Papers

The dynamical formation of a complete Bose-Einstein condensate and the asymptotic purification of trapped bosons coupled to a quantized radiation field are rigorously established. By tracking the macroscopic one-particle density matrix $\kappa_T$ through a nonlinear resonance cascade, it is demonstrated that radiative relaxation irreversibly depletes all excited trap modes, even when the initial macroscopic state $\kappa_0$ is genuinely mixed. Here $\kappa_T$ is an average, with respect to a de Finetti measure $\mu_0$, over mean-field particle-field orbits, so that the mixedness of $\kappa_0$ reflects a statistical ensemble of orbits. Under the assumptions of a non-vanishing initial ground state component and strictly positive downward Fermi-Golden-Rule rates, strong trace-norm convergence to a pure condensate is proven, $$ \lim_{T\to\infty} \| \kappa_T-|\chi_0\rangle\langle\chi_0| \|_{\mathfrak{S}_1(\mathfrak{h})} = 0 \, . $$ Furthermore, the von Neumann entropy of the macroscopic one-particle density matrix is shown to vanish asymptotically, $$ \lim_{T\to\infty}S[\kappa_T]=0 \, , $$ without any claim of monotonicity in $T$. Thus, any initially mixed, positive-entropy state ($0<S[\kappa_0]<\infty$) is purified to the ground state $|\chi_0\rangle\langle\chi_0|$. For the proof, we derive the Hartree hierarchy of the coupled system of bosons and photons from the finite-$N$ BBGKY hierarchy of the microscopic quantum many-body system, establishing trace-norm convergence in the mean-field limit as $N\to\infty$. The limits are taken successively; first $N\to\infty$, then the weak-coupling limit, and finally $T\to\infty$. Although the global particle-field evolution remains unitary, the macroscopic one-particle state converges to a pure state as energy and correlations are continuously radiated into the dispersive field.

In stochastic control problems involving optimal portfolio choice, one often seeks an optimizer of some local criterion (dynamic programming value function) generated by some global criterion (e.g., expected utility). The local optimizer then typically acts as a stochastic integrand with respect to some multivariate stochastic process (e.g., asset prices). This paper takes a closer look at local criteria in the form of “local expected utility”. Since the existence of the local optimizer can be established under very mild conditions, it is crucial to understand under what circumstances the local optimizer has good integrability properties. It is shown that the reasonable asymptotic elasticity condition of Kramkov and Schachermayer (1999) applied to the local utility function plays a key role in this context.

We prove that network coding and fractional routing have the same capacity region in undirected networks whose demand graph has matching number at most two, that is, when no three demand edges have pairwise disjoint endpoints. The metric-packing reduction of Liu et al. reduces this problem to a distance-weighted communication inequality on the fixed graph $\Gamma_{3,3}$. We first prove two entropy inequalities for sessions between designated terminal pairs on $\Gamma_{3,3}$. We then extend this result to sessions with arbitrary endpoints to obtain the distance-weighted communication bound required by the reduction. The proof combines conditional entropy inequalities with two graph contractions and an information inequality for deterministic interaction. Combining this inequality with the existing reduction establishes routing optimality for this class of demand graphs.

We study the geometry and resulting properties of Fréchet functions and barycenters in metric spaces with curvature bounded above by a positive constant in the sense of Alexandrov. Existing results typically rely on the assumption that the support is small enough that the squared distance, and thus the Fréchet function, is strongly geodesically convex. We work in the sharper regime where existence and uniqueness of the barycenter are guaranteed, but where convexity typically fails. We develop new geometric properties of these spaces, including a quantitative four-point inequality and a geodesic stability result. Using these tools, we establish new finite sample error bounds for two statistical estimators: the empirical barycenter and the iterated barycenter, as well as new results on sparse representations and approximations of barycenters.

