Papers

We develop a Poincaré duality theory for defects of nondegenerate sesquilinear pairings over Laurent polynomial rings. A key ingredient is a novel flat resolution of the character module, constructed from a triangulation of the sphere at infinity associated with a fan. The cup product on this resolution turns Poincaré duality on the sphere into canonical pairings between the resulting defect modules. In middle degrees, we construct distinguished quadratic refinements using equivariant cohomology of the sphere with the antipodal action. The effective replacement of the sphere with a projective space provides a geometric substitute for division by two. Applied to translation-invariant Pauli stabilizer codes, our results establish the nondegeneracy of higher-dimensional braiding pairings. They extend the two-dimensional T-junction formula for topological spin to higher dimensions, while giving it a geometric interpretation.

We extend the global existence results for critical parameters of [Bianchini and Leccese, J. Math. Anal. Appl. (2024)] to the multidimensional problem \begin{equation*} \partial_t u + \text{div } (b(t,x) u^{1+k}) = \Delta u \end{equation*} for $b:(0,\infty)\times\mathbb{R}^d\to\mathbb{R}^d$ non-autonomous fields, in the case \begin{equation*} b\in L^\infty_{\mathrm{loc}} \bigl((0,\infty),L^{p,\infty}(\mathbb{R}^d)\bigr), \qquad p>d\ge 2. \end{equation*} For the critical case, we study the difference between two ordered solutions whose conserved small mass absorbs the drift. Finitely many increments then reach an arbitrary solution.

Let $X$ be a compact Brody-hyperbolic complex manifold admitting a strictly positive real $(1,1)$-current $T$ with smooth $dT$ and $dd^cT\le0$. We prove that the canonical bundle $K_X$ of $X$ is nef. If every positive Lelong level set of $T$ has complex dimension at most two, then $K_X$ is ample and $X$ admits a smooth plurinegative Hermitian metric. Strictly negative Chern holomorphic sectional curvature implies Brody hyperbolicity and, when constant, forces the given metric to be Kähler under the latter current hypotheses. We also establish Kählerness for nonpositive constant Chern holomorphic sectional curvature in Fujiki's class, and for nonzero constant curvature on compact complex threefolds and compact balanced fourfolds. In every complex dimension at least seven, we construct compact balanced metrics with zero Chern holomorphic sectional curvature, nonzero Chern curvature, vanishing of all four Chern–Ricci contractions, and a nowhere-vanishing Chern-parallel holomorphic volume form. Finally, for compact Hermitian manifolds with zero Chern holomorphic sectional curvature and vanishing first Chern–Ricci tensor, a smooth semipositive plurinegative $(1,1)$-form, positive definite at one point, forces the metric to be Kähler flat.

We determine the Gabriel spectrum of a category of sheaves of vector spaces in purely topological terms. For every topological space $X$, we prove that the Gabriel spectrum of ${\mathbf{Sh}}(X)$ is homeomorphic to ${\mathrm{Sk}}({\mathrm{Sob}}(X))$, where ${\mathrm{Sob}}(X)$ denotes the sobrification of $X$ and ${\mathrm{Sk}}$ indicates passage to the Skula topology. The proof is based on a classification of indecomposable injective sheaves and a characterization of localizing subcategories in terms of Skula-open subsets. We show that the Gabriel spectrum is always Hausdorff, zero-dimensional, and totally disconnected, and that it is compact if and only if $X$ is Noetherian. This leads to computations of Gabriel spectra arising in persistence theory. In particular, the Gabriel spectrum of the category of persistence modules over ${\mathbb R}^n$ is homeomorphic to the space of ideals of ${\mathbb R}^n$ with a natural topology.

