Papers

In these lecture notes, we discuss various `hydrodynamic LLN' and `CLT' scaling limits, among others, in types of stochastic interacting particle systems, connecting `microscopic' behaviors to continuum laws. Via `short stories', the aim is to present some of the `basics' for students and those entering the field, as a complement to books such as Kipnis-Landim 1999, Komorowski-Landim-Olla 2012, Liggett 1985, Liggett 1999. To be concrete, attention is restricted to a few `mass conservative' systems on discrete spaces, namely exclusion and zero-range processes, that have proved robust in the study of different phenomena. After preliminaries, we discuss the `entropy' and `relative entropy' methods to prove hydrodynamic limits of the bulk mass in finite volume, as well as other items such as construction of systems in infinite volume and the structure of their invariant measures, and scaling limits of local functionals, such as occupation times of sites and the motion of a tagged particle. In the last part, we also discuss equilibrium fluctuations of the bulk mass when the process starts from an invariant measure.

A notable feature of confined active matter systems is the tendency for motile particles to accumulate near solid boundaries. In various linear models with no-flux boundary conditions, this accumulation is realized through the development of sharp boundary layers at small particle diffusivity $\kappa$. In this paper, we present the first rigorous investigation of nonlinear boundary layers in the context of confined active matter. Specifically, we consider a family of 1D run-and-tumble models with nonlinear advection and tumbling on the half-line $\mathbb{R}_+$. We rigorously prove the vanishing diffusivity limit with quantitative convergence rates. In the limiting system, the boundary mass enters as a new variable which solves a nonlinear ODE, coupled to the PDE through a dynamic boundary condition. Interestingly, the nonlinearity on the boundary at $\kappa = 0$ cannot be obtained without reference to the boundary layer analysis at $\kappa \ll 1$. Numerically, these models exhibit rich behavior, including phase transition and hysteresis in the boundary layer.

Let $\mathrm{Sym(n,k)}$ denote the set of permutations on $\{1,2,\ldots,n\}$ with exactly $k$ cycles. A family $\mathcal{F}\subset\mathrm{Sym}(n,k)$ is said to be intersecting if $\sigma^{-1}\tau$ has a fixed point for all $\sigma,\tau\in\mathcal{F}$. In this paper, we investigate the size and structure of maximum-sized intersecting families of permutations in $\mathrm{Sym}(n,k)$. In the regime $k\leq n^{0.25}$, we show that every maximum-sized intersecting family is a star, meaning it consists of all permutations in $\mathrm{Sym}(n,k)$ that agree at a given point in $[n]$. We establish this result by proving a stronger stability result that bounds the maximum possible size of a non-centred intersecting family. Specifically, in the regime $k\leq n^{0.25}$, the size of any non-centred intersecting family is at most $\left(2/3+o(1)\right)$ times the maximum possible size of a star. In the tighter polylogarithmic regime $k\leq (\ln n)^{d}$, we improve this bound to $\left(1-1/e+o(1)\right)$ times the maximum possible size of a star; we show that this bound is asymptotically sharp. Thus, we establish both an Erdős–Ko–Rado theorem and its corresponding stability version for $\mathrm{Sym}(n,k)$.

Frankston, Kahn and Narayanan proved that every regular increasing $3$-wise intersecting family of subsets of $[n]$ has cardinality $o(2^n)$ using Friedgut's junta theorem. We give a short quantitative proof using elementary tools from the analysis of Boolean functions and entropy. More precisely, if $\mathcal{A}\subseteq\mathcal{P}_n$ is a nonempty $3$-wise intersecting family that is both regular and increasing, then $$ \log\frac{2^n}{|\mathcal{A}|}\ge \frac{n}{2}\left(\frac{|\mathcal{A}|}{2^n-|\mathcal{A}|}\right)^2, $$ and consequently $|\mathcal{A}|\le 2^n\sqrt{W(n)/n}$, where $W$ is the principal Lambert function defined by $W(x)e^{W(x)}=x$ for $x\ge0$. We also give a purely Fourier-analytic proof of the weaker estimate $$ |\mathcal{A}|\le \frac{2^n}{1+n^{1/3}}. $$

Directional subdifferentials of the value function provide a quantitative measure of optimal value response to perturbations. While existing results are largely limited to finite-dimensional settings, this paper develops a comprehensive variational framework in Asplund spaces. We establish essential directional calculus rules, extending directional nonsmooth analysis to infinite dimensions. To address the lack of compactness of bounded sets in infinite-dimensional spaces, we introduce a new directional condition, under which we derive upper estimates for directional limiting and singular subdifferentials of the value function. These results provide a refined analytical foundation for sensitivity analysis in infinite-dimensional hierarchical systems.

