Papers

We study a bi-infinite system of interacting Brownian particles on the real line with singular asymmetric interactions mediated by the collision local times. Particles perform Brownian motions, and when neighboring particles collide, the associated local time is split in proportions $p$ and $q=1-p$. We first develop well-posedness theory for the particle system, proving pathwise uniqueness and strong existence under natural growth assumptions on the initial configuration and local times. We also identify a family of stationary distributions for the infinite-dimensional process of gaps between successive particles: for every $\lambda>0$, the product measure with i.i.d. Exp$(\lambda)$ gaps is invariant. Our main result concerns the equilibrium fluctuations of the associated particle-count (height) function in a weakly asymmetric regime. Taking $p=p_\varepsilon$ so that $p_\varepsilon^{-1}-1=\exp\{\sigma \varepsilon^{1/4}\}$, initializing the interparticle gaps with i.i.d. Exp$(1)$ random variables, and applying a microscopic Hopf-Cole transform to the diffusively rescaled count function, we prove convergence, as $\varepsilon \rightarrow 0$, to the multiplicative stochastic heat equation (SHE) with Brownian exponential initial data. Equivalently, the logarithm of the limit is the Hopf-Cole solution of the KPZ equation with two-sided Brownian initial data. The proof combines localization to finite particle subsystems via chains of collisions, Brownian last-passage percolation estimates, a key local time cancellation property, and a martingale problem for a scale-adapted mollification of the Hopf-Cole field, whose space-time regularity is tuned to match that of the limiting SHE. The resulting fluctuation theorem places these Brownian particle systems with asymmetric singular collision dynamics within the KPZ universality class.

We consider the dynamics of inertial particles in unsteady fluid flows. At low Reynolds numbers, where the drag force is linear in the relative velocity, it is well-known that the dynamics admit an attracting, invariant, slow manifold which emerges as the perturbation of a normally hyperbolic critical manifold. However, at high Reynolds numbers, where the drag force is quadratic in the relative velocity, the critical manifold is no longer normally hyperbolic, and therefore its persistence has remained an open problem. Here, we resolve this issue by a particular application of the blowup method, which transforms the equations of motion to a generalized weighted cylindrical coordinate system, thereby desingularizing the dynamics on the critical manifold. We subsequently prove that the critical manifold persists under sufficiently small perturbations and derive the reduced equations of motion on the perturbed slow manifold to arbitrary accuracy. Our reduced equation differs from its linear-drag counterpart in its asymptotic expansion as well as its convergence rate. Using two examples, we demonstrate the validity of our slow manifold reduction. We also showcase an application of the reduced equations to the problem of source inversion in a turbulent dispersion model.

We prove that if a closed, indecomposable, properly convex real projective $4$-manifold is geometric or admits a geometric decomposition in the sense of Thurston, then every piece is real hyperbolic. This extends a theorem of Benoist to dimension four. Moreover, we build orientable (non-hyperbolic) $4$-manifolds of the above type, with arbitrary positive, even, Euler characteristic. Finally, we characterise the compact locally symmetric spaces that virtually support properly convex real projective structures in terms of their geometry.

The integration kernels for polylogarithm functions on a compact Riemann surface of arbitrary genus $h$ are shown to close as the surface undergoes a non-separating degeneration to one of genus $h{-}1$. Explicit formulas are obtained for the non-separating degeneration of the multivariable Enriquez connection for genus $h$ with an arbitrary number of variables to the Enriquez connection for genus $h{-}1$ with two additional punctures whose Lie algebra generators are related to the original ones by the characteristic Bernoulli generating functions known from the degeneration at $h=1$. Analogous degeneration formulas are obtained for the single-valued DHS kernels at the leading order in the real degeneration parameter that is adapted to relating modular tensors at genus $h$ and $h{-}1$.

