Papers

We extend the recent results of Arnold on estimates for Kreiss positive semigroups on $L_p$-spaces to the non commutative setting, $1<p<\infty$. We also provide a sharp example. All the results are achieved thanks to the use of the Schur test, including a new version for von Neumann algebras.

We classify the monotone $\SOn d$-invariant valuations, $d\ge2$, on the space of all closed convex cones. They are precisely the linear combinations of the conic intrinsic volumes with nondecreasing coefficients, or equivalently, the nonnegative linear combinations of intrinsic volume tails, up to an additive constant. On nonzero pointed cones we obtain the corresponding characterisation by Grassmann angles, settling a conjecture of McMullen. Every such valuation is automatically continuous in the spherical Hausdorff topology and is $\On d$-invariant. Our proof builds on the signed simplex and averaging arguments of Wang and Wu, but does not rely on the continuous classification theorem.

Fermionic quantum cellular automata (QCA) are automorphisms of local fermionic operator algebras ($\mathbb Z_2$-graded superalgebras) that have bounded spread: they map local operators to nearby operators. We prove that every 2-dimensional fermionic QCA on a locally finite-dimensional algebra is a composition of local automorphisms and a fermionic shift. This is implied by our result that every locally finite-dimensional fermionic invertible subalgebra in a one-dimensional lattice is Brauer trivial, i.e., it is stably bounded-spread isomorphic to a tensor product fermionic algebra.

We prove a sharp mean-width inequality for monotone zonal operators and derive global and differential bounds for Euclidean and hyperbolic-harmonic self-maps of the unit ball. Let $k:[0,\pi]\to\mathbb R$ be continuous and nonincreasing, and let $T_k$ be the associated zonal integral operator, \[ (T_kF)(\xi)=\int_{\mathbb S^{n-1}} k\!\left(\arccos\langle\xi,\eta\rangle\right) F(\eta)\,d\sigma(\eta), \] where $\sigma$ is normalized surface measure. For every measurable $F:\mathbb S^{n-1}\to\overline{\mathbb B^n}$, we prove \[ w\!\left(\operatorname{co}T_kF(\mathbb S^{n-1})\right) \le2\lambda_1(k), \] where $w$ denotes mean width, normalized by $w(\overline{\mathbb B^n})=2$, and $\lambda_1(k)$ is the eigenvalue of $T_k$ on the space of spherical harmonics of degree one. For strictly decreasing kernels, equality holds exactly for $F(\eta)=Q\eta$ almost everywhere, with $Q\in O(n)$. For the ordinary Poisson kernel, the multiplier is $r$, yielding sharp mean-width, intrinsic-volume, and image-volume contraction. In particular, $|f(r\mathbb B^n)|\le\omega_n r^n$ without injectivity assumptions, answering the area and higher-dimensional volume question of Koh and Kovalev.

We provide a complete classification of Fano fourfolds of Picard rank 1 and index 1 obtained as zero loci of completely reducible homogeneous vector bundles in generalized Grassmannians, i.e. Grassmannians of Dynkin type different from A. This extends the classification done by Küchle in standard Grassmannians. In doing so, we exhibit three new families of Fano fourfolds of Picard rank 1 and index 1. For each of the studied Fano fourfolds, we compute several numerical invariants, such as volume and Hodge numbers.

We prove a modified invertibility property for the parabolic defects introduced in arXiv:2102.12283, arXiv:2505.14836. Namely, we show that the Borel defect cancels its dual, up to insertion of an invertible Weyl defect and up to restriction to an open-subcategory. The main result holds for an arbitrary reductive group $G$ – for illustration we give extended computations for $G=\mathrm{SL}_3$. Along the way, we introduce a defect skein theory with defects in both codimension one and two, which is compatible with gluing of defect 3-manifolds and defect surfaces. Our results provide a potential defect skein theoretic construction of certain standard charts which appears frequently in quantum cluster varieties associated to surfaces.

