Papers

We develop a theory of affine algebraic monoids over connected base schemes whose unit groups are split reductive groups. Our main result is a classification theorem for such objects, generalizing the work of Vinberg and Rittatore over a field. As applications, we obtain combinatorial descriptions and normality properties of orbit closures, prove a Steinberg-type theorem on adjoint quotients of split reductive monoids, and construct finite type integral models of the Vinberg monoids.

We establish approximation and learning guarantees for Fourier neural operators (FNOs) applied to time-$T$ solution operators of dissipative evolution equations. The analysis builds on the premise that FNOs can efficiently approximate and learn solution operators whenever these operators admit stable and accurate spectral discretizations. To formalize this idea, we introduce classes of evolution operators defined through spectral methods and derive FNO approximation bounds and polynomial sample complexity guarantees for these classes. For equations with polynomial nonlinearities, the learning rates depend primarily on the smoothness of the input space and the dimension of the physical domain. Our results hold uniformly over broad families of dissipative equations, rather than for a single fixed PDE, and apply in particular to the Navier–Stokes, Allen–Cahn, and Cahn–Hilliard equations. For equations with non-polynomial smooth nonlinearities, we prove that polynomial sample complexity still holds with rates that now additionally depend on the smoothness of the nonlinear terms and the dissipation strength. Overall, we connect classical spectral approximation theory with modern operator learning and explain when FNOs can learn nonlinear evolution operators efficiently.

A family of permutations is called $t$-intersecting if any two permutations in the family agree on at least $t$ elements. We prove that there exists $n_0 \in \mathbb{N}$ such that for any $n>n_0$ and any $1 \leq t \leq n$, the maximum size of a $t$-intersecting family in $S_n$ is obtained by one of the families $\mathcal{F}_{n,t,r}=\{\sigma \in S_n: |\mathrm{Fixed}(\sigma) \cap \{1,2,\ldots,t+2r\}|\geq t+r\}$, where $\mathrm{Fixed}(\sigma)$ is the set of fixed points of $\sigma$. This proves an analogue of the classical Complete Intersection Theorem for large permutation groups, thus providing an essentially complete solution of the Deza-Frankl intersection problem for permutations (1977).

We study Gibbs measures on high-dimensional Haar-random unimodular lattices, where the energy of a lattice vector is its squared Euclidean norm. The random lattice is viewed as quenched geometric disorder, and $c>0$ denotes the scaled inverse temperature. We first analyze the edge window of vectors whose length is within the factor $e^{a/n}$ of the shortest length, with fixed $a$ as $n\to\infty$. For the full sign-class Gibbs ensemble, we prove a Poisson point process limit theorem for the Gibbs mass of this window. The mass vanishes in probability for $0<c\le1$, while for $c>1$ it has a nontrivial Poisson limit, and the ranked Gibbs weights converge to the Poisson-Dirichlet distribution with parameter $1/c$. We then pass to a primitive-direction Gibbs ensemble and consider a fixed approximation factor $\gamma>1$. For this modified ensemble, we prove a weighted moment formula and a quenched thermal concentration result in the high-temperature range $0<c<1$. This yields the primitive fixed-factor visibility curve $c=\gamma^{-2}$ for approximate shortest directions. More precisely, the primitive Gibbs mass of the fixed-factor window tends to zero for $c<\gamma^{-2}$, to one for $\gamma^{-2}<c<1$, and to $1/2$ at the critical boundary $c=\gamma^{-2}$. Thus the fixed-factor theorem is a visibility statement for an idealized primitive target measure, not for the original full lattice Gibbs measure. The results provide a random-lattice thermodynamic reference model for Gibbs targets related to approximate shortest vectors, without implying an efficient algorithm for the shortest vector problem.

