Papers

Baker, Huh, Kummer, and Lorscheid define the triangular-hyperfield threshold q(n)=q(U_{2,n}) and conjecture exact values for all n. Using their identity q(n)=P(n-1), where P(m) is the universal negative-type exponent of m-point Ptolemaic metrics, we prove q(6)=\log_2(9/4) and q(7)=1. For five points, we separate zero-sum coefficient vectors by sign pattern. A sharp two-summand inequality proves the inequality for the 1+4 pattern. For the 2+3 pattern, a sharp four-point partial-correlation bound and an exact copositivity identity prove the required inequality. For six points, we construct an involutive coefficient transport under metric inversion that preserves the negative-type quadratic form. Applying the transport at an index with positive local contribution converts a hypothetical 3+3 counterexample into a 2+4 counterexample. Complete split graph metrics attain both bounds.

Let $G$ be a group, let $\mathbb{K}$ be a field, and let $V$ be a $\mathbb{K}$-vector space. We prove that there exists a bijective linear cellular automaton $V^G\to V^G$ whose inverse is not a cellular automaton if and only if $G$ is not locally finite and $\dim_{\mathbb{K}}V \geq |\mathbb{K}|^{\aleph_0}$. In particular, $|\mathbb{K}|^{\aleph_0}$ is the exact dimension threshold for non-reversibility. This answers an open problem proposed by T. Ceccherini-Silberstein and M. Coornaert.

We investigate the asymptotic behavior of Jacobi-Piñeiro polynomials of degree $2n$ orthogonal on $[0,1]$ with respect to weights $w_j(x) = x^{\alpha_j}(1-x)^{\beta}$, $j=1,2$ where $\alpha_1,\alpha_2, \beta>-1$, and $\alpha_1-\alpha_2 \notin \mathbb{Z}$. These polynomials are characterized by a Riemann-Hilbert problem for a $3 \times 3$ matrix-valued function. We use the Deift-Zhou steepest descent method for Riemann-Hilbert problems to obtain strong uniform asymptotics in the complex plane. The local parametrix around the origin is constructed using Meijer G-functions. We match the local parametrix around the origin with the global parametrix with a double matching, a technique that was recently introduced.

We discuss the almost stability theorem of Dicks and Dunwoody in the context of probability measure preserving equivalence relations.

In this paper we develop a Li-Yau-Hamilton (LYH) type differential Harnack estimate for positive solutions to the FitzHugh-Nagumo equation on $\mathbb{R}^n$. We then use our LYH-differential Harnack inequality to prove several properties about positive solutions to the equation, including a classical Harnack inequality and a lower bound for the speed of traveling wave solutions.

In this article, we show that flips and divisorial contractions preserve the Kähler condition (for strongly $\mathbb{Q}$-factorial compact Kähler generalized klt pairs with $B+\beta_X$ big), and we give a criterion for varieties in Fujiki's class $\mathcal C$ to be Kähler. We also prove the existence of small $\mathbb Q$-factorializations for generalized klt pairs and of dlt modifications for generalized pairs.

We extend, and fully implement, in Maple, Jim Propp's charming way of estimating Pi via tossing a fair coin, and compute many other constants, even going beyond the far more general set-up of Bruss and Paindaveine.