Principal Component Analysis or PCA-like properties (orthogonality, variance ranking) are seldom realized in deep autoencoder architectures. In this work, we present ODIN (Orthogonal Dendritic Intrinsic Network), a novel autoencoder architecture that recovers PCA-like latent structure in a fully non-linear regime. By incorporating a set of geometric constraints directly into the training objective, ODIN encourages latent dimensions to be mutually orthogonal and ordered by explained variance, mirroring the interpretable decomposition of PCA while retaining the expressive power of deep networks. We provide theoretical grounding for these constraints and demonstrate their compatibility with standard encoder-decoder frameworks. We also establish empirical results for both synthetic and real world datasets, establishing a principled path toward interpretable, structured feature learning and dimensionality reduction.

Metastability is the tendency of a system to dwell for a very long time near an apparently stable equilibrium before a rare fluctuation drives it, on a comparatively short time scale, towards another. Among the rigorous frameworks developed to capture this phenomenon, the pathwise approach proceeds by identifying the “typical” trajectories of the stochastic dynamics at hand and estimating their probabilities. In this article we review the pathwise approach and its application to the Galves–Löcherbach (GL) class of stochastic models of spiking neural networks. After recalling the conceptual and historical roots of the theory – which goes from chemistry to rigorous probability theory, with fundamental ideas coming mainly from statistical physics – and illustrating them on two classical examples, we give a general definition encompassing the known variants of the GL model and survey the metastability results already established for some of these variants. As far as we can, we do so in a self-contained fashion, and we sketch the proofs when possible, highlighting their common structure. We close with a discussion on open problems and point to possible further directions.

For nonlinear control systems on normed vector spaces, we characterize an incremental input-to-state stability (ISS) type property in which the overshoot constant multiplies both the initial-condition and the input terms. Working through the associated variational system, we show that two properties are equivalent: an ISS-type bound on the variational system, and the incremental ISS-type bound on the original system. We further establish the equivalence between an infinitesimal contraction condition, expressed through a Lyapunov-type function, and an incremental Lyapunov condition. Each of these equivalent conditions yields a necessary condition and a sufficient condition for the ISS-type bounds, differing only in the input Lipschitz constant of the vector field. When the overshoot constant equals one, the infinitesimal contraction condition reduces to the standard norm-based contraction conditions. We establish these implications under mere continuous differentiability of the vector field, and we illustrate the results through sensitivity matrices and Lyapunov characteristic exponents. Moreover, we develop similar implications for discrete-time systems.

The goal of the minimal model program for the Hilbert scheme of points on a surface aims is to describe the (stable) base loci of all divisors, their associated birational models, and the maps between them. We answer all of these questions for the Hilbert scheme of points on the blowup of the affine plane at the origin. The birational models are Brill-Noether loci in a larger Hilbert scheme, and nested variants thereof, and the wall-crossing maps are described as explicit projections. We also establish several new facts about the homogeneous coordinate ring of this Hilbert scheme, including finding the minimal set of line bundles whose sections generate the ring.

We discuss a free boundary model for tumor invasion that describes a cloud of cells that diffuse and, at the same time, are drifted along the vector field of the chemotactic direction. The model captures the evolution of a solid tumor, including the process of angiogenesis, which consists in the formation of new blood vessels that supply the tumor with oxygen and other nutrients, thereby promoting its spread and growth. We prove that, once formed, the tumor survives through time, maintaining strictly positive thickness. An explicit expression in terms of the initial data is derived. Moreover, we distinguish two regimes depending on the ratio $\kappa$ between the spreading of tumor cells and the growth of the tumor mass. If $\kappa$ is sufficiently large, then the tumor grows exponentially in time and invades the entire host tissue. In contrast, if $\kappa$ is small enough, then either the tumor remains bounded in size over time or may experience a fast contraction.

We construct flat, projective morphisms that are $\mathbb Z$-homology fiber bundles, have a smooth total space, but are not smooth. This disproves one of the conjectures of the second author and Fernández de Bobadilla.

