Papers

We prove a function-field analogue of Bourgain's $L^2$ pointwise ergodic theorem. Let $q$ be a power of a prime $p$, let $\mathbb{F}_q[t]$ be the ring of polynomials over the finite field $\mathbb{F}_q$, and let $\mathbb{F}_q[t][u]$ be the ring of polynomials over $\mathbb{F}_q[t]$. Let $T^{(1)},\ldots,T^{(\ell)}$ be commuting, measure-preserving $\mathbb{F}_q[t]$-actions on a $\sigma$-finite measure space $(X,\mu)$, and let $P_1,\ldots,P_\ell\in \mathbb{F}_q[t][u]\setminus\{0\}$. Define a sequence of operators $(A_n)_{n\in \mathbb{N}}$ by \[ A_n g(x):=\frac{1}{q^n}\sum_{\substack{f\in \mathbb{F}_q[t]\\\deg f<n}} g\left(T^{(1)}_{P_1(f)}\cdots T^{(\ell)}_{P_\ell(f)}x\right) \qquad \left( g\in L^2(X),\,\,x\in X\right). \] We prove that $(A_n)_{n\in\mathbb{N}}$ satisfies an $L^2$ oscillation ergodic theorem: \[ \sup_{\substack{n_1<\cdots <n_{t_0}\\ t_0\in \mathbb{N}}} \left( \int_X \sum_{j=1}^{t_0-1} \sup_{n_j\leq n<n_{j+1}} |A_ng(x)-A_{n_{j+1}}g(x)|^2 \,d\mu(x) \right)^{1/2} \leq C_1\|g\|_{L^2(X)}\qquad \left( g\in L^2(X)\right), \] where the constant $C_1>0$ depends only on $P_1,\ldots,P_\ell$ and $q$. This in particular implies that the sequence $(A_ng(x))_{n\in\mathbb{N}}$ converges for almost every $x\in X$ and that $(A_n)_{n\in\mathbb{N}}$ satisfies an $L^2$ maximal inequality: \[ \big\|\sup_{n\in\mathbb{N}}|A_ng|\big\|_{L^2(X)} \leq C_2\|g\|_{L^2(X)} \qquad \left( g\in L^2(X)\right), \] where the constant $C_2>0$ depends only on $P_1,\ldots,P_\ell$ and $q$. Our tools include the circle method in function fields and refinements of Weyl sum estimates in this setting, further developing the work of Lê-Liu-Wooley and Champagne-Ge-Lê-Liu-Wooley. These refinements are of independent interest.

We study free products, that is, coproducts, in the category of Banach lattices and contractive lattice homomorphisms. We give a concrete construction of the free product of an arbitrary family of Banach lattices as a quotient of a free Banach lattice, and prove its basic structural properties. We also establish stability results for sublattice embeddings and projective Banach lattices, and also analyze the behavior of quotient maps. For compact Hausdorff spaces $K_1$ and $K_2$ we identify $C(K_1)\ast C(K_2)$ lattice isomorphically with $C(K_1\ast K_2)$, where $K_1\ast K_2$ denotes the topological join, and we derive an explicit formula for the free product norm in this representation. We further discuss free factors of free Banach lattices, and exploit the existence of non-trivial homological spheres to show that a free Banach lattice can have free factors which are not isomorphic to free Banach lattices.

In this paper, training a neural network is identified, exactly, as a search through Hamilton–Jacobi initial-value problems: each gradient step selects the initial data of a viscous Hamilton–Jacobi equation whose Hopf–Cole propagator best fits the observations; at inference, the input is the spatial point at which that solution is evaluated and the initial condition is already encoded in the weights. The correspondence is exact for log-sum-exp layers, with ReLU, sigmoid, SiLU, and GELU each an exact limit, gradient, or moment of the same object, and exact in composition across depth and width, with a quantified error at finite depth that vanishes in the joint limit. It is structural for residual networks, transformers, and recurrent networks (RNNs, LSTMs, SSMs), each discretizing the same class of equations, at a named and quantified approximation error. A single deformation parameter $\varepsilon$ unifies all four perspectives (network, tropical algebra, viscous PDE, convex optimization) in a commutative diagram closed under Lipschitz conditions. Quantitative consequences include: the minimax optimal generalization rate $O(n^{-1/(d+2)})$ for fixed $t$; adversarial robustness controlled by $\varepsilon$; backpropagation as the co-state equation of the Hamiltonian system for residual networks (Pontryagin Maximum Principle); scaling exponents consistent with data intrinsic dimension via PDE quadrature; and a closed-form $O(N)$ influence function (softmax attribution weights $\pi_j$) whose entropy landscape undergoes fold bifurcations as $\varepsilon$ increases, each merging attribution basins.