We develop a unified treatment of scattering and discrete spectra for the one-dimensional Dirac equation with scalar and vector interactions. Under the spin-symmetry condition, the coupled first-order Dirac system maps exactly onto an effective Sturm–Liouville (Schrö\-din\-ger-like) problem for a single spinor component. This mapping provides a convenient framework for analyzing transmission, reflection, and analytic continuation. As an explicit application, we consider effective interactions of hyperbolic Pöschl–Teller type and exploit supersymmetric quantum mechanics and shape invariance to obtain a closed-form expression for the transmission probability. The bound-state spectrum is then recovered from the poles of the analytically continued transmission amplitude, reproducing known results and offering a unified description of scattering and bound states. For the barrier configuration, we briefly comment on the resulting pole pattern in the complex momentum plane and its connection with resonance and quasi-normal-mode behavior. Moreover, we use the chiral transformation to relate the spin- and pseudospin-symmetry sectors and translate results between them without repeating the full derivation.

We complete the investigation begun in a previous paper to find unitary representations of the non-trivial real $K$-theory elements for the sphere $S^d$ with an involution. Here we consider all involutions except the antipodal involutions. We write down explicit unitaries representing the generators in all cases for $d \leq 3$, and for $d > 0$ we describe a recipe for generating such unitaries.

Let $\Pi_{0}$ be a cuspidal automorphic representation of $\mathrm{PGL}_{3}(\mathbb{A}_{\mathbb{Q}})$. In this paper, we use Levinson's method to prove that, as $Q\to \infty$, at least $1/9$ of the zeros of the $L$-functions $L(s, \Pi_{0}\,\times\, \chi)$ lie on the critical line, where $\chi$ ranges over the family of primitive Dirichlet characters of conductor up to $Q$. This result is unconditional when $\Pi_{0}$ is self-dual, and otherwise holds under a mild condition. The key technical input is a new asymptotic formula with a power-saving error term for the mean square of the product of $L(s, \Pi_{0}\times \chi)$ and a Dirichlet polynomial with arbitrary coefficients in both the $T$- and $Q$-aspects for the range $Q^{\epsilon}\le T \le Q^{1/3-\epsilon}$. When $T=Q^{\epsilon}$, our asymptotic formula allows Dirichlet polynomials of length $\theta <1/2-\epsilon$; when $\theta=0$, it gives a strong error term of size $O_{\epsilon}(Q^{7/4+\epsilon})$. Furthermore, our result provides evidence for the CFKRS conjectures for large twists and large vertical shifts. We also obtain corresponding results for $\mathrm{PGL}_{2}(\mathbb{A}_{\mathbb{Q}})$, which are fully unconditional, quantitatively stronger, and also appear to be new. This work develops a refined, flexible, and uniform version of the Asymptotic Large Sieve for $L$-functions that does not require any unproven progress toward the Generalized Ramanujan Conjecture. The arithmetic of $\Pi_{0}$ plays a crucial and delicate role in our argument. This work also makes extensive use of Mathematica to handle various elaborate Hecke algebra computations. Our mean value theorem is readily applicable to many other problems in analytic number theory.

Simulating incompressible Stokes flow is essential for studies in microfluidics and low-Reynolds number hydrodynamics. However, the computational cost of resolving the associated saddle-point problem grows prohibitively with the dimensionality of the problem. In this work, we present a quantum algorithm based on the Schrödingerisation technique for the Stokes equations, incorporating an artificial compressibility regularization. The core of our approach is the design of an explicit quantum circuit that encodes the resulting regularized system. The artificial compressibility formulation provides a unified framework for the system, which is then efficiently mapped to a quantum circuit via the Schrödingerisation procedure. A rigorous complexity analysis demonstrates the quantum computational advantage of our algorithms in high-dimensional settings, notably an exponential speedup in problem dimensionality. The validity and scalability of the proposed method are corroborated by numerical simulations performed on Qiskit.