Bernard and Letac (1971) introduced a method for uniform random sampling among $m$ outcomes from an unknown biased source of independent and identically distributed symbols. The process terminates when the multinomial coefficient of the cumulative symbol counts equals zero modulo $m$. This study extends the computational and information-theoretic analysis of their construction by presenting five algorithms with formal correctness guarantees and comprehensive complexity analyses. For prime $m=p$, the Bernard-Letac framework is analyzed in greater detail. The Rényi entropies of the source yield an exact product formula for the expected number of draws. A first-order approximation consistently overestimates this value, and the entropy lower bound is never attained. As $p$, treated as a continuous parameter, approaches 1, the expected cost converges to a constant greater than 1, determined by the entire source distribution. Furthermore, for every prime modulus $p$ and every finite alphabet $I$, an explicit automaton with $p + |I| + 3$ states computes the mod-$p$ first-passage kernel of the walk from the base-$p$ digits of its arguments, reducing the fair assignment cost from quadratic to nearly linear.

Let $\text{F}$ be a non-archimedean local field of characteristic zero with residual characteristic $p$. In this paper we give an explicit description of the local Langlands correspondence for middle supercuspidal representations of $\text{GL}(2n,\text{F})$, under the tameness condition $p\nmid 2n$, in terms of the maximal simple types that define them. We achieve this by explicitly computing and comparing the gamma factors on the automorphic and Galois sides of the local Langlands correspondence. The computation on the automorphic side requires neither the tameness condition nor the characteristic zero condition.

Many estimators are ratios of coprime polynomials in a sample covariance matrix, and their accuracy depends on the relative fluctuation of the sample denominator. Under Gaussian sampling in fixed dimension, we call a nonconstant polynomial denominator self-normalizing if the first-order variance of its relative error does not depend on the population covariance. We prove that these denominators are exactly the flag powers, nonzero constant multiples of products of positive integer powers of nested generalized variances. Equivalently, the denominator's sample-to-population ratio has a covariance-independent finite-sample law, which we determine explicitly. Sufficiency is classical; the new converse shows that a first-order variance condition forces an exact sampling law. We show that relative stability, meaning bounded first-order relative variance, characterizes uniform tightness of scaled relative errors over positive-definite covariances. It permits replacing the sample denominator by its population value in the limit theory of the ratio. Self-normalization is its rigid core. We locate these classes in applications, where regression on predecessors in a fixed order yields only constant or self-normalizing denominators, instrumental-variable formulas yield relatively unstable ones, and nonparametric identifiability does not guarantee relative stability.

Let $S_n$ denote the smallest constant such that \[ |\det T|\|T^{-1}\| \leq S_n \|T\|^{n-1} \] for every invertible operator $T$ on every $n$-dimensional complex Banach space. In Hilbert space the optimal constant is $1$. For arbitrary Banach spaces, J. J. Schäffer proved in 1970 that \[S_n\leq \sqrt{en}. \] Subsequent work showed that $S_n$ grows like $\sqrt n$, but the sharp asymptotic constant has remained open for more than five decades. We resolve this problem by proving \[ \lim_{n\to\infty}\frac{S_n}{\sqrt n}=\sqrt e. \] Thus Schäffer's upper bound is asymptotically sharp, including its constant. Our proof is constructive, providing explicit Banach-space norms through duality and explicit matrices through the theory of model operators. At the analytic core of the argument, an extremal formulation of Schäffer's problem in the Wiener algebra reduces the matching asymptotic lower bound for $S_n$ to uniformly controlling the Taylor coefficients of products $QB_n$, where $B_n$ is a finite Blaschke product of degree $n$ and $Q$ is a polynomial factor. We optimize simultaneously the zero distribution of $B_n$ and the choice of $Q$. The resulting zeros follow a logarithmic asymptotic distribution, and a sharp uniform asymptotic analysis of the Taylor coefficients of $QB_n$ yields the constant $\sqrt e$. The corresponding model operators then yield matrices with these spectra that asymptotically attain Schäffer's bound.