We develop a general first-order theory of Schur apolarity for the study of secant varieties of flag varieties in arbitrary homogeneous embeddings. Extending the classical apolarity–fat-point correspondence for Veronese varieties, we show that in the Schur setting the algebraic square of the apolar ideal need not coincide with the geometric double-point conditions. We introduce a geometric Schur square whose relevant component is the conormal space, yielding a Schur Dual Terracini Lemma. Our construction recovers classical apolarity in the symmetric case. A slot-by-slot Consistency Theorem realizes these intrinsic conditions as multigraded double points. As an application, we determine the dimensions of all secant varieties of $\operatorname{Fl}(1,2;V_n)$ embedded by $\mathcal{O}(1,1)$: the only defective cases are $\sigma_2(\operatorname{Fl}(1,2;V_3))$ and $\sigma_3(\operatorname{Fl}(1,2;V_4))$, both of defect one.

In this paper, we study non-perturbative vortex partition functions of four-dimensional N=1 supersymmetric gauge theories on the space-time geometry $T^2 \times D^2$, and we propose that these partition functions offer a notion of quantum elliptic cohomology. For particular U(1)-gauge theories we calculate these partition functions explicitly by applying equivariant localization methods to Handsaw quiver varieties that realize for this particular class of U(1)-gauge theories the moduli spaces of the non-perturbative vortex sectors. The determined vortex partition functions are annihilated by difference operators, which are interpreted as Ward identities among N=(0,2) BPS surface defects. Compared to lower dimensional gauge theories with four supercharges, anomalies play an essential role for a consistent formulation of the four-dimensional partition functions. In our proposal towards a mathematical formulation of the vortex partition functions in terms of equivariant elliptic cohomology, the gauge theory anomalies relate to geometric properties of the Thom sheaves corresponding to the relevant quasimap moduli spaces. Motivated by the explicit computations we reflect on the existence of a `virtual structure sheaf' on the moduli space of quasimaps for a general mathematical theory of quantum elliptic cohomology.

The fixed-charge network flow problem (FCNFP) couples continuous flow allocation with discrete arc-activation decisions, making it a canonical but computationally challenging model for a variety of network design and resource allocation problems. Exact mixed-integer linear programming formulations capture the fixed-charge structure faithfully, but often become difficult to solve on large networks. We propose a scalable continuous-optimization algorithm for large-scale single-commodity FCNFP based on an iteratively reweighted least-squares (IRLS) framework. The method replaces the discontinuous fixed-charge and linear arc cost objective with a smooth nonconvex Lasry–Lions surrogate and solves a sequence of weighted quadratic flow subproblems. Each subproblem is solved by a warm-started dual semismooth Newton method whose Newton systems have weighted graph-Laplacian structure, enabling the use of modern Laplacian solvers. To further improve the discovered arc supports of the challenging underlying combinatorial problem, we also develop an algorithmic variant that incorporates objective-driven perturbation restarts and an anchor-union restricted search that jointly leverages supports discovered by IRLS and by complementary FCNFP heuristics. Computational experiments on 410 benchmark, synthetic, and large-scale instances show that our method obtains the best objective quality among the evaluated scalable FCNFP algorithms, with a mean gap of $1.316\%$ to a time-limited MILP reference and a win-or-tie rate of $90.0\%$ among the non-MILP methods. The results indicate that combining smooth continuous optimization with support-level search is an effective strategy for producing high-quality feasible solutions to large-scale FCNFP.

We study feasible stable equilibria of Lotka-Volterra systems and their higher-order extensions. We complete the classification of impossible ecological interaction networks with at most four species and extend several of these impossibility results to families with arbitrarily many species. We then show that these sign-pattern obstructions are specific to the pairwise Lotka-Volterra model: arbitrary prescribed growth rates and pairwise coefficients can be supplemented by higher-order interactions so as to admit a feasible asymptotically stable equilibrium. Through the correspondence with replicator dynamics, we interpret feasible equilibria of higher-order Lotka-Volterra systems as totally mixed symmetric Nash equilibria of symmetric multiplayer games, derive bounds on their number, and study their robustness under perturbations of the payoff tensors. We conclude by showing that every impossible ecology determines a nonempty open class of symmetric two-player games with no totally mixed evolutionarily stable strategy.