This paper aims both to provide a unified introduction to d-convexity and to contribute new structural results on metric segments and convexity in metric spaces. After developing the basic theory of metric segments, we study their geometric and topological properties, establishing several structural results, and show that metric segments realize arbitrary closed subsets after the metric is replaced by a topologically equivalent and bounded one. We then examine metrically, Menger, and strictly convex spaces, obtaining new characterizations of strict convexity in terms of an order structure on metric segments and their isometric embeddability into R. Additionally, we offer an exposition of the relationship between d-convexity and other notions of convexity in metric spaces from the standpoint of axiomatic convexity. Finally, we characterize metric spaces in which metric segments are trivial via a "local snowflaking" condition.

Bayesian Optimization is widely used for expensive black-box optimization, yet its success often depends on choosing a kernel that matches the objective's unknown structure. In this work, we propose ALAS, a flexible Gaussian Process kernel family built from symmetric $\alpha$-stable spectral components. By learning the stability parameter $\alpha$, ALAS adapts its effective smoothness from data, capturing both smooth trends and sharp irregularities. We present two parameterizations: ALAS, a single stationary component with joint spectral modulation, and ALAS-Sep, a separable variant that learns dimension-wise tail behavior to improve robustness on approximately decomposable objectives. Experiments on standard benchmarks and real-world surrogates demonstrate strong and robust performance across diverse settings.

We review the classical theory of principal bundles, with particular emphasis on frame bundles and $G$-structures. We then develop the noncommutative framework by introducing the necessary notions of differential calculi, Hopf algebras, quantum principal bundles, and associated quantum vector bundles. Within this setting, we review Majid's notion of a quantum frame resolution. Building on the theory of reductions of quantum principal bundles, we introduce a notion of quantum $G$-structure as a reduction of a quantum frame resolution and prove that every such reduction naturally inherits the structure of a quantum frame resolution.

Mean-field approximations for dynamical processes on networks are widely used, but existing derivations often rely either on moment closures or on idealised assumptions about network structure, leaving the nature of the underlying averaging unclear. Here we present a mathematically principled framework for deriving edge-based mean-field approximations for a broad class of Markov processes on networks using approximate lumping. We consider models in which each vertex is in one of a finite number of vertex states and transitions depend on the number of neighbours in each state. Our approach partitions the full Markov chain state space according to the number of vertices and edges in each possible state, and averages transition rates between partitions. This yields density-dependent population processes that, in the limit of large system size, reduce to a low-dimensional system of ordinary differential equations. We demonstrate the method on single graphs and graph ensembles, such as Erdős-Rényi random networks, and show that well-known edge-based mean-field approximations arise as special cases of our approach. Our approximate lumping framework clarifies the nature of the averaging underlying mean-field approximations, providing a basis for future work on assessing their accuracy.

Let $p$ be a binary word of length $\ell$ with $r\geq2$ runs. Previously known only for $k\leq4$, we show for $n$ sufficiently large that the number of binary words of length $n$ with exactly $k$ subsequences equal to $p$ is polynomial in $n$ of degree at most $\ell-r+1$ for any positive integer $k$. We also prove a sharp upper bound on the number of subsequences equal to $p$ of a binary word $w$ in terms of the runs of $p$ and $w$.

Unit-interval parking functions are counted by the Fubini numbers and are in explicit bijection with ordered set partitions. We transport the deranged ordered set partitions of Belbachir, Djemmada, and Németh through this bijection and obtain the deranged unit-interval parking functions $\mathrm{DUPF}_n$. The equality $|\mathrm{DUPF}_n|=\widetilde F_n$, the Stirling-transform formula, the exponential generating function $e^{1-e^x}/(2-e^x)$, and the dominant asymptotics are therefore not presented as new enumerative discoveries; they are consequences of the known deranged Bell-number theory. The new material of this note is the parking-side structure: leader and lucky-car characterizations, a fixed-block stratification of all unit-interval parking functions, rencontres-type generating functions and a Poisson limit law for fixed blocks, a bijective fixed-block decomposition of the Fubini numbers, a multivariate block-size refinement, a fully deranged $r$-start extension, and a Cayley-permutation model based on first appearances.