Demand for wireless communication is growing, and commercial satellite services are expanding with it, increasing pressure on a finite radio spectrum. Allocation efficiency directly limits how many users a satellite can serve. We study how to assign frequency sub-channels to the carriers of a multi-beam satellite, maximising the total priority of admitted carriers while keeping each within its interference tolerance. Earlier work on this interference-limited channel allocation problem placed each carrier in a single continuous frequency block. Removing this restriction can admit more carriers, but a direct integer programming model cannot solve the largest test problems in practical time, so new methods are needed. We prove the problem computationally hard, reformulate it around how carriers share sub-channels, and derive three bounds on the optimum. We develop heuristics, including matheuristics combining local search with optimisation models, and an orchestrated portfolio that schedules them, recombines their solutions and stops early once its result is proven optimal or no longer improves. On 24 problems with up to 500 carriers, the best heuristics come within about 1% of the best known solution on average. The portfolio matches or exceeds the best heuristic for each problem on 23 of 24 with equal weights and 22 with priority weights, and is up to about 5% better on larger, more congested problems. It requires about a quarter of the time to run every heuristic and keep the best, and under half that of its heuristics run in a fixed order, which return slightly worse solutions on average.

Does there exist an 8-variable balanced Boolean function with nonlinearity 118? This is a famous and long-standing open problem in Boolean functions. In this paper, we give a negative answer to this open problem, and prove that the maximum nonlinearity of balanced Boolean functions in 8 variables is 116, establishing the 8-variable instance of Dobbertin's conjectured recursion for the minimum Walsh peak of balanced functions. Equivalently, the covering radius of the Reed–Muller code RM(1,8) in 0-resilient Boolean functions is 116. We deduce the result by obtaining a quadratic form of polar rank 6, and reducing the problem to the existence of a 6-variable Boolean function with nonlinearity greater than 26 whose Hamming distance to a bent function is between 4 and 6.

We study the recovery of a three-dimensional volume in single-particle cryo-electron microscopy (cryo-EM) from noisy tomographic projections acquired at unknown viewing rotations, drawn independently and uniformly from $\mathrm{SO}(3)$. We show that, in a spherical-shell representation satisfying an explicit shell-separation condition, the first three moments generically determine the underlying real volume up to its intrinsic global $\mathrm{O}(3)$ ambiguity. The proof is constructive and yields a frequency-marching procedure that successively recovers the volume. Consequently, consistent recovery in the high-noise regime has sample complexity $n=\omega(\sigma^6)$, where $n$ is the number of observations and $\sigma^2$ is the noise variance. Numerical experiments validate the constructive recovery scheme under finite sampling and additive noise, and indicate that direct third-moment optimization can succeed beyond the structural assumptions required by the constructive proof.

We prove that $r(4,t) \ge \frac{t^3}{(\log t)^{2+o(1)}}$. This improves the lower bound of Mattheus and Verstraëte (Ann. of Math., 2024) by a factor of $(\log t)^{2-o(1)}$.

We show that every symplectic manifold admits a symplectic form whose metaplectic-c prequantizations give quantizability conditions on all its Lagrangian submanifolds. Metaplectic-c prequantizations descend to prequantizations over the Lagrangian Grassmannian bundle that are fibrewise flat with holonomies primitive fourth roots of unity on the generators of the fundamental groups. This fibrewise holonomy is consistently interpreted as that of both Leray two-orientations and normalized Blattner–Kostant–Sternberg half-forms. The prequantizations pull back to flat circle bundles on each Lagrangian submanifold whose holonomy character defines a quantizability condition. The first and second Stiefel–Whitney classes are identified as splitting obstructions to the Lagrangianwise holonomies. These holonomies split into a product of holonomies on any orientable Lagrangian submanifold, and split consistently for all Lagrangian submanifolds of a manifold with metaplectic structure. When the first Chern class vanishes, the holonomy splits into the Bohr–Sommerfeld factor and a quarter phase Maslov correction, recovering the classical Einstein–Brillouin–Keller condition. We develop the underlying obstruction theory for circle extensions associated to general Lie group double coverings with first hypercohomology groupoids.

Let $F$ be a non-Archimedean local field of characteristic zero. We prove multiplicity one for every irreducible smooth admissible representation in the split forms of Wan-Zhang Models 6-10. Combining the Gelfand-Kazhdan method with nilpotent-wavefront restrictions on Bernstein-finite equivariant distributions, we establish the required symmetries through rational orbit analysis, parabolic descent, stabilizer characters, and conormal arguments. The type-$D_6$ analysis treats both the vector-similitude group of Model 9 and the semispin form arising in the exceptional $E_7$ model. For Model 10, the distribution theorem gives a product bound for periods; the individual multiplicity bound follows from Langlands classification and a vanishing theorem for standard modules induced from nongeneric tempered data on the maximal $E_6$ Levi.

