Papers

We study classical string commitment over quantum channels with limited preshared entanglement. For noninteractive protocols, we determine the commitment capacity of a class of channels with input dimension $d$ that, at each use, sample a pair of classical random variables $(F,Z)$, apply one of the $d^2$ Heisenberg–Weyl operators indexed by $Z$ to the input, and deliver the transformed quantum system together with $F$ to the receiver. If $E$ is the available entanglement rate in bits per channel use, then the capacity is $\min\{H(Z|F),\log_2d+E\}$. This class of channels encompasses quantum erasure and depolarizing channels, as well as families of Pauli channels. Additionally, for interactive protocols, we show that the commitment rate cannot exceed $\log_2d+E$ bits per channel use, so that when $H(Z|F)\geq\log_2d+E$, interactive communication does not increase the capacity. As a consequence, for interactive protocols, we determine the capacity of the quantum erasure channel.

We study a coupled Stokes-reaction-diffusion-advection system in a three dimensional domain perforated by a large number of evolving spherical inclusions with radii of order $\varepsilon^\alpha$ for $\alpha\in(1,3]$. Their centers are separated on the order of $\varepsilon$ but are not assumed to form a periodic lattice. The inclusions grow or shrink through an interfacial adsorption-desorption mechanism, so that the evolution of the geometry is coupled to the Stokes-reaction-diffusion-advection system. We first establish well-posedness on arbitrary large time intervals for sufficiently small $\varepsilon$. We then derive quantitative homogenization limits in terms of the limiting empirical measure describing the spatial distribution and size of the inclusions. Its evolution is governed by a continuity equation in the radius variable, and the convergence of the empirical measures is controlled in the $2$-Wasserstein distance. In the critical case, $\alpha=3$, the Stokes system converges to a Brinkman system, while non-vanishing concentration boundary layers modify the microscopic exchange law for the reaction-diffusion-advection equation. For $\alpha\in(1,3)$, the limit is of Darcy type and the original exchange law is retained. The homogenization results are quantitative and provide explicit error estimates.

We define an invariant of alternating links—a homogeneous, four-variable Laurent polynomial—that encodes the symmetrized Alexander polynomial, the signature, and other topological data. Along the way, we extend a spanning tree formulation of the Alexander polynomial due to Murasugi and Stoimenow from special alternating links to all alternating links. This project is motivated by Fox's trapezoidal conjecture; accordingly, we prove certain sequences associated to our invariant are trapezoidal for all alternating links. We also conjecture our polynomial has $M$-convex support, and that it satisfies symmetry and log-concavity properties. We prove a partial symmetry result.

We examine how the standard proof that left adjoints preserve colimits behaves in the setting of wild categories, a natural setting for synthetic homotopy theory inside homotopy type theory. We show that the proof may fail for adjunctions between wild categories and even produce a wild left adjoint that fails to preserve colimits. Our core contribution, however, is a sufficient condition on the left adjoint for the proof to go through. The condition, which we call 2-coherence, expresses that the naturality structure of the hom-isomorphism commutes with composition of morphisms. We present two useful examples of this condition in action. First, we use it, along with a new version of a known trick for homogeneous types, to show that the suspension functor, as well as a generalization thereof, preserves graph-indexed colimits. Second, we show that every modality, viewed as a functor on coslices of a type universe, is 2-coherent as a left adjoint to the forgetful functor from the subcategory of modal types, thereby proving this subcategory is cocomplete. We have formalized our main results in Agda.

We generalize the homological sieve method, developed by Das, Lehmann, Tosteson, and the author, to study certain quartic del Pezzo fibrations, and we prove a version of Manin's conjecture over global function fields in these cases. Our proofs combine the $3$-dimensional positive-characteristic minimal model program, the geometry of the space of sections and the Abel–Jacobi mapping, and the homological sieve method. Our quartic del Pezzo surfaces have Picard rank $2$, and admit two birational morphisms to non-split quadric surfaces. In particular, they do not possess any conic fibrations.

