Papers

The recently introduced framework of $\alpha$-potential games facilitates the analysis of finite-player dynamic games by reducing the search for approximate Nash equilibria to the minimization of a single $\alpha$-potential function. In this work, we show that the approximate potential structure of finite-player games persists in the large-population limit, leading naturally to a notion of $\alpha$-potential mean field games (MFGs). Specifically, we show that both the optimal values and the minimizers of normalized $N$-player $\alpha_N$-potential functions converge to those of a mean field control (MFC) problem with measure-valued controls. We identify the objective of this limiting MFC problem as an $\alpha_\infty$-potential function for an MFG, and characterize the parameter $\alpha_\infty$ analytically. We further show that $\lim_{N\to\infty}\alpha_N= 0$ is equivalent to standard conditions for potential MFGs, in which case the limiting MFC objective becomes a potential function for the MFG. Together, these results provide new constructions of potential MFGs from finite-player games through the asymptotic condition $\lim_{N\to \infty}\alpha_N= 0$, and establish propagation of chaos of potential-selected equilibria from $N$-player games to MFGs with general controlled diffusions, common noise, and non-separable state-control interactions.

We investigate the high Mach number limit for the scaled compressible Navier–Stokes system in the critical Besov framework. In the scaled momentum equation, the pressure force is represented by the term \(\varepsilon^2\nabla a^\varepsilon\), where $\varepsilon$ is the inverse Mach number; as \(\varepsilon\to0\), the formal limiting system is the compressible pressureless Navier–Stokes system. The analysis is complicated by the absence of density dissipation in the limiting model and by the highest-order coupling created by the viscous terms. For \(d\geq2\), we prove the global well-posedness of the scaled system for small initial data and obtain estimates that are uniform with respect to $\varepsilon$. A crucial ingredient is a parameter-dependent lower-order estimate for \(\varepsilon a^\varepsilon\), which compensates for the purely transport nature of the density equation and allows the uniform bounds to be closed. Based on these estimates, we justify the high Mach number limit and recover a global strong solution to the pressureless Navier–Stokes system. For \(d\geq3\), we further derive quantitative error estimates between the scaled solutions and the pressureless limiting solution. More precisely, on each fixed finite time interval, if the initial discrepancy is of order \(\mathcal{O}(\varepsilon)\), then the corresponding lower-order critical Besov error satisfies the same rate, which yields a quantitative justification of the pressureless limit.

We construct a de Branges-Rovnyak space containing the polynomials as a dense subspace, together with a function in the space that lies outside the closed linear span of its Taylor partial sums. The space has the further property that the monomials are bounded in norm. The construction is carried out by reducing it to an elementary problem about series, which we then solve.

For graphs $F$ and $H$, the generalized Turán number $\mathrm{ex}(n,F,H)$ denotes the maximum number of copies of $F$ in an $H$-free graph on $n$ vertices. We prove that if $s\in \{2,3\}$, $s< a\leq b$ and $t$ is sufficiently large, then $\mathrm{ex}(n,K_{a,b},K_{s,t})=\Theta(n^s)$. The $s=2$, $a=b=3$ case of this result answers a question of Spiro. Proving another conjecture of Spiro, we show that for every graph $F$ with at least one edge, there exist infinitely many real numbers $r$ such that $\mathrm{ex}(n,F,H)=\Theta(n^r)$ holds for some graph $H$.

We generalize to the fibres of the fibration $\mathcal{O}\colon\mathbf{Cat}\rightarrow\mathbf{Set},$ defined by mapping a small category $\mathbb{X}$ to its set of objects $X_0=ob(\mathbb{X}),$ the classical notions of action and semidirect product of monoids. We prove that the equivalence between monoid actions of a monoid $Y$ and Schreier split extensions on $Y,$ which is well known to generalize the equivalence between actions and split extensions for groups, is an instance of a broader adjunction between Schreier points and actions in the fibres $\mathcal{O}^{-1}(B).$ This adjunction is an equivalence if and only if $B=1,$ i.e., for the category $\mathbf{Mon}$ of monoids. Similarly, we prove that there is an adjunction (which, in the case of monoids, results in a known equivalence due to Patchkoria) between Schreier internal categories in the fibres $\mathcal{O}^{-1}(B)$ and the category of crossed semimodules in $\mathcal{O}^{-1}(B).$ The latter are defined by translating in $\mathcal{O}^{-1}(B)$ the notion of crossed semimodule in $\mathbf{Mon}.$ Eventually, we prove that, by defining crossed modules appropriately, this last adjunction yields an equivalence between crossed modules and Schreier internal groupoids in the fibres of $\mathcal{O}.$

