationary stabilizing functionals of i.i.d.fields on polynomial-growth Cayley graphs; the functionals may depend on the field outside the observation set.

We determine two infinite families of poles on the positive real axis for classical single-variable Witten zeta functions. In type $A_r$, for $r \geq 5$, the point $q_r^A = 2(r-4)/(r^2+r-4)$ is a simple pole except in ranks $12$ and $20$, where it is a double pole. In type $D_r$, for $r \geq 4$, the point $q_r^D = (r-3)/(r(r-1)-1)$ is a simple pole except at $D_8$, where it is double. In root-product normalization, we express the simple residues and the leading coefficients of these three double poles in terms of gamma, trigonometric, and Riemann zeta values. For $r \geq 4$, the functions of types $B_r$ and $C_r$ are holomorphic at $q_r^{BC} = (r-3)/(r^2-1)$ except possibly in ranks $7$ and $11$, where any pole is simple. These pole and holomorphy statements arise from quadratic normal Taylor coefficients. For every fixed higher even normal degree, we also determine exactly when the associated continued $B/C$ chamber sum is nonzero; this auxiliary result does not by itself classify poles of the full Witten function. The proofs combine exhaustive support classifications, explicit integration-by-parts identities, and finite chamber relations. No numerical nonvanishing estimate is used.

Gaussian graphical models (GGMs) describe the dependence structure among jointly Gaussian random variables. However, the most common parameterisation of GGMs, the precision matrix, describes both the dependence and scale of the variables. This has been shown to lead to model selection methods that depend on the scale of the variables, despite graphical models being scale invariant. Even after standardising data to have unit sample variances, entries of the precision matrix can be on different scales leading to poor model selection. This paper decomposes the precision matrix into marginal variances and interpretable scale-invariant parameters - the partial correlations and variance inflation factors. This decomposition gives new insights into the precision matrix entries and the dynamics of model selection methods such as penalised likelihoods. In particular, it explains the observed phenomenon that penalties on the precision matrix perform poorly at selecting hub variables and motivates the necessity of data standardisation. It also shows why methods based on partial correlations have better hub detection properties. The effect of penalisation of different quantities on the estimation of marginal variances is then investigated and an interesting simplification of the log-likelihood is shown when using maximum likelihood estimation of the marginal variances.

The multicrossing homology of a knot can be considered as a generalization of the homology of its fundamental quandle. We show that certain sums of knot secants define invariant classes in the multicrossing homology.

A graph $G$ is $k$-choosable if it has a proper coloring for every $k$-list assignment. While every $C_3$-free planar graph is $4$-choosable, some of them are not $3$-choosable, as constructed by Voigt. Hu and Zhu conjectured that if $G$ is a $C_3$-free planar graph and $X \subseteq V(G)$ induces a bipartite subgraph, then $G$ has a proper $L$-coloring whenever $|L(x)| = 3$ for $x \in X$ and $|L(v)| = 4$ for $v \in V(G) \setminus X$. As evidence, they proved the conjecture when $X$ is an independent set. We provide further evidence by proving the conjecture when the induced subgraph $G[X]$ is an induced sparse matching. This is the first result supporting the conjecture in which the set $X$ receiving smaller lists may induce a subgraph with edges.

Modern data systems are commonly studied through computational and information-theoretic methods, while their algebraic properties remain largely unexplored. This paper introduces a mathematical framework for modelling data systems using algebraic codes over finite groups, organized by an axiom signature: each algebraic axiom of the data domain is shown to be sufficient for a specific coding-theoretic capability, and each implication is given its precise reading (MDS parameters do not require a field; quotients require normality, not commutativity). Three types of results are presented. Proved: a Gilbert-Varshamov existence bound over arbitrary finite groups, commutative or not; explicit direct-sum codes over the elementary divisors of any finite abelian group; that transactions preserving a validity predicate form a submonoid of the full transformation monoid, which is not a group; that schema-preserving transformations form a finite group whose orbits are counted by Burnside's lemma; and that the constraint-preserving bijections of a data system form a finite group whose orbits partition the valid configurations. Demonstrated: an explicit code construction over $\mathbb{F}_5$, with known codes over $\mathbb{Z}_4$ and $(2^U,\triangle)$ cited to situate the framework. Proposed: an encoding efficiency measure Ef and an optimization functional $\Phi$ with tunable weights, both evaluated at declared coding instances. Weight-independent comparison holds under componentwise dominance; in general the weight simplex decomposes into polyhedral chambers on which the induced order is constant.

For complex tori with algebraic normalized period matrix, in particular for Abelian varieties of CM-type, the map from the second cohomology group of the analytic sheaf of the second Milnor K-groups to the rational cohomology classes of Hodge type (2,2) is surjective (up to denominators).

We develop a theory of Thom series for contact function singularities. Quadratic stabilization $\sigma_q$ of a contact singularity $\eta\in J^k(n,1)$ gives contact singularities $\sigma_q^{i}\eta\in J^k(n+i,1)$. We study the stable Thom polynomials (i.e.\ the Thom polynomial in quotient variables) of $\sigma_q^{i}\eta$ as $i$ increases. To this end, we define the Thom series of a contact function singularity $\eta$, which for any given $i$ is a linear combination of Schur polynomials $s_\lambda$. For large enough $i$ the value of the Thom series at $1-(n+i)$ is the stable Thom polynomial of $\sigma_q^i\eta$ and for small $i$ it determines its coefficients outside the kernel of a specialization map. We prove that the Thom series has two main properties: 1) as $i$ increases, the partitions $\lambda$ follow a simple stabilization pattern, and there is a finite set of $\lambda$ which generates the support of the entire Thom series via this stabilization; 2) the coefficients of the $s_\lambda$ are polynomials in $i$ with explicit degree bounds. We describe the precise relationship between unstable and stable Thom polynomials of contact function singularities and Legendre Thom polynomials and we carry out computations of all three, as follows. We compute the complete family of stable Thom polynomials for function singularities with $\gamma\leq 6$. We compute unstable and Legendre Thom polynomials for several families of binary and ternary singularities. We define a class of multi-binary singularities, and compute their Thom polynomials. We discuss second order Thom-Boardman classes, and indicate difficulties that arise beyond function singularities. The results of the paper will be used in a companion paper, where Thom polynomials will be applied to problems in enumerative geometry.