What is the least recurrent memory needed to reproduce a specified expert under partial observability? The instantaneous requirement is the conditional entropy of the expert's behavioral quotient, but recurrence must also preserve distinctions that future observations will not restore before use. We characterize this minimal recurrent behavioral memory by a compatibility relation: under transitivity its classes attain the exact minimum, while the general case is an entropy minimization over closed compatible state assignments, with exact certificates on finite instances. A sole-carrier measurement protocol separates behavioral sufficiency, excess code rate, and information carried by observations or other memory paths; experimental bit requirements refer to the induced symbolic behavioral model under the stated occupancy. Across manipulation tasks, learned code rates remain near zero- and two-bit requirements as hidden modes grow to $512$, and anticipatory memory follows a $2\to1\to0$ requirement despite zero instantaneous demand during waiting. Learning this representation remains difficult: event-agnostic future-behavior supervision yields $36/40$ sufficient seeds with one frozen configuration and improves the longest-horizon pixel setting from $0/8$ to $6/8$ sufficient held-out seeds (closed-loop success from $0.08$ to $0.57$). On unmodified community benchmarks, the protocol certifies delay-independent requirements, which sufficient codes match at mid-delay. The supervision aids commitment but can induce predictive surplus; annealing it lets imitation and rate training reduce that surplus, separating the information-theoretic target from the ability to learn it.

In this paper we propose first-order methods for a class of nonsmooth composite strongly convex–strongly concave and nonconvex–concave minimax optimization. We first develop an inexact proximal method and an accumulative regularizated method for strongly convex–strongly concave problems. The latter achieves the optimal dependence on the curvature parameters and places the smaller curvature inside the accuracy logarithm. Using this method as a subsolver, we propose a proximal point method for nonconvex–concave problems. Under suitable assumptions, it finds an $\epsilon$-stationary point with an operation complexity of $O(\epsilon^{-5/2})$, which improves the best-known $O(\epsilon^{-5/2}\log(1/\epsilon))$ bounds by removing the logarithmic factor. We further develop parameter-free variants for both problem classes, and achieve the same complexity without knowledge of any problem constants. All proposed methods are equipped with verifiable termination criteria.

We prove global well-posedness of the periodic unidirectional Euler-alignment system in every dimension for supercritical dissipation of order $0<\alpha<1$ and arbitrary non-vacuum initial data in the Sobolev class of the local theory. The key idea is to propagate simultaneously two moduli of continuity at critical scales for the density $\rho$ and the potential gradient $\Gamma=\nabla\Lambda^{-\alpha}u$, both of which have scaling-invariant amplitudes. We also establish exponential alignment of the velocity and exponential convergence of the density to a traveling profile.

We study a geometric singular perturbation theory (GSPT) approach to optimal control problems involving entry-exit phenomena in planar fast-slow dynamical systems. In contrast to previous work, we assume that the control acts only on the slow dynamics and we consider minimum-time optimal control problems (i.e., we aim at minimizing the time to reach a certain target). We apply this method to a simplified fast-slow toy model, characterized by a repetition of entry-exit processes, enabling a transparent analytical exploration of the control dynamics and allowing us to derive the key results explicitly. First, we analyze each non-autonomous entry-exit phenomenon separately, and we derive the optimal way to control them. Second, by artificially generating a stable limit cycle, we determine under which conditions it is possible to reach the target. Finally, under such conditions, we combine all entry-exit processes together and we demonstrate the existence of an optimal control. Our approach mainly exploits techniques of GSPT and dynamic programming. Indeed, we show that our problem can be formulated in a way resembling the Bellman equation, even though not in its standard setting. Numerical simulations, built on such Bellman-like formulation, illustrate our analytical results. In particular, the optimal control is derived with an algorithm that converges in a finite number of steps. Lastly, we extend this novel approach to more complex planar fast-slow dynamical systems.

We study the zero-run structure of the $(3,2)$-Raney numbers modulo a prime $p$. For every prime $p$, we determine the left-to-right maxima of the zero-run lengths, the complete zero-run spectrum, and the exact number of nonzero entries in $0\le n<p^m$. Interestingly, these results fall into three cases: $p=2$, $p=3$, and $p\geq5$, and the behaviors in these three cases are very different. For $p=2$, we characterize exactly the odd terms and determine the positions of the left-to-right maxima and the results involve Fibbinary integers, Fibonacci numbers, and Jacobsthal numbers. For $p=3$, we characterize exactly the nonzero terms and determine their residues. For $p\ge 5$, the zero runs are governed by a multiscale system of residue intervals modulo powers of $p$, from which both the record values and the complete zero-run spectrum are obtained.