Extending the notions of Kneser graphs and Schrijver graphs, we define the $G$-stable Kneser graph $KG(G,k)$ (where $k\geq 2$ and $G$ is a graph on $n\geq 2k$ vertices) as the graph whose vertices are the $k$-element independent sets in $G$ and whose edges join pairs of disjoint independent sets. The well-known result of Lovász on the chromatic number of Kneser graphs implies that $\chi(KG(G,k)) \leq n-2k+2$ for any $G$. We call $G$ $k$-tight if equality holds, and we study the following question: Which graphs are $k$-tight? A result of Daneshpajouh et al.\ implies a characterization of $2$-tight graphs. We prove a necessary condition for a graph to be $k$-tight for $k\geq 3$ which extends this characterization, and we conjecture it to be sufficient. Another result gives a lower bound to the chromatic number of the $G$-stable Kneser graphs when $G$ is a disjoint union of cycles. We use this result to give a short proof of the fact that $4$-regular graphs on $3k$ vertices consisting of two vertex-disjoint cycles and $k$ cycles of length $3$ admit an independent set of size $k$.

Spin wave theory plays a central role in the analysis of low-energy properties of the quantum ferromagnetic Heisenberg model. In this work, we present a first step towards a rigorous justification of this approximation in the context of quantum dynamics in one dimension. According to spin wave theory, the evolution of a spin wave state should be well approximated by its free evolution with the appropriate energy of free quasiparticles called magnons. Our main result establishes this idea rigorously for a class of spin wave packets. We focus on spin-$\frac{1}{2}$ systems, and consider wave packets composed of two magnons. We argue our result is optimal with respect to the timescales.

We propose a uniform moduli-theoretic description of the v-sheaf local model $\mathbb{M}_{\mathcal G,\mu}$ for an arbitrary parahoric group scheme $\mathcal G$ and an arbitrary dominant cocharacter $\mu$. Our approach is Tannakian and builds on a new theory of parahoric induced modules. When $\mu$ is minuscule, this framework explains the linear-algebraic conditions arising in the theory of local models of Shimura varieties.

This article investigates random Dirichlet series within the framework of Hardy and mixed norm spaces of Dirichlet series. We completely characterize when the random symbol space of the Hardy space $\mathscr{H}^p$ coincides with $\mathscr{H}^2$. In the setting of mixed norm spaces $\mathscr{H}^{p,q}_{\alpha}$, the corresponding characterizations for the random symbol space of $\mathscr{H}^{p,q}_{\alpha}$ coinciding with $\mathscr{H}^{2,q}_{\alpha}$ are given when $q\geq4$, and some sufficient and necessary conditions are established when $q<4$. We also obtain some integrability results when Dirichlet series belong to the corresponding random symbol spaces.

We relate several nonlinear flattenings of structure tensors of finite algebras and modules to their algebraic and homological invariants. Since smoothability of algebras is equivalent to their structure tensors having minimal border rank, this shows equivalence of several tensor-theoretic nonsmoothability certificates to known classical ones. Particularly, we prove two conjectures of Doležálek and Michałek relating the tangency flattening and the $3\times3$ minor flattening to dimensions of tangent spaces to the Hilbert scheme of points. In the process, we generalize the tangency flattening to a mixed regime with two tensor inputs, relating this to dimensions of $\operatorname{Ext}$ modules, and introduce the dual notion of the cotangency flattening, related to dimensions of $\operatorname{Tor}$ modules. In the case of algebras, we also realize the kernel of the dualized tangency flattening within the setting of Harrison cohomology, making the connection to the tangent space of the Hilbert scheme explicit.

What is the covering radius of the Reed–Muller code RM(1,9) ? This is a long-standing open problem in cryptography and coding theory. In this paper, we resolve this problem, and prove that the covering radius of RM(1,9) is 242. We exclude nonlinearity 244 by obtaining a cubic Boolean function achieving a sharp Walsh $\ell^1$ bound. Its equality cases reduce either to an impossible cubic hyperplane restriction or to two homogeneous cubic forms in the Brier–Langevin classification. To exclude nonlinearity 243, we deduce the inequality ${\Vert W_g\Vert }_{1}+2|W_g(0)|\le 10368$, and then reduce either to a spectrum bounded by 24 or to the same expression to be at least 10496.