For a countable family of hyperplanes $H_j\subset \mathbb{P}^m$, $j\in\mathbb{N}$, in general position and a linearly nondegenerate holomorphic curve $f\colon \mathbb{C}\to \mathbb{P}^m$ of finite lower order, we prove that the Nevanlinna defects $\delta_f(H_j)$ satisfy $$ \sum_{j=1}^{\infty}\delta_f(H_j)^{1/3}<\infty. $$ This resolves a long-standing open problem in Nevanlinna theory and extends Weitsman's celebrated scalar endpoint theorem (the case $m=1$) as well as Krutin's results for exponents strictly greater than $1/3$. The same uniform finite-family estimate yields the corresponding endpoint theorem for divisors cut out on a projective variety by ambient hypersurfaces of uniformly bounded degree, assuming that the divisors are in general position with respect to the variety and that the curve is not contained in the support of any divisor.

Let $(M^n,g)$ be a connected, closed, smooth Riemannian manifold with dimension $n\geq 3$. There exists a positive constant $\varepsilon_n<1$ such that, if Ricci curvature $\operatorname{Ric}_g\geq \varepsilon_n(n-1)g$ and the scalar curvature $R_g\geq n(n-1)$, then the volume $V_g(M^n)$ is less than or equal to the volume of standard $n$-sphere. This confirms a conjecture by Bray in 1997.

We extend the analysis of the class of quantum phase transitions (QPTs) that can be interpreted as condensations in state space, first introduced in [M. Ostilli and C. Presilla, J. Phys. A 54, 055005 (2021)], by generalizing the arguments of [M. Ostilli and C. Presilla, Phys. Rev. Lett. 127, 040601 (2021)] to prove the existence and determine the location (via simple bounds) of QPTs in general one-parameter lattice Hamiltonians. Unlike our original formulation, this extension also encompasses second-order QPTs, for which we provide the explicit example of the transverse-field Ising model. Our analysis suggests that, under conditions typically satisfied in physical contexts, any QPT taking place in lattice systems can be interpreted as a condensation in state space.

A central methodological question in data-driven control is whether to adopt a direct or indirect approach. Direct methods infer a controller or certificate directly from data, while indirect methods first identify a system model and then apply model-based control techniques. Recent developments of the direct method have led to finite-sample guarantees for the data-driven stability analysis of switched linear systems under various settings. However, for the indirect method, such guarantees remain largely elusive. In this paper, we provide a novel framework for the stability analysis of switched linear systems from noisy state measurements, using the indirect approach and quadratic Lyapunov analysis. Our framework comes with finite-sample guarantees on the convergence rate of the system. For that, we combine generalization bounds from machine learning and system identification with sensitivity analysis from quadratic Lyapunov analysis. To enable comparison, we also extend existing direct data-driven methods to handle measurement noise beyond the bounded noise case currently available in the literature. Finally, we compare the two approaches through numerical experiments, revealing that under moderate-to-high noise levels the indirect approach yields tighter probabilistic guarantees as well as greater robustness to noise and outliers than the direct approach

We study embeddings and operators on the Möbius-invariant spaces $Q_K$ using two dyadic capacities. One capacity is defined by analytic packets; the other is defined by domination of positive semidefinite matrices. We prove that they are equivalent. They give criteria for bounded and compact embeddings $Q_K\to L^2(\mu)$, bounded and compact Volterra operators, and pointwise multipliers. For the weights $K(t)=t^p/\log^\beta(e/t)$, $0<p<1$ and $\beta\ge1$, we prove $\mathcal M(Q_K)\subsetneq H^\infty\cap Q_K^{\log}.$ This disproves the logarithmic multiplier conjecture recorded in Bao and Wulan's 2021 survey.