The goal of this short note is to prove qualitative stability for a family of trace Sobolev inequalities first proven by Carlen & Loss for $p=2$ and by Maggi and the author for $p\in (1,n)$. This answers an open problem raised in a recent paper of Fan, Li & Zhang, and in conjunction with their local analysis, yields sharp quantitative stability for this family of inequalities when $p=2$.

We introduce edgewise jump invariants and gradient-type structures for the partition graph $G_n$, whose vertices are the partitions of $n$ and whose edges correspond to elementary transfers of one unit between parts. Previous work on $G_n$ has focused mainly on vertex-level invariants such as degree, local simplex dimension, and support size. Here we study how such invariants change along edges. For an oriented edge $e=(\lambda,\mu)$ and a vertex invariant $F$, we define the signed jump $\Delta_e F=F(\mu)-F(\lambda)$ and focus on the basic jump signature \[ J(e)=(\Delta_e d,\Delta_e\delta,\Delta_e\sigma), \] where $d$ is degree, $\delta$ is local simplex dimension, and $\sigma$ is support size. We prove that support jumps are universally bounded by $2$ and describe them in terms of local multiplicity data. We also develop a taxonomy of active, neutral, pure, and mixed transitions, relate nonzero jumps of integer-valued invariants to threshold-layer crossings, and discuss strict gradient orientations associated with real-valued vertex invariants. Finally, we formulate a reproducible protocol for a computational atlas of jump spectra, transition ranks, large-jump edges, and localization patterns. No large-scale computations are carried out here; the atlas is presented as a framework for subsequent work.

Given a matrix $X$, and two ranks $r_1$ and $r_2$, the Hadamard decomposition (HD) looks for two low-rank matrices, $X_1$ of rank $r_1$ and $X_2$ of rank $r_2$, both of the same size as $X$, such that $X\approx X_1\circ X_2$, where $\circ$ is the Hadamard (element-wise) product. In most cases, HD is more expressive than standard low-rank approximations such as the truncated singular value decomposition (TSVD), as it can represent higher-rank matrices with the same number of parameters; this is because the rank of $X_1 \circ X_2$ is generically equal to $r_1 r_2$. In this paper, we first present some theoretical insights for HD, in particular a useful reformulation $X\approx WH^\top$ where $W$ and $H$ have $r_1 r_2$ columns and belong to certain manifolds. These allow us to develop three new algorithms for computing HD. The first one uses the representation $X\approx X_1\circ X_2$ and relies on the Manopt toolbox. The other two rely on the reformulation $X\approx WH^\top$: one is a block projected gradient method, and the other is a manifold-based gradient descent algorithm that does not require projection onto the feasible set. The last two algorithms are particularly effective for handling large sparse data. We also propose new initializations that allow us to improve the accuracy of the HD. We compare our algorithms and initialization strategies with the TSVD and with the state of the art. Numerical results show that the new methods are efficient and competitive on both synthetic and real data.

This work addresses the mean-field limit of inertial particle systems with singular interactions in a perturbative regime around Gibbs equilibrium. We prove that small fluctuations around equilibrium are asymptotically governed by the linearized Vlasov equation. The result applies to a broad class of singular interaction kernels, including the Coulomb case in dimensions $d\le3$. In particular, this provides a rigorous derivation of the linearized mean-field dynamics near equilibrium in settings where the corresponding nonlinear mean-field limit remains out of reach.