Dedekind's problem counts monotone Boolean functions, equivalently downsets of a Boolean lattice. We recast this enumeration as a finite layer-ratio reconstruction problem for the Whitney numbers of the ranked ideal lattice. An exact adjacent-layer double count expresses each layer ratio through local averages of the number of addable elements and the number of removable elements. Reversible fixed-layer Markov chains estimate these averages and hence estimate the Dedekind number $M(n)$. Backtests at $M(8)$ and $M(9)$ calibrate seed-level variability under the fixed protocol and measure the observed Monte Carlo budget scaling. The resulting estimate probes the Whitney-number sequence of the ideal lattice. Although these rows have previously been described empirically as unimodal, the high-precision $n=9$ estimate has a shallow two-shoulder feature around the central rank, contrary to that empirical description; $n=11$ and $n=13$ center-window estimates show a larger-contrast analogous pattern. The protocol estimate for $M(10)$ is \[ \widehat M(10)=(8.9360\pm0.0010)\times 10^{78}, \] where the displayed uncertainty is the budget-based forecast scale from the cross-$n$ scaling law under the production budget.

A biclique decomposition of a graph is a partition of its edges into complete bipartite subgraphs. We consider graphs whose vertices can be ordered such that the neighborhood of every vertex is the union of a sublinear number of intervals. We observe that these graphs admit compact representations in the form of biclique decompositions of small size. Here, the size of a decomposition is measured as the sum of the number of vertices of its bicliques. Combining this result with the existence of suitable vertex orderings for graphs of low neighborhood complexity, as proven by Welzl in 1988, we recover and extend several known results up to logarithmic factors. These results include upper bounds on the Zarankiewicz problem, matrix multiplication, quantum circuit complexity, and shortest path algorithms in "well-structured" instances.

We introduce a class of continuous time finite horizon mean field games where the objective function of the representative player depends on a hidden state, in addition to position, control, and the population distribution. While acting on the position dynamics, the agent has the option to pay for seeing the hidden state. We connect the original formulation of our model with a mean field model of optimal control with discretionary stopping, characterize solutions, and give a simple explicitly solvable example. For a class of $N$-player games with compatible information structure, we show that approximate Nash equilibria can be constructed starting from a solution to the limit model.

We propose a geometrization of compact metric spaces that is based on ideas of S. Banach and J. Mycielski. Then we prove the following conjecture of S. Banach and S. Ulam from 1935: in every compact metric space there exists a finitely additive probability measure, invariant under congruences. Moreover, our techniques allow us to solve a problem of M. Talagrand related to the Marczewski problem and the Banach-Tarski paradox with pieces having the property of Baire. We give also a very simple proof of the conjecture of Ulam about the product Lebesgue measure in the Hilbert cube and explain the existing results about congruence-invariant Borel measures in the language of our geometrization.

We describe the linearized dynamics near the Lamb-Chaplygin dipole, a classical traveling solution of the two-dimensional Euler equations. Exploiting the Hamiltonian structure of the system together with its symmetries, we identify all possible sources of linear instability. For general perturbations in $L^1\cap L^p$, $p>2$, growth can occur through two explicit mechanisms triggered by: (i) a nonzero circulation on the core of the dipole, and (ii) a nontrivial component along the generalized eigenvectors associated with the eigenvalue $0$. In particular, we completely classify the spectrum and the Jordan chains of the operator associated with the linear dynamics. Both mechanisms hint for a nonlinear dynamics that may drift along the symmetry-generated family of traveling dipoles without moving away from it.

We propose and study all-multiplicity building blocks for tree-level string amplitudes in AdS. These are worldsheet integrals obtained by dressing the corresponding flat-space disc and sphere integrals with multivariable multiple polylogarithms and their single-valued analogues, respectively. We derive monodromy relations for the open-string building blocks and a KLT factorisation for their closed-string counterparts. This extends the non-commutative AdS uplift of lower-point flat-space structures to general $n$-point kinematics.

