Papers

We show that, for the words $r=\mathtt{aabABabABBAbaabABBAb}$ and $r'=\mathtt{aabABabABabABBAbaBAb}$, both in the commutator subgroup of the free group on $\mathtt{a}$ and $\mathtt{b}$, the one-relator groups $\langle \mathtt{a,b}\mid r=1\rangle$ and $\langle \mathtt{a,b}\mid r'=1\rangle$ are isomorphic, whereas $r$ and $r'$ have distinct stable commutator lengths. This provides a negative answer to a question posed by Heuer and Löh.

We develop a new direct accelerated Newton method for minimizing convex functions with Lipschitz continuous Hessian. The algorithm uses only primal variables and performs just one linear solve per iteration. With a simple predetermined choice of parameters, it achieves the global convergence rate of $O(1/k^3)$ in terms of the functional residual. To the best of our knowledge, this is the first second-order method for this problem class attaining this rate while relying solely on one linear system solve per iteration (without solving auxiliary nonlinear regularized subproblems, such as cubic regularization, performing nonlinear parameter searches, or using dual extragradient corrections). Our method can be implemented in a Hessian-free way, using an inexact linear system solver, while preserving the fast global rate. We further extend our construction to arbitrary geometry through Bregman divergence, and to composite optimization problems.

Let $f$ be a Steinhaus or Rademacher random multiplicative function. We use methods from the theory of critical chaos to improve on the best known upper bound for partial sums of random multiplicative functions. In particular, our results imply that for any $\varepsilon>0$, almost surely $$ \Big|\sum_{n\le x}f(n)\Big| \ll_{\varepsilon}\sqrt{x}(\log_2x)^{1/4}(\log_3x)^{1+\varepsilon}. $$ This proves in a strong form a conjecture of Harper on large fluctuations of partial sums of random multiplicative functions, and determines the exact corresponding logarithmic exponent.

We develop a reduction and transfer framework for cohomological, variational, and thermodynamic problems in fiber-contracting extensions of local homeomorphisms. Under uniform contraction along the fibers and the existence of a continuous global section, every Hölder potential admits the explicit decomposition \[ \varphi=\psi\circ\pi+u-u\circ F, \] where $\psi$ is defined on the quotient and $u$ is obtained by a uniformly convergent fiberwise series. We give quantitative criteria for continuity and Hölder regularity and use the reduction to transfer subcohomological and Livšic-type statements from the quotient to the extension. We prove that invariant measures on the extension and quotient are in bijection, invariant averages are preserved, and the fibers have zero relative topological entropy. Consequently, metric entropy, ergodic optimization, pressure, minimizing measures, and equilibrium states correspond. We further establish weak Gibbs transfer under subexponential comparison of Bowen-ball masses and prove blockwise Bowen sequential weak Gibbs transfer along prescribed point-dependent Gibbs times; under uniformly bounded sequential mass distortion, the strong Bowen sequential Gibbs property is preserved. Concrete criteria for Bowen-ball comparability are provided. Finally, affine and symbolic contracting skew-products illustrate the framework, including the explicit reduction \[ \varphi(x,y)=g(x)+by \quad\;\;\;\;\;\;\;\;\quad \psi(x)=g(x)+\frac{b}{1-a}\rho(x) \] for $F(x,y)=(f(x),ay+\rho(x))$, $|a|<1$.

A canon permutation is a $k$-regular word over $[n]$ in which, for each $j$, the $j$-th copies of the letters form the same permutation $\sigma$. These were introduced by Elizalde as a generalization of nonnesting multipermutations, which are the case $k = 2$. We study classical pattern avoidance in them for arbitrary $k$. We show that avoiding any one of $112$, $122$, $211$ or $221$ is counted by the $k$-Catalan numbers $\frac{1}{n}\binom{kn}{n-1}$. We enumerate the classes obtained by forbidding one of these together with any $\tau \in \mathcal{S}_3$, and we give a bijection with $k$-ary trees that we use to generalize a theorem of Gabriel, Peske, Pudwell and Tay. We then show that avoiding a set of patterns closed under relabeling reduces, up to a factor of $n!$, to avoidance in $k$-regular lattice words. We use this to enumerate the canon permutations avoiding some natural generalizations of the nonnesting and noncrossing patterns, as well as the family $\{1^a21^b, 2^a12^b\}$. We close with several conjectures and questions.