Geometric Goppa codes are constructed by evaluating elements of an algebraic function field at the places of degree one. Extending this, generalized geometric Goppa codes were defined in [XNL99] and [Hey02] by allowing evaluation at places of arbitrary degree. In this paper, we establish necessary and sufficient conditions for equality of generalized geometric Goppa codes, following the approach of [MP93]. These conditions serve as a key tool in our characterization of iso-dual and self-dual generalized geometric Goppa codes. Given a separable extension M/F of algebraic function fields and an iso-dual generalized geometric Goppa code over F, we extend the construction of [CPQT24] to obtain iso- dual generalized geometric Goppa codes over M. Furthermore, we study self-duality of generalized AG codes ([XNL99], [DM03]).

Let $\nu_2(m)$ denote the $2$-adic valuation of a positive integer $m$ and set $N_m=m\bigl(1+\nu_2(m)/2\bigr)$. We prove four conjectures of Merca on the nonnegativity of truncated pentagonal number series weighted by the infinite products $\prod_{m\ge 1}(1-q^{2N_m})$ and $\prod_{r\ge 1}\bigl(q^{2^r r};q^{2^{r+1}r}\bigr)_\infty$. Our method is to regard the exponent map $m\mapsto m(\nu_2(m)+2)$ as a dynamical system on the positive integers: the factors of the associated quotient link into chains along its forward orbits, and the three orbits seeded at $1$, $4$ and $5$ are pairwise disjoint and telescope to exactly $1/\bigl((1-q)(1-q^4)(1-q^5)\bigr)$. The exponent triple $(1,4,5)$ is admissible in the sense of earlier work by Liu, which yields the desired factorization into two series with nonnegative coefficients. This orbit telescoping technique appears to be a mechanism complementary to the Pólya–Szeg\H o criterion that underlies most existing positivity results of this kind in the literature.

We study the problem of designing optimal damping for a system of $n$ coupled linear oscillators. Using the Weyl–Horn theorem, we construct an explicit damping matrix that realizes the optimal asymptotic decay rate of the system. We prove that this decay rate is sharp and is determined by the geometric mean of the natural frequencies. The resulting damping strategy is shown to outperform the classical Rayleigh (proportional) damping approach. We conclude by characterizing the parameter regimes in which the optimal damping matrix necessarily ceases to be positive definite and discuss the implications of this phenomenon.

We consider the superposition $S_n$ of $n$ i.i.d. simple point processes on $\mathbb{R}^d$ and apply a dependent thinning $T_n$, where the retention probability of a point depends on its local neighbour count within a radius $r_n$. While superpositions under independent thinning converge to Poisson processes, we show that under a critical geometric scaling $n v_d r_n^d \to \tau \in (0,\infty)$, the local interactions are transformed in the limit into an inhomogeneous Poisson point process with modified intensity. We prove that the thinned sequence $T_n(S_n)$ converges to a Poisson process with a non-linearly modified intensity $\tilde\lambda(x) = \lambda(x)\alpha(\tau\lambda(x))$.

Repeated averaging on graphs is a natural prototype of local stochastic exchange, modeling consensus formation, information spreading, and wealth redistribution. Its convergence has attracted longstanding interest, from Bourgain's complete-graph $L^1$ cutoff question in the 80s (answered in recent years by Chatterjee, Diaconis, Sly, and Zhang) and the systematic study of Aldous and Lanoue. Addressing the still outstanding and widely asked $L^1$-cutoff question on general graphs, we establish a criterion based on size-biased sampling, and develop a general framework for sparse expanders. Specifically, we prove $L^1$ cutoff with a Gaussian profile on random $d$-regular graphs, for any $d\ge 3$ and uniformly over starting vertices. Remarkably, although the expected mass profile agrees exactly with the corresponding random-walk distribution, cutoff occurs strictly later than both the random walk cutoff time and the universal lower bound attained on complete graphs, revealing an unexpected entropy gap. The framework is robust, involving ideas of polymer representations, random geometry, universal-cover lifting, and adaptive revealing. Our companion work will extend it to Ramanujan graphs and bounded-degree configuration models.

We prove that every sufficiently large finite planar point set contains either four collinear points or seven points with at most one non-visible pair. More generally, we show that for every fixed graph $H$ with chromatic number at most five, or with chromatic number six and a color-critical edge, the visibility graph of every sufficiently large finite planar point set with no four collinear points contains a copy of $H$. These results extend the recent breakthrough of Bonnet (2026), guaranteeing six pairwise visible points, and come within one visibility edge of the next open case of the big-line-big-clique conjecture.

We establish exact thresholds for the failure of simultaneous quantum typicality: three parties for mixed states and four for pure states. For the four-qubit Higuchi–Sudbery state, every sequence satisfying the expected purity bounds on three overlapping marginals, with sufficiently small fixed positive entropy slack, is asymptotically perfectly distinguishable from the tensor powers of the input: its trace distance from them tends to the maximum value $2$. This maximal separation persists, with a common positive slack, throughout an open neighborhood in the full four-qubit state space. Tracing out one party yields a mixed tripartite example. The proof connects simultaneous quantum smoothing with polynomial concentration on $\CP^1$. Symmetry reduces the problem to a frame inequality on $\Sym^n(\mathbb C^2)$, where a linear entropy deficit competes with a quadratic cost of polynomial concentration. Bernstein–Markov estimates and a subharmonic barrier control this cost, yielding an exponential obstruction to simultaneous typicality. These results disprove Dutil's multiparty quantum typicality conjecture, formulated in 2011, and the one-shot simultaneous min-entropy-smoothing conjectures of Drescher–Fawzi and Colomer–Winter. Together with the positive results for smaller systems, they determine the smallest numbers of parties at which simultaneous typicality can fail.

In this article, we consider a general version of the classical group ring isomorphism problem, called the twisted group ring isomorphism problem (TGRIP), which determines an isomorphism between the twisted complex group algebras of finite groups. Although for isoclinic groups $G$ and $H$ their complex group algebras are isomorphic, i.e, $\mathbb C G \cong \mathbb C H$, their twisted complex group algebras need not be isomorphic. Continuing this line of investigation, we establish sufficient conditions under which two isoclinic groups have isomorphic twisted complex group algebras, thereby providing solutions to (TGRIP) for isoclinic groups. As applications, we study (TGRIP) for special $p$-groups of rank $2$, unicentral groups, and nilpotent groups of class $2$ with elementary abelian Schur multipliers, all considered up to isoclinism. We also present several examples illustrating the main results, including a complete solution of (TGRIP) for special $p$-groups of rank $2$ of order $p^6(p \ge 3)$

The first author and Dong \cite{DD} proposed three conjectures on antichain generating polynomials. Jiang \cite{Jiang} recently proved Conjectures 4.3 and 4.5, concerning real-rootedness and \(\gamma\)-positivity. We prove Conjecture 4.2 by adapting his method from \([k]\times [2] \times [n]\) to \([k]\times P_{n,s} \), where \(P_{n,s}\) is a two-row Ferrers shape. We establish real stability for a family of bivariate polynomials associated with adjacent shapes, then apply the Chudnovsky-Seymour compatibility criterion to obtain a common interleaver. As noted in \cite{DD}, Conjecture 4.2 also implies Conjecture 4.3.

Let $A\subset\mathbb{R}$ be a finite set. Combining the multiplicative slope estimate of Rudnev–Stevens, Cushman's higher-energy regularization, Shakan's $d^+$–$d^\times$ decomposition, and Solymosi's classical sum–product estimate, we prove \[ |AA|^{204}|A+A|^{301}\gtrsim |A|^{675}, \] where $\gtrsim$ suppresses a fixed polylogarithmic factor in $|A|$. Consequently, for every $\varepsilon>0$, \[ \max\{|A+A|,|AA|\}\gg_\varepsilon |A|^{135/101-\varepsilon}. \] The proof is organized around two intermediate estimates. For every nonempty finite set $B\subset\mathbb{R}_{>0}$, \[ d^\times(B)|BB|^{12}|B+B|^{16}\gtrsim |B|^{38}, \] whereas for every nonempty finite set $U\subset\mathbb{R}$, \[ d^+(U)^{17}|U+U|^{29}\gtrsim |U|^{46}. \]

Many statistical procedures compute an entire solution path rather than a single estimator. For one class the path is piecewise linear, traced corner to corner by an active-set homotopy. Two of its members coincide exactly: under $\Sigma=X^\top X$ and $\mu=X^\top y$ the gross-exposure-constrained mean–variance program and the constrained LASSO trace the same piecewise-linear curve, and they keep doing so under arbitrary linear equality and inequality constraints. Which parameter is swept, a leverage budget or a return tilt, is not a choice the curve notices: under a homogeneous mandate the two sweeps differ by a scalar. A mandate that holds the portfolio invested costs a radial rescaling instead, and the efficient frontier is that curve rescaled; its corners pass unnoticed because the frontier is continuously differentiable where they fall. We give the map between the two parametrisations and the single way it degenerates. That mean–variance selection and the LASSO instantiate one parametric quadratic program is prior art (Gärtner, Jaggi and Maria, 2012). The Critical Line Algorithm itself lies off that route, and the correspondence established here is new. The identity is one of curves, not of statistical experiments. The map between the parametrisations is computed from the data, so the geometry of the path transfers while quantities averaged over the response do not. We cross that boundary deliberately and give the degrees of freedom of the constrained fit under arbitrary linear constraints. In the long-only, fully invested case it is the expected number of holdings away from a bound, less one.

We prove an $S$-integrality theorem for special divisors on GSpin Shimura varieties in positive characteristic. Let $C$ be a generically ordinary curve in such a Shimura variety not contained in any special divisor. Then, for any increasing sequence of prime-to-$p$ positive integers $m_i$ and any finite set of closed points $S\subset C$, we prove that $C\setminus S$ meets the special divisor $Z(m_i)$ for all but finitely many $i$. The key new input is an algebraicity theorem for formal special endomorphisms which allows us to use techniques from Diophantine approximation. We prove this algebraicity theorem using a punctual monodromy theorem and a positive-characteristic analogue of the Mumford–Tate conjecture.

Despite its physical importance, the stability of Prandtl boundary layers under wall suction remains relatively unexplored in rigorous PDE analysis. In this paper, we establish the global-in-$x$ asymptotic stability of a suction profile. For any prescribed algebraic decay rate in $x$, we prove that sufficiently small inflow perturbations with sufficiently rapid spatial decay yield downstream decay at that rate. The von Mises transformation reduces the Prandtl system to a degenerate parabolic equation with a favorably directed convection term. We first identify the downstream decay mechanism in an associated uniformly parabolic problem, then control the boundary degeneracy using a sharp Hardy-type estimate. Together, these arguments yield stability and decay for the original Prandtl system.

The paper studies open optimization programs, that is, parametric optimization programs viewed as open systems, and their interconnection. Each open optimization program is assigned a behavior and interconnection is defined through variable sharing. For a class of open quadratic programs (QPs), we characterize closure and well-posedness under interconnection and, for modules interconnected by adding their objectives, analyze the convergence of a sequential module-by-module solution method, deriving error bounds for the resulting approximation. We further illustrate that the variables through which modules are interconnected indicate an appropriate behavioral description, and each description requires dedicated interconnection rules. This perspective suggests a route toward compositional analysis and design of optimization architectures.

We consider an action of an inverse semigroup on a C*-algebra $A$ and use it to construct a groupoid of germs with unit space the spectrum of $A$. Motivated by the representation theory of C*-algebras of groupoids, we construct a concrete family of covariant representations for the action. We use this family to give new definitions of the reduced and essential crossed product C*-algebras that avoid, respectively, passing to the double commutant and local multiplier algebra of $A$. Our reduced crossed product is isomorphic to the one defined by Exel, Buss and Meyer, and when the inverse semigroup is quasi-countable our essential crossed product is isomorphic to the one defined by Kwaśniewski and Meyer.

The Scott-Vogelius element is a classical divergence-free mixed finite element for the Stokes problem that has attracted decades of research attention, yet its theoretical framework remains incomplete. In two dimensions, the inf-sup stability on Freudenthal and other regular meshes has been rigorously established. In three dimensions, Zhang established inf-sup stability on Freudenthal meshes for $k\ge6$ in 2011, while numerical evidence indicates that inf-sup stability remains true for $k=4,5$. In this paper, we strengthen Zhang's approach and prove that the Scott–Vogelius element is inf-sup stable on Freudenthal meshes for every velocity degree $k\ge 4$, thereby resolving a conjecture proposed by Farrell, Mitchell, and Scott in 2024. The proof proceeds by explicit constructions on local patches and does not rely on computer verification.

Generalised $T$-links provide a simple description of all links in $S^3$ as closures of products of standard twisting blocks, parametrised by finite sequences of integers. We prove that every link admits infinitely many pairwise distinct generalised $T$-link presentations. Starting from any such presentation, we give explicit parameter transformations that preserve the represented link and generate families of pairwise distinct presentations depending on arbitrarily many independent integer parameters.

Let $g_t$ be a smooth family of Riemannian metrics on an oriented manifold of even dimension $n\geq4$, and let $D_t=d+\delta_{g_t}+\Psi_t$ on the fixed exterior bundle, with $\Psi_t$ self-adjoint. We evaluate a scalar-weighted residue functional obtained from the operator expression in Hawkins' Hamiltonian action using the coefficient derivative $\partial_tD_t$. The velocity contribution depends only on $\partial_tg_t$. A direct symbol calculation reduces the lapse correction to a divergence and a Dirichlet term; for a nonconstant test function, integration by parts retains a mixed gradient term. For compact manifolds with a warped collar, we specify compatible factorizations of the weighted KKW term and of the commutator correction. Their boundary residues are computed explicitly. The collar-derivative and perturbation contributions cancel in the weighted KKW factorization, and the remaining boundary terms involve normal derivatives of the scalar weights. The closed formula is recovered when the boundary is empty.

We prove a colorful quantitative Helly theorem for volume with the optimal number $2d$ of colors. If every rainbow intersection from $2d$ finite families of convex sets in $\R^d$ has volume at least one, then the intersection of one family has volume at least $d^{-O(d^2)}$. We also prove a colorful quantitative Steinitz theorem for origin-centered ellipsoids of different shapes. The proof uses a common normalization of positive operators and a lift that produces two rainbow bases with large determinants.

We recast the Lange-Stuhler theorem as the universal property of a $2$-cartesian square of classifying stacks. This formulation shows that every Frobenius-periodic vector bundle of rank $r$ on an arbitrary fibered category in characteristic $p>0$ is trivialized by a finite étale torsor, without geometric hypotheses or restrictions on the ground field beyond its characteristic. The construction extends from $\mathrm{GL}_n$ to group schemes over finite fields that are connected affine or of finite type.

This paper is devoted to the global dynamics of a nonlocal Lagrangian-averaged Navier-Stokes-$\alpha$ (LANS-$\alpha$)-type system (also known as the viscous Euler-Poincaré system) in one, two, and three spatial dimensions. In the unforced periodic setting with zero mean, we establish global well-posedness for the initial-value problem and prove that the corresponding solution semigroup is dissipative and possesses a compact global attractor consisting of the singleton $\{0\}$. Furthermore, we obtain the following global-in-time algebraic asymptotic stability estimate: \[ \|u(t,\cdot)\|_{H^3}\lesssim t^{-1/2},\qquad \forall\, t>0, \] which is derived from higher-order energy inequalities combined with a uniform Gronwall-type argument. Our results provide a complete description of the long-time behavior of the unforced periodic dynamics for this nonlocal LANS-$\alpha$-type system.