We study $\mathbb{Z}$-Ramsey ultrafilters, ultrafilters containing witnesses to every shift-invariant instance of Ramsey's theorem. We prove that it is consistent that there are no $\mathbb Z$-Ramsey ultrafilters. We also prove that every $(\mathbb{Z},3)$-Ramsey ultrafilter, as well as every $\mathbb{Z}$-Ramsey P-point, is selective. Further, we exhibit a generic extension – using quotient algebras of the form \(\mathcal{P}(\mathbb{Z})/\mathcal{I}\) for certain \(F_{\sigma }\)-ideals – that contains P-points that are not \(\mathbb{Z}\)-Ramsey ultrafilters, thereby addressing open questions raised by Petrenko and Protasov.

We introduce a family of homotopy theories for generalized metric spaces with natural number distances, via the magnitude-path spectral sequence (MPSS). The first page of the MPSS is known as magnitude homology; the second page is known as bigraded path homology, and contains GLMY path homology as its top row. For each natural number r, we define a class of maps of metric spaces called r-quasi-isomorphisms: those maps that induce a quasi-isomorphism at page r of the MPSS. We show that every page of the spectral sequence satisfies a suitable metric analogue of each of the Eilenberg-Steenrod axioms. In particular, we introduce the notion of r-cofibration and prove a Mayer-Vietoris theorem for page r with respect to r-cofibrations. We establish a family of Brown category structures on generalized metric spaces which allow us to explicitly compute homotopy colimits. We apply this to describe r-suspension and r-spheres of dimension n, and compute their spectral sequences. Finally we prove that for r = 1 the entire theory restricts to directed graphs.

The goal of this paper is to establish polynomial Fourier decay for images of self-similar measures $\mu$ on $\mathbb{R}^k$ under sufficiently nonlinear real-analytic maps $f \colon \mathbb{R}^k \to \mathbb{R}^d$. For example, we prove that if $f$ is analytic on $\mathbb{R}^k$, its graph does not lie in an affine hyperplane in $\mathbb{R}^{k+d}$, and $\mu$ is not supported in an affine hyperplane in $\mathbb{R}^k$, then the image measure has polynomial Fourier decay. Key steps in the proof include establishing a uniform Lojasiewicz-type inequality for self-similar measures, and using the decay of the Fourier transform of $\mu$ outside a very small exceptional set of frequencies. As an application of our results, we prove polynomial Fourier decay for self-conformal measures on $\mathbb{C}$ for a large class of complex analytic IFSs which are not self-similar but are conjugate to a linear IFS via an analytic map.

Lovász' famous Hamiltonicity conjecture (1969) states that every connected vertex-transitive graph has a Hamiltonian path. A stronger version of the conjecture, often attributed to Thomassen (1978), states that every sufficiently large such graph even has a Hamiltonian cycle. Despite the great amount of attention these conjectures have attracted over the past decades both in the combinatorial and algebraic communities, for more than 40 years the best known lower bound for the maximum length of a cycle (path) in a connected vertex-transitive graph of order $n$ remained of the form $\Omega(\sqrt{n})$, due to Babai (1979). A series of recent works has successively improved the exponent in this lower bound further. In this paper, improving the previous state-of-the-art bound $\Omega(n^{9/14})$ due to Norin et al.~(2025), we prove that every connected vertex-transitive graph of order $n$ contains a cycle of length at least $n^{2/3-o(1)}$. This hits a natural barrier for several existing approaches from previous work. Our proofs combine recent embedding techniques for paths in sublinear expanders, sublinear expander decompositions of almost-regular graphs, and several additional combinatorial ideas.

In this paper, we study the monogeneity of a special class of composed polynomials of the form $ (f \circ g)(x) = (x^m + c)^n + a(x^m + c)^{n-1} + d(x^m + c)^{n-2} + b,$ where \( f(x) = x^n + a x^{n-1} + d x^{n-2} + b \in \mathbb{Z}[x] \) satisfies \( a^2 = 4d \) and \( g(x) = x^m + c \in \mathbb{Z}[x] \). Assuming that \( (f \circ g)(x) \) is irreducible over \( \mathbb{Q} \), we obtain necessary and sufficient conditions on the parameters \( a, b, c, d, m, n \) for the polynomial to be monogenic. These conditions help to identify when the set \( \{1, \theta, \dots, \theta^{mn-1}\} \) forms an integral basis of the number field \( \mathbb{Q}(\theta) \), where \( \theta \) is a root of \( (f \circ g)(x) \). We also provide lower bound for the counting of such monogenic polynomials. Furthermore, we study the behaviour of solutions to certain related differential equations and present a class of polynomials with non-square-free discriminants as an application of the main results.

