Papers

For $n\in\mathbb{N}$, we consider the set of lattice paths from $(0,0)$ to $(n,n)$ using only unit north and east steps. Given a point $B$ to be avoided, we ask: at which point $A$ on the grid with corners $(0,0)$ and $(n,n)$, different from the endpoints, does the largest number of $B$-avoiding lattice paths pass through? We show that for $n\ge 9$, regardless of the location of $B$, the maximum is attained at one of ten specific points clustered near the two endpoints of the grid. This stability, however, conceals an interesting anomaly. When the obstruction $B$ lies on the antidiagonal $x+y=n$, the points of maximal traffic migrate from the near-corner points $(1,1)$ and $(n-1,n-1)$ to boundary points in the set of possible maximizers. The migration occurs for every $8\le n\le 375$, and intermittently up to $n=495$. We conjecture that the anomaly disappears for $n\ge 496$.

We construct a bounded pseudoconvex complete Reinhardt domain $\Omega$ with smooth boundary in $\mathbb{C}^3$ such that the $\bar\partial$-Neumann operator $N_1$ is compact although $b\Omega$ contains an analytic disc and thus also fails Catlin's Property $(P)$ and McNeal's Property $(\tilde P)$. This example solves an open problem on compactness of the $\bar\partial$-Neumann operator in the negative.

In this paper, we study an Erlang-type loss system with energy-constrained servers. Each server is equipped with a finite battery and becomes temporarily unavailable for service when its energy is depleted, entering a charging phase before returning to operation upon full recharge. Customers who arrive to find all servers unavailable are blocked and lost immediately. We characterize the steady-state behavior of this two-dimensional Markov process by extending classical truncation results for Jackson networks to incorporate energy dynamics. This yields a closed-form product-form stationary distribution, from which we derive explicit expressions for the steady-state moments and the blocking probability. Finally, we establish that the corresponding M/G/$k$/$k$ queue with stochastic charging is insensitive to both the service-time and charging-time distributions, depending only on their means. Thus, we extend the insensitivity property of the Erlang loss queue to the stochastic server setting.

Let $H\in C^\infty(\mathbb R^n\times\mathbb T^n)$ be a periodic Tonelli Hamiltonian with critical value $c$. For each $k\in\mathbb N$, let $u_k$ be the normalized minimizer of the variational functional introduced by Evans[7], \[ I_k[w]=\int_{\mathbb T^n} e^{kH(Dw,x)}\,dx, \qquad \int_{\mathbb T^n}w\,dx=0. \] If $u_\infty$ is a uniform limit of a subsequence of $\{u_k\}$ and the Mather quotient $({A}_M,\delta_M)$ satisfies $H^1( A_M,\delta_M)=0$, then $u_\infty$ is a critical subsolution that is strict outside ${{A}}$ and \[ {A} = \{x\in\mathbb T^n\,:\,Du_\infty(x)\ \text{exists and }H(Du_\infty(x),x)=c\}=\{x\in\mathbb T^n\,:\,u_\infty(x)=u_{-}(x)\}, \] where ${A}$ is the projected Aubry set and $u_{-}$ is the backward weak KAM solution associated with $u_\infty$. In particular, by the theorem of Fathi–Figalli–Rifford[10], this conclusion holds for all smooth Tonelli Hamiltonians on $\mathbb T^n$ when $n\leq3$. This characterization also suggests a natural numerical localization principle for approximating the entire Aubry set through near-contact sets between $u_k$ and its large-time backward Lax–Oleinik evolution.

Let $R$ be a commutative $\Q$-algebra and let $HR$ denote its Hurwitz series ring. We introduce a one-parameter convolution $\hconv{t}$ on $HR$ for which the specialization $t=-1$ is the intrinsic Hurwitz product. Positive integral parameters act on finite-support truncations and reproduce finite free convolution in coefficient coordinates. A logarithmic transform yields additive and homogeneous cumulants, an explicit inversion formula, binomial, Hermite, Laguerre, and hypergeometric families, and coefficientwise analogues of the law of large numbers and the central limit theorem. At $t=-1$, addition of independent random variables becomes multiplication of their moment sequences in $HR$; this gives applications to classical cumulants, beta–gamma products, and self-decomposable laws. Weighted coefficient norms turn $\hconv{t}$ into a commutative Banach algebra product. Norm-continuous convolution semigroups then have generators conjugate to multiplication operators. Explicit formulas are obtained for the Hermite and Laguerre semigroups, the finite free heat generator, and the Lévy–Khintchine generator.