We study structural completeness in the infinitary sense (strong structural completeness) in an algebraic setting. A variety is structurally complete (SCpl) if it is generated, as a quasivariety, by its free algebras, and it is strongly structurally complete (SSCpl) if it is generated, as a prevariety, by its free algebras. A quasivariety is SSCpl if it is generated, as a prevariety, by its free algebras. We prove that every quasivariety of finite type with the CEP that is generated by finite algebras and contains an infinite irreducible algebra is not SSCpl. Moreover, every congruence meet-semidistributive variety of finite type generated by finite algebras is SSCpl if and only if it is tabular. Thus, Dummett's and Medvedev's logics are SCpl but not SSCpl. A variety is primitive if it is SCpl and all its subvarieties are SCpl; it is strongly primitive if it is SSCpl and all its subvarieties are SSCpl. We prove that in primitive congruence-distributive varieties of finite type, the tabular subvarieties, and only those, are strongly primitive. This observation also yields a criterion for strong primitivity.

In the Josephus problem with stepsize four, the participants in a circle are eliminated one by one, every fourth person leaving, until a single survivor remains. A fixed point occurs when the survivor turns out to be the person who began in the last seat. The circle sizes with this property form the sequence 1; 21; 38; 51; 122; 163; 689; 919; 2,906; and so on, whose gaps fluctuate erratically. This paper explains the fluctuation and turns it into a recurrence. Between consecutive fixed points, the circle sizes at which the survivor falls exactly one or two seats short of the last one, the near-misses, group into alternating blocks of the two kinds, and the length of every block is the number of times three divides a simple quantity built from the circle size that precedes the block. Iterating these divisibility counts carries each fixed point to the next. Stepsize four is the first case in which two kinds of near-miss coexist, and the alternation they force is what separates it from the solved cases of stepsizes two and three. As a byproduct, the survivor's position for an arbitrary circle size can be computed by walking the near-misses of a single interval, in a number of steps proportional to their count, rather than stepping through every smaller circle as the defining recursion does.

The sequence of Mersenne numbers $\{M_n\}_{n\geq 0}$ is defined as $M_n = 2^n-1.$ In this study we introduce the Mersenne-Bernoulli and Mersenne-Euler polynomials. Using the generating functions and $M$-calculus we find some identities associated with them. Moreover, we define the corresponding matrices with these polynomials, factorise them and find their inverses.

In this work, we study a normalized remainder $T_{n,\lambda}[\e_\lambda]$ for the degenerate exponential $\e_\lambda(u)=(1+\lambda u)^{1/\lambda}$ ($\lambda>0$). We establish an integral representation, an exact monotonicity threshold at $\lambda=1/(n+1)$, and rigorous conditions for the local failure of logarithmic convexity at the origin. We then prove a sharp asymptotic result: for every $\lambda$ in the increasing regime $(0,1/(n+1))$, the second logarithmic derivative satisfies $u^2L(u)\to -\alpha<0$ as $u\to\infty$, showing that global logarithmic convexity on $(0,\infty)$ fails throughout this regime. We further give a necessary and sufficient condition for absolute monotonicity, showing it holds only on a countable, measure-zero set of parameters, and we derive explicit two-sided truncation-error bounds that are pointwise sharp at the origin.

We reformulate the bounded-length-distortion condition for maps between metric spaces in a certain relaxed form that requires the presence of a reference measure on the source space, which makes the new approach more natural from the perspective of maps from metric measure spaces to metric spaces. Within this framework, we establish some mapping results in an entirely singular setting of the following general structure: a metric measure space of finite Hausdorff dimension admits a map with the relaxed bounded-length-distortion condition into a finite-dimensional normed space.