Being encouraged by [AKRS] that provides an amazing bridge between Statistics and Invariant Theory, and especially by [FM], where quiver semi-invariant techniques apply to verify the existence of MLE for a recent iPCA model, we provide an enhancement to [FM]. Our Theorem 5.2 yields necessary and sufficient conditions for MLE to exist generically for any dimension vector. The conditions can be easily checked with either of our software [T] or [FF] based on Derksen-Weyman algorithm. This simplifies the application for statistics practitioners and non-specialists in quivers. For those deep in quiver Representation Theory, Theorem 5.2 relates the MLE existence to the local semi-simplicity of representations as introduced in [Sh07]. Moreover, we implemented in [FF] the Flip-Flop algorithm from [TA] and checked that it does work in all cases guaranteed by Theorem 5.2. We hope that our elementary and short text can serve for the experts in both domains as a warm start in a new category. And we welcome practitioners to use our code at [FF] for their own experiments.

Let $K$ be a number field, $k\geq 2$ an integer, $(K^*)^k$ the $k$-fold direct product of $K^*$ with coordinatewise multiplication, and $\Gamma$ a finitely generated subgroup of rank $r$ of $(K^*)^k$. Further, let $H(\alpha )$ denote the absolute exponential height of an algebraic number $\alpha$. Fix non-zero elements $a_1,\ldots , a_k\in K$. We give asymptotic formulas for the number of $\mathbf{x}=(x_1,\ldots , x_k)\in\Gamma$ with $H(a_1x_1+\cdots +a_kx_k)\leq X$ as $X\to\infty$ such that no non-empty subsum of $a_1x_1+\cdots +a_kx_k$ vanishes. By the same method of proof, we obtain an asymptotic formula as $X\to\infty$ for the number of non-negative integers $n$ with $H(u_n)\leq X$, where $\{ u_n\}$ is a linear recurrence sequence.

In this article, we study mixed Killing vector fields, defined by the condition $L_V L_V g = f\,L_V g$, on Cigar Ricci–Bourguignon solitons. While conformal vector fields are always mixed Killing, the converse fails in flat and open cylinders with base manifold geometries, where the mixed Killing class is infinite-dimensional. We establish a rigidity phenomenon for Cigar Ricci–Bourguignon solitons: any complete steady almost gradient Ricci–Bourguignon soliton on a surface with positive curvature is, up to homothety, Hamilton's Cigar soliton. We then characterise complete mixed Killing fields and show that locally any mixed Killing field is the sum of a rotational Killing field and a mixed Killing radial field. Finally, we establish that the dimension of the vector space of complete mixed Killing fields of Cigar Ricci–Bourguignon solitons is $5$. Moreover, we explicitly determine a basis. Our results show that Cigar Ricci–Bourguignon solitons exhibit behaviour completely different from that of Euclidean space. Finally, we provide a complete description of the geodesic structure of Cigar Ricci–Bourguignon solitons.

We propose the systematic study of presentations that can be generalised over a continuous open group monomorphism. Presentations with this property can turn well-known presentations such as those for as orientable surface groups, Artin groups, and some Thompson groups, into topological groups with a prescribed open subgroup. Later we focus on right-angled Artin groups (RAAGs) and introduce a notion of topological RAAGs. Our approach differs from lattice envelopes and produces examples of locally compact (LC) groups that contain RAAGs as discrete subgroups, but generally not as lattices. We investigate some geometric aspects of topological RAAGs, with a special emphasis on compactness properties of LC ones. This includes a study of universal Salvetti-type complexes which may be of independent interest. These complexes share some properties with buildings. Although in some cases they are CAT(0) cube complexes and provide models for classifying spaces, in other cases they are not even uniquely geodesic. For a large class of examples we establish high connectivity properties for these complexes. This yields novel examples of LC groups with prescribed compactness properties or rational cohomological dimension. We note that the Bestvina-Brady machinery does not automatically generalise to this setting; nevertheless, we extend the Bieri-Stallings construction to obtain totally disconnected locally compact (TDLC) groups of type $FP_n$ but not $FP_{n+1}$. Along the way we record counterparts of cohomological results, such as a Mayer-Vietoris sequence and Künneth formula in discrete (co)homology for TDLC groups, which have not appeared elsewhere in the literature. Despite our non-discrete LC focus we obtain, as by-product, new examples of discrete groups with controlled finiteness properties including, for every $n \geq 1$, a Thompson-like Bieri-Stallings group of type $F_n$ but not $F_{n+1}$.