We study regularisation and irreducibility properties of Gaussian quantum Markov semigroups (GQMSs) acting on continuous-variable quantum systems. We first identify a natural notion of regularity for operators in this setting, which allows us to formulate and characterise the smoothing effects of GQMSs in terms of algebraic conditions involving the drift and quantum diffusion matrices. These conditions establish a connection with the controllability theory of quantum linear systems and with the structure of decoherence-free subsystems. We then characterise irreducibility through several equivalent algebraic criteria: one formulated in terms of the drift and quantum diffusion matrices, one in terms of the operators appearing in the generalised GKLS representation of the generator, and a third given by a quantum analogue of Hörmander's condition. A central and somewhat surprising consequence is that, in contrast with the classical case, irreducibility is strictly stronger than conditions ensuring regularisation. Our results provide an algebraic framework for analysing these properties and lay the groundwork for a broader study of reducible Gaussian quantum Markov semigroups and, more generally, more general relevant quantum Markov semigroups on continuous-variable systems.

This paper establishes necessary and sufficient conditions for locally flat knotted surfaces in simply-connected $4$-manifolds to be equivalent. For surfaces with knot group $\mathbb{Z}_d$, we extend results of Lee-Wilczynski from spheres to surfaces of arbitrary genus; the surfaces are permitted to be nonorientable and have boundary. We prove that most projective planes with knot group $\mathbb{Z}_2$ and the same Euler number are determined by the equivariant intersection form of their exterior. We also prove that knots with prime power determinants bound at most one Moebius band in $D^4$ with knot group $\mathbb{Z}_2$ and a given Euler number. Cancellation results lead to new criteria for homologous discs to be equivalent rel. boundary. Finally, we determine the topological extendable mapping class group of knotted surfaces with abelian knot group.

We prove that a coaction of a Hopf algebra on another preserving the latter's associative-algebra structure and comultiplication automatically also preserves the counit, antipode, and tensor-interchange map and in fact factors through the coaction dual to an action of an affine group scheme by Hopf-algebra automorphisms. This generalizes and unifies a number of results to the effect that quantum groups have only classical symmetries, due to Kasprzak et al., Budziński-Kasprzak and Brannan et al. The stated automatic classicality follows from the fact that the antipode, counit and tensorand-interchange of a Hopf algebra internal to any symmetric monoidal category are contained in the (non-symmetric) monoidal subcategory generated by the associative-algebra structure and the comultiplication, assuming a weak form of arrow-retraction invariance.

We prove stability of the Riemannian positive mass theorem in all dimensions, extending the Dong-Song stability theorem. For a sequence of complete asymptotically flat manifolds with nonnegative scalar curvature and ADM masses tending to zero, excising domains whose boundary areas tend to zero yields exterior regions converging to Euclidean space in the pointed measured Gromov-Hausdorff topology. The proof constructs global coordinates from minimal graphs and controls their Hessians using scalar solutions of the conformal Laplace equation on the associated graph metrics.

Generalized semi-infinite programs (GSIPs) arise in robust optimal control whenever the admissible uncertainty depends on the state or controls. Existing GSIP methods either impose restrictive structural assumptions or require global optimization that scales poorly to control problems. We present a general framework that reformulates any GSIP with mild regularity as an existence-constrained semi-infinite program, smoothing its disjunctive feasibility condition into differentiable existence constraints over a fixed index superset. The resulting program is solved by established adaptive discretization (cutting-plane) methods using only off-the-shelf nonlinear-programming solvers, and converges under standard assumptions. Treating the state trajectory as part of the uncertainty extends the framework to robust nonlinear optimal control with state-dependent uncertainty. We demonstrate it on a nonconvex benchmark GSIP and a satellite de-tumbling problem with dynamically varying inertia.

