Papers

The Renormalization Group (RG) is a powerful and versatile framework for analyzing complex physical systems. Here, we reinterpret it as a stochastic process that explores physical phase spaces, scale by scale, progressively revealing finer details of small-scale structures. This perspective establishes a natural connection to other random exploration processes, such as the Schramm-Loewner evolution, and is more suited for a probabilist audience. It links RG concepts such as RG transformations, effective actions, etc, to usual probabilistic tools such as conditional expectation values, martingales, etc, but also makes contact with stochastic quantization.

In this note, we prove that an arbitrary product of $W$-spaces is a $\kappa$-Fréchet-Urysohn space. We also show that the boundary tightness of a product $X \times Y$ is at most $\kappa$ provided that $X$ is compact with tightness $t(X) \leq \kappa$ and $Y$ has boundary tightness $tb(Y) \leq \kappa$. The first result fully resolves, and the second partially addresses, two questions recently raised by Tkachuk.

Coset incidence geometries are an important tool which connect group theory and geometry. While the representations of the unitriangular group and its pattern subgroups have been studied extensively, the underlying geometric structures of these groups has remained largely unexplored. In this article, we construct the natural coset geometry associated with each pattern group over a finite field $\FF_q$, and prove geometrical properties of theses structures. We show how combinatorial features of their defining closed sets directly correspond to properties of their parabolic subgroups, in particular the intersection of parabolics and normality of the Borel subgroup. Finally, we characterize the automorphisms of these geometries and examine how pattern geometries behave under change of field within the same characteristic $p$.

We investigate local and global bifurcation of periodic solutions for two classes of nonlinear fractional differential equations with symmetry. For the one-sided periodic Weyl equation, the nonzero temporal Fourier modes give rise to complex characteristic functions, leading naturally to a two-parameter bifurcation problem. The associated local invariant is expressed in terms of winding numbers and twisted equivariant degree. In contrast, the periodic Riesz equation is governed by real spectral quantities, and its local bifurcation invariant is obtained from the jump of the equivariant degree across isolated critical values. In both settings, the decomposition into spatial isotypical components and temporal Fourier modes determines the critical representations and the possible symmetries of bifurcating solutions. A nonvanishing local invariant yields the existence of nearby nontrivial periodic solutions, while the corresponding global bifurcation theorems describe the continuation of connected solution components away from the trivial branch.

We will provide an example of a Banach space with the Daugavet property which does not possess the weak operator Daugavet property. We will also show that every separable Banach space with the Daugavet property contains a non-poor subspace isomorphic to $\ell^1$ while noticing that every separable subspace of $L^\infty[0,1]$ is poor.

We construct examples of fibrewise Anosov flows over the Fenley–Mann–Potrie example of dimension 4. This is a family of non-algebraic Anosov flows with unstable dimension $1+4n$ and stable dimension $2+4n$, for every positive integer $n$.

We address the problem of solving large-scale tensor-structured linear systems in the Tucker format. In this setting, standard iterative solvers such as GMRES face a fundamental bottleneck: the multilinear ranks of the Krylov basis vectors grow with the iteration count, leading to rapidly increasing tensor operation costs and memory requirements. To overcome these challenges, we propose two randomized algorithms within the sketched GMRES framework that replace full Arnoldi orthogonalization with short recurrences. The first, RHOSVD-Tucker sGMRES, uses randomized HOSVD with per-iteration rank selection, providing robustness across a wide range of problems. The second method, MLN-Tucker sGMRES, leverages the multilinear Nyström approximation with a fixed rank, enabling streaming computations; the streamability of the approximation further allows, at no additional cost, a memory-efficient reconstruction of the solution from a compact sketched representation of the Krylov basis. Both methods outperform standard low-rank Tucker solvers in symmetric and non-symmetric settings. Applied to inverse problems, the low-rank Tucker constraint acts as an implicit regularizer; combined with adaptive projected Tikhonov penalization and automatic regularization parameter selection, the methods yield stable reconstructions.