For fixed integers $k\ge3$ and $1\le d\le k-1$ and sufficiently large $n\in k\mathbb N$, we establish the sharp minimum $d$-degree thresholds that forces perfect matching in every $n$-vertex $k$-uniform hypergraphs. This was conjectued by Treglown and Zhao, and the $d=1$ case was conjectued by Kühn, Osthus and Treglown.

We prove new bounds on complete sums of products of k additively shifted Kloosterman sums to odd high prime power moduli q=p^n, which feature substantially stronger power savings (about q^(-1/ceil(k/2)) in generic configurations) and a novel quantification of the alignment among the shifts. We prove our bounds by developing a method to estimate complete exponential sums with a broad class of phases well-controlled by a specified number of initial terms in a property reflecting p-adic differentiability.

We prove that stable nonlocal minimal cones in three dimensions are flat, for $s$ sufficiently close to zero. Our approach develops new structural properties of stable $s$-minimal surfaces in every dimension as $s \downarrow 0$, notably a compactness theory and pinching theorems near a half-space.

We parametrize the isometry and equivalence classes of skew constacyclic codes over a finite field by classifying the corresponding classes of their ambient rings, and present algorithms for the parametrizations. We achieve a tight classification by taking all possible Hamming-weight preserving isomorphisms between their ambient Petit rings into account. We also count the number of equivalence classes of these rings. We present many examples where isometry is a strictly stronger relation than equivalence, that is, codes that are isometric but not equivalent.

We study finitely generated maximal right ideals of the Banach algebra $\mathcal{B}(E)$ of bounded operators on a complex Banach space $E$. Every maximal right ideal is either fixed by a non-zero functional or contains the ideal of finite-rank operators; when $E$ is infinite-dimensional, each non-fixed maximal right ideal in fact contains the ideal of inessential operators. Using the elementary representation of finitely generated right ideals as lifting ideals $\operatorname{Lift}(T)=\{TU:U\in\mathcal{B}(E,E^n)\}$, where $T\in\mathcal{B}(E^n,E)$ for some $n\in\mathbb{N}$, we identify the exact operator-theoretic obstruction. The ideal $\operatorname{Lift}(T)$ contains the finite-rank operators precisely when $T$ is surjective, and it equals $\mathcal{B}(E)$ precisely when $T$ is right invertible. If $T$ is surjective but not right invertible, then $\operatorname{Lift}(T)$ is maximal exactly when the row operator $[T\ S]$ is right invertible for every $S\in\mathcal{B}(E)\setminus\operatorname{Lift}(T)$. We apply this framework, together with duality, pullback, lattice-theoretic and cardinality arguments, to obtain maximal right ideals which are not finitely generated for large classes of Banach spaces. These include the following infinite-dimensional spaces: reflexive spaces, separable spaces with an unconditional Schauder decomposition into a countably infinite sequence of non-zero subspaces, spaces containing a complemented copy of $\ell_1$, KB-spaces, Lebesgue spaces $L_p(\mu)$ for $1\leqslant p<\infty$, full Orlicz spaces with order-continuous norm, and scalar-plus-compact spaces. We obtain the stronger conclusion that every finitely generated maximal right ideal is fixed for Hilbert spaces, $\ell_1(\Gamma)$-spaces, reflexive spaces with the bounded approximation property, and the mixed spaces $\ell_1(\Gamma)\oplus H$ with $H$ a separable Hilbert space.