We study the Schrödinger evolution generated by the isotropic Ornstein–Uhlenbeck operator \[ \mathscr L=\Delta-\langle x,\nabla\rangle \] in $L^2(\mathbb R^d,d\gamma)$, where $d\gamma$ is the invariant Gaussian measure. Combining the classical lens correspondence between the free Schrödinger equation and the harmonic oscillator with Gaussian conjugation, we identify the free Schrödinger structure hidden in the Ornstein–Uhlenbeck evolution. The main consequence is a dynamic restriction theorem of spacetime type. After an explicit transformation of spacetime variables, the adjoint Ornstein–Uhlenbeck extension operator becomes an ordinary spacetime Fourier transform restricted to the classical Schrödinger paraboloid. On the Ornstein–Uhlenbeck side this corresponds to an explicitly time-dependent Gaussian spacetime measure. Thus a restriction geometry which is neither suggested by the symbol of $\mathscr L$ nor selected by a scaling symmetry of the Ornstein–Uhlenbeck equation emerges dynamically from the free Schrödinger representation. The same correspondence identifies the weighted Gaussian spaces in the homogeneous and inhomogeneous Ornstein–Uhlenbeck Strichartz estimates and transports the mass-critical nonlinear Schrödinger equation to \[ i\partial_tu+\mathscr Lu = \mu e^{-|x|^2/d}|u|^{4/d}u. \] We also obtain a global description of the unitary Ornstein–Uhlenbeck group, including its oscillatory representation away from the caustics and its exact values at the caustic times, together with the associated sharp weighted dispersive estimate and Hardy-type uncertainty principles.

We construct a finitely generated parafree Lie ring with nonzero second homology and, over every nonzero commutative ring, a finitely generated parafree augmented associative algebra with nonzero second homology. These examples disprove the analogues of the Parafree Conjecture for Lie rings and for augmented associative algebras. Our constructions also provide finitely generated parafree examples that are not finitely presented, both in the Lie-ring and augmented-algebra settings.

Let $N\geq 3$. The class of $N$-Koszul AS regular algebras, or more generally, that of $N$-Koszul AS Gorenstein algebras, has attracted much attention from algebraists. Nevertheless, there have been no known examples of $N$-Koszul AS regular algebras of finite global dimension other than the ones of global dimension $3$. A recent work by Kabbaj showed that, such an $N$-Koszul algebra $A$ of finite global dimension has to have a large global dimension and that $N$ has to be prime, under the assumptions that $(1)$ $A$ has a Hilbert series of weighted polynomial rings and that $(2)$ the trivial $A$-module $\mathbb{K}$ has a finite free resolution. All AS regular algebras satisfy the latter assumption and are expected to do the former as well. In this paper, we prove that such an $N$-Koszul algebra $A$ must be one of the known $3$-Koszul AS regular algebras of global dimension $3$ if the order of the pole of its Hilbert series $h_A(t)$ at $t=1$ is greater than $\frac{21d+1}{22}$, where $d$ is the global dimension of $A$. As a corollary, we prove that any $N$-Koszul AS regular algebra $A$ must be one of the known $3$-Koszul AS regular algebras of global dimension $3$ if $A$ has a Hilbert series of weighted polynomial rings and if the GK dimension of $A$ coincides with the global dimension of $A$, both of which have been conjectured to hold for any AS regular algebras.

For an integer $q\ge2$, a matroid is $q$-entropic if its rank function, multiplied by $\log q$, is the joint-entropy function of random variables on a $q$-element alphabet. We prove that a matroid is $4$-entropic if and only if it is representable over $\F_4$. The corresponding statements for alphabet sizes two and three were known. The proof combines minor closure and the excluded-minor characterization of quaternary matroids with structural properties of quasigroups of order four. Thus arbitrary four-symbol partition representations yield no matroids beyond the quaternary ones. As an application, every access structure admitting an ideal perfect scheme with a uniform four-symbol secret and four-symbol active shares also admits an ideal $\F_4$-linear scheme.

We study exponential directed last passage percolation conditioned on the last passage time to a specified macroscopic point being atypically large. We determine the one-point fluctuations throughout two of the three spatial regions arising under this conditioning, as well as on the boundaries between these regions, extending the work of Baik-Cordaro-Tripathi. Depending on the location of the observation point, the limiting fluctuations are governed by the GUE Tracy-Widom distribution, a Gaussian distribution, or the one-spike BBP distribution on the boundaries between these regions. Through the correspondence with tandem queues, our results also describe how an atypically late departure at one customer-station pair affects departure epochs elsewhere in the network.