We study transitive dynamical systems on the Gehman dendrite $\mathcal{G}$ for which the endpoint set $\mathrm{End}(\mathcal{G})$ is invariant. Our goal is to approximate such systems by maps whose measure-theoretic behaviour at maximal entropy is governed by an explicit countable Markov structure. We introduce a class of Markovian maps, encode their dynamics by countable Markov graphs, and use the criteria of Vere-Jones, Gurevich, Salama and Ruette to control the existence of measures of maximal entropy. The main theorem gives two arbitrarily close mixing Markovian perturbations of any given system in the considered class: one has a unique measure of maximal entropy, while the other has none.

We investigate local invariants and geometric phenomena for polynomial spaces of low degree on the $q$-ary Hamming scheme $C_q^N$, where $C_q$ denotes the cyclic group of order $q$. Our main analytic tool is a support-sensitive Bohnenblust–Hille inequality for spherical polynomial spaces, showing that the relevant complexity parameter is the support size of the monomials rather than their total degree. Equivalently, in the corresponding toroidal formulation, this leads to estimates for polynomials whose coordinate degrees are bounded by $q-1$, while the growth of the constants is governed by the interaction order of the variables. These inequalities yield applications to the learning theory of spherical low-level functions and also provide the basis for dimension-free comparisons between several classical local invariants, including Sidon constants, unconditional basis constants, and Gordon–Lewis constants. As a consequence, we obtain sharp asymptotic estimates for these invariants in the spherical setting, with analogous comparison and asymptotic results for homogeneous and tetrahedral polynomial spaces. We also study projection constants and the associated reproducing kernels. In the spherical case, suitably normalized Krawtchouk polynomials converge to Hermite polynomials under central-limit scaling, leading to explicit Gaussian limits and sharp asymptotic formulas. By contrast, in the homogeneous and tetrahedral settings a dichotomy appears between the Boolean case and the regime $q\ge3$, where the limiting behaviour is governed by moments of a circular complex Gaussian.

We construct an acyclic augmented chain complex of abelian groups whose entry in degree $n > 0$ is free on the set of noncrossing partitions of degree $n-1$ equipped with a $\{0, 1\}$-labelling of their gaps. The definition of the differential in this complex is related, via a restricted Leibniz rule, to the gap-insertion operad of Ebrahimi-Fard, Foissy, Kock, and Patras. We conjecture that this augmented chain complex is the linearisation of a polygraph presenting a self-dual model of the coherent walking $\omega$-equivalence constructed by the author, Loubaton, Ozornova, and Rovelli, and provide evidence for this conjecture.

Let $A$ be an arbitrary $2\times 2$ matrix. In \cite{Cisneros:PhD,Cisneros:I2x2M} I gave a formula for the trace of the $k$-th symmetric power of $A$ in terms of the anti-diagonal entries of $A^{k+1}$ and $A$. This was based on formulae that I found for the entries of the $k$-th power $A^k$ of the matrix $A$ in terms of its entries but I only sketched the idea of how I obtained such formulae. In this article I give the full proof of those formulae by counting some walks of length $k$ over the complete digraph of order $2$. I compare them with formulae for $A^k$ given by Mc Laughlin in \cite{McLaughlin:CIDnP2x2M} and by Williams in \cite{Williams:nthP2x2M}. This leads to combinatorial identities, in particular expressions for Fibonacci and Lucas numbers.

We develop the representation theory of vertex algebras over arbitrary commutative rings as part of a relative theory, suitable for various applications such as the study of modular forms, modular tensor categories, and sheaves of coinvariants and conformal blocks. Towards this end, we construct Zhu algebras (and higher analogues) and mode transition algebras over arbitrary rings via the universal enveloping algebra. We prove that these constructions are compatible with base change and give rationality criteria for Möbius vertex algebras over arbitrary fields satisfying mild assumptions. We apply this framework to study Virasoro vertex operator algebras over arbitrary fields. Using integral forms and base change, we extend the rationality of the discrete series Virasoro VOAs from $\mathbb{C}$ to arbitrary fields of characteristic zero. In positive characteristic, the Virasoro theory exhibits surprising new phenomena: the so-called restricted Virasoro VOA is $C_2$-cofinite and has a semisimple Zhu algebra, but it is not rational. We also show that this VOA is not simple because it admits nontrivial singular vectors in positive characteristic for every central charge. We calculate the singular vector of degree $2p$ in characteristics $p=5,7$ using Mathematica.