This paper develops the first distributed cross-layer algorithm for joint congestion control, routing, scheduling, and power control in covert multi-hop, multi-modal wireless networks, where adversarial wardens (Willies) monitor radio modalities via energy detection. The Detection Error Probability (DEP), the probability that a Willie fails to reliably detect ongoing transmissions, is generally non-concave in the transmit powers, making DEP-based covert network optimization challenging. We resolve this by constructing the tightest concave lower bound on the log-DEP, yielding a conservative convex problem that guarantees satisfaction of the original DEP constraints and unifies hard covertness constraints and covertness-utility maximization in a single problem. We develop a Parallel Proximal Alternating Direction Method of Multipliers (PP-ADMM) algorithm for the resulting cross-layer problem and prove global Q-linear convergence, i.e., exponentially fast convergence, to the set of optimal solutions under standard regularity conditions. Numerical results confirm linear convergence and demonstrate robust tracking performance under channel fading and Willie mobility.

We prove uniform exponential bounds for the compactly supported Betti numbers of spaces of morphisms from curves of fixed genus to projective varieties. For targets in a fixed projective space cut out by a prescribed number of equations of fixed degrees, the bound is exponential in the degree of the morphism and is independent of the ground field, the source curve, and the target. The proof constructs bounded-degree affine presentations involving only linearly many variables and equations, and then applies Katz's estimate. As an application, we establish a higher genus function field version of Manin's conjecture for split quartic del Pezzo surfaces, generalizing a recent theorem of Das–Lehmann–Tanimoto–Tosteson. The passage from $\mathbb{P}^1$ to arbitrary source curves requires uniform bounds for compactly supported Betti numbers over all curves of fixed genus, together with new geometric results on higher genus morphism spaces, including irreducibility and dimension estimates for the relevant incidence strata. These results, combined with the necessary control of the configuration space contributions, allow the homological sieve and virtual height zeta function argument to yield, over sufficiently large finite fields and within a slightly shrunken nef cone, the predicted asymptotic with the expected leading constant.

We establish a weighted Wasserstein spectral gap for the three-dimensional damped cubic wave equation with genuinely finite rank Brownian forcing. Under a saturation condition, the gap holds with respect to the negative phase topology $\mathcal E_s=H^{-s}\times H^{-1-s}$ for every $0<s<1/2$, from which we deduce unique ergodicity and exponential mixing in the energy topology. Sharp geometric characterizations of the saturation condition are also obtained. The method we develop is a stable–compact asymptotic coupling mechanism for hypoelliptic dissipative SPDEs beyond the parabolic setting. Instead of relying on positive time smoothing or asymptotic gradient estimates, it reduces the infinite-dimensional obstruction to contraction to a compact defect in a weaker coupling topology. Dense Malliavin range then permits this defect to be compensated by a finite dimensional Cameron–Martin shift, producing a finite distance contraction on bounded Lyapunov cores. Together with a separate high energy contraction from dissipation, this yields a global weighted Wasserstein spectral gap.

A conjecture of Komlós states that the combinatorial discrepancy of any matrix $A\in\mathbb R^{m\times n}$ whose columns have Euclidean norm at most one is bounded by a universal constant. We prove that the combinatorial discrepancy of every such matrix is at most $O((\log n)^{1/4}(\log\log n)^{7/4})$. This is the first asymptotic improvement over the $O(\sqrt{\log n})$ bound established by Banaszczyk [Banaszczyk, Random Struct.\ Algorithms, 1998], and it refutes a conjecture of Hajela [Hajela, European J.\ Combin., 1988] that a lower bound of order $\Omega(\sqrt{\log n})$ should hold.

Markovian open-system dynamics have widespread applications throughout quantum information science, including algorithmic state preparation. Their convergence is commonly quantified using the worst case global trace distance between the evolving and stationary states. However, this criterion can be unnecessarily stringent when only physically relevant observables are of interest. Here we introduce and study observable-specific mixing times. We prove that, for quasi-local, rapidly mixing Lindbladians, sums of geometrically local observables equilibrate in a time independent of system size, in contrast to the logarithmic dependence of global state mixing. This separation reduces the runtime of dissipative quantum algorithms, including quantum Gibbs samplers, for estimating quantities such as the Gibbs state energy and local order parameters, yielding an overall scaling that is linear in system size. Complementing this quantum result, we develop a quantum-inspired classical algorithm for estimating the same quantities. Its runtime is likewise linear in system size, but scaling exponentially in $\mathcal{O}\big(\log(1/\epsilon)^D\big)$, where $D$ denotes the spatial dimension of the lattice. We further analyse non-interacting Lindbladians over qudits, fermions, and bosons, demonstrating that locality of observables is not always necessary for a qualitatively faster mixing. Small-scale simulations of quantum Gibbs samplers reveal no large hidden constants in our asymptotic analysis and show that the theoretical predictions closely capture the finite-size dynamics.