We define and study the arithmetic de Rham stack $X^{\operatorname{arith}}$ of a scheme $X$ over a field of characteristic $p$, and analyze its relation with related stacks such as the Hyodo–Kato stack. We show that $X^{\operatorname{arith}}$ gives a stack-theoretic approach to rigid cohomology and its coefficients, known as overconvergent isocrystals and arithmetic $D$-modules, which avoids the long-standing problem of frame-choosing in earlier approaches. We finally give some arithmetic applications of our formalism, such as a proof of Berthelot's conjecture on the preservation of overconvergent isocrystals by smooth proper pushforward.

We present a note on two related questions on 4-forms in 8-dimensions. In particular, we clarify that the Cayley form is not unique in determining a metric via the formula of Karigiannis. Additionally, we discuss and prove part of a conjecture of Salamon and Walpuski.

The refined Diophantine exponent, recently introduced by the author, is a quantity that measures the periodicity of an infinite word. In this article, we study this exponent from combinatorial and topological viewpoints. First, we show that, over a ternary alphabet, the spectrum of the refined Diophantine exponent is $[1,\infty]$. Second, we show that this exponent has topological properties similar to those of the set of Liouville numbers. Finally, we provide concrete examples with the Champernowne, Rudin–Shapiro, and Thue–Morse words, words coming from coding a rotation by intervals, and bracket words.

Sublinear expansion is weak enough to be extracted from arbitrary graphs while retaining nearly all of their average degree, yet it has proved strong enough to force global structures in many sparse extremal problems. Letzter, Methuku and Sudakov [JLMS 2026] proved the existence of nearly Hamilton cycles in sufficiently dense regular sublinear expanders. Montgomery [ICM 2026] subsequently conjectured that, every $d$-regular sublinear expander with $d$ sufficiently large (but constant) is Hamiltonian. We disprove this conjecture in a strong form by constructing $n$-vertex $d$-regular sublinear expanders with $d=\left(\frac12+o(1)\right)\log^2 n$, which can forbid any cycle covering an arbitrarily small given positive constant portion of its vertices. The construction blows up one side of a biregular Ramanujan graph into almost-complete blocks and keeps the other side as a sparse vertex separator. We also prove a similar statement for a closely related notion of edge expanders. For every sufficiently small $\gamma>0$, there is an infinite family of $n$-vertex $d$-regular $\gamma$-edge-expanders with $d=\Theta(\gamma^{-1})$ and circumference $O(\gamma n)$, matching the standard lower bound $\Omega(\gamma n)$. The construction also comes from an expanding regular core such that each vertex has an almost-complete graph attached to it. The core guarantees edge expansion, while the single-vertex attachments confine every cycle.

We present a one-dimensional model for water infiltration coupled with a hydration reaction, relevant to coupled transport and chemical processes in the Earth's subsurface. In this model the sharp interface separating the saturated and dry regions evolves over time, leading to a moving free boundary problem. Consistent with recently presented numerical treatments in the literature, this solution indicates an interesting dynamic for the free boundary. At early time, for a given domain porosity, the infiltration front advances with a specific square-root-in-time behavior. At later time, depending on the consumption of the hydration reaction, the front advance can exhibit square-root, linear, or exponential forms. The presented closed solution provides an analytical tool that can be used to quantify important behavior in coupled transport and reaction systems.

In this note, we prove a priori interior curvature bounds for convex solutions to an elliptic Weingarten curvature quotient equation. The proof avoids integral estimates, doubling inequalities and compactness arguments. It relies on a concavity inequality for the equation operator and a novel choice of the auxiliary function to carry out an elementary maximum-principle argument. As an application, we obtain interior curvature bounds for convex solutions of a prescribed special Lagrangian curvature equation in dimensions three and four.

Classical information criteria such as the Bayesian Information Criterion (BIC) rely on regularity assumptions that break down for singular models, leading to incorrect model selection in settings such as deep learning. The Widely Applicable Bayesian Information Criterion (WBIC) relies on local learning coefficients $\lambda$, which in the analytic case coincides with local Real Log Canonical Thresholds (RLCT) of the Kullback-Leibler divergence of the model, to capture correct marginal likelihood asymptotics. Exact computation of the learning coefficients has been limited to special cases, and only sampling-based estimation methods are generally applicable. We present the first deterministic algorithm that computes local RLCTs exactly for any two-dimensional model whose Kullback-Leibler distance is contact equivalent to a polynomial, derive a bound on its complexity, and demonstrate its effectiveness for a broad class of models, with applications including polynomial neural networks. Beyond providing ground truth to calibrate sampling-based estimators, exact computation reveals algebraic structure in learning coefficients that sampling cannot and out-speeds it in the shallow regime.