Recent work of Lam, Bychkov-Gorbounov-Kazakov-Talalaev, and Chepuri-George-Speyer gave a stratification of the totally nonnegative Lagrangian Grassmannian into electroid cells parameterized by cactus networks, paralleling Postnikov's stratification of the totally nonnegative Grassmannian by positroid cells. Electroid varieties arise as an algebro-geometric extension of electroid cells. The combinatorics of these varieties was studied by Lam in 2018. We build on this work and study the geometric properties of electroid varieties. In analogy to results of Knutson, Lam, and Speyer on positroid varieties, we show that electroid varieties are reduced, irreducible, regular in codimension one, compatibly Frobenius split, and form a stratification. We also show a decomposition of certain electroid varieties as a product of two electroid varieties. As a consequence, the grove measurement map that embeds electroid cells can be extended algebraically to embed an algebraic torus.

We express integrals over virtual cycles of GIT quotients $X/\!\!/G$ in terms of integrals over virtual cycles of fixed loci $X^T$. The results hold for both perfect obstruction theories and $(-2)$-shifted symplectic structures, in cohomology and in $K$-theory, and for noncompact Deligne-Mumford stacks acted on by a reductive group with compact quotient.

In survey sampling, the goal is to estimate finite population parameters such as totals, means, and proportions. At the estimation stage, it is common to have access to auxiliary information in the form of covariates known either in aggregate form or for each population unit. These covariates are often used, through models relating them to the variable of interest, to improve efficiency; this approach is known as model-assisted estimation. Modern applications increasingly involve settings where a large number of covariates are observed, sometimes of the same order as the sample size. While this setting offers greater modeling flexibility, it also creates important challenges for inference. In this article, we study variance estimation for the generalized regression (GREG) estimator in high-dimensional regimes. We derive new theoretical results that characterize the high-dimensional asymptotic bias of commonly used variance estimators, including those based on Taylor linearization. Furthermore, under suitable distributional assumptions on the covariates, we show that a cross-validated variance estimator is naturally asymptotically unbiased.

The flip graph of an origami crease pattern has the locally flat-foldable mountain-valley assignments as vertices, and an edge joins two of them that differ by a single face flip. A basic invariant of this graph is the degree sequence, which counts the vertices of each degree. On the $m\times n$ Miura-ori, this sequence is known to be a bivariate polynomial only for small degrees, each count obtained by a separate argument. This paper gives one uniform construction that expresses, for every degree $d$, the number of degree-$d$ vertices as a single symmetric polynomial in $(m,n)$ for all sufficiently large $m,n$. Its degree in each variable is $d-2$ unconditionally. Subject to a single degree bound, its total degree is $d-2$ as well, with top-degree part an explicit multiple of $m^{d-2}+n^{d-2}$ for $d\ge5$. The bound is proved here when the count splits into independent row and column factors, and open otherwise. The region is $m,n\ge\max(d-1,2)$. The polynomials are given in closed form through $d=10$, unconditional through $d=8$, where the degree bound holds in every case, and conditional on it beyond. Below this region the count departs from the polynomial. One step below, this departure has leading coefficient $-4$ times a Baxter number through $d=11$. Each such polynomial thus counts the Miura-ori's locally flat-foldable assignments admitting exactly $d$ single face flips.