The Chromatic Sum problem asks, given a graph $G$ and an integer $k$, whether $G$ admits a colouring $c$ with sum $\sum_{v\in V}c(v) \leq k$. We study the complexity of Chromatic Sum on graph classes defined by some set of forbidden graphs. First, we show that three known frameworks fully classify the complexity of Chromatic Sum on $HH$-minor-free graphs and $HH$-topological-minor-free graphs for any set of graphs $HH$, and on $HH$-subgraph-free graphs for any finite set of graphs $HH$. To show this, we prove a new NP-completeness result for Chromatic Sum on certain subdivisions of planar subcubic graphs. Next, we consider other containment relations. We formalise a novel framework of problems that are NP-complete for planar graphs as well as for graphs of bounded independence number. For every problem in this framework, we obtain an almost complete complexity classification on $H$-induced-minor-free graphs, $H$-induced-topological-minor-free graphs, and $H$-free graphs for every graph $H$. We show that Chromatic Sum belongs to this framework, as do several other problems. We also define a more fine-grained framework for the induced subgraph relation. We apply this to obtain a complete complexity classification for Chromatic Sum on $H$-free graphs, as well as for several other problems. We justify the choice of this framework by proving that Chromatic Sum is NP-complete for graphs of clique-width at most $3$. This result complements a known polynomial-time result for graphs of clique-width at most $2$.

Multi-level selection and senescence do not at first sight have much in common. Here, we demonstrate that the emergent mortality patterns generated by demographic senescence can be understood as the product of multi-level selection. We formulate a two-level Moran type process and use its scaling limits to illustrate that a simple mathematical framework that models multi-level selection in group-structured populations also models damage accumulation patterns and resultant mortality curves in ageing organisms. To verbally make the connection, observe that defectors spread within a group consisting of cooperators and defectors; when groups compete against each other, defector-rich groups suffer, and between-group selection causes such groups to be systematically under-represented. Exactly analogously, senescing individuals accumulate damage to physiological sub-systems, and `damage begets damage'; individuals who are more damaged are more likely to die, hence damage-rich individuals are systematically under-represented in later age classes. Thus, emergent senescence patterns in complex, integrated organisms are formally equivalent to the patterns generated by a within-generation multi-level selection process in which intra-organismal sub-systems play the role of particles, organisms play the role of collectives, and selective disappearance plays the role of group selection.

This paper establishes finite-sample worst-case maximal inequalities for averages of independent centered heavy-tailed random vectors. The object of interest is the expected top-$k$ Euclidean norm of the sample average, which includes the expected coordinate-wise maximum as the special case $k=1$. Under coordinatewise variance constraints and tail-envelope constraints, the worst-case value is characterized up to universal constants over the class of distributions satisfying a finite $q$:th envelope moment condition. Analogous bounds are obtained for the sub-Weibull envelope class and the marginal sub-Weibull class.

Accurate prediction of complex dynamical systems from noisy measurements remains a significant challenge in scientific computing. Kernel ridge regression learning strategies are often effective when applied to clean data, but have limited success with noisy data. Recent work has observed that a weak formulation can act to filter noisy data, and different learning strategies have achieved increased noise robustness with a weak-form framework. In this manuscript, we give an overview of the filtering mechanism behind the weak formulation and provide a bias-variance error decomposition. Using these insights, we combine a weak formulation with a kernel learning strategy to propose Weak-form Kernel Ridge Regression (WKRR) for learning dynamical systems. The proposed framework is simple to implement, effective for both clean and noisy data, and outperforms several baseline methods. We demonstrate the performance of WKRR on chaotic benchmark systems in up to 64 dimensions, as well as 15,000-dimensional real-world fluid data.