Augmented Lagrangian methods are among the most effective approaches for solving constrained convex optimization problems. However, classical complexity analyses of first-order methods applied within the augmented Lagrangian framework usually rely on the assumption that the objective function has a Lipschitz continuous gradient. This assumption excludes an important class of generalized smooth functions whose gradients may grow unboundedly. In this paper, we study an inexact augmented Lagrangian method for solving linearly constrained convex optimization problems with $(L_0,L_1)$-smooth objective functions. We show that the augmented Lagrangian subproblems preserve the $(L_0,L_1)$-smooth structure, with parameters depending on the penalty coefficient. This property allows us to employ recent accelerated first-order schemes designed for generalized smooth optimization instead of classical smooth optimization methods. In particular, we combine the inexact augmented Lagrangian framework with a two-stage acceleration procedure based on clipped gradient descent and accelerated optimization.

The Kneser graphs $\text{KG}(n,k)$ are a classically studied family of graphs. One known invariant of graphs is gonality (also called divisorial gonality), which is the minimum degree of a rank 1 divisor on the graph. Using known bounds on gonality of simple, connected graphs, one may obtain that the gonality of $\text{KG}(n,k)$ is bounded above by $\binom{n-1}{k}$. In 2014, Harvey and Wood showed that the treewidth (a lower bound on gonality) for $\text{KG}(n,k)$ is $\binom{n-1}{k}-1$ for $n\geq 4k^2-3k+2$. In this paper, using scramble number, another lower bound on gonality, we improve this polynomial bound and show that the gonality of $\text{KG}(n,k)$ is exactly $\binom{n-1}{k}$ for $n\geq \frac{3k^2+k+2}{2}$, and conjecture an even stricter polynomial bound using the uniform edge scramble. We then extend our argument to the family of generalized Kneser Graphs, computing the scramble number and gonality using the same polynomial bound.

The initial-to-final-state inverse problem for the Schrödinger equation consists in determining uniquely the Hamiltonian that generates the evolution, assuming the knowledge of the initial-to-final-state map. This functional maps each initial state $f\in L^2(\mathbb{R}^n)$ to the corresponding final state at a fixed time $T$. The problem was formulated by Caro and Ruiz in the case of Hamiltonians arising from time-dependent bounded electric potentials that exhibit super-exponential decay at infinity. Caro, Parissis and the authors of this article established that uniqueness also holds for time-independent bounded potentials with super-linear decay at infinity. In this paper, we quantify the above uniqueness results establishing that the potentials are stable under small changes of the initial-to-final-state maps. In the case of time-dependent potentials we get a stability of logarithmic type. A notable improvement is achieved when the potentials are time-independent, where under this assumption we prove Hölder stability estimates.

We compute the large-$N$ asymptotics of the determinant of the identity plus or minus a generalized Hilbert matrix. For $N \in \mathbb{N}$ and $\delta \in \mathbb{R} \setminus \{-1, -2, \ldots\}$, let \[ H_N^\delta := \left( \frac{\sin((1+\delta)\pi)}{\pi(j+k+1+\delta)} \right)_{j,k=0}^{N-1} \] be the generalized Hilbert matrix. For $\delta$ not a half-integer, we prove the large-$N$ power-law asymptotics of the determinant \[ \log \text{det}(I_N \pm H_N^\delta) = -\frac{\theta_\delta^2 \pm \theta_\delta}{2} \log N + O(1) \] where $\theta_\delta := \delta$ if $\delta < -1/2$, and $\theta_\delta := \frac 1 \pi \arcsin(\sin(\delta\pi))$ if $\delta \ge -1/2$ and $\arcsin \colon [-1,1] \to [-\frac{\pi}{2}, \frac{\pi}{2}]$ denotes the principal branch of $\arcsin$. For $\delta \geq -\frac 12$ the asymptotics is known. The novelty of the present paper is the regime $\delta < - \frac 12$ and understanding the dichotomy in the decay exponent. This stems from the $\pm 1$ edge eigenvalues of the limiting operator appearing for $\delta < - \frac 12$. Most notably, we obtain for $\delta < -\frac12$ \[ \log \det\bigl(I_N - (H_N^\delta)^2\bigr) = -\delta^2 \log N + O(1). \] Such determinants arise in the study of Anderson's orthogonality catastrophe.

We study left-invariant generalized complex structures on almost abelian Lie groups $G_A$ with Lie algebra $\mathfrak{g}_A=\mathbb{R} e_0\ltimes_A\mathfrak{h}$, where $\mathfrak{h}$ is an abelian ideal of codimension one, and on their compact quotients. First, when $A$ is diagonalizable over $\mathbb{R}$, we characterize all admissible types by pairings of its eigenvalues and characterize the structures admitting a closed invariant pure-spinor generator. For general $A$, we obtain Jordan-theoretic type bounds and a construction using a complex quotient and a symplectic ideal. Finally, in dimension six, we establish nonexistence results and give explicit intermediate-type constructions.