We study the Cauchy problem in $\mathbb{R}^{3}$ for the repulsive compressible Navier-Stokes-Riesz system with Riesz exponent $0<s<1$ and viscosity $0<\varepsilon\leq1$, where the Riesz interaction $\nabla(-\Delta)^{-s}(\rho-\bar{\rho})$ is a generalization of the Coulomb interaction for electrons. For small perturbations of a constant equilibrium, with the solenoidal component of the initial velocity of order $\mathcal{O}(\varepsilon)$, we prove the global existence and uniqueness of smooth solutions. We derive time-decay estimates in $L^{2}$ norms and $L^{\infty}$ norms that capture both uniform-in-$\varepsilon$ dispersive behavior and viscosity-dependent dissipation. We further establish a global-in-time inviscid limit to the irrotational global solution of the compressible Euler-Riesz system whose initial data consist of the same density and the curl-free component of the velocity, with an explicit convergence rate in $W^{k,p}$ norms. The proof combines viscosity-adapted dispersive estimates, normal-form analysis and nonlinear energy estimates with control of both negative and positive Sobolev norms.

This paper introduces the topology-independent distributed multichannel Wiener filter (TI-dMWF), a novel algorithm for distributed node-specific signal estimation in wireless acoustic sensor networks (WASNs) with unconstrained topologies. The TI-dMWF enables each node in the network to compute its centralized multichannel Wiener filter solution by exchanging only low-dimensional fused signals, without requiring iterative estimation, unlike state-of-the-art approaches such as the topology-independent distributed adaptive node-specific signal estimation (TI-DANSE) algorithm. The TI-dMWF is proven optimal when each source is observed by either all nodes or only one node. Theoretical analysis and numerical simulations confirm that it achieves centralized estimation performance in a single run. Its latency as a function of the pruned-tree depth and its computational complexity are also analyzed. Its robustness is assessed in reverberant-room simulations under estimated second-order statistics, various network topologies, and deviations from the assumed observability model.

Let $\Lambda$ denote the von Mangoldt function; we prove that for each $r > 2$, there exist constants \[ r' < \mathbf{c}(r) < 2 < \mathbf{C}(r), \qquad \lim_{r \to \infty} \mathbf{c}(r) = 1, \ \lim_{r \to \infty} \mathbf{C}(r) = \infty \] so that the discrete variational Carleson operator along the primes \begin{align} \mathcal{V}^r \Big( \sum_{n \neq 0} f(x-n) \Lambda(|n|) \frac{e^{2\pi i \lambda n}}{n} : \lambda \in \mathbb{T} \Big) \end{align} is bounded on $\ell^p$ for all $\mathbf{c}(r) < p < \mathbf{C}(r)$, while the variation is unbounded when $p \leq r'$. At the non-variational endpoint, the same argument gives the sharp maximal result: the prime Carleson operator \[ \sup_{\lambda\in\mathbb T} \Big|\sum_{n\neq0} f(x-n)\Lambda(|n|)\frac{e^{2\pi i\lambda n}}{n}\Big| \] is bounded on \(\ell^p(\mathbb Z)\) for the full expected range \(1<p<\infty\). The proof gives a new mechanism for treating modulation-invariant singular integrals after arithmetic sparsification. It combines higher-order Fourier uniformity, a variable-coefficient multi-frequency principle in the spirit of Bourgain, and an additive-combinatorial inverse argument. A key step is a reduction to finite periodic models, where the Ramanujan structure of the major arcs is converted into a sharp estimate for structured atoms by elementary number theory.

We study the interplay between graded families of ideals in $\mathbb{K}$-domains and their associated convex regions. These regions, called Newton-Okounkov regions, arise naturally from graded families of ideals associated to a valuation with one-dimensional leaves. Our main focus is to compute asymptotic resurgence number of a pair of graded families of ideals. By combining techniques from Attouch–Wets topology and convex-geometric properties of Newton-Okounkov regions, we characterize the asymptotic resurgence number through containment relations between the pair of corresponding Newton-Okounkov regions.