Euler sums, also called alternating multiple zeta values (MZVs), have been shown to play important roles in many research areas in mathematics and theoretical physics. Motivated by a similar conjecture for MZVs by Kaneko and Zagier, the author proposed a conjectural isomorphism between the space of finite Euler sums and the space of classical ones modulo $\zeta(2)$-products. In this paper, we explicitly construct those finite elements (for which we call effective Euler sums) that correspond to the classical Euler sums under this conjecture in depth one and two. In the appendix, we offer another group of plausible candidates of effective Euler sums when depth is two and weight is even, by extending the heuristic argument of Kaneko and Zagier for the double zeta case. Kina recently obtained independently some the same results in the MZV setting. In particular, using one of his results we prove in the appendix that Kaneko and Zagier's heuristically defined effective double zeta values coincides with ours.

In this note we unveil some subfields of the field of complex numbers which are well known to specialists, but seem to remain hidden for a wider mathematical audience. The existence and construc- tions of these fields follow directly from famous results of the 20-th century, however standard graduate textbooks avoid these topics.

We present a method for recovering the moral graph of a causal DAG from a probability distribution over discrete variables, using fully connected tensor networks (FCTNs) with nuclear-norm-regularized bond corrections. Each bond matrix is parameterized as a baseline all-ones matrix plus a low-rank correction $C_{ij} = U_{ij}V_{ij}^\top$, and the nuclear norm of the correction implemented via the variational Frobenius norm penalty on the factors drives unnecessary bonds to zero. We prove that under faithfulness, positivity, and a no-implicit-rerouting assumption on the local tensor architecture, every optimal FCTN with zero reconstruction error $\varepsilon = 0$ has effective graph exactly equal to the moral graph. For the approximate regime ($\varepsilon > 0$), we provide explicit recovery bounds using the Fannes-Audenaert continuity of conditional mutual information, and derive a sufficient condition on the regularization parameter $\beta$. The effective graph is read directly from the optimized bond matrices.

Higher genus Gromov-Witten invariants of Calabi-Yau threefolds are encoded in a generating function which is an asymptotic series in a formal parameter $\lambda$. Using resurgence, analytic functions in this formal parameter were uncovered. In this paper we focus on the resolved conifold and study the enumerative meaning of the strong-coupling asymptotic expansion, in powers of $ 1/\lambda $, of the resurgent analytic functions. We show that this expansion contains both a closed curve-counting contribution in dual variables and a contribution governed by relative Gromov-Witten invariants, naturally interpreted in logarithmic geometry.

Graph-data workloads such as diffusion estimation, ranking, semi-supervised learning, and network optimization often solve many Laplacian or symmetric diagonally dominant M-matrix (SDDM) systems with the same coefficient matrix. Approximate Cholesky preconditioners eliminate vertices one at a time and store the resulting sparse approximate factorization, the factor, whose construction cost is amortized across these solves. But eliminating a vertex, the pivot, creates a dense Schur-complement clique among its $d$ active neighbors. We introduce CAST (Canonical Approximate Schur Tree), which replaces this clique with a weighted random spanning tree sampled directly from it. Every realization is connected and contains exactly d-1 edges, while reweighting each selected edge by the reciprocal of its tree-inclusion probability makes the update unbiased. The distribution is independent of the ordering of the pivot neighbors, and we prove that its leverage-score marginals minimize the largest normalized reweighted-edge contribution among unbiased inverse-marginal one-tree estimators. We also introduce CAST-$\rho$, which replaces each pivot neighbor with $\rho$ copies, each carrying a $1/\rho$ share of that neighbor's incident weight, samples a weighted random spanning tree on the expanded clique, and contracts the copies back to the original neighborhood. The resulting update remains unbiased and connected, can be sampled exactly in $O(\rho d)$ time, and satisfies a $1/\rho$ bound on the second moment of the normalized local Schur error. Increasing $\rho$ therefore reduces certified local sampling variability, but may increase construction cost and downstream fill. Empirically, we observe that CAST-1 is the faster default, whereas CAST-2 is preferable when its additional edge contributions remain inexpensive.