We consider a competition between two independent random walks on a cycle of length $N$. Each vertex is claimed by the walker that visits it first, and remains claimed thereafter. We prove that if the initial distance between the walkers is $d$, then the expected number of edges whose endpoints are claimed by different walkers is of order $\ln(1+N/d).$ This confirms the logarithmic dependence on $N/d$ predicted in Gomes Jr. et al. [Coloring of a one-dimensional lattice by two independent random walkers. Physica A: Statistical Mechanics and its Applications 225.1 (1996): 81-88].

We prove that for the spin-1/2 Heisenberg XXZ chain in the range $\Delta\in[-1,1/2]$, the ground state on the torus of length $L$ converges to an infinite volume ground state $\langle\cdot\rangle$ as $L\to\infty$, and that the spin-spin correlation $\langle S_0^{(1)}S_x^{(1)}\rangle$ decays polynomially fast in $x$. In the range $\Delta\in[-1,0]$ we have the stronger results: that the convergence holds for several finite-volume, finite temperature states, that $L$ and $\beta=1/T$ can be taken to infinity in any order, that the convergence holds on (Euclidean) dynamic correlators, and that $\langle S_0^{(1)}S_x^{(1)}(t)\rangle$ decays to zero in $|(x,t)|$, and cannot decay exponentially fast. The convergence of the ground state in infinite volume is known rigorously by Bethe Ansatz methods, while our correlation decay results (apart from the points $\Delta=0,-1$), the convergence on dynamic correlators and the interchangeability of limits are new at the rigorous level. Moreover our methods are new and do not use any Bethe Ansatz techniques, using only the following related probabilistic models. In the Lorentz mirror model with loop weight 2, a model of random loops on the square lattice, in a large range of parameters we prove that connection probabilities tend to 0, and do so polynomially fast in the symmetric case of the model. Our proof uses two couplings of this mirror model with the six-vertex model, one of which is new. The main input is the delocalisation of the height function of the six-vertex model proved by several authors. We further prove that the height function of a certain space-time version of the six-vertex model delocalises and then prove through analogous couplings to those mentioned above that connection probabilities converge to 0 in the loop representation of the XXZ model introduced by Ueltschi.

The retrieval dynamics of a modern Hopfield network is the gradient flow of a log-sum-exp energy, while the attention update is its exact difference-of-convex minimization step. We study which time discretizations preserve not only energy decay and equilibria but also basins of attraction. We introduce energy cells, connected components of sublevel sets containing one attractor and no other critical point. Our main theorem shows that every finite energy cell below the escape energy is contained simultaneously in the basin of the continuous flow, every relaxed attention map $\Psi_\theta=(1-\theta)\,\mathrm{id}+\theta\,\mathrm{attention}$ for $0<\theta<2$, and implicit Euler throughout its uniqueness regime. A parameter-uniform unit-curvature majorant yields unconditional dissipation and a monotone interpolation of each discrete step. We also derive explicit local contraction bounds near well-separated patterns, with a certified optimal slight overrelaxation; characterize proximal tunneling and overshoot beyond the preservation regimes; compare first-order error constants; establish an order barrier for scalar reparametrizations of the relaxed family; construct a second-order scalar-auxiliary-variable scheme; and extend cell preservation to damped difference-of-convex iterations in Bregman geometry, including a certified overrelaxed window under bounded asymmetry. Nine numerical campaigns test the bounds and failure mechanisms. In two-dimensional basin experiments, all observed disagreements between continuous and discrete retrieval occur above the attractor-specific numerically inferred escape level.

Hyperelliptic theta divisors, and more generally, Brill-Noether loci, are shown to be locally catalecticant determinantal varieties. This has many consequences for their local and global structure.

We solve the problem, posed some twenty years ago, of explicitly constructing a continuous function whose unique maximizing measure for the doubling map is Lebesgue measure. This is a special case of a more general result: given a nontrivial compact metrizable abelian group and a continuous endomorphism for which normalised Haar measure is ergodic, we explicitly construct a continuous function on the group for which Haar measure