Hypergraphs encode rich multiway interactions, but not all structural information is equally accessible through the dynamics. By analyzing pattern-forming instabilities in reaction-diffusion systems on directed hypergraphs, this work develops a theory of structural visibility that characterizes which features of higher-order structure survive successive levels of dynamical reduction. It is established that higher-order structure is not automatically dynamically relevant. Linearization destroys most higher-order information. Meanwhile, nonlinear reduction recovers only specific higher-order marginals of the adjacency tensor, and projection along critical directions further filters what is dynamically visible. First, we show that the linearized dynamics depends on the hypergraph only through its first-tail-moment statistics, termed exposure. Consequently, exposure-equivalent hypergraphs are linearly indistinguishable in the sense that they exhibit identical dispersion relations and instability thresholds. Next, we define a hierarchy of hyperedge tail-moments that captures progressively detailed co-occurence, and we prove a structural decomposition theorem describing how contractions of these tensors, termed packing effects, influence the reduced amplitude dynamics. This leads to a visibility hierarchy in which successive asymptotic orders reveal increasingly richer structural information. More specifically, exposure governs linear onset while packing effects control post-onset dynamics. Finally, we establish results on nonlinear distinguishability, characterizing when linearly indistinguishable higher-order systems may exhibit different post-onset behaviors. In addition, we formalize when higher-order systems become dynamically indistinguishable from pairwise systems, leading to the notion of dynamical graph surrogacy. Numerical simulations support the theoretical predictions.

Unmanned Aerial Vehicles (UAVs) have become key enabling platforms for low-altitude economic networks, yet achieving efficient and adaptive optimization under resource-constrained and dynamic environments remains challenging. This paper investigates language models for UAV-enabled Wireless Power Transfer (WPT) systems. First, a lightweight Small Language Model (SLM)-based solution is developed using a pre-trained BERT backbone, enhanced UAV embeddings and contextual features, a geometry-aware path decoder, and ensemble inference to achieve low complexity, low latency, and high energy efficiency. Second, an Agentic AI-based framework is designed to exploit the reasoning and interactive capabilities of Large Language Models (LLMs). It integrates four collaborative agents-Initializer, Actor, Critic, and Reflector-to form a closed loop of generation, optimization, evaluation, and reflection for iterative UAV path and energy optimization. Finally, simulations compare the SLM-, LLM-, and Agentic AI-based approaches.

We present an algorithm for the group distributionally robust (GDR) least squares problem. Given $m$ groups, a parameter vector in $\mathbb{R}^d$, and stacked design matrices and responses $\mathbf{A}$ and $\mathbf{b}$, our algorithm obtains a $(1+\varepsilon)$-multiplicative optimal solution using $\widetilde{O}(\min\{\mathsf{rank}(\mathbf{A}),m\}^{1/3}\varepsilon^{-2/3})$ linear-system-solves of matrices of the form $\mathbf{A}^{\top}\mathbf{B}\mathbf{A}$ for block-diagonal $\mathbf{B}$. Our technical methods follow from a recent geometric construction, block Lewis weights, that relates the empirical GDR problem to a carefully chosen least squares problem and an application of accelerated proximal methods. Our algorithm improves over known interior point methods for moderate accuracy regimes and matches the state-of-the-art guarantees for the special case of $\ell_{\infty}$ regression. We also give algorithms that smoothly interpolate between minimizing the average least squares loss and the distributionally robust loss.

This paper focuses on identifying defective units in unbounded periodic arrays of point sources using boundary data. The study is motivated by the noninvasive evaluation of large-scale periodic source systems. Unlike classical inverse source problems in free space, the key challenge here lies in the disruption of periodicity caused by defective sources in the infinite array. To address this, we employ the Floquet - Bloch transform to reformulate the original inverse source problem as a quasi-periodic inverse source problem. We first establish uniqueness theorems for both the original and the quasi-periodic formulations. Then, we develop a new numerical method for identifying defective sources. This method combines a sampling indicator function with an algebraic technique to determine not only the number of defective sources, but also their locations and intensities. Numerical experiments are presented to validate the effectiveness of the proposed method.