An infinitely divisible nonnegative matrix is an entry-wise nonnegative matrix that admits an entry-wise nonnegative $m$th root, with respect to usual matrix multiplication, for every positive integer $m$; it is called strongly infinitely divisible when it is, in addition, invertible. The study of such matrices has its origins in the theory of infinitely divisible probability distributions and the embedding problem for Markov matrices and is closely connected with continuous one-parameter semigroups; in the invertible case, this connection admits a natural description in terms of the exponential map. In this paper, we characterize the bijective linear maps on $M_n(\mathbb{R})$ that preserve strongly infinitely divisible matrices and infinitely divisible nonnegative matrices. In the former case, we combine the Inverse Function Theorem with Zariski-density techniques, using the Zariski-density approach recently developed in linear preserver theory by Fallat and Mondal [Proc. Amer. Math. Soc., 2026]. In the latter case, we first show that preserving infinite divisibility forces preservation of the cone of entry-wise nonnegative matrices and then exploit the additional structure afforded by infinite divisibility to complete the characterization.

The subset $A = \{a_i:i \in I\}$ of a normed vector space is a $B_h$-set if every element of the sumset $hA$ has a unique representation as a sum of $h$ elements of $A$. Let $\varepsilon = \{\varepsilon_i:i \in I\}$ be a set of positive real numbers. An $\varepsilon$-perturbation of $A$ is a set $A' = \{a'_i:i\in I\}$ such that $|a_i'-a_i|<\varepsilon_i$ for all $i \in I$. Let $\Delta_{hA} = \inf\{|x'-x| : x,x' \in hA \text{ and } x\neq x'\}$. It is proved that if $A$ is finite or countably infinite set with $\Delta_{hA}>0$, then there is a $B_h$-set $A'$ that is an $\varepsilon$-perturbation of $A$.

Rajam$\unicode{228}$ki and Koivunen asked whether minimum-redundancy symmetric Kl$\unicode{248}$ve-Mossige arrays with contiguous sum co-arrays are always Kl$\unicode{248}$ve arrays. We give a computer-assisted proof that, at every fixed sensor count, every maximizing sensor set belongs to the Kl$\unicode{248}$ve class. The argument classifies the overlaps between a generator and its shifted reflection, then bounds the aperture of a hypothetical optimizer outside the Kl$\unicode{248}$ve class. Comparing these bounds with a classical Kl$\unicode{248}$ve construction settles all sensor counts at least 330. An exact integer certificate covers the remaining counts from 2 through 329 and matches every equality case to a Kl$\unicode{248}$ve array as a complete set. Consequently, optimization within the full symmetric Kl$\unicode{248}$ve-Mossige class reduces to the previously known search over Kl$\unicode{248}$ve parameters. The theorem concerns this specified class, rather than unrestricted sparse arrays or all restricted additive bases.

The McVittie metric describes a self-gravitating compact object embedded in an expanding universe. It inherits two singularities: a curvature singularity at $r \to 0$ and a cosmological singularity at $a(t) \to 0$. Motivated by the construction of black-bounce geometries, we regularize both of these singularities and propose a regularized McVittie metric. The central singularity is regularized by replacing $r \to \sqrt{r^{2}+b^{2}}$, leading to a geometry that interpolates between a cosmological black hole, a black bounce, and a traversable wormhole. Similarly, the cosmological singularity is regularized by introducing a non-vanishing scale factor $a \rightarrow \sqrt{a^{2} + a_b^{2}}$. The resulting spacetime is supported by an effective imperfect fluid with finite anisotropic stress. We investigate the formation of the apparent horizon, analyze the circular geodesic structure, and argue that the geometry can also be interpreted as a conformally evolving Morris-Thorne wormhole embedded in a regular cosmological background.