Derangements are permutations without fixed points, and are in bijection with desarrangements: permutations whose first non-descent is even, or equivalently, permutations without “pixed points”. Bsila, Cox, Hugo, Styron, and Zhuang recently proved a theorem characterizing all $\Pi\subseteq\mathfrak{S}_{3}$, such that $1\leq\left|\Pi\right|\leq3$, for which the number of derangements avoiding all patterns in $\Pi$ is equal to the number of desarrangements avoiding all patterns in $\Pi$. They left finding a bijective proof of this theorem as an open problem, and posed a related conjecture concerning the distributions of fixed points and pixed points over pattern avoidance classes. In this paper, we give bijective proofs of this theorem and conjecture.

This paper establishes sharp dimension-free Frobenius-norm concentration inequalities for sample moment tensors. Our bounds are optimal over the sub-Gaussian class, while for Gaussian data we obtain matching two-sided estimates. We also identify a parity effect: the intermediate Gaussian and sub-Gaussian scales coincide at odd tensor orders but differ at even orders. Our second main result, which supplies the weak-moment estimate behind these bounds, proves a dimension-free moment bound for Hilbert-valued polynomials under Gaussian convex domination. The proof of this bound combines a recently developed structured martingale coupling theorem with new estimates for moment tensors of uniformly log-concave distributions.

We consider a bounded set $P \subset \mathbb{R}^d$ and the lattice-point enumerator $L_P(t) = |tP \cap \mathbb{Z}^d|$ for real $t > 0$. We show that if two bounded measurable sets with boundary of measure zero have the same real-parameter lattice-point enumerators for all integer translates, then their indicator functions agree almost everywhere. As a corollary, any convex body is uniquely determined by this data. Our proof is short and Fourier-analytic, where the key device is a periodic point-counting function whose Fourier coefficients recover the Fourier transform of the indicator function on a dense set. This recovers and extends, with a unified argument, the uniqueness results for rational polytopes and symmetric convex bodies established by Royer [arXiv:1712.01973, arXiv:1712.03937], whose proofs relied on intricate case-specific geometric constructions.

In a multi-agent system, there are often stubborn followers of specific opinions or beliefs. Motivated by this observation, in this paper, we aim to understand how stubborn agents affect the distribution of opinions in a network where both stubborn and non-stubborn agents interact with each other. To do so, we assume that all agents have an opinion in the set $\{0,1\}$ and each non-stubborn agent updates its opinion according to the $2k$-choices rule, where the agent samples $2k$ neighbours (including both stubborn and non-stubborn neighbours) uniformly at random and adopts the majority opinion among the sampled group of neighbours and itself. We assume that a proportion of agents, $\gamma_i$, are stubborn followers of opinion $i\in \{0,1\}$. It is natural to expect that the steady-state distribution of the opinions in the network will be dominated by the opinion with the larger proportion of stubborn followers. We show that while this is true, the time to reach steady-state depends heavily on the values of the parameters $\gamma_0$ and $\gamma_1$. When the individual values of these parameters, as well as their difference, are small, it can take an exponentially long time (in the network size) to reach the steady-state. In sharp contrast, when at least one of the parameters $\gamma_0$ and $\gamma_1$ is large, the network reaches the steady-state in a time that is only logarithmic in the network size. Hence, there exists a sharp phase transition in the network dynamics based on the proportions of stubborn agents. We also characterise the behaviour of the system when the parameters $\gamma_0$ and $\gamma_1$ lie on the boundary of the phase transition. In this boundary region, we show using Stein's method that the dynamics are driven by a diffusion process which takes polynomial time to mix.