We study an online version of the robust mean change point detection problem under a dynamic Huber contamination model with arbitrary contamination distribution and inlier distribution possessing exponentially- or polynomially-decaying tails. This robustness framework is systematically studied for the first time in the change point literature. For univariate data, we characterise the detection delay by partitioning the parameter space into four regimes, in terms of the true change location, signal size and contamination level. Efficient detection procedures are accompanied by matching lower bounds, up to poly-logarithmic factors. For the multivariate setting, we devise an efficient robust mean testing procedure and apply this to the robust online change point problem. The theoretical analysis of the robust mean testing procedure is the first in dealing with both Huber contamination and heavy-tailedness, and is thus of independent interest. Extensive numerical experiments are conducted to support our theoretical findings.

The triangle packing number $\nu(G)$ of a graph $G$ is the maximum size of a set of edge-disjoint triangles in $G$. Tuza conjectured that in any graph $G$ there exists a set of at most $2\nu(G)$ edges intersecting every triangle in $G$. We show that Tuza's conjecture holds in the random geometric graph for a large range of densities. We also study the problem of covering almost all edges of the random geometric graph with edge-disjoint copies of some fixed graph $F$. In particular, we show the existence of almost-perfect packings for an infinite family of $F$, and state some negative results as well.

For a clique $K_r$, a graph is $K_r$-saturated if it contains no copy of $K_r$ and the addition of any edge from its complement creates a $K_r$. A classical result of Erdős-Hajnal-Moon and Zykov shows that the number of edges of an $n$-vertex $K_r$-saturated graph is at least $(r-2)n-\binom{r-1}{2}$. In this paper, we focus on the number of edges of the $K_r$-saturated graphs with a fixed matching number. Let $G$ be an $n$-vertex $K_r$-saturated graph with matching number $\nu(G) = s$. For sufficiently large $n$, we prove that the number of edges \begin{equation*} e(G)\geq \left\{\begin{array}{cl}{(r-1)n-\frac{r}{2}(r-1)-1,}&{\quad\mathrm{if}~s=r-1;}\\{(r-1)n + (s-r)^2 - \frac{1}{2}(r+2)(r-3) - 5,}&{\quad\mathrm{if}~s>r-1.}\\\end{array}\right. \end{equation*} Moreover, we completely characterize the graphs attaining the equality.

We introduce a stochastic probing problem with correlated items. In our model, which we call Bayesian Probing, the correlations are modeled by an underlying graph $G$. Each vertex is independently active with a known probability. Each item corresponds to an edge in the graph. Probing an edge has some cost, gives some reward if both endpoints are active, and also reveals the state of its endpoints. Hence a probe induces a Bayesian update on the remaining edges. The goal is to adaptively probe items/edges subject to a knapsack constraint to maximize the expected total reward obtained from the probed edges. Bayesian Probing generalizes stochastic knapsack and stochastic probing by allowing correlations between items. Moreover, it gives a tractable model for the Bayesian Active Search problem, a popular problem considered in the machine learning community. In Bayesian Active Search, the goal is to find items in a particular class by adaptively probing at most, say $k$, items. Given a prior distribution over items, we want to compute a Bayesian policy to maximize the number of such items found. For this general problem with arbitrary priors, there are strong lower bounds on efficiently computing good policies. In this paper, we design efficient approximation algorithms for Bayesian Probing. These results give the first efficient approximation algorithms for Bayesian Active Search, for a class of practically-relevant prior distributions.

We study nearest zero–critical point distances for the Gaussian SU(2) polynomial $p_n$. For each zero $z_i$ of $p_n$, let $D_n(z_i)$ denote its Fubini–Study distance to the nearest critical point. We prove that the empirical measure of the rescaled distances $nD_n(z_i)$ converges weakly in probability to the law of $2/|Z|$, where $Z$ is distributed according to the Fubini–Study probability measure in the affine chart.