This paper provides sharp rates of invasion in nonlinear integro-differential equations of ignition type. The framework covers both integrable convolution kernels and singular fractional-type kernels. While the emergence of a threshold on the decay $1+2s$ of the jump kernel is expected from earlier works, far fewer quantitative results were available in such a general framework. In particular, in addition to the spreading rates, our study provides information on the shape of the invasion profiles. Importantly, we obtain the first sharp spreading estimate in the critical case $s=\frac12$, which separates linear-in-time and accelerated regimes.

For finite-dimensional linear inverse problems where the variables are Gaussian, it is well-known that the minimum-mean-square error estimator takes the form of a regularized least-squares data fit. In this chapter, we show that this equivalence extends to a much broader infinite-dimensional setting where generalized splines take the role of linear regressors and generalized Gaussian processes on a nuclear space $S$ are the counterpart of Gaussian random vectors. The scope of this extension is of the same nature as the switch from the classic notion of function to that of a distribution, also known as a "generalized function." Our formalism involves a whitening/regularization operator $L: S\to S'$ whose continuous extension induces a native Hilbert space $H\subset S'$ that plays a central role in our characterization. The presentation is self-contained for the most part and remarkably general and powerful. It allows for the recovery of all known instances of such equivalences; in particular, the methods involving innovations and reproducing-kernel Hilbert spaces developed by Kailath and his students, and the mathematical correspondence between fractional splines and Mandelbrot's fractional Brownian motion (fractals), with the former being the optimal estimators of the latter. It also covers general Bayesian methods for the resolution of infinite-dimensional inverse problems.

Given a pair of commuting, ergodic, trace-preserving and trace-integrable actions of locally compact unimodular groups on a common semifinite von Neumann algebra we prove a form of associativity for convolutions of triples of operators. We view this as a basis for a general version of quantum harmonic analysis.

In this paper, we study a significance-driven cross- layer semantic communication design problem. Based on sta- tistical decision theory, we introduce an information-theoretic measure of per-sample data significance that quantifies the task-specific value of each individual observation. Using this metric, we formulate a cross-layer optimization problem that simultaneously optimizes (i) physical-layer semantic encoding and inference and (ii) MAC-layer resource allocation, with the objective of maximizing semantic spectrum efficiency, defined as the semantic value delivered per unit bandwidth per unit time. At the physical layer, we develop Meta-Learning Variational Information Bottleneck (Meta-VIB), a new semantic transceiver that employs a meta-learned hypernetwork to compress high- dimensional observations into semantically significant latents, enabling instantaneous adaptation to dynamic channel conditions and varying symbol budgets without online retraining. At the MAC layer, we model channel allocation as a Multi-Action Restless Multi-Armed Bandit (MA-RMAB) and adopt the Q- Maximization algorithm, which dynamically allocates channel resources to sensors based on their semantic value of information. Experimental results on a real-world pedestrian safety dataset demonstrate that our joint design achieves substantial gains in semantic spectrum efficiency over baselines, reaching up to 1000 times gain at an average SNR of 0 dB and 40 times gain at an average SNR of 5 dB.

Let $X$ be a smooth projective toric variety of dimension $n\geq1$ over an algebraically closed field of characteristic zero, let $\mathcal E$ be a toric vector bundle of rank $r\geq2$, and let $\pi\colon Y=\mathbb P_X(\mathcal E)\to X$ be the projective bundle of one-dimensional quotients. Write an ample line bundle on $Y$ as $A=\mathcal O_Y(a)\otimes\pi^*L$, with $a\geq1$. We record a blow-up argument proving that $K_Y+mA$ is globally generated whenever an integer $m$ satisfies $ma\geq r$ and $m\delta(A)>n$, where $\delta(A)$ is a positive integer obtained from the degrees of $A$ on the invariant quotient sections over the torus-invariant curves of $X$. In particular, $K_Y+mA$ is globally generated for $m\geq n+1$ and $ma\geq r$. Consequently every projectivized toric vector bundle satisfies Fujita's freeness conjecture. The uniform bound is sharp. We also formulate the result as a global-generation theorem for adjoint symmetric powers of $\mathcal E$ and explain its relation with the Seshadri-constant results of Hering–Mustaţă–Payne and Fulger–Murayama. ChatGPT (OpenAI) was used to assist with mathematical discussion, language, and bibliographic searches.