Let $K\subset\mathbb{R}^n$ be a convex body and $C=\frac12(K+(-K))$ its central symmetral. We study the defects $\mathcal A_k(K)=W_{n-k}(C)-W_{n-k}(K)$ through the even–odd decomposition of the support function. We derive exact even mixed-volume expansions, identify their Krawtchouk transform with the mixed difference-body coefficients, and obtain a Kubota formula expressing each defect as the averaged symmetrization gain of projections. These identities yield strengthened isoperimetric inequalities. The quadratic defect is the average of the absolute oriented areas of projected Wigner caustics and obeys a sharp spherical-harmonic stability estimate. We give explicit formulas in dimensions $2$, $3$, and $4$. The four-dimensional expansion includes a sign-indefinite quartic Wigner volume; on a plane of cubic harmonics we find nonspherical constant-width bodies for which it vanishes. Finally, we prove a projection-transfer bound and an arbitrary-dimensional expansion for constant-width bodies.

This paper develops a path-space theory of entropy production for a class of stochastic evolution equations on infinite-dimensional Hilbert spaces. Since such spaces have no canonical Lebesgue reference measure, the usual finite-dimensional density formulas do not extend directly. We instead work relative to the invariant Gaussian measure of a reversible Ornstein–Uhlenbeck reference process. Combining an infinite-dimensional Girsanov transform, time reversal of the reference process, and the stationary Fokker–Planck equation relative to the Gaussian measure, we derive an explicit entropy-production formula in terms of an irreversibility field. On the natural test class, this field represents the difference between the forward and reversed nonlinear drifts. Under the standing assumptions, vanishing entropy production is equivalent to vanishing stationary probability current, self-adjointness of the generator in the invariant Hilbert space, detailed balance, and invariance of the stationary path law under time reversal. The reversible case is therefore characterized by a Gaussian-reference gradient structure for the nonlinear drift.

This paper presents a classification of induced centrosymmetric alternant codes through the study of the automorphism structures inherited from Generalized Reed–Solomon (GRS) codes. We introduce the twisted conjugation action naturally associated with projective semilinear transformations and establish its correspondence with ordinary conjugacy in the projective semilinear group. This correspondence enables the application of Shintani's theorem to classify the $\gamma_{p^j}$-similarity classes of involutions. As a consequence, we obtain necessary and sufficient conditions for an alternant code to admit a centrosymmetric structure induced by a projective semilinear automorphism. The resulting classification unifies the different families of induced centrosymmetric alternant codes within a common automorphism-based framework.

We study high-dimensional numerical integration with respect to complex target measures using diffusion-based transport maps and randomized quasi-Monte Carlo (RQMC). Score-based diffusion models induce a deterministic probability flow ODE that transports a simple prior to the target, suggesting a principled way to transform low-discrepancy points on the unit cube into informative samples. We construct a cube-to-target map by composing a Gaussian base transformation (the component-wise inverse Gaussian CDF) with an Euler-discretized probability flow ODE. To retain unbiasedness under transport approximation, we formulate integration as importance sampling (IS) on the cube. Our main result provides verifiable conditions under which the resulting IS integrand satisfies the boundary growth condition, implying an $O(N^{-1+\epsilon})$ RMSE for scrambled nets. We then establish these conditions for diffusion probability-flow transport under mild bounded-derivative assumptions on the learned vector field, explicitly controlling the boundary singularities introduced by the inverse Gaussian CDF. Experiments range from a 2D mixture to 784D images and a 40,960D conditional vorticity-assimilation task; in the latter, blocked scrambled Sobol' sampling reduces the randomization standard deviation of nonlinear accuracy metrics at essentially unchanged online denoising cost. Together, these results give a theoretical and empirical foundation for combining diffusion generative modeling with high-precision RQMC integration.

In this note, we prove that there exist two families of bi-rotational asymptotically conical self shrinkers with boundary called `trumpets', and two families of bi-rotational self shrinkers with boundary which close up called `disks', so that these families together with two generalized self shrinking cylinders, and a minimal cone foliate the whole space except some compact set containing the origin.

Following recent developments on matrix-valued orthogonal polynomials (MVOPs), we construct the matrix Szegő factorisation of the matrix weight function when it is supported on multiple intervals. We find that on the gaps the matrix function has unitary multiplicative jumps. In the next part, performing the Deift-Zhou steepest descent analysis we study the associated global parametrix which comes from the Riemann-Hilbert problem of the MVOPs. A solution is constructed visualising the rows of the parametrix as sections of vector bundles on a hyper-elliptic Riemann surface. An alternate view point to the global parametrix is also presented using a vanishing lemma. As an application we obtain strong asymptotics of MVOPs for multiple cuts which extends the works of Deaño, Kuijlaars, and Román.