A knot in $S^3$ is said to be not $SU(2)$-abelian knot, if every non-trivial surgery along it yields a $3$-manifold whose fundamental group admits an irreducible $SU(2)$-representation. We provide examples of knots in $S^3$ that are not $SU(2)$-abelian. A knot $K$ is said to be $SU(2)$-clean if whenever the fundamental group of the integer $r$-surgery $\fund{K(r)}$ has no $SU(2)$-irreducible representation, then the Alexander polynomial of $K$ does not vanish at any $r$-th root of unity. We show that if $K$ is a non-trivial $SU(2)$-clean knot, the connected sum $K\#K$ is never $SU(2)$-abelian. Finally, combining known results on $SU(2)$-abundant knots and a classical epimorphism between knot groups, we show that every non-torus knot with at most $9$ crossings is not $SU(2)$-abelian, with at most the two exceptions of $9_{47}$ and $9_{49}$.

Let $G$ be a finite simple co-chordal graph with at least one edge, and let $I(G)$ be its edge ideal. Over an arbitrary field, we prove that the symbolic powers of $I(G)$ satisfy $\operatorname{reg} I(G)^{(s)}=2s$ for every $s\ge1$. Thus every symbolic power of $I(G)$ has a degree resolution. Using Takayama's formula and clique trees, we reduce the regularity problem to a topological one. Finite convex geometry then shows that nontrivial homology implies a suitable set of leaves, and a weighted counting argument gives the required regularity formula. Finally, we show that symbolic powers of co-chordal edge ideals are not necessarily componentwise linear.

We study angular variation in the frontier of branching Brownian motion (BBM) in multiple dimensions. The time average of this frontier is tied to the long-time limit of the critical derivative martingale, which is a random integrable function on the sphere. We show that this function is nowhere locally bounded. As a consequence, BBM exhibits strong persistent anisotropy: there are dense arbitrarily large gaps between the BBM frontier in different directions.

A sequence $a_1,\ldots,a_n$ of integers is nearly consecutive if $a_{i+1}-a_i \in \{1,2\}$ for $1 \leq i \leq n-1$. We prove that there are nearly consecutive sequences of length $\Omega(2^k/k^2)$ that contain no $k$-term arithmetic progression. This improves on the previous best known bound of Alon and Zaks from 1998. We also prove a generalization for sequences with bounded gaps.

The Nyström method approximates $A\approx A(\,:\,,I)A(I,I)^{\dagger} A(\,:\,,I)^{\top}=CA(I,I)^{\dagger} C^{\top}$, where $C:=A(:,I)\in\mathbb{R}^{n\times r}$ is a column subset matrix of $A$. When applied to symmetric but indefinite matrices, the Nyström method can fail because the core matrix $A(I,I)$ may severely underestimate the eigenvalues of $A$ and may become (nearly) singular. We address this issue by developing and analyzing an algorithm that carefully chooses $\widehat{M}\in\mathbb{R}^{r\times r}$ in place of $A(I,I)^\dagger$ by solving the two-sided sketched least-squares problem $\min_{M}\|X(A-CMC^{\top})X^{\top}\|_F$, where $X\in\mathbb{R}^{t\times n}$ is a random sketch matrix. We study in detail the cases where $X$ is a Gaussian or a leverage score sampling (LSS) matrix, and show that with oversampling $t>r$ the residual $\|A-C\widehat{M}C^{\top}\|_*$ is comparable to $\min_{M}\|A-CMC^{\top}\|_*$. For the Gaussian sketch, we require $t=\mathcal{O}(r)$ samples; for LSS, we show that $t=\mathcal{O}(r \log r)$ samples suffice for the theoretical guarantee, with the LSS approach carrying the advantage that once a set of $t$ row indices is identified, the approximation requires only $t^{2}$ matrix-entry evaluations to find $\widehat{M}$, given $C$. We illustrate our results with synthetic examples and applications to kernel methods.