The singular difference graph, denoted by $\Gamma$, of the vector space of square matrices over a field is a graph whose vertex set is the set of all elements of the vector space, where two distinct vertices are adjacent if and only if the difference of the corresponding matrices is singular. In this paper, we investigate fundamental graph-theoretic properties of $\Gamma$, including connectivity, diameter, regularity, the Eulerian property, independence number, clique number, and domination number. We show that $\Gamma$ is a connected regular graph with diameter two. Over finite fields, we obtain an explicit formula for the degree of each vertex and characterize precisely when $\Gamma$ is Eulerian. We determine the independence number and clique number and provide explicit constructions attaining these values using companion matrices of irreducible polynomials. We also construct an explicit dominating set, yielding an upper bound for the domination number.

In this paper, we consider the slow passage through a Hopf in $\mathbb R^3$ in a certain class of real-analytic systems. Classically, the (simplified) Shishkova problem: $\epsilon \frac{dz}{d\mu} = \lambda(\mu) z+\mathcal O(\epsilon)$, $\lambda(\mu) = \mu-\i$, has been used to illustrate the delay associated with slow passage through a Hopf at $\mu=0$. In this paper, we consider a general $\lambda(\mu)$ with $\lambda(0)\in i\mathbb R\setminus \{0\}$, $\real[\lambda'(0)]\ne 0$, and add an equation for the complex conjugate. It is a basic fact that these systems are local normal forms for any real-analytic $(1,2)$ slow-fast system in $\mathbb R^3$ experiencing slow passage through a Hopf. In this paper, we study these systems under additional (global) assumptions on $\lambda$. In particular, we suppose that $\lambda$ has a simple zero away from the real axis. The remaining assumptions then relate to invariant manifolds of the so-called elliptic system $\dot \mu = -\i\overline{\lambda(\mu)}$ as well as properties of the level set $\operatorname{Re}[\i{\lambda(\mu)}{\lambda(\overline \mu)}]=0$. Under these assumptions, we then provide an asymptotic formula for the exponentially small splitting of attracting and repelling slow manifolds. As a corollary, we also obtain an asymptotic formula for the maximal delay. Our approach is geometric and is inspired by the work of Hayes et al (2016) (using blowup) and Neishtadt (1987,1988) (using so-called elliptic paths) but also by recent work of the second author on exponentially small splitting in unfoldings of the zero-Hopf bifurcation.

We present a unified Geronimus-type construction that leads to various orthogonal polynomials including indefinite orthogonal polynomials, exceptional and, more generally, lacunary orthogonal polynomials, and multiple orthogonal polynomials. We use generalized Hermite polynomials to demonstrate the idea and obtain various results for the Geronimus-type transformation such as zero location, recurrence relations, and differential equations.

Let $M^n$ be a $C^2$ closed convex hypersurface in Euclidean space, and $\sigma_m$ be its $m$th mean curvature. We show that $M$ is symmetric with respect to a hyperplane orthogonal to a given direction $e$, if $\sigma_m(p)\leq\sigma_m(q)$ whenever $p-q$ is parallel to $e$. For convex hypersurfaces, this settles a conjecture of Li, extends the mean-curvature theorem of Li-Yan-Yao to all $\sigma_m$, and strengthens some earlier results of Li-Nirenberg by removing nondegeneracy assumptions. The proof is based on the theory of mixed volumes, specifically the rigidity of quermassintegrals under Steiner symmetrization.

We study the quantum transport of Dirac electrons in graphene subjected to a polyadic Cantor-structured electrostatic potential. Using the superperiodic potential formalism, we obtain a closed-form expression for the transmission probability. As the Cantor stage increases, the transmission spectrum evolves from a sparse set of superlattice resonances to a near-transparent regime, with the polyadic order setting the rate of this evolution. The angular response depends on the doping configuration, showing distinct behavior in the $n$–$n$–$n$, $n$–$p$–$n$, and Dirac-point cases. In the near-transparent regime, the transmission follows double-logarithmic scaling laws with respect to four independent control parameters: the Cantor stage, the potential height, the initiator length, and the angle of incidence. By combining these individual scaling relations, we establish a unified scaling law governing quantum transport. These results show that the hierarchical self-similarity of the potential governs the transport properties of such systems.