Let $X$ be a compact Riemann surface and $G$ a connected reductive complex Lie group with centre $Z$. Consider the moduli space $M(X,G)$ of polystable $G$-Higgs bundles on $X$. The group of isomorphism classes of $Z$-bundles on $X$, which is isomorphic to $H^1(X,Z)$, acts on $M(X,G)$ via extension of structure group by the multiplication homomorphism $Z\times G\to G$. The group $\text{Aut}(G)$ also acts on $M(X,G)$ by extension of structure group, and so does the group $\text{Aut}(X)$ of holomorphic automorphisms via pullback. Finally, $\mathbb{C}^*$ acts by multiplying the Higgs field. Combining these provides an action of the semidirect product of $H^1(X,Z)$ and $(\text{Aut}(G)\times\text{Aut}(X))\times\mathbb{C}^*$ on $M(X,G)$, where $\text{Aut}(G)$ and $\text{Aut}(X)$ act on $H^1(X,Z)$ by extension of structure group and pullback, respectively. Let $H$ be such semidirect product. Let $\Gamma$ be a finite subgroup of $H$. The goal of this thesis is to find a Prym–Narasimhan–Ramanan-type construction to describe the fixed points of the action of $\Gamma$ on $M(X,G)$. More precisely, we show that fixed points correspond to twisted equivariant Higgs pairs over certain étale covers of $X$. Our results generalize García-Prada–Ramanan, where $\Gamma$ was considered to be cyclic, and Narasimhan–Ramanan, who only consider actions of cyclic subgroups of $H^1(X,\mathbb{C}^*)$ for $G=\text{GL}(n,\mathbb{C})$.

The second homology group is of central importance in the study of profinite rigidity of $3$-manifold groups. Although general and deep results imply that the integral homology of cocompact hyperbolic $3$-orbifold groups is computable in principle, the resulting algorithm is not practical. We develop an effective method for computing $H_2$ in the case of orbifold groups arising as finite extensions of the fundamental group of hyperbolic rational homology $3$-spheres. As a special case, this yields explicit computations of the second homology groups of all cocompact lattices between $\pi_1(\mathcal{W})$ and its normalizer in $\mathrm{PSL}_2(\mathbb C)$, where $\mathcal{W}$ is the Weeks manifold. We also show that these lattices are absolutely profinitely rigid, completing work by Bridson, McReynolds, Reid & Spitler in this setting. As a special case, we determine that $H_2(\Gamma_{\mathcal{O}}, \mathbb Z) \cong \mathbb{Z} / 2\mathbb{Z}$, where $\Gamma_{\mathcal{O}}$ is the normalizer of the group of units $\Gamma_{\mathcal{O}}^1$ in a choice of maximal order $\mathcal{O}$ of the quaternion algebra associated to $\mathcal{W}$, thereby answering a question of Bridson & Reid. Although this non-vanishing obstructs one possible construction of Grothendieck pairs in $\Gamma_{\mathcal{O}}^1 \times \Gamma_{\mathcal{O}}^1$, we use our computations to show the vanishing of the second homology of another lattice whose derived subgroup is $\Gamma_{\mathcal{O}}^1$, which then yields Grothendieck pairs in this direct product by a theorem of Bridson & Reid. Finally, to showcase the generality of the techniques, we also compute the second homology of some finite extensions by orientable isometries of the fundamental group of some Fibonacci manifolds $M_n$.

We study the problem of optimally sending a real-valued source through multiple uses of a channel with feedback. First, we state a set of conditions that are sufficient for an encoder to achieve maximal mutual information between the source and all the channel outputs. This set of conditions are also necessary when the channel is input-identifiable, a condition widely satisfied by common channel models. More notably, we further study the information maximization-distortion minimization problem, where the mutual information between the source and all channel outputs still needs to be maximized, while at each step, the MMSE of estimating the source from the channel outputs so far also needs to be minimized. We derive a solution to this problem for discrete channels with certain symmetries, e.g. $k$-ary symmetric or $k$-ary erasure channels. We show that for such channels the famous posterior matching scheme, while not necessary for information maximization alone, is sufficient and essentially necessary for achieving both information maximization and distortion minimization. This work also provides a new perspective of regularizing distortion-minimizing feedback communication through information maximization, which enables us to find the optimal solution that otherwise would be intractable.