Existing theory of momentum assumes that gradients arrive at every parameter at a roughly constant rate, an assumption violated in practice by heavy-tailed data distributions and modern architectures. We theoretically analyze the dynamics of two tractable models of momentum under sparse updates: a least squares model with sparse inputs and a logistic regression model with a rare class. Both admit exact closed-form second-moment dynamics whose high-dimensional limits we characterize across three scaling exponents for sparsity, batch size, and momentum decay. The phase structure on both problems is governed by the ratio of two intrinsic timescales: a momentum retention timescale (how many active updates the buffer survives) and a learning timescale (how many active updates it takes to reduce the squared error). When learning is much slower than retention, the limit matches SGD; when learning is faster, the system is unstable; where the timescales coincide, we recover classical heavy-ball dynamics. The oscillatory dynamics occur at different momentum values for different token sparsity, creating a spectral conflict for global momentum across token frequencies.

We give a necessary and sufficient condition for the existence of invariant Gibbs measures for sequences of continuous functions on one-sided subshifts and, more generally, for the existence of Gibbs measures. These extend the results of Kim [7] and Baker and Ghenciu [2], respectively, to sequences of continuous functions on one-sided subshifts. In particular, we characterize the existence of invariant Gibbs measures for subadditive sequences. For superadditive sequences on subshifts with the strong specification property, such a characterization gives a necessary and sufficient condition for the uniqueness of invariant Gibbs measures. We apply these results to study some problems in the theory of relative pressure and relative equilibrium states.

Recent cold atom experiments have realized one-dimensional anyons and enabled the tuning of 1D~statistics between bosons and fermions. Here, we analyze the symmetries, integrability, and resulting degeneracies of the underlying anyon-Hubbard model of finite length. Our results reveal a switching between symmetry classes AI, BDI, and CI in dependence on system size, particle number, and boundary conditions, and show that two anyons with periodic boundaries are integrable, while two anyons with open boundary conditions are not. We include a comprehensive analysis of all model limits, especially of interacting bosons and pseudofermions and resolve spectral signatures. We additionally reveal an exactly solvable doublon state that hides in the continuum of scattering states and the exact solution of the nullspace of two noninteracting anyons. The uncovered symmetries shape the fundamental properties of the one-dimensional anyons at hand, and the predicted states are accessible in state-of-the-art experiments.

E-values have attracted considerable interest in recent years as flexible tools for enabling anytime-valid and adaptive data analysis. Hypothesis testing is at the core of many of these applications, which can often involve private or sensitive data. In this work, we answer a simple but important question: given two distributions $\mathbb{P}$ and $\mathbb{Q}$, what is the maximum achievable e-power when testing $X\sim \mathbb{P}^n$ against $X\sim\mathbb{Q}^n$ with e-values that satisfy $\varepsilon$-differential privacy? We characterize the optimal rate for this problem and provide an algorithm which matches it exactly. In the sequential setting, when observations arrive one-by-one and the analyst chooses when to halt, we give matching upper and lower bounds on the stopping times of any private e-process. Numerical experiments confirm the practicality of our algorithms, which require less data than the recently proposed DP-SPRT across a range of sequential testing problems and privacy levels.

We review the notion of Yang-Baxter integrability for spin chains that have Hilbert spaces with constraints, such as a Rydberg blockade. We focus on anyonic chains, whose constraints arise from the fusion rules of the fusion categories on which they are based. We discuss the emergence of Temperley-Lieb algebras and present a new result on which types of anyonic chains exhibit them. We then give an overview of known results for integrable anyonic chains and extend them to several fusion categories up to rank $7$. Using a modification of the boost operator formalism, we find several new integrable anyonic chains and discuss some of their properties. These include spin-$\frac32$ models for $\mathfrak{su}(2)_k$ fusion categories, anyonic chains based on the Tambara-Yamagami fusion categories TY$(\mathbb{Z}_n)$, and product fusion categories Fib$\times$Fib and Fib$\times$Ising. We review recent results for spin chains based on the Haagerup-Izumi fusion category HI$(\mathbb{Z}_3)$, and present preliminary numerics for a HI$(\mathbb{Z}_5)$ model.