We study continuous-time dynamic portfolio optimization under a Conditional Value-at-Risk (CVaR) constraint on the investor's terminal loss. For a general class of convex trading objectives, we exploit the auxiliary-threshold representation of CVaR to establish the existence of an optimal strategy and strong duality without requiring market completeness. These results motivate a dual-based nested bisection–golden-search algorithm over the threshold and Lagrangian multiplier, where the inner iterations reduce to standard unconstrained stochastic control problems. We prove that the resulting strategies converge to the optimal control as the number of iterations tends to infinity. Numerical experiments recover the Merton policy when the risk constraint is nonbinding. When the constraint is binding, the optimal strategy becomes state dependent: the investor reduces risky exposure following adverse outcomes but preserves, and near maturity may increase, exposure following favorable outcomes. Thus, a terminal CVaR constraint produces an asymmetric reallocation across states rather than uniform de-risking. Nontraded endowment risk amplifies the conservative adjustment, whereas price impact lowers desired positions and adjustment speeds.

We compute the Hofer-Zehnder capacity of the unit disk cotangent bundle of every ellipsoid in $\mathbb R^3$. The capacity is determined by the smaller of two distinguished quantities in the geodesic length spectrum: twice the systole and the length of the shortest simple closed geodesic of Morse index 3. For the lower bound, we use Riemannian billiards on a suitable cut of the ellipsoid. For the upper bounds, we develop two complementary methods. The first combines an argument by Hofer-Viterbo with neck-stretching and yields, more generally, an upper bound for positively curved Riemannian two-spheres in terms of closed geodesics of prescribed index. The second uses the pair-of-pants product in symplectic homology and the Viterbo isomorphism to bound the Hofer-Zehnder capacity of any disk cotangent bundles of Riemannian two-spheres by twice the diastole; for positive curvature, the diastole agrees with the systole.

We solve the open problem of determining the maximal value of Spearman's rho when Spearman's footrule is prescribed, thereby completing the exact attainable region of these two quantities. To prove this result, we reformulate the underlying copula optimization problem as an optimal transport problem with a linear moment constraint and construct the unique optimal coupling through a matching feasible dual potential and its contact set. Equivalently, this coupling minimizes the variance of $|U-V|$ among all couplings of $U,V\sim\mathcal{U}(0,1)$ with prescribed mean $\mathbb{E}|U-V|$. Our main result admits several applications: First, in the context of finite rankings, we obtain an improved Cauchy–Schwarz inequality between Spearman's footrule distance and the associated quadratic rank difference. Second, in the framework of generalized mixability, we characterize the attainable constant values of $|U'+V'|$, for $U',V'\sim\mathcal{U}(-1/2,1/2)$, and determine the minimal quadratic deviation for a given mean. Third, we derive explicit bounds relating Chatterjee's rank correlation $\xi(X,Y)$, which can detect complex functional dependence of $Y$ on $X$, to the copula correlation ratio–a rank-based fraction of explained variance–by exploiting their conditional i.i.d. representations in terms of Spearman's footrule and Spearman's rho.

Let $R$ be a commutative Noetherian ring, let $\mathbf{x}=x_1,\ldots,x_n$ be an $R$-regular sequence, and let $\mathbf{y}=y_1,\ldots,y_m$ be a sequence of elements of $R$. Put $I=(\mathbf y)$. Let $\mathcal S$ be a Serre subcategory of the category of $R$-modules. We consider the double complex obtained from the Koszul co-complex with respect to $\mathbf x$ and the Čech complex with respect to $\mathbf y$. Using the two spectral sequences associated with this double complex, we prove that \[ Ext_R^i(R/(\mathbf x),H_I^j(R))\in\mathcal S \quad\text{for all }i,j\in \mathbb N_0 \] implies \[ H_I^j(R/(\mathbf x))\in\mathcal S \quad\text{for all }j\in\mathbb N_0. \]

We prove that the orbit equivalence class of every free ergodic probability-measure-preserving (pmp) action of a free group contains a totally weak mixing action. Equivalently, every ergodic treeable pmp equivalence relation of cost $n\in\mathbf{N}\cup\{\infty\}$ is generated by a free totally weak mixing action of $\mathbf{F}_n$. This answers a question of Miller and Tserunyan. The proof goes by considering a Polish space of edge slidings along a fixed mixing transformation and proving that for every $w\not=e\in\mathbf{F}_n$ the set of edge slidings that produce an action with $w$ weakly mixing forms a comeager set.