We prove the uniqueness, asymptotics, symmetry, and orbital stability of energy ground states for 3D magnetic Gross-Pitaevskii equations under mild conditions on the electric potential and magnetic field. In particular, both the electric potential and the magnetic field are allowed to have singularities, and we cover in a unified approach the physically relevant Aharonov-Bohm magnetic field as well as the constant magnetic field. Since we are in the 3D case, the unique energy ground state (up to a phase factor) is obtained as a local minimizer, rather than a global one, by restricting the kinetic energy of candidate critical points within a suitable range. The qualitative analysis of the energy ground state we carry on is mainly based on a related Pohozaev identity, the implicit function theorem, and variational methods.

We develop a continuum theory for a fluid of elongated particles that advance along their own axes. The same kinematics is ideally shared by flocks of sheep, self-propelled rods, vehicular traffic, and turning flocks of birds. Within the framework of continua with vectorial microstructure, in which each point also carries an orientation, we impose the no-side-slip (skate) condition $\vv=u\,\nn$ as an ideal, non-integrable internal constraint. We derive the pure equations of motion and the equation governing the constraint reaction from the principle of virtual power. Using a constitutive closure provided by the Ericksen-Leslie theory of nematic liquid crystals, we show how the constraint also shapes collective effects. It turns parabolic orientational diffusion into hyperbolic orientation-density waves and forbids any steady simple shear. It fixes the empirical Toner-Tu convective coefficient to the flow-alignment ratio measured in colloidal rollers, thereby providing a mechanical foundation for angular sound in micropolar active hydrodynamics. It also recovers the inertial spin model of bird flocks, with the addition of a banking force and a turn-density coupling. Finally, it recovers the classical macroscopic models of one-dimensional traffic flow, while describing steering, lateral tyre forces, and road geometry in two dimensions.

Let $\alpha>0$ and restrict a finite inverse-power potential with arbitrary real coefficients to a line. Combine terms having the same projected centre and squared height $\left(a, b^2\right)$, delete classes whose coefficient sum is zero, and let $m$ be the number of remaining classes. We prove the following dichotomy. If $m=0$, exact cancellation occurs if and only if every class sum vanishes, and every ordinary point of the original domain is critical. If $m \geq 1$, there are at most $2 m-1$ critical points: when all effective heights are positive the zeros are counted with analytic multiplicity, while in the presence of effective sources on the line the assertion is one global distinct-point bound. This proves the signed line conjecture of Gabrielov-Novikov-Shapiro, valid throughout their range and in fact for every $\alpha>0$. The structural input is a projective paired Haar theorem: for finite $\beta>1$, the full $2 m$-dimensional space $\sum L_j / Q_j^\beta$, with pairwise nonproportional positive-definite binary quadratics and arbitrary real linear numerators, has at most $2 m-1$ projective zeros counted with multiplicity. For positive charges, the substitution $p=2 \alpha$ proves Conjecture 3 of Edelsbrunner-Fillmore-Oliveira throughout its stated range $p \geq 1$ and extends the same conclusion to every $p>0$. For every $n$, an explicit positive $n$-charge configuration attains $2 n-1$ simple critical points.

We obtain inequalities for isolated $cD_n, cE_6, cE_7$ and $cE_8$ singularities, analogues of the $4\mu^2/(n+1)$-inequality for isolated $cA_n$ singularity in \cite{KOP}. These inequalities are sharp. This enables us to prove the birational rigidity of certain families of Fano $3$-fold weighted hypersurfaces, which contain only terminal quotient singularities and one $cD_n$ singularity.

I propose that mathematicians who use AI in research maintain a research profile: a record of what the system has learned about working with them, where those lessons came from, and whether they remain useful. I describe how such a profile grew out of my own mathematical work, through corrections, failed approaches, and repeated attempts to keep difficult ideas from drifting toward easier ones. The profile is not a portrait of the mathematician and does not replace mathematical judgment. It belongs under the mathematician's control and must remain open to revision. I invite others to develop profiles of their own and report what they learn.