We study the interaction between gravitational waves and elastic bodies within the framework of relativistic elasticity. Starting from the Lagrangian formulation of relativistic elasticity, we derive the linearized equations governing the response of a homogeneous and isotropic solid to a weak gravitational wave by expanding the action to quadratic order in the elastic field derivatives and to linear order in the metric perturbation. In this way, Dyson's interaction term from the effective potential approach emerges naturally from the relativistic theory. We then apply the formalism to a thin rectangular elastic plate aligned with the direction of propagation and polarization of a plus-polarized gravitational wave. For a material with vanishing Poisson ratio, the equations decouple and admit simple explicit solutions. We obtain closed-form expressions for the induced displacements and for the energy deposited on the plate by both short gravitational wave bursts and continuous harmonic waves. Finally, we compute the gravitational wave emission generated by the oscillating plate itself under continuous harmonic excitation. These results provide a fully relativistic derivation of the elastic response to gravitational waves together with explicit solvable examples relevant to resonant gravitational wave detectors.

Given a data set with a notion of distance, such as a point cloud in Euclidean space, topological data analysis (TDA) uses techniques from algebraic topology and metric geometry to infer the topology of a hypothetical manifold from which the data are sampled. This inference is achieved by calculating topological invariants, some of which are difficult to compute classically. Meanwhile, quantum TDA utilizes quantum processes to extract the invariants used in making such inferences in an attempt to speed up the computations. Because applying transformations to the original classical dataset could alter the associated topological invariants, we investigate which quantum encodings would best preserve the invariants of the original dataset. This line of inquiry is distinct from standard approaches in quantum TDA, whose typical starting point is not from the classical dataset directly, but rather from the associated combinatorial objects, such as simplicial complexes, which typically demand a lot of resources to construct. We take the first step at a more direct approach by focusing on which quantum encodings acting directly on the data are admissible for applying quantum algorithms to extract topological features from classical datasets.

In mixed states of quantum systems, symmetries come in two types: strong and weak. Furthermore, it has been argued that in quantum many-body systems, strong symmetries can be "spontaneously broken" down to weak symmetries. An issue is that as previously formulated, such "strong-to-weak symmetry breaking" appears to be a fairly non-local effect. In this paper, we show how to understand and diagnose strong symmetries and strong-to-weak symmetry breaking in an explicitly local way. Our main technical tool is a rigorous definition of strong symmetry in the limit of infinite volume, which generalizes the conventional finite-volume definitions, and for which we give several equivalent formulations, including one involving the concept of "local charge coherence". Finally, we introduce von Neumann systems, which in infinite-volume symmetries are intermediate between strong and weak symmetries. We derive a Lieb-Schultz-Mattis type anomaly constraint for von Neumann symmetries (and therefore, in particular, strong symmetries) in quantum spin chains.

Let $f(k)$ be the maximum possible chromatic number of a graph whose edge set can be partitioned into at most $k$ complete bipartite graphs. Alon, Saks, and Seymour conjectured that $f(k)=k+1$ for all $k$. While the conjecture was verified for $k \leq 9$ by Gao et al., it was disproved by Huang and Sudakov, and further Balodis et al. proved that $f(k) \geq 2^{\widetilde{\Omega}((\log k)^2)}$. In this note, we give a simple proof of the recursive upper bound $f(k+1) \leq f(k)+f(\lfloor k/4 \rfloor)$. Consequently, $f(k) \leq 2^{(\log_2 (4k))^2/4}$ for $k \geq 1$. This improves the previous best known upper bound of Mubayi and Vishwanathan in the exponent by a factor which is asymptotically two. Note that these bounds are sharp up to a lower order factor in the exponent by the result of Balodis et al.

This article investigates $k$-regular factorizations of characteristic functions associated with completely non-coisometric row contractions. In this setting, a one-to-one correspondence is established between chains of joint invariant subspaces \[ \mathcal{M}_1 \subseteq \cdots \subseteq \mathcal{M}_{k-1} \] and $k$-regular factorizations of the characteristic function of a completely non-coisometric row contraction. A functional model corresponding to a given $k$-regular factorization of a purely contractive multi-analytic operator satisfying the Szegő condition is further constructed, and the associated chain of joint invariant subspaces is characterized in terms of the underlying multi-analytic factors. Finally, it is shown that any such chain of joint invariant subspaces induces a block upper-triangular decomposition of the underlying row contraction, and that the characteristic function of each diagonal block coincides with the purely contractive part of the corresponding factor in the $k$-regular factorization.