We study the axisymmetric partial engulfment of a small rigid sphere by a fluid Helfrich membrane. When the size ratio $\eps$ between the particle and the membrane is small, the neck that joins the wrapped cap to the surrounding membrane is an elastic boundary layer, and we analyse it by matched asymptotic expansions. The leading inner surface is a catenoid, a minimal surface that stores no Helfrich energy, so that the energy of partial engulfment is carried by the first correction and appears only at order $\eps^{2}\ln(1/\eps)$. We obtain it by solving the inhomogeneous Jacobi equation of the catenoid, forced by the spontaneous curvature and by the ambient mean curvature that the neck must match, and we give its coefficient in closed form. The boundary layer can then be integrated out, and the neck replaced by a scalar self-energy carried at the pole, so that the outer field can be closed independently of the inner one. For a membrane coupled to a tension reservoir we derive the binding threshold, which turns out to be independent of both the tension and the spontaneous curvature, the complete-wrapping threshold, and the hysteresis of the envelopment transition. The neck self-energy law is confirmed against the full nonlinear shape equations.

We study the graphic $s$-$t$ path TSP on subcubic graphs (maximum degree 3): given distinct vertices $s,t$, find a shortest $s$-$t$ walk that visits every vertex. We prove an upper bound with the asymptotically optimal leading coefficient $5/4$ for every terminal pair, even when $G-\{s,t\}$ is disconnected. Specifically, every simple 2-connected subcubic graph $G$ on $n$ vertices has a spanning $s$-$t$ walk of length at most $\lfloor(5n+n_2(G))/4\rfloor$, where $n_2(G)$ counts its degree-2 vertices. An $O(n^2)$-time algorithm attains this bound. Combining an edge-rooted even-cover theorem of Wigal, Yoo, and Yu (WYY) with an even-cover-to-walk lemma proved here yields this bound for adjacent terminals, a consequence not stated explicitly in their paper. We extend the bound to arbitrary terminal pairs. For cubic graphs, it becomes $\lfloor 5n/4 \rfloor$, to our knowledge the first direct $5/4$ bound for cubic path TSP that does not use the general path-to-tour reduction.

In the unbiased Chooser-Picker (also known as Client-Waiter) game played on the edge set of a graph, Picker offers a pair of unclaimed edges in each turn, Chooser claims one, and the remaining edge goes back to Picker. We study the Chooser-Picker (C-P) degree game played on $d$-regular graphs, where Chooser aims to maximize the maximum degree of their induced subgraph, and Picker's objective is to defend every vertex by securing a certain minimum degree in Picker's own subgraph. While classical static pairing strategies guarantee a minimum degree of at least $\lfloor d/4 \rfloor$ for Breaker on general $d$-regular graphs in Maker-Breaker (M-B) games and for Picker in C-P games, outperforming this threshold has been a major open challenge in both frameworks. According to the foundational monograph of J. Beck, this challenge stands as the first among the seven most humiliating problems in combinatorial game theory. Our main result is that Picker can beat the $d/4$ bound. First, we prove that Picker can always guarantee a degree of at least one at every vertex on any $3$-regular graph. Based upon this we introduce a direct strategy to prove that Picker can secure a degree of at least $\lfloor d/3 \rfloor$ at every vertex for any $d$-regular graph. This highlights a fundamental structural advantage that Picker usually possesses over Breaker in sparse local games.