We study the marked simplicial set obtained as the Roberts-Street nerve of the walking coinductive equivalence. We show that it is a non-contractible saturated complicial set for which all of its finite truncations are contractible. When regarding saturated complicial sets as a model for right $(\infty,\infty)$-categories, it represents a concrete and explicit example of a right $(\infty,\infty)$-category that is itself non-contractible, but whose reflection to a left $(\infty,\infty)$-category is contractible.

We study the combinatorial (red-blue) discrepancy of finite point sets with respect to hyperplanes and, more generally, bounded-complexity affine algebraic sets. We prove that every $n$-point set in a real Euclidean space admits a red-blue coloring for which every affine algebraic set of dimension at most $D$ and degree at most $k$ has discrepancy at most $n^{\frac12-\frac{1}{2(D+1)}-\varepsilon}$ for some $\varepsilon=\varepsilon(D,k)>0$. This gives a polynomial improvement over the straightforward VC-dimension bound $\tilde O(n^{\frac12-\frac{1}{2(D+1)}})$. In the opposite direction, we construct $n$-point sets in $\mathbb R^d$ whose discrepancy with respect to hyperplanes is $\tilde\Omega(n^{\frac12-\frac{1}{d+1}}),$ extending the point-line discrepancy lower bound of Chazelle and Lvov. We present further applications of our methods in communication complexity, concerning separation between randomized communication cost and deterministic communication cost with access to equality oracle.

An operator–theoretic formulation is developed for the synthesis of parameter-dependent controllers and observers for the Vlasov–Maxwell system. The linearized dynamics are modeled as a non-autonomous evolution system whose generators depend on measurable plasma quantities. Well-posedness of the associated evolution family is established together with uniform growth bounds. Parameter-dependent Lyapunov operators yield operator differential LMIs ensuring uniform exponential stability and observer convergence. An $H_\infty$ extension provides disturbance attenuation conditions consistent with the intrinsic energy structure of the coupled Vlasov–Maxwell equations. Galerkin projections lead to finite-dimensional LMIs consistent with the operator inequalities, enabling reliable numerical synthesis while preserving the analytical structure of the original model. Numerical results on a reduced Vlasov–Maxwell benchmark confirm the predicted convergence properties.

We prove Mühlherr's Twist Conjecture: any two angle-compatible Coxeter generating sets of a Coxeter group differ by a finite sequence of elementary twists and a conjugation. Combined with earlier work of Howlett-Mühlherr and Marquis-Mühlherr, this completes the resolution of the Isomorphism Problem for Coxeter groups. A further consequence is that ${\rm Aut}(W)$ is finitely generated for every Coxeter group $W$, and there is an algorithm producing a finite set of generators for ${\rm Aut}(W)$ starting from any Coxeter matrix. Of the vast literature on the Twist Conjecture, we utilise only two results in an essential way: strong rigidity of $2$–spherical Coxeter systems, due to Caprace and Mühlherr, and the framework of markings and hierarchies developed by Caprace and Przytycki for the twist-rigid case. We also exploit in a fundamental way some soft ideas from JSJ theory and an observation of Mihalik-Tschantz on splittings of Coxeter groups. No form of AI was used in the writing of this manuscript, nor in the research that it presents.

Bonk and Kleiner proved that if $G$ is a Gromov hyperbolic group whose boundary $\partial_{\infty}G$ is homeomorphic to an Ahlfors $Q$-regular metric $2$-sphere $Z$, and the Ahlfors regular conformal dimension of $Z$ is attained and equal to $Q$, then $G$ acts discretely, cocompactly, and isometrically on $\mathbb{H}^3$. In this article, we extend the Bonk-Kleiner theorem to the setting of relatively hyperbolic groups. More precisely, we prove that if $(G,\mathcal{H})$ is a relatively hyperbolic group whose Bowditch boundary is homeomorphic to an Ahlfors $Q$-regular metric $2$-sphere $Z$, with the Ahlfors regular conformal dimension of $Z$ attained and equal to $Q$, then $G$ acts discretely and isometrically on $\mathbb{H}^3$, and every subgroup in $\mathcal{H}$ is virtually $\mathbb Z^2$.