For every even Galois representation in a family of prime conductor $\ell$, we prove that if $k \in \mathbb{Z}_{\geq 1}$ is maximal such that \[\ell \equiv 1 \bmod 3^k\] then the even mod $3$ deformation ring $R_S \otimes_{\mathbb{Z}_3} \mathbb{F}_3$ of level $S=\{\ell,3\}$ does not have a quotient isomorphic to $\mathbb{F}_3[T]/(T^{2+3^{k-1}})$. As a corollary, we deduce that the global even deformation rings $R_S$ are finite and flat in the category of $\mathbb{Z}_3$-modules.

For every integer $d\geq 3$ and real number $Q\in \left(d,\frac{d(d-1)+2}{2}\right]$, we construct a $Q$-Ahlfors regular metric measure manifold $(M,\mathbf{d},\mu)$ of topological dimension $d$ and analytic dimension $2$ that satisfies a Poincaré inequality. For every $Q>\frac{d(d-1)+2}{2}$, the construction yields a metric measure manifold $(M,\mathbf{d},\mu)$ that is doubling, satisfies a Poincaré inequality, and for which $M$ has $\mu$-essential Hausdorff dimension $Q$. The manifold is $\mathbb{R}^d$ equipped with Lebesgue measure and a sub-Riemannian distance given by vector fields that are locally Hölder continuous but not Lipschitz continuous. These metrics give an uncountable family of new models that are genuinely distinct from the only known families of metric manifolds satisfying a Poincaré inequality: weighted and deformed Carnot groups and smooth sub-Riemannian spaces. The construction is modeled on a singular variant of a filiform Carnot group.

In this paper, we prove that every Hermitian conformal class on a connected noncompact complex manifold of dimension at least two contains a Gauduchon metric. We give a sufficient condition for uniqueness of such Gauduchon metrics up to scaling in terms of functions bounded from above and their Levi forms. This condition is necessary when the manifold has at least two ends. In that case, uniqueness in a fixed conformal class is equivalent to the assertion that every positive solution of the complex Laplace equation for a metric in that class is constant. For a connected noncompact complex Lie group of dimension $n$, uniqueness in every conformal class is equivalent to the assertion that, for every Hermitian metric, every positive solution of the complex Laplace equation is constant. These properties hold exactly when the complex tangent space of a maximal compact subgroup has dimension $n-1$; such a group is abelian. In a conformal class containing a balanced metric, the same equivalence holds without a restriction on the number of ends. We also prove uniqueness for manifolds with a suitable potential on a compactification with compact fibers over a point, the real line, or a parabolic Riemann surface. The examples include compact pluripolar deletions, partially deleted compact fibers, and semipositive line bundles. We construct conformal Gauduchon metrics which are not related by scaling on Stein manifolds and on the total space of $\mathcal O_{\PP^1}(1)^{\oplus2}$.

Let $\ell$ and $p$ be distinct prime numbers, $K$ a local field with residue characteristic $p$, $C$ a complete algebraically closed extension of $K$, and $X$ a quasi-compact quasi-separated rigid space over $K$. When $\mathop{\mathrm{char}} K = 0$, we prove that the alternating sum of traces of elements in the positive part of the Weil group acting on $R\Gamma(X_C,\mathbb{Q}_{\ell})$ is independent of $\ell$ and is an integer, extending a result of Mieda to the singular case. We also give optimal bounds for the $p$-adic valuations of the eigenvalues of geometric Frobenius liftings acting on $H^i(X_C,\mathbb{Q}_{\ell})$, proving a conjecture of Lu–Zheng. In the general case, we give optimal bounds for the $p$-adic valuations of the eigenvalues of geometric Frobenius liftings acting on $H^i_c(X_C,\mathbb{Q}_{\ell})$, and we show that these eigenvalues are algebraic integers and Weil numbers, thereby extending results of Lu–Zheng and Mieda to the positive characteristic case.