For every $n\ge3$, we show, by a family of explicit entire real-analytic solutions, that, for every subcritical phase and every $0<\alpha\le1$, oscillation-based interior $C^\alpha$ estimates fail for the special Lagrangian equation in $\mathbb{R}^n$. In fact, this family has no common interior modulus of continuity.

We establish multiplicative comparisons between Rényi entropies of different orders for weighted sums of independent Bernoulli random variables. In particular, we prove a logarithmic comparison between the zeroth- and infinity-order Rényi entropies, which yields a polynomial improvement over the square-root bound of Jain, Sah, and Sawhney. As an application, this leads to an improved parameterized running time for the randomized bin-packing algorithm of Nederlof, Pawlewicz, Swennenhuis, and Wȩgrzycki. We also obtain explicit dimension-free, constant-factor comparisons between Rényi entropies of positive orders.

We exhibit three conics with 144 tritangent circles, disproving the conjectured maximum of 136.

We derive a stable limit theorem for stochastic differential equations driven by multiplicative $\alpha$-stable processes. A key ingredient is the $L^1$-exponential contractivity estimate for the SDEs. The limiting process is a non-degenerate symmetric $\alpha$-stable process with an averaged Lévy measure.

The symmetric Macdonald polynomial at peculiar values of parameter $t=q^{-m}$, $m\in\mathbb{Z}_{\ge 0}$ is naturally split into non-symmetric parts, which are the (quasi)polynomial Baker-Akhiezer (BA) functions. One may think this is due to symmetricity, and one just picks up this way non-symmetric parts already containing all the information. However, we demonstrate that, in the case of non-symmetric Macdonald polynomials, it still works, though each single BA function splits into $N!$ distinct (quasi)polynomial BA functions. The sum of these functions gives rise to the universal solution of the eigenstate problem for the Cherednik Hamiltonians. Extending to arbitrary values of $t$ is also immediate giving rise to counterparts of the Noumi-Shiraishi power series. Altogether, this power series and its reductions to non-symmetric Macdonald polynomials and to BA functions form a non-symmetric triad. There are $N!$ different branches of the non-symmetric triad, each branch being split into $N!$ distinct triads, and of these $(N!)^2$ triads $N!(N-1)!$ are independent. We describe in detail the simplest $N=2$ case.

An orthodox dominating set on an $n\times n$ chessboard occupies every row of one parity and every column of a possibly different parity. For $n\ge400001$, every such set contains more than $(1/2+1/80000)n-2$ queens, on odd and even boards and with attacking or boundary queens allowed. Second-moment estimates and an exact rational line-weight certificate give a stronger bound for $p$-covers; finite diagonal completion and board extension transfer it to orthodox covers. The cost of extending an arbitrary dominating set to an orthodox cover yields an inequality with an explicit defect term. The previously constructed independent, border-free Type-A $1$-cover of $Q_{221}$ with $111$ queens supplies a finite seed. Classical amplification gives ordinary and independent domination upper bounds with coefficients $112/221$ and $113/221$, respectively. Its order 221 is below the density threshold 400001. For admissible seeds whose orders tend to infinity, the lower limit of these coefficients is at least $1/2+1/16000$.

Combinatorial-topological methods for characterizing dynamics are rigorous, generalizable, computable, and they only require approximations, but the dimension of the phase space is a computational bottleneck to their wider application. Motivated by the growing number of machine learning techniques for obtaining lower-dimensional latent representations of dynamics, we present an initial study of combinatorial-topological techniques in the dimensionality reduction setting. We establish bounds under which an algebraic structure that organizes dynamics can be lifted from the latent space to the original system. As a corollary, one can conclude the existence of attractors within certain regions of the original phase space. The hypothesis of these results is expressed in terms of an approximate semiconjugacy between the original and latent dynamics. To demonstrate the ideas, we combine autoencoder-based models with Conley-Morse graph computations for Leslie population models, a thirteen-dimensional Mediterranean red coral population model, and the Chafee–Infante equation. While the lift of the Conley index is still an open question, the examples recover the expected algebraic topological invariants in several settings.