We describe a version of positive macroscopic scalar curvature motivated by the work of Alpert, Balitskiy, and Guth, and prove that this condition on a manifold implies a bound on its 1-width in terms of its first Betti number. A key tool in the proof is a decomposition of any closed manifold into a family of chains with convenient combinatorial structure, which was inspired by Nabutovsky, Rotman, and Sabourau's work on sweepouts. Finally, using techniques developed by Chodosh, Li, and Liokumovich, we show that for a sufficiently connected manifold, if the universal cover satisfies this curvature condition, then the manifold has a finite cover homotopy equivalent to $S^n$ or connected sums of $S^{n-1} \times S^1$.

In this paper, we study the derivations of an irreducible plane algebroid curve $R:=\frac{k[[X,Y]]}{(f)}$, which is not regular. Since the normalization of $R$ is isomorphic to $k[[t]]$, every $k$-derivation of $R$ is induced from a $k$-derivation of $k[[t]]$, which are of the form $a(t)\frac{d}{dt}$ for some $a(t)\in k[[t]]$. We establish a lower bound on the $t$-order of a nonzero power series $a(t)$ for $a(t)\frac{d}{dt}$ to induce a derivation of $R$ in terms of the multiplicity of $R$. We also prove a related result for the value semi-group of such curves.

We determine the sharp exponential growth rate of the minimum number of abelian subgroups required to cover a group with bounded pairwise noncommutativity. Let $\omega(G)$ denote the largest size of a pairwise noncommuting subset of a group $G$, let $a(G)$ be the least size of an abelian cover, and define $h(n)=\sup\{a(G):\omega(G)\le n\}$. We prove the quantitative estimate $\log_2 h(n)=n/2+O(\sqrt{n}\,(\log(n+2))^3)$, and hence $h(n)^{1/n}\to\sqrt{2}$. Extraspecial $2$-groups give the matching lower bound. For the upper bound, we reduce to finite groups by isoclinism, analyze finite $p$-groups through a central series of the derived subgroup and alternating commutator forms, control interactions between central factors, and then pass through Sylow decomposition and the Fitting subgroup at polynomial cost. The argument also determines the same sharp exponential rate for the least possible index of an abelian subgroup and shows that asymptotic extremality is concentrated in $2$-groups. This result resolves Erdős Problem #117 at the level of its sharp exponential asymptotics.

We study Ricci solitons on the Riemannian manifold $\mathbb H^2\times\mathbb R$ equipped with the standard metric. A complete classification of soliton vector fields is obtained: they form a four-dimensional affine space, namely a translate of the Killing algebra $\mathfrak{isom}(\mathbb H^2\times\mathbb R)$. All corresponding solitons are expanding. In addition, gradient solitons are fully characterized and shown to form a one-parameter subfamily of the complete family of soliton fields. As a byproduct, every soliton vector field turns out to be affine, preserving the Levi-Civita connection, the curvature tensor, and the Ricci tensor.

In this paper, we study the submanifolds of holomorphic statistical manifolds with vanishing Bochner curvature tensor (Bochner holomorphic statistical manifolds). We study the Lagrangian statistical submanifolds and discuss their conformal flatness under the assumption of vanishing Bochner curvature tensor. Next we study the holomorphic submanifolds of Bochner holomorphic statistical manifolds and prove that the doubly autoparallel holomorphic submanifold of a Bochner holomorphic statistical manifold has vanishing Bochner curvature tensor.