We prove that, under suitable assumptions, the space of order continuous orthogonally additive operators between Riesz spaces is a Riesz space. As a consequence, we give a negative answer to an open problem posed by Mykhaylyuk and Popov concerning the order continuity of the modulus of an order continuous orthogonally additive operator. We also show that, under an appropriate condition on the domain, every order continuous orthogonally additive operator is order bounded, obtaining an orthogonally additive analogue of the corresponding result for linear operators. Moreover, we establish the order boundedness result in ZF for atomic Riesz spaces and in ZF with dependent choice (DC) for Dedekind complete Riesz spaces. Finally, we characterize when every orthogonally additive operator is an abstract Uryson operator.

This paper develops theoretical and numerical extensions of Birkhoff pseudospectral transcription for aerospace optimal control problems. Building on the recently proposed body of knowledge, a Birkhoff integration operator is derived for flipped Legendre Gauss Radau (fLGR) grids, drawing connections to regular LGR grids and to the favorable conditioning properties that make Birkhoff interpolants particularly appealing. We also propose the enforcement of differential constraints via pseudospectral operators, and the primal and dual equivalence of this approach to state augmentation is established for control-rate limits. The numerical developments include the formulation and solution of heterogeneous multiphase optimal control problems, parameterized non-affine time mappings, composition of affine scaling and equilibration methods, and a hybrid derivative strategy to combine linear pseudospectral operators with numerical derivatives for nonlinear functions. These methods are implemented in the RAPTOR (Rapid Algorithm for Pseudospectral Trajectory Optimization and Robustification) transcription framework and assessed through feature-oriented benchmarks: a one-dimensional landing problem, a planar orbit-raising scenario, an atmospheric entry problem, and a multiphase reusable launch vehicle problem with simultaneous booster-recovery and upper-stage-insertion trajectories.

For a graph $H$, the suspension $\widehat H$ of $H$ is a graph obtained from $H$ by joining a new vertex to every vertex of $H$. The generalized Turán number $\textrm{ex}(n,K_3,\widehat H)$ is the maximum number of triangles in an $n$-vertex $\widehat H$-free graph. For fixed $k \ge 4$, Mubayi and Mukherjee conjectured that \[ \textrm{ex}(n,K_3,\widehat P_k) =\left\lfloor\frac{k-2}{2}\right\rfloor\frac{n^2}{8}+o(n^2), \] as $n \to \infty$, where $P_k$ is the path on $k$ vertices. They verified the conjecture for $k=4,5,6$. In this paper, we determine the asymptotic value of $\textrm{ex}(n,K_3,\widehat P_k)$ for every $k\ge 4$ via a unified approach and thus prove the conjecture of Mubayi and Mukherjee.

We study the recovery of a Hamiltonian from discrete trajectory observations using structure-preserving kernel methods. When exact-flow data are fitted through a symplectic one-step scheme, the fixed-step learning target is generally not the physical Hamiltonian itself, but a scheme-dependent inverse modified Hamiltonian. For the implicit midpoint and $p$-implicit symplectic Euler methods, we establish the existence of analytic inverse targets and show that their discrepancies from the physical Hamiltonian are second and first order in the observation step, respectively. Within an RKHS framework, we derive finite-sample reconstruction bounds for random trajectories, allowing dependence among observations from the same trajectory, and separate the error into estimation, approximation, and integrator-induced components. We further establish stability under noisy trajectory observations and develop a multistride Richardson correction that systematically raises the order of the integrator-induced error without requiring additional trajectories. Numerical experiments confirm the predicted scheme-dependent orders, the effect of Richardson correction, and the applicability of the method across several Hamiltonian systems.

Any pair of sufficiently generic Hamiltonians on $\mathbb{C}^d$ generates the full unitary group, i.e., any $W\in U(d)$ can be synthesized to arbitrary precision by alternating between the two evolutions for suitably chosen durations. For any given pair of Hamiltonians, the minimal number of switches required to cover the entire group is at least $d$ and unbounded in general. Despite this, we show that it scales linearly in $d$ in expectation. Assuming that the two spectra are sufficiently generic, we derive constructive upper bounds on the minimal switching number as a function of the change-of-basis matrix between the two eigenbases. In particular, if this matrix has a column whose entries all have equal moduli, as for a complex Hadamard matrix, then $9d$ switches suffice to reach every unitary. To obtain these results, we construct a short exact representation of any unitary as a product of unitaries that are alternately diagonal in two given bases. All bounds are extended to non-generic spectra and bounded evolution times.