We consider the stability problem of fork-join systems with redundancy (FJR) and heterogeneous servers under both static and dynamic capacity-allocation policies. In an $(n,k)$ FJR system, each arriving job is split into $n$ independent tasks, with one task assigned to each of $n$ parallel servers. Once $k \le n$ tasks have been processed, they are joined and the corresponding job departs the system; the remaining $n-k$ unprocessed tasks are then removed and are therefore termed redundant. We first identify the nominal traffic intensity and characterize the maximal stability region, defined as the set of traffic intensities for which there exists an admissible policy that stabilizes the system. We then establish conditions under which this maximal stability region is attained for two classes of policies: static and dynamic. Specifically, we show that for static allocation policies, in which service capacities remain fixed over time, maximality is achieved whenever the fastest server is allocated no more than $1/k$ of the total service capacity. For dynamic allocation policies, in which a fixed total service capacity may be repeatedly reallocated among the servers, we show that maximality is achieved whenever the cumulative capacity allocated to the $j$ shortest queues does not exceed $j/k$ of the total capacity for every $j=1,\ldots,k-1$. Our analysis is based on a projection of the $(n,k)$ FJR system onto a simpler $(k,k)$ system that has no redundancy, together with a novel sample-path comparison argument for multidimensional processes based on the generalized Schur-convex order.

Simple affine vertex operator algebras $L_k(\mathfrak{g})$ at non-integral admissible levels $k$ are generally neither $C_2$-cofinite nor rational. Despite the absence of these standard finiteness conditions, we prove that the sheaf of coinvariants (and, dually, of conformal blocks) of ordinary modules in the category $\mathcal{O}_k$ forms a vector bundle over the moduli space $\overline{\mathcal{M}}_{0,n}$ of stable $n$-pointed genus-zero curves. The proof relies on establishing the finite-dimensionality of genus-zero coinvariants, a vanishing theorem that isolates ordinary admissible modules at the boundary, and a smoothing theorem enabled by partial strong identity elements in the mode transition algebra. Consequently, admissible affine vertex operator algebras supply a rich class of non-rational, non-$C_2$-cofinite examples in which vector-bundle behavior is preserved for a well-behaved subcategory of modules.

We study how far gradient descent (GD) can be accelerated by predetermined stepsizes in smooth convex optimization. Writing $p_{\mathrm{sil}}=\log_2(1+\sqrt{2})$, we prove an $\Omega\left(n^{-p_{\mathrm{sil}}-O(\sqrt{\log\log n/\log n})}\right)$ non-anytime lower bound. In the anytime setting, every infinite schedule has infinitely many horizons with error $\Omega\left(n^{-\frac{2p_{\mathrm{sil}}}{1+p_{\mathrm{sil}}}-O(\sqrt{\log\log n/\log n})}\right)$. Together with the silver-schedule upper bound [Altschuler and Parrilo, 2025] and the anytime upper bound [Zhang et al., 2025], our results determine the optimal polynomial convergence exponents in both settings.

As a generalisation of the Erdős-Sós conjecture about graphs, Addario-Berry, Havet, Linhares Sales, Reed and Thomassé conjectured that every digraph on $n$ vertices with more than $(k-1)n$ arcs contains every antidirected tree with $k$ arcs. We prove a dense, approximate version of this for trees with bounded maximum degree, as well as for trees whose layers are evenly distributed. We use a regularity based approach, centred around finding a copy of a given tree in the blow up of a caterpillar.

In this paper, we show that the border Waring rank of $x_1x_2\cdots x_n$ over fields of characteristic zero is exactly $2^{n-1}$. As a consequence, the classical polarization identity is an optimal Waring decomposition even if we allow limits. As a symmetric tensor, this monomial is identified with the $n\times n$ permanent tensor. However, the lower bound is proved via the higher-order Koszul flattening of the $n\times n$ determinant tensor, which is not symmetric.

Let u be a morphism of algebraic varieties. Given m integer or infinite, we prove that the set of points whose fiber has size m is a constructible set.

We show that the Tate module of the $\ell$-adic unramified cohomology $H^3_{\nr}(X, {\Q_\ell}/{\Z_\ell})$ of a smooth projective surface $X$ over a local or strictly local field $k$ of residue characteristic prime to $\ell$ vanishes under suitable conditions. We apply this to answer some questions concerning the left and right kernels of the Brauer–Manin pairing for open subvarieties of $X$. To prove the vanishing theorem, we establish Bloch's formula for the Chow group of codimension-two cycles on a semistable model of $X$, and derive applications to the restriction map for such cycles to the closed fiber of the model.