In this paper we prove matrix weighted Sobolev and Poincaré inequalities using techniques derived from the theory of Rubio de Francia extrapolation. Given weights $w,\,v$ and a symmetric non-negative definite matrix valued function $Q$ defined on a connected open subset $\Omega$ of $\mathbb{f}R^n$ that satisfies the lower ellipticity condition \[ w(x)^p \leq |\sqrt{Q(x)}\xi|^p,\quad \xi\in \mathbb{R}^n, \] we give Lebesgue integrability conditions on the weights $w,v$ that ensure there exists $\tau\geq 1$ so that Sobolev and Poincaré inequalities of the form \[\bigg(\int_\Omega |u|^{\tau p} \,vdx\bigg)^{\frac{1}{\tau p}} \leq C(v,w) \bigg(\int_\Omega |\sqrt{Q}\nabla u|^p\,dx\bigg)^{\frac{1}{ p}},\textrm{ and}\] \[\bigg(\int_\Omega |u-\langle u\rangle_{\Omega,v}|^{\tau p} \,v dx\bigg)^\frac{1}{\tau p} \leq C(v,w)\bigg(\int_\Omega|\sqrt{Q}\nabla u|^p \, dx\bigg)^{\frac{1}{p}}\] hold for smooth $u$. We explore these and related results in the context of several examples that include John domains, the Heisenberg group, and CR manifolds.

We introduce the notion of a total inner product on an $A_\infty$-algebra, and a cyclic Chern-Simons form associated to a topologically nilpotent element. The total inner product applied to the cyclic Chern-Simons form gives a superpotential function that is gauge invariant on solutions of the Maurer-Cartan equation known as bounding cochains. The derivative of the superpotential is computed. The definition of total inner product replaces strict symmetries that appear in previous notions of homotopy inner products with symmetries up to an infinite family of coherent homotopies. We show how to recover previous notions of homotopy inner products as special cases of total inner products. Total inner products are designed to facilitate the definition of descendent open Gromov-Witten invariants. The definitions of the total inner product and the cyclic Chern-Simons form use the total complex of cyclic codifferential forms, which gives a chain model for cyclic homology. We give explicit formulas for homotopy equivalences with other known models. We discuss also the notion of an $\infty$-trace, which arises from Connes' cyclic complex, and the associated $\infty$-modulus, which gives another gauge-invariant function on bounding cochains. In open Gromov-Witten theory, the $\infty$-modulus is used to normalize the bounding cochains to which the superpotential is applied. We develop our definitions for general curved Banach $A_\infty$-algebras, and give both unital and non-unital versions of the main results. In the unital setting, we work with weak bounding cochains, solutions of an inhomogeneous Maurer-Cartan equation involving the unit. Weak bounding cochains arise in the open Gromov-Witten theory of Lagrangian submanifolds with non-vanishing Maslov class. The derivative of the superpotential at a weak bounding cochain is related to the derivative of the $\infty$-modulus.

We consider the problem of recovering functions defined on the half-line from inexact input information. For the proposed regularizing summation methods based on Laguerre polynomials, we analyze their approximation properties on functions from Wiener classes. We establish conditions under which these methods are stable with respect to small perturbations of the input data and order-optimal not only in accuracy in the weighted $\sup$-norm, but also in the number of Fourier-Laguerre coefficients used.

We study the Green-Griffiths-Lang Conjecture for complements of smooth plane curves. We develop an effective method for computing a family of negatively twisted invariant logarithmic 2-jet differentials. By realizing the first logarithmic jet space as a hypersurface in $\mathbb{P}^2 \times \mathbb{P}^2$, we encode these jet differentials in a finitely generated bigraded module that can be computed explicitly. We use this description to give a computational criterion for the Green-Griffiths-Lang Conjecture and verify it for several families of smooth plane curves. In examples with sufficiently many independent jet differentials, we determine the exceptional locus explicitly.