We present the knot polynomial associated to the 12-dimensional sporadic Nichols algebra $\mathrm{FK}_3$ with automorphism, compute it for all knots with $\leq$ 16 crossings and discuss the found patterns. This knot polynomial is not a Vassiliev power series. We ponder how it compares to the knot polynomials that come from Lie algebras and their representations and what the other knot polynomials associated with the higher dimensional sporadic Nichols algebras are.

We determine the torsion part of the higher smooth surgery structure sets of complex projective spaces (up to some extension problems) and complete the description of the forgetful map to their topological versions in low dimensions. In addition, an application to mapping class groups of complex projective spaces in low dimensions is presented.

We introduce the class of star cluster algebras and classify their singularities. Then we focus on the combinatorial structure of the desingularization by determining the number of irreducible centers that are blown up.

We consider integer-valued $\nabla\phi$ height functions, with general convex interactions, placed on a slope. Sheffield (2003) conjectured that for all slopes, such height functions are localized in dimensions $d\ge3$, in the sense of having tight fluctuations in finite volume, and admitting infinite-volume limits. We establish this conjecture when there are two coordinates on which the slope vector is zero and on which the interactions are even and low temperature. In this setting, we also prove the uniqueness, in the appropriate sense, of the infinite-volume limit and classify its extremal components. We further study the structural properties of the infinite-volume limit. We establish the entropic repulsion between macroscopic domain walls (boundaries of level sets), showing that they are maximally separated in a precise sense described by a rotation-of-the-circle dynamical system. Lastly, we show exponential decay of correlations in the extremal components. Our setup includes, as a special case, infinitely many zero-slope integer-valued surfaces (of dimension two or higher) conditioned not to cross. We deduce the existence and structural properties of the bulk Gibbs measure over such non-crossing surfaces with any given average spacing.

The Erdős–Stone Theorem asserts that if a graph has edge density $1-1/r+\delta$ then it contains a complete $(r+1)$-partite graph with $b$ vertices in each part, where $b=b_n(r,\delta) \gg 1$. The celebrated Chvátal–Szemerédi theorem determined the exact order of $b_n(r,\delta)$ for every $\delta < 1/r^3$. Their bound, however, is not tight when $\delta=1/r-\epsilon$, that is, when the graph has edge density $1-\epsilon$ for small $\epsilon$. Our main result in this paper determines the correct order in this remaining regime, thereby enabling us to give a tight bound for the Erdős–Stone problem for all edge densities. More precisely, we prove that for every integer $r\geq 2$ and $0< \delta < 1/r$ we have $$ b_n(r,\delta)=\Theta\left(\frac{\log n}{(1/r-\delta)r\log(1/\delta)}\right)\;. $$ The lower bound is obtained using a Kövari-Sós-Turán-type argument combined with a variant of Nikiforov's method of constructing large blow-ups, while the upper bound is proved using a correlated random graph construction, related to tensor powers.

Krauel, Shafiq and Wood (\KSW{}) realized torus \(1\)-point functions for the simple affine \voa{}s \(L(k,0)\) built from \(\alg{sl}(2)\) as vector-valued modular forms attached to a cyclic \(R\)-module structure and to the modular tensor category \(\rep{L(k,0)}\). We extend this to torus \(n\)-point functions, defined as traces of chains of \(n\) intertwining operators winding once around the \(\tau\)-cycle. Every coefficient of their local (Laurent, or Puiseux) expansion about a diagonal stratum is again a \vvmf{}, lying, whenever a mild lowest weight hypothesis holds, in the very \(R\)-modules of \cite{KSW}. Giving a torus-primarity hypothesis verified below for \(\alg{sl}(2)\), the operator product expansion then reduces leading short-distance behaviour to the classified \(1\)-point theory. For \(\alg{sl}(2)\) we classify the fusion-chain conformal block spaces, identify torus \(n\)-point primary vectors, and treat \(n=2\) in detail, obtaining a \((k+1)\)-dimensional family of vector-valued Jacobi-type forms generalizing the level-\(k\) forms \(\eta^{3k/2}\) of \cite{KSW}. We also extend \KSW{} categorical picture, representations of \(\Bthree=\PMod(\surf{1}{1})\) built from a modular tensor category, to representations, on fusion-chain block spaces, of a mapping-class subgroup of the \(n\)-punctured torus generated by \(S\), \(T\), and adjacent braidings, with a Verlinde-type dimension formula and categorical \(S\)-, \(T\)-operators. The \(\alg{sl}(2)\) matrix form of \(S\) beyond this abstract construction remains open.