Consider the set of isoenergetically degenerate integrable Hamiltonians with two degrees of freedom. We show that a cusp-generic perturbation of a generic Hamiltonian in this set gives rise to meandering invariant tori - embedded Lagrangian tori which are not graphs. Moreover, an exponentially dense subset of perturbations admits higher order meandering tori, of all orders from two to infinity. These infinite order meanders have an endless nested structure.

Field quantization is a central feature of modern physics, that underpins the concept of photons and forms the foundation of quantum electrodynamics as well as much of solid-state theory. Classical linear wave equations are not generally expected to reproduce the quantization arising in quantum systems without introducing additional ingredients such as ad hoc nonlinear constraints, resonant particle-wave couplings or stochastic background fields. Here, we show that appropriately engineered linear wave media can recover fundamental features evocative of energy quantization in quantum mechanics. The key is to tailor periodic media in which wave propagation is strongly suppressed, except over a discrete set of narrow pass bands. In this regime, stationary wave solutions exhibit discrete energy and frequency spectra analogous to those arising in quantum mechanics despite the underlying dynamics remaining linear. Owing to the universality of the proposed mechanism, these effects may be realized experimentally using mechanical, electrical, or electromagnetic waves in appropriately designed periodic media. This work opens new avenues for designing metamaterials that enable control over discrete wave states while strengthening the conceptual bridge between classical and quantum wave physics.

Let $X$ be a quasiprojective subscheme of $\mathbb{P}^n_{\mathbb{F}_q}$. We prove a Bertini theorem for Hilbert–Samuel multiplicity of $X$; that is, there exists a positive-density set of hypersurfaces $H_f$ such that for every point $\xi\in X\cap H_f$, one has $\operatorname{ord}_\xi(f)=1$ and $e_\xi(X\cap H_f)=e_\xi(X)$. Furthermore, we extend this result on hypersurfaces to complete intersections, hypersurfaces containing a prescribed subscheme, and semiample linear systems.

We study degenerate controlled diffusions and Hamilton–Jacobi–Bellman equations posed on genuine continuum percolation clusters. The diffusion is constrained to evolve inside the infinite cluster and degenerates according to the distance to the irregular random boundary. Under suitable regularity and degeneracy assumptions on the diffusion matrix, we prove that the corresponding controlled diffusion never reaches the boundary and therefore admits a unique global strong solution. Using this result, we establish a stochastic control representation formula for viscosity solutions of degenerate Hamilton–Jacobi–Bellman equations posed on the random cluster, without imposing boundary conditions. We further verify that the structural assumptions introduced in this work, including quantitative integrability properties of the distance-to-the-boundary function and the associated degeneracy, hold for concrete continuum percolation models such as the Boolean model. In particular, although the diffusion never reaches the boundary, the boundary geometry remains a fundamental ingredient of the theory through the admissible degeneracy regime. The results establish a quenched framework for stochastic control and degenerate partial differential equations on genuine continuum percolation geometries, and identify a link between the analytic structure of the diffusion and the stochastic geometry of the underlying cluster. They also provide the foundational framework for the homogenization theory developed in the companion article [BMM26].

We develop a numerical algorithm for the efficient solution or approximation of solutions to the Helmholtz equation on a structured grid in two dimensions. We make use of the Operator Fourier Transform (OFT) and a low-rank cross approximation scheme (Cross-DEIM) to decompose the problem into an integral over a pseudo-time of solutions to the Schrödinger equation. The OFT is a framework for solving operator equations like fractional Laplacian equations or the Helmholtz equation, when the latter is written as a product of two paraxial operators. The main computational cost in the OFT is the solution to the Schrödinger equation, especially when the dimension or mesh resolution is high. In this work, we alleviate this cost by utilizing a low-rank method. Such methods aim to beat the curse of dimensionality when low-rank structures are present in the solution. We show that the combination of these two approaches can have large cost reductions for certain classes of problems.

We develop a fast space-time numerical scheme for approximating solutions to the linear wave equation. The approach is based on the sparse-grid combination technique applied to a coercive space-time discretization. Designed for tensor-product space-time discretizations, the method enables efficient parallelization of the resulting solver. We provide a rigorous theoretical analysis establishing convergence rates and computational complexity estimates. Numerical experiments validate the theoretical estimates and demonstrate the efficiency of the proposed method.