Let $f\colon{\mathbb Z}^2\to{\mathbb Z}$ be a Riemann function whose weight $W$ is a perfect matching. Then there is a family of sheaves of $k$-vector spaces $\{{{M}}_{W,{\bf d}}\}_{{\bf d}\in{\mathbb Z}^2}$ on a five-point topological that models $f$ in that $f({\bf d})=b^0({{M}}_{W,{\bf d}})$ and that $$ b^1({{M}}_{W,{\bf d}})= f^\wedge_{\bf K}({\bf d}-{\bf K}) $$ for any ${\bf K}\in{\mathbb Z}^2$. Hence a Riemann-Roch formula for $f$ is equivalent to an Euler characteristic computation of ${{M}}_{W,{\bf d}}$. If $f$ and $W$ are $r$-periodic, then the sheaves ${{M}}_{W,{\bf d}}$ become ${{O}}_r$-modules of finite type for a natural sheaf of rings ${{O}}={{O}}_r$. We show that in this case there is a “canonical ${{O}}$-module” $\omega=\omega_W$ and a pairing for $i=0,1$, $$ H^i(M_{W,{\bf 0}}\otimes F) \times {\rm Ext}^{1-i}(F,M_{W^\wedge_{\bf L},{\bf K}})\to H^1(\omega)\cong k $$ that is perfect when ${\bf L}={\bf K}+{\bf 1}$ and ${{F}}$ is a certain type of line bundle or a certain type of skyscraper sheaf. In particular when ${{F}}$ is a line bundle, we realize the above formula for $b^1({{M}}_{W,{\bf d}})$ as a duality theorem akin to Serre duality. We show that canonical ${{O}}$-module $\omega_W$ is a rather exceptional element in a family of tensor products of two modules ${{M}}\otimes_{{O}}{{M}}'$, where ${{M}}$ and ${{M}}'$ vary over ${{O}}_r$-modules of the form ${{M}}_{W',{\bf d}}$. This article doesn't assume any background in sheaf theory; rather we describe all our sheaves as a “diagrams of vector spaces,” where each diagram is essentially a sheaf of vector spaces on a fixed topological space of five points.

Nonlinear algebraic (polynomial) differential equations that govern fluid-structure interactions, such as those modeling vortex-induced vibrations, and shock waves, often lack analytical solutions, creating significant challenges to efficient prediction and control. While Physics-Informed Neural Networks (PINNs) offer a mesh-free numerical alternative, they frequently suffer from convergence stagnation when optimizing over chaotic landscapes or stiff singularities. This paper introduces a hybrid methodology that integrates tropical differential algebraic geometry with deep learning. Using tropical algebra, we algorithmically determine a hard constraint, which we use to restrict the neural network's hypothesis space to the exact support of the valid formal power series solution. We establish a theoretical Valuation-Support equivalence between classical Briot-Bouquet indicial analysis and the fundamental theorem of tropical differential algebraic geometry, proving that tropical methods accurately identify singularity structures. Numerical experiments on the Van der Pol and Burgers' equations demonstrate that embedding these tropical constraints directly into the network architecture drastically reduces the search space, overcoming optimization stagnation and improving both accuracy and convergence speed in non-homogeneous physical regimes.

We study homology manifolds through the eyes of the six functor formalism of spectral sheaves on locally compact Hausdorff spaces. As main results, we characterize cohomologically smooth objects by adapting an argument of Scholze, deduce that any hypercomplete locally compact ANR homology manifold is cohomologically smooth, show that compact ANR homology manifolds $X$ are Poincaré duality complexes whose Spivak tangent fibration identifies with the dualizing sheaf of $X$, and prove a generalization of Wilder's monotone mapping theorem about cell-like maps. Moreover, we introduce the notion of homotopy manifolds for which we prove an unstable analog of Wilder's orientability conjecture and show that hypercomplete ANR homology manifolds are homotopy manifolds. As a consequence, we show that for a compact $d$-dimensional ANR homology manifold, the Spivak tangent fibration of its associated Poincaré duality complex canonically destabilizes to a pointed $S^d$-fibration. Finally, we introduce homotopy manifolds with conical singularities, a generalization of Cohen's triangulated homotopy manifolds, and show that such objects are in fact topological manifolds, generalizing a result of Siebenmann. Along the way, we obtain comparisons between sheaf and singular cohomology and between the shape and the weak homotopy type of a topological space, explore the relation between various notions of cohomological dimension and hypercompleteness, and study six functor formalisms satisfying the Künneth formula.