We consider the three-dimensional full compressible magnetohydrodynamic(MHD) system on the periodic torus $\mathbb T^3$ in the regime where the only dissipative mechanism acting on the system is the viscosity of the fluid: the magnetic field is non-resistive and the flow is non-heat-conducting. We prove that this system admits a unique global smooth solution, together with explicit algebraic decay rates, provided that the perturbation $(\mathbf u_0,\,P_0-\bar P,\,\mathbf H_0-\mathbf n)$ of the equilibrium state $(\mathbf 0,\bar P,\mathbf n)$ is sufficiently small in a high-order Sobolev space and the background magnetic field $\mathbf n\in\mathbb R^3$ satisfies a Diophantine condition. No smallness whatsoever is imposed on the initial density: it is only required to be bounded away from vacuum and from infinity, and may exhibit arbitrarily large variations. The proof uncovers a hidden dissipation mechanism. Although neither the density, nor the pressure, nor the magnetic field is endowed with any diffusion or damping of its own, the coupling of these quantities with the velocity through the background field $\mathbf n$, combined with a Poincaré-type inequality of Diophantine origin, generates effective dissipation for both the pressure and the magnetic field perturbations. The large variations of the density are handled by a two-tier energy argument, in which weighted time-decay estimates for the intermediate-order energy compensate exactly for the linear-in-time growth of the highest-order norm of the density.

For a finite graph $G$ on $n$ vertices, let $\eta(G)$ denote the least order of a finite abelian group $\Gamma$ for which $G$ is an induced subgraph of some Cayley graph of $\Gamma$. Babai and Sós (1985) settled the worst-case order of magnitude: it is $\Theta(n^2)$. We treat $\eta$ instead as an invariant of the individual graph, minimised over all finite abelian groups rather than over the cyclic groups alone, which is the restriction implicit in the literature on representation numbers modulo $n$. We prove a local order floor: $\eta(G)$ is at least the maximum of $n$ and twice the largest independence number of a neighbourhood of $G$. This localises at an arbitrary vertex the correspondence of Babai and Sós between induced stars and sum-free sets; a corollary of the classification of maximum sum-free sets in abelian groups does not lower this floor, but restricts which host orders are admissible and so prunes the search. We determine $\eta$ exactly for paths, where it equals $n+1$, and for complete bipartite graphs $K_{a,b}$, where it equals $2\max(a,b)$ and meets the floor. A Cartesian product bound gives $\eta(P_m \,\square\, P_m) = (1+o(1))n$. We report certified exact values of $\eta$ for $22$ graphs, computed over all abelian groups. Seventeen of the $22$ optimal hosts are cyclic, so on most of these graphs the cyclic restriction costs nothing; where it bites, however, it is expensive. A search restricted to cyclic groups returns $36$ for the Petersen graph against the true value $16$, and $59$ for the Frucht graph against $27$. The cost of the restriction is concentrated rather than diffuse, and we identify the graphs on which it is paid. We also determine $\eta$ exactly for the double stars $D_{q,q}$ with $2 \le q \le 6$, obtaining $5q$ in each case. Since $\eta(D_{6,6}) = 30$ exceeds $2n = 28$, no constant below $15/7$ can bound $\eta(T)/n$ over all trees.