Let $L$ be a Lie algebra acting by derivations on an associative algebra $A$ over a field $F$ of characteristic zero. The polynomial identities satisfied by $A$ with respect to this action are called differential identities, or $L$-identities. In this paper, we study the differential identities of the algebra $UT_k$ of $k\times k$ upper triangular matrices and take a first step toward the classification of minimal $L$-varieties of differential exponent $3$. We first prove that, whenever $UT_k$ generates a minimal variety of algebras with derivations, the $L$-action can be replaced by its semisimple part. More precisely, it is enough to consider inner derivations induced by diagonal elements. We then apply this reduction to $UT_3$ and explicitly determine the $T_L$-ideal of differential identities and the corresponding differential codimension sequence for every such action on $UT_3$. Finally, we show that every $L$-variety generated by $UT_k$, with $k\geq 3$, contains $UT_3$ endowed with one of these $L$-actions.

We give a new proof of the Bernoulli theorem, conjectured by Talagrand and proved in the seminal work of Bednorz and Latała. Our approach is based on information-theoretic ideas: lower bounds on the supremum of a Bernoulli process are translated to the fundamental limits of Bayesian estimation in a Cauchy additive channel. This leads to a new information-theoretic functional that characterizes Bernoulli-process suprema and plays a role analogous to Fernique's majorizing-measure functional for Gaussian processes. The same viewpoint yields a distributional strengthening: for any prescribed law of the index, we characterize the largest expected value attainable over all couplings of that index with the Bernoulli process. This extends to Bernoulli processes a phenomenon previously understood for Gaussian processes through the work of Fernique and Talagrand.

For a discrete topological space $D$, let $\mathrm{Met}(D)$ denote the set of metrics on $D$ that are compatible with the discrete topology, equipped with the topology induced by the supremum distance. Ishiki's Conjecture 5.1 asserts that $\mathrm{Met}(D)$ is not completely metrizable when $\lvert D\rvert = \aleph_1$. We prove this in ZFC by constructing a set $A \subseteq [0,1]$ of cardinality $\aleph_1$ that is not $F_\sigma$ and embedding its complement as a closed subspace of $\mathrm{Met}(D)$. The proof has also been formalised in Lean 4.

We systematically investigate finite-difference (FD) derivative computation in Physics-Informed Neural Networks (PINNs) as an alternative to automatic differentiation (AD). On three benchmark PDEs we show that, with a properly calibrated step size, FD matches AD in accuracy on every problem while running faster across the full tested batch-size range and using substantially less GPU memory, and that a stochastic variant we propose outperforms AD on a stationary problem. We further show that for neural architectures with inter-sample dependencies (e.g. BatchNorm, self-attention) the standard PyTorch autograd idiom is silently incorrect; the correct per-sample alternative is computationally infeasible at PINN-relevant batch sizes, while FD provides a forward-only approximation that is empirically an order of magnitude closer to the true per-sample derivative.

This paper addresses the integrated Master Surgical Scheduling Problem and Surgical Case Assignment Problem under weekend bed capacity constraints. We propose a joint optimization framework that simultaneously determines the assignment of specialties and surgeons to Operating Room (OR) blocks, the selection and assignment of patients to these blocks, and the sequencing of surgeries, while accounting for reduced bed availability during weekends. The model is formulated as a multi-criteria Mixed-Integer Linear Program (MILP) that prioritizes the total priority-weighted amount of scheduled patients, minimizes excess weekend bed occupancy, and maximizes OR utilization. To solve large instances, we develop a Column Generation (CG) algorithm in which the pricing subproblem is formulated as a Resource-Constrained Shortest Path Problem. The framework accommodates block, open, and modified block scheduling policies. Computational experiments show that the CG approach outperforms the direct MILP solution in terms of solution quality and computational time, particularly for larger instances. We further analyze the trade-offs between weekend bed availability, OR utilization, and total patient priority. Our results provide managerial insights into how hospitals can balance patient access, operational efficiency, and staff workload when resources are limited.