It is shown that harmonic signals incorporating a type of weak non-standard frequency modulation (wNSFM) have interesting spectral properties, namely, time-dependent bandwidths that become increasingly broader with increasing time. As such, they represent a class of signals with frequency-time coupling in their spectra. Specifically, the weakly damped oscillator exhibits always two transient resonance captures involving two distinct harmonics possessing relatively high amplitudes over finite time intervals, while the overall response decays as $~t^{-1/2}$ as $t\rightarrow\infty$. Considering the undamped oscillator, it possesses two types of resonances, referred to as simple and non-simple resonances. Simple resonances correspond to finite-amplitude steady-state responses caused by two sustained resonance captures, in the form of two distinct modulated quasi-periodic responses, which, however are "activated" at different time instances. The necessary and sufficient conditions for non-simple resonances are given in the form of a theorem which predicts the existence of resonant harmonics and specifies the special phase conditions that the resonant harmonics must satisfy for constructive interference; the resulting undamped non-simple resonance grows unboundedly as $~t^{-1/2}$ as $t\rightarrow\infty$, in contrast to the classical resonance growth of the linear resonator with unmodulated harmonic excitation whose response grows as $~t$ as $t\rightarrow\infty$. These resonant responses are persistent to changes in the parameters of the wNSFM. Our results reveal interesting infinite sets of resonances in linear SDOF resonators under frequency-modulated excitations.

We develop a unified function space theory of deep fully connected neural networks. Functions in our spaces are defined recursively as $\ell^1$-bounded linear combinations of activated functions from preceding layers, with a dictionary of affine functions at the first layer. Unlike existing theories that are largely specialized to homogeneous activations such as the ReLU, our framework provides a meaningful notion of functional complexity for deep networks with a broad range of homogeneous and non-homogeneous activation functions commonly used in practice. This simple construction unites several seemingly disparate ideas from the literature, including norm-based complexity bounds and variational characterizations of depth, and facilitates novel analyses of what kinds of functions deep norm-constrained networks can represent. To this end, we prove a novel representer theorem for our spaces and establish novel function-space complexity bounds showing that the associated function classes remain qualitatively small at arbitrary depth. In the univariate ReLU case, we prove a "depth saturation" result: depth in this setting yields only a small constant rescaling of the function class, with no added functional diversity. As a consequence, we show that deep norm-controlled ReLU functions in any dimension cannot exhibit high frequencies along any direction. This finding reveals that some commonly cited expressivity benefits of depth disappear once network complexity is controlled by an appropriate function space norm, rather than parameter count or other representational costs that permit compounded rescaling across layers. Overall, our results illustrate how a function space perspective yields new structural insights into the relationship between depth and complexity.

This paper is a prequel to our recent work, "Equivariant Borel liftings in complex analysis and PDE" (arXiv:2507.12058). While the results presented here were established in that work in a more general and abstract setting, the purpose of this paper is to provide a direct proof of the equivariant Weierstrass theorem. It states that there exists a Borel map assigning to each non-periodic positive divisor $\Lambda$ an entire function $F_\Lambda$ such that the divisor of zeroes of $F_\Lambda$ is $\Lambda$ and such that $F_{\Lambda-w}(z) = F_\Lambda (z+w)$, $w\in\mathbb{C}$. In general, non-periodicity cannot be omitted, and Borel measurability cannot be strengthened to continuity. The two key ingredients are the Runge approximation theorem and the existence of "Borel toasts", which are Borel counterparts of Rokhlin towers from ergodic theory. We do not assume prior knowledge of descriptive set theory and have aimed to make the exposition self-contained, aside from several results taken from graduate textbooks.