Let $E/\mathbb{Q}$ be a rational elliptic curve whose torsion group grows over a quadratic field $K$. In a previous paper, a relation between the primes of the conductor of $E$ and the primes that divide the discriminant of $K$ was shown. In the present paper, we go further in this study and we compute the ramification of the field $K$, when a torsion point of prime order $\ell$ is added, in terms of invariants of the curve. Concretely, the conductor, the prime $\ell$ and the Kodaira symbol at the primes of bad reduction.

We construct additive rank-metric and sum-rank-metric codes from Drinfeld modules by restricting bounded-degree morphisms to prime-to-characteristic torsion. For supersingular Drinfeld modules of rank $r$ in characteristic $\mathfrak{p}$ of degree $d$, the stabilization formula for morphism spaces yields rank-metric codes of $\mathbb{F}_q$-dimension $mrt-c$ and minimum distance $r-t+1$, where $c=r(r-1)(d-1)/2$. Simultaneous restriction to $\ell$ distinct degree-$m$ torsion modules gives additive sum-rank codes of the same dimension and minimum distance at least $\ell r-t+1$. Their normalized Singleton defects tend to zero, while in characteristic $(T)$ the module $\phi_T=\tau^r$ makes the defect vanish and produces an explicit MSRD family. We identify this family with a skew Chinese remainder theorem code supported on central skew polynomials and prove that its poly-skew weight is exactly $m$ times its sum-rank weight. This gives a specialized Singleton-type bound and a polynomial-time unique decoder up to the full sum-rank unique-decoding radius. We also derive a Welch-Berlekamp-type filter equation for the general supersingular sum-rank construction; it becomes an effective decoder whenever bases of the relevant morphism spaces and the restriction maps are computable.

A pair $(\mathcal{A}, \mathcal{B})$ of families of subsets of $[n]$ is cancellative if whenever $A, A' \in \mathcal{A}, B \in \mathcal{B}$ satisfy $A \cup B=A' \cup B$, then $A=A'$, and whenever $A \in \mathcal{A}, B, B' \in \mathcal{B}$ satisfy $A \cup B=A \cup B'$, then $B=B'$. We show that for every cancellative pair $(\mathcal{A}, \mathcal{B})$, the inequality $|\mathcal{A}||\mathcal{B}| \le 2.25^n$ holds, matching Tolhuizen's $(2.25-o(1))^n$ lower bound construction.

We study linear codes whose coordinates are the ordinary edges and self-loops of complete undirected graphs; a node erasure removes all coordinates incident with a failed vertex. The construction results are binary. For triple-node erasures, we extend the published cyclic construction by allowing a suitable cyclic check slope to depend on the prime graph length. An explicit determinant test proves that one of three fixed slope choices works at infinitely many prime lengths, unconditionally, and gives redundancy $3n-2$, one bit above the graph Singleton bound. We also give Singleton-optimal triple-node codes at $n=6,8,10,12$, together with a general ordinary-edge framework that isolates the remaining loop-completion problem. When $2$ is primitive modulo an odd prime $n$, a binary multi-slope construction corrects every $\rho$-node erasure for $2\leq\rho<n$, with redundancy $\rho n-(\rho-1)$ in the range $2\leq\rho\leq(n+1)/2$. Returning to arbitrary prime powers, we derive exact generating transforms and inclusion–exclusion formulas for node-metric ball volumes, fixed-radius asymptotics, and packing, existence, and covering bounds. Finally, for the complementary clique-erasure metric, we obtain an exact weight enumerator and a Singleton-optimal node–clique duality.

We study a family of fibre-cofibre sequences in the $\infty$-category of exact $\infty$-categories, which we call Keller sequences, generalising the notion of Verdier sequences of stable $\infty$-categories. We show that algebraic K-theory is localising with respect to this family, and use this to give a categorical proof of Quillen's resolution theorem. Our arguments hinge upon a mapping space characterisation of special inclusions as studied by Keller, from which we deduce another proof of Keller's theorem that such inclusions induce fully faithful functors on stable envelopes.