We study perfect state transfer on Cayley graphs from the point of view that state transfer is a property of a graph and not of a group. This paper is a bridge between the classical question about isomorphic Cayley graphs of non-isomorphic groups and quantum walks on graphs. We show that a Cayley graph of a group with an abelian subgroup of index two is a Cayley graph of an abelian group under any one of three hypotheses, two drawn from the theory of isomorphic Cayley graphs. A statement of the same kind holds for extraspecial groups: every Cayley graph of an extraspecial $p$-group of order $p^{2n+1}$ with a conjugacy-closed connection set is a Cayley graph of $Z_p^{2n+1}$. From these results we deduce that every explicit construction of perfect state transfer in the six papers we survey, on dihedral, dicyclic, generalized dihedral, $V_{8n}$ and extraspecial $2$-groups, is a non-abelian presentation of an abelian Cayley graph. Moreover, we show that a non-abelian group with an abelian subgroup of index two admits a connected Cayley graph with perfect state transfer if and only if its order is divisible by four. Genuinely non-abelian examples do exist. We prove that, for every odd prime power $q\ge 5$, the $SL(2,q)$ graph of Pantangi and Sin, which they showed to admit perfect state transfer, is a Cayley graph of no abelian group; to our knowledge, this is the first infinite family of Cayley graphs with perfect state transfer provably admitting no abelian Cayley presentation. We also construct an infinite family of Cayley graphs with peak state transfer and determine all regular subgroups of the automorphism group of every member. An appendix records a census of the connected vertex-transitive graphs with perfect state transfer on at most $30$ vertices.

The Bingham distribution is a family of antipodally symmetric distributions on the unit sphere, characterised by an exponential-of-quadratic change of measure with respect to the uniform distribution. Kent, Ganeiber and Mardia proposed a rejection sampler for generating samples from Bingham distributions using proposals from an angular central Gaussian (ACG) distribution. Their empirical results suggest that the least efficient regime is the high-concentration limit, where the acceptance probability is of order $d^{-1/2}$ in dimension $d$, implying a polynomial complexity guarantee. In this note, we verify this dimension-dependent prediction, establishing the uniform guarantee $\inf\{\alpha_D:D=D^\top\in\mathbb{R}^{d\times d}\}\ge c_\star/\sqrt{d}$, where $c_\star=0.759\ldots$. A one-dimensional high-concentration limit demonstrates that the $d^{-1/2}$ rate is unimprovable and that even the constant $c_\star$ cannot be improved beyond $0.857\ldots$. The proof relies on a novel interpretation of the acceptance probability and a comparison principle for weighted sums of chi-squared random variables, which may be of independent interest.

Let $k$ be a commutative ring, $H$ a bialgebra over $k$, $A$ an $H$-algebra and $\mathcal C$ an $(H,A)$-coring, i.e a left $(H,A)$-bimodule which is an $A$-coring in a compatible way. Left $(H,A)$-bimodules and their deformations play a fundamental role in the study of the differential geometry of noncommutative manifolds. Let $\Lambda$ be a right $(H,{\mathcal C})$-comodule. The $k$-module $_HEnd^{\mathcal C}(\Lambda)$ of right $(H,{\mathcal C})$-colinear maps from $\Lambda$ to $\Lambda$ is a ring. Let us assume that $\mathcal C$ is flat as a left $A$-module. If $\Lambda$ is finitely generated (finitely presented) as a right $(H,{\mathcal C})$-comodule, we give necessary and sufficient conditions for projectivity and flatness of a module over $_HEnd^{\mathcal C}(\Lambda)$. If $\mathcal C$ contains a fixed $H$-grouplike element, we can replace $\Lambda$ with $A$.

We prove that for every non-elementary torsion-free hyperbolic group $H$ and sufficiently large odd $n>1$, the quotient $H/H^n$ has cost $1$. In particular, when $H$ is the free group of rank $m\ge 2$, we obtain that the free Burnside group $B(m,n)$ has cost $1$ for every odd exponent $n\geq 665$. It follows that $B(m,n)$ is anti-treeable, and that its first $\ell^2$-Betti number $\beta_1^{(2)}(B(m,n))$ vanishes. The latter recovers and extends a result of Feldkamp and Kionke [Proc. Amer. Math. Soc., 2023], who proved that $\beta_1^{(2)}(B(m,p))=0$ for all sufficiently large prime exponents $p$.