Saddle-point models arise throughout optimization, optimal transport, robust learning, and control. In many applications, the relevant function f(x,y) is convex in x and concave in y, and preserving this geometry is essential for obtaining tractable min–max formulations and reliable certificates. We introduce a structured separable decomposition that preserves the convex-concave geometry and prove a complete one-dimensional approximation theorem under a mixed Monge-type convexity condition. We then describe practical saddle network architectures that preserve convexity in x and concavity in y by construction. The proposed architectures require only convexity-preserving neural networks, together with simple output transformations enforcing sign and concavity constraints. Finally, we report numerical benchmarks in dimension 1 and 5, showing that the proposed saddle networks achieve high accuracy on smooth, nonsmooth, and high-rank convex–concave test functions.

We construct a continuous-time, positively divisible non-Markovian process with memory of the initial state that satisfies the differential Chapman–Kolmogorov equation. In the stationary state, the correlation function exhibits exponential decay, a behavior typically regarded as characteristic of Markovian dynamics. Nevertheless, the memory is preserved throughout the evolution of the process, manifesting itself in observable statistical quantities. We further demonstrate that mutual information serves as a reliable measure of the non-Markovian character of the process. As an application, we study a random walk driven by the constructed process and show that the memory effect breaks ergodicity and modifies transport properties such as the diffusion coefficient.

Let $\Gamma_g$ be the fundamental group of a closed orientable surface of genus $g\geqslant 2$. The outer automorphism group $\mathrm{Out}(\Gamma_g)$ naturally acts on the character variety $\mathcal{X}(\Gamma_g,G)$ for any Lie group $G$. We consider the set of simple-stable representations which are modelled on Minsky's primitive-stable representations. We prove that the set of conjugacy classes of simple-stable representations of $\Gamma_g$ in $\mathrm{PU}(2,1)$ is a domain of discontinuity for this action, strictly larger than the set of conjugacy classes of convex cocompact representations.

The hydrodynamic formulation of the Dirac equation has historically been hindered by the inability to close the system of physical variables without resorting to infinite moment hierarchies. We resolve this longstanding issue by developing a fully covariant chiral-hydrodynamic formulation of the Dirac field in curved spacetime. Working in the Weyl representation, we introduce two independent null vectors, $P_L^\mu$ and $P_R^\mu$, which decouple the left and right chiral components. This allows us to define chiral geodesic and stochastic velocities, yielding a closed system of exactly eight real equations that corresponds directly to the Dirac field degrees of freedom. Remarkably, this formulation naturally isolates the spin-orbit coupling $(q/2)\sigma^{\mu\nu}F_{\mu\nu}$ while demonstrating the vanishing of the spin-gravity coupling in torsion-free general relativity. To demonstrate the analytical power of this framework, we specialize to the Schwarzschild geometry. We obtain exact radial solutions in terms of confluent Heun functions and directly compute the quasi-bound state spectrum (fermionic resonances), quasinormal mode frequencies, and greybody factors. Furthermore, by establishing an exact energy balance equation – representing the first law of thermodynamics for Dirac fields – we derive the Hawking radiation flux purely from chiral flux conservation at the event horizon. This work not only provides a closed hydrodynamic theory for spin-1/2 fluids but also establishes a unified framework for analyzing quantum information and fermion dynamics in strong gravitational backgrounds.