Bregman Proximal Gradient methods (BPGM) exploit the underlying geometry of the objective function through a carefully chosen mirror map. In this work, we introduce a novel notion of strong convexity, termed Restricted Relative Strong Convexity, and establish linear convergence rates for BPGM under this condition. We then exploit the proposed theoretical framework to provide an in-depth analysis of the convergence of BPGM for (regularized) Kullback–Leibler regression problems, covering scenarios with both unique and non-unique minimizers, as well as regularized and unregularized formulations. Specifically, we demonstrate that using the popular Burg's entropy as a distance-generating function may only yield linear convergence for certain KL regression problems. In contrast, we show that employing a smoothed version of the Burg's entropy induces the suitable geometry required to guarantee linear convergence. We conclude with numerical experiments that nicely align with our theoretical findings.

Several open inequalities for ratios and logarithmic derivatives of the modified Bessel functions $I_\nu$ of the first kind and $K_\nu$ of the second kind reduce to sign questions for quadratic Riccati expressions. We isolate this reduction and use it in two directions. First, for the quotient $W_\nu(z)=zI_\nu(z)/I_{\nu+1}(z)$, the canonical product for $I_{\nu+1}$ yields the partial fraction $W_\nu(\sqrt{s})=2(\nu+1)+2\sum_{n\ge1}s/(s+j_{\nu+1,n}^2)$, where $j_{\nu+1,n}$ is the $n$-th positive zero of $J_{\nu+1}$. Consequently $x\mapsto W_\nu(x^\tau)$ is a Bernstein function for $\nu>-1$ and $0<\tau\le1/2$, and this positive exponent range is sharp. Second, an exact rational certificate at $(\nu,u)=(0,10)$ places $I_1(10)/I_0(10)$ below 0.949. This refutes the log-concavity question of Baricz, Ponnusamy, and Vuorinen for $u\mapsto \sqrt{u} I_\nu(u)$ and its displayed Riccati reformulations. The same framework completes the monotonicity classification of $K_\nu'/K_\nu^2$, refutes Baricz–Ponnusamy–Vuorinen Question 7 at $\nu=1/2$, and gives an entire counterexample to Baricz's coefficient-ratio complete-monotonicity transfer problem.

We provide details for a Theorem which is only available on a unfinished preliminary draft of P. Nyikos: the existence under $\clubsuit_C$ of a hereditarily collectionwise normal countably compact non-compact manifold which does not contain a copy of $\omega_1$. This shows in particular that MA + $\neg$CH does not imply the existence of a copy of $\omega_1$ in a countably compact non-compact manifold (it is known that PFA does imply it). The said manifold is obtained from a principal $\mathbb{S}^1$-bundle over the long ray, and some structural theorems for these spaces (due to Nyikos as well) are also proved. We show that the same type of theorems hold for $n$-to-$1$ closed preimages of $\omega_1$ and $\mathbb{Z}_n$-“bundles” over $\omega_1$, where $\mathbb{Z}_n$ is the additive group of integers modulo $n$. A small generalization of $\clubsuit_C$ is also quickly investigated.

Our aim is to consider what the exact sequence for rt-categories is. For this, we introduce the notion of exact sequence of rt-categories, modeled on exact sequences of finite tensor categories. Our central result explores the relationship of exactness at different levels. Specifically, let $H_1\xrightarrow{f}H_2\xrightarrow{g}H_3$ be a sequence of finite-dimensional Hopf algebras. We prove that $H_1\xrightarrow{f}H_2\xrightarrow{g}H_3$ is strictly exact if and only if $H_1\text{-}\mathrm{comod}\xrightarrow{f_*}H_2\text{-}\mathrm{comod} \xrightarrow{g_*}H_3\text{-}\mathrm{comod}$ is an exact sequence of finite tensor categories and $g_*$ admits an exact left adjoint, if and only if $D^b_{H_1\text{-}\mathrm{comod}}(H_2\text{-}\mathrm{comod}) \to D^b(H_2\text{-}\mathrm{comod}) \to D^b(H_3\text{-}\mathrm{comod})$ is an exact sequence of rt-categories and $f_*$ is fully faithful.