Clustering is a fundamental class of data analysis techniques with the most important representatives being centroid-based methods like $k$-means. Such methods are strongly connected to quantization problems, which aim to approximate general probability measures with discrete ones. For example, $k$-means corresponds to quantization with respect to the Wasserstein distance. While Wasserstein quantization clusters points within a fixed space, this paper studies Gromov-Wasserstein (GW) quantization, which additionally aims at clustering the ambient geometry of the space. We show existence of solutions to the GW quantization problem and give a characterization that justifies an analogue to the $k$-means algorithm (Lloyd's algorithm) to approximate them numerically. We further calculate the quantization rate for usual Euclidean geometries that are used in the GW context, and relate it to standard Wasserstein quantization rates. Finally, numerical experiments show that GW quantization opens up many modeling possibilities beyond normal clustering methods (e.g., for geodesic distances of 3D shapes or structured pruning of neural networks) and that the introduced algorithm leads to useful numerical solutions with approximation quality often in line with theoretically optimal rates.

The main difficulty in 3D gravity inversion is that surface observations lack vertical wavenumber information, making the problem underdetermined and depth resolution poor. Building on the author's spectral-domain pseudo-inverse theory for unitary diagonalizable systems, this paper presents a true 3D inversion method. Surface data are analytically continued upward via Laplace's equation to form a 3D data volume, providing the vertical wavenumber sampling for the 3D Fourier transform. A general analytical expression for the half-space spectrum is derived, separating the horizontal spectrum from the vertical propagation kernel. The Green's function of the 3D Poisson equation is unitarily diagonalized, yielding the forward spectral response $\lambda(\kk) = -i 4\pi G k_z / K^2$, and a stable inverse filter $q_\alpha(\lambda) = \bar{\lambda}/(|\lambda|^2 + \alpha)$ is constructed. This filter is proved to have bounded stability and consistency (reducing to the exact inverse as $\alpha \to 0^+$). Validation with a homogeneous sphere model shows correct recovery of the anomaly location and singular behavior at the source. The contributions are twofold: (1) upward continuation constructs a 3D volume from 2D surface data, which does not add physical observations but unfolds the 3D spatial variation information (data innovation) derived through the governing physical equation; (2) the inversion is reduced to one forward 3D Fourier transform, one spectral scaling, and one inverse transform. The framework extends to any linear potential-field inverse problem with a translation-invariant forward operator, including magnetic inversion.

The study of coordinator polynomials of Weyl group lattices was initiated by Conway and Sloane. In 2013, by using a trigonometric substitution approach, Wang and Zhao proved the real-rootedness of the type $D$ coordinator polynomials. Subsequently, Xie and Zhang proved the compatibility between the types $C_n$ and $D_n$ coordinator polynomials. In this paper, we discover the interlacing pattern between the types $C_n$ and $D_n$ coordinator polynomials, which may be named as the second pattern of interlacing zeros.

In this paper, we prove that a smooth, properly embedded ancient mean curvature flow that is asymptotic to an $O(n)\times O(m)$ symmetric minimal quadratic cone for $n +m \geq 10$, and lies on one side of the cone has to have unique asymptotics in the parabolic region. When additionally assuming mean convexity, we upgrade unique asymptotics to full uniqueness, and show that such flow has to be a stationary flow given by one of the leaves of the Hardt-Simon foliation. This is the first rigidity/unique asymptotics theory for ancient mean curvature flow with a $\textbf{singular minimal cone as the asymptotic model}$.

We introduce a general monoidal category of $\mathbf{N}$-root crystals and then study the special case of square root $\mathfrak{gl}_n$-crystals. The latter objects include Yu's crystals on semistandard set-valued tableaux. Prior work of the first author, Tong, and Yu showed that regular square root $\mathfrak{gl}_n$-crystals can be a useful tool for proving Grothendieck positivity results. The objects studied here go beyond the regular case and allow us to construct a square root analog of the direct limit crystal $B(\infty)$. We give several descriptions of our square root of $B(\infty)$, using marginally large tableaux, the Lusztig or PBW parameterization, and the Nakashima–Zelevinsky polyhedral model. We show that this crystal has a simple character formula, exhibits a nontrivial Demazure filtration, and recovers Yu's semistandard set-valued tableau crystals after taking appropriate tensor products. We also investigate a number of differences between square root crystals and classical crystal constructions.