Motivated by several open problems in the Afham-Ferrie theory of generalized fidelity, we study polar fidelities, Holevo bases, and unitary factors on \(\Pd\), the cone of \(d\times d\) complex positive definite matrices. We prove that, for every fixed pair \(P,Q\in\Pd\), the one-sided polar fidelities \(x\mapsto F_{P^x}(P,Q)\) and \(x\mapsto F_{Q^x}(P,Q)\), as well as their symmetrization, are nondecreasing on \((-\infty,1]\) and nonincreasing on \([1,\infty)\). Hence the polar paths realize exactly the interval \([\FM(P,Q),\FU(P,Q)]\) on \([-1,1]\), yielding pointwise, generally pair-dependent realizations of all \(z\)-fidelity values with \(z\ge1/2\) and of the Log-Euclidean fidelity as generalized fidelities. We also show that for \(d\ge2\) and \(0<z<1/2\), such realization fails in general, even for interior fidelities. We further solve the fixed-pair Holevo-base equation \(F_R(P,Q)=\FH(P,Q)\), classify all Holevo bases, and classify exactly which unitary factors of generalized fidelity can arise from a base \(R\). These are precisely the unitaries \(W\) for which \(P^{-1/2}Q^{1/2}W\) is similar to a positive definite matrix. This recovers the special-unitary constraint and disproves the global reverse inclusion \(SU(d)\) for \(d\ge2\).

In this work, we present the logistic branching Brownian motion with selection (Log-BBM), a modification of the N-BBM defined by Groisman et. al (2020), in which birth and competition events are decoupled to allow for a variable population size that follows the branching process with logistic growth defined by Lambert (2005). We study the representation of the Log-BBM as a microscopic model for a FKPP-type equation. In the large population limit, the renormalised empirical measure of the Log-BBM converges weakly to a probability measure whose density solves a nonlocal version of the FKPP equation, while its cumulative distribution function solves the classical FKPP equation. We also show that this model exhibits behaviour that is characteristic of the Brunet-Derrida family of systems, in particular the so-called weak selection principle. Indeed, we show that, in the Log-BBM, the particles select the minimal propagation speed, in contrast to the FKPP equation, where the selected front speed depends on the initial condition.

For an algebraic group $Q$ with $\mathsf{Lie\,} Q=\mathfrak q$, we develop a method for estimating the index of a subalgebra $\mathfrak h$ in $\mathfrak q$ via the use of coadjoint $Q$-orbits in $\mathfrak q^*$. Let $\mathfrak q^\xi$ denote the stabiliser of $\xi\in\mathfrak q^*$. In the special case when $\mathfrak q^\xi\oplus\mathfrak h=\mathfrak q$, our estimate implies that $\mathsf{ind\,}\mathfrak h=0$. Using our theory, we also answer a question of Duflo. An orbit $Q{\cdot}\eta\subset\mathfrak q^*$ is said to be strange, if $\mathfrak q^\eta\oplus\mathfrak h=\mathfrak q$ for some $\mathfrak h$. In the second part of the paper, we study strange orbits for a semisimple algebra $\mathfrak g$. It is shown that an orbit ${\mathcal O}\subset\mathfrak g\simeq\mathfrak g^*$ is strange whenever the complexity of ${\mathcal O}$ is at most 1. Furthermore, if ${\mathcal S}\subset\mathfrak g$ is a sheet containing a strange nilpotent orbit, then all orbits in ${\mathcal S}$ are strange. We also show that strange orbits in $\mathfrak{sl}_n$ are not as sparse, as one might expect, and discuss some conjectures on strange orbits.

We propose a definition of a non-invertible symmetry enriched topological order (NI-SETO), and we implement our definition for string net models. We do so in two ways, using full inclusions of unitary fusion categories (UFCs), as well as anyon condensation. In both cases, the NI-SETO is a relative center of UFCs. All NI-SETOs can be realized in either model, where we can use enriched UFCs to get chiral examples on the boundary of a 3D Walker-Wang model representing the anomaly. We describe several examples of NI-SETOs and compute the qualitative symmetry action on anyons and symmetry defects using tube algebra techniques.

For positive integers $s$ and $k$, the Ramsey number $r(s,k)$ is the minimum integer $n$ such that any graph on $n$ vertices contains a clique of size $s$ or an independent set of size $k$. We prove that for any fixed $s \ge 3$ and $k$ tending to infinity, the off-diagonal Ramsey numbers satisfy \[ r(s, k) \ge \Omega \left(\frac{k^{s-1}}{(\log k)^{2s-4}} \right), \] which matches, up to polylogarithmic factors, the upper bound established over 90 years ago by Erdős and Szekeres. For $s \ge 5,$ this improves the best known lower bound of the form $r(s, k) \ge k^{\frac{s+1}{2} + o(1)}$ which was first established by Spencer in 1977 and has since only seen polylogarithmic improvements.