This paper studies controllability and minimal norm control problems (NP problems) for mean-field stochastic differential equation (MF-SDE) systems. We first establish an equivalent rank criterion for terminal controllability, namely, that the diffusion matrices $D$ and $D+\bar D$ are both of full row rank. We then derive an explicit Riccati-based representation of the minimal norm control by applying a parameterization technique. A key step is to identify the value $P(0)$ of an associated Riccati solution with the controllability Gramian of a controllable auxiliary system, which yields the parameter required in the construction. The resulting representation provides a basis for the future development and analysis of numerical methods for NP problems.

We use microlocal tools to define distinguished incoming and outgoing vacuum states of the quantum theory of massive Klein–Gordon fields in non-trapping asymptotically flat spacetimes, including radiative perturbations of Minkowski space. The asymptotic vacuum states satisfy the Hadamard condition, and the incoming and outgoing particle representations are unitarily equivalent, or in other words the quantum S-matrix is well-defined. These results are based on a description of scattering of classical fields which keeps track of the asymptotic distinction between positive- and negative-frequency solutions of the Klein–Gordon equation. Our emphasis is on developing the theory without recourse to explicit asymptotics for classical waves.

We study several aspects of deformation invariance of plurigenera in Kähler families over a smooth curve. First, for a relatively flat coherent sheaf, we characterize local constancy of fiberwise cohomology dimension by equivalent conditions in terms of extension, base change, torsion freeness of higher direct images, and the vanishing of jumping classes. Then, combining this criterion with the Kähler minimal model program, we prove invariance of plurigenera for Kähler families of threefolds whose central fiber has canonical singularities, and for smooth Kähler families of maximal Albanese dimension. We also establish deformation invariance of log plurigenera when the central fiber is projective, and is of log general type. We prove the invariance of higher cohomological plurigenera assuming that the canonical bundle of the total space is semipositive. Finally, we extend Z.~Li–Z.~Wang's deformation invariance result for generalized polarized pairs with abundant nef parts from projective families to Moishezon families.

We establish quantitative hypocoercivity for singular stochastic damped wave equations arising in stochastic quantization. Building on the finite-dimensional argument of Cao, Lu, and Wang [Arch. Rat. Mech. Anal. 2023], we derive a dimension-free criterion that replaces uniform semiconvexity by a log-Sobolev inequality together with exponential lower-tail bounds for the bottom of the Hessian spectrum. We verify this criterion for the two-dimensional $\Phi^4_2$ and sine-Gordon models and for the three-dimensional $\Phi^4_3$ model, through estimates on the lowest eigenvalue of associated singular random Schrödinger operators. Combined with the continuum limit established in a companion paper, this yields exponential $L^2$ convergence to equilibrium for the $\Phi^4_2$ dynamics and for sine-Gordon with $\beta^2<2\pi$. For $\Phi^4_2$, we verify that in a weak-coupling, small-mass regime, optimizing the damping yields a kinetic relaxation rate of at least order $m$. This yields a genuine acceleration over the order-$m^2$ relaxation rate of the overdamped dynamics. The lattice estimates are uniform in the discretization for sine-Gordon throughout $\beta^2<4\pi$, and for $\Phi^4_3$ at fixed volume, including regimes beyond those for which the continuum dynamics is vpresently constructed.

For a Bruhat interval $[x,y]$ in an arbitrary Coxeter group which is a lattice, we give a type-uniform construction of a convex polytope whose complete face lattice realizes $[x,y]$. This gives a polytopal realization of the coincidence between classical Kazhdan–Lusztig polynomials for lattice Bruhat intervals and Stanley's toric $g$-polynomials. Consequently, the Kazhdan–Lusztig polynomial of a lattice Bruhat interval cannot have internal zeros, satisfies a multiplicative strengthening of monotonicity, and obeys the generalized upper bound theorem for polytopes. For lattice Bruhat intervals in finite Weyl groups, we explain the relation with the toric geometry of Richardson varieties and derive formulas for their local and global intersection cohomology, as well as a Boolean criterion for rational smoothness. We conjecture that the coefficients of these Kazhdan–Lusztig polynomials form an $M$-sequence.