We study a stochastic model for epitaxial thin-film growth driven by spatially rough additive noise in a regime where the noise is singular and regularization via truncation in Fourier space is used to give a meaning to the solution, leading to a vanishing nonlinearity in the limit. In order to study this phenomenon numerically, the nonlinearity is discretized by a spectral Galerkin projection, while time integration is performed with an exponential Euler scheme. For roughness stronger than space-time white noise, we derive strong error estimates in $L^p(\Omega;C([0,T];\mathcal H^1))$ that display explicitly the interaction between the spatial cut-off, the time step, and the decay of the nonlinear current. We also quantify the growth of the truncated stochastic convolution and the corresponding vanishing rate of the nonlinearity. Numerical experiments illustrate the transition from persistent hill formation to noise-dominated dynamics as the roughness parameter increases.

For a closed symplectically aspherical manifold, we introduce cap-decorated Floer persistence by refining the action-filtered Hamiltonian Floer persistence module with layers defined by iterated actions of homogeneous cohomology ideals. We prove functoriality and stability of these layers, and construct a universal cap-profile pseudodistance bounded above by Hofer distance. As an application, we obtain new invariants of Hamiltonian conjugacy classes, detecting symplectic mapping-class displacement phenomena invisible to ordinary Floer barcodes and spectral invariants. In particular, for a genus-two surface we construct Hamiltonian diffeomorphisms with identical ordinary Floer data but positive separation in the Hamiltonian-conjugacy quotient.

Random access is a central requirement in DNA-based storage systems: one would like to recover selected information symbols without sequencing the whole encoded object. A recent combinatorial model associates to a generator matrix $G\in F_q^{k\times n}$ the random variable $\tau_i(G)$, measuring the number of sampled columns needed to recover the information vector $e_i$. We study the cardinality-based extremal and finite-geometric aspects of simultaneous multi-symbol recovery. For a nonempty set $I\subseteq[k]$, let $\tau_I(G)$ denote the number of random column samples needed until all vectors $e_i$, $i\in I$, lie in the span of the observed columns. This variable interpolates between the singleton random access problem and the full-recovery problem underlying coverage depth. For each $m$, we introduce uniform worst-case and average parameters over all requested sets $I$ with $|I|=m$. Using the known subset-counting formula for $E[\tau_I(G)]$, we establish general upper and lower bounds for these parameters. In particular, the lower bounds are expressed through order statistics of the singleton recovery variables and specialize to the known singleton bounds when $m=1$. For systematic MDS encoders, we record an equivalent form of the known multi-symbol expectation formula and derive monotonicity and asymptotic consequences. For simplex encoders in arbitrary dimension, we obtain closed formulae in terms of Gaussian binomial coefficients; the full-recovery endpoint agrees with the known coverage-depth formula for simplex codes. Finally, in dimension three we study balanced quasi-arcs and compare their values with the simplex and MDS benchmarks.

We explore quantum Turing pattern zoology, where the same Lindblad equation supports a morphology atlas of stripes, spots, holes, labyrinths, and defects. The stable stripe species provides a quantitatively controlled case in which morphology and Gaussian witness loss separate parametrically. In particular, visible Turing stripes can remain after two Gaussian witness margins associated with the same $k_*$ mode cross zero in a completely positive Lindblad lattice. The witness thresholds on the exact shell fall as $\mathcal{N}^{-1}$. The stripe nematic threshold tends to a nonzero value at fixed lattice size, time window, and morphology criterion. The ratio of the morphology threshold to either witness threshold therefore grows with $\mathcal{N}$. Imaging and momentum-resolved covariance measurements probe these sectors separately.