We give a self-contained exposition of metric differentiability for Lipschitz mappings from sets in Euclidean spaces into arbitrary metric spaces, together with the corresponding area and co-area formulas. The novelty is a new metric implicit function theorem for mappings into metric spaces, which is then used to give a direct and geometric proof of the co-area formula for Lipschitz mappings into metric spaces.

Classically, the places of $\mathbb{Q}$ are often regarded as a one-point compactification of $\mathrm{Spec}(\mathbb{Z})$, with the real place corresponding to a formal “prime” added at infinity. Re-examining this picture from a topos-theoretic perspective reveals a subtler geometry: while the non-Archimedean places are identified with singletons indexed by the non-zero prime ideals of $\mathbb{Z}$, the Archimedean place is represented by the space of upper reals $\overleftarrow{[0,1]}$, which may be informally thought of as the unit interval equipped with a non-Hausdorff topology. On a technical level, our analysis brings together geometric logic and descent techniques from topos theory, distinguishing standard descent from lax descent toposes both at the level of sheaves and of the geometric theories they classify. More broadly, this paper brings into conversation two parallel distinctions: on the number-theoretic side, between Archimedean and non-Archimedean phenomena, and on the topos-theoretic side, between standard and lax descent. Looked at from a high level, these perspectives begin to converge on a common theme: how should the connected and the disconnected interact?

We define symplectic Dirac operators on homogeneous spaces and study their representation-theoretic role. For an invariant polarization, the symplectic Dirac operator decomposes into two symplectic Dolbeault operators. We compute their commutator as the natural symplectic analogue of the square of the classical Dirac operator. Our first main result gives a necessary and sufficient condition for this commutator to satisfy a Parthasarathy-type formula. We further prove that, whenever this condition fails, no cubic perturbation of the symplectic Dolbeault operators can yield such a formula, in contrast with Kostant's cubic Dirac operator in the orthogonal setting. As applications, we establish an ${\mathfrak s}{\mathfrak l}_2$-structure generated by the symplectic Dolbeault operators and derive Dirac-type inequalities for unitary representations of Hermitian symmetric spaces labelled by the levels of the symmetric powers of the antiholomorphic tangent space at the identity. The level-zero inequality recovers the standard Parthasarathy-Dirac inequality, while the higher levels inequalities yield new constraints. For $SU(1,n)$, we show that, for representations with a specific Kraljević corner, the level-one inequality strengthens all basic Parthasarathy inequalities of the first kind for particular $K$-types, precisely those satisfying an explicit highest-weight condition.

For $n\ge 3$ and $c\in(0,1)$, let $\mathcal{GT}(n,c)$ denote the class of geodesic metric trees of valence at most $n$ whose branch points are uniformly relatively separated with constant $c$. We prove that $\mathcal{GT}(n,c)$ has no bi-Lipschitz universal element. More precisely, we construct a family $(T_a)_{a\in[1/4,1/3]}\subset\mathcal{GT}(n,c)$ such that, for every $n_M\geq 3, c_M\in(0,1)$ and every $M\in\mathcal{GT}(n_M, c_M)$, there are at most countably many parameters $a$ for which $T_a$ admits a bi-Lipschitz embedding into $M$, whereas each $T_a$ admits a bi-Lipschitz embedding into $\mathbb R^2$. Thus the obstruction is neither dimensional nor caused by a failure of planar embeddability. This gives a negative answer to a question of Chrontsios-Garitsis, Ioannidis, and Vellis~\cite[Question~1.11]{CGIV2024}. Furthermore, we show a complementary positive result for ultrametric spaces: every bounded ultrametric space $X$ admits a bi-Lipschitz embedding into every complete metric space $Y$ satisfying $\dim_A X<\dim_{LA} Y$ where $\dim_A X$ and $\dim_{LA} X$ are Assouad and lower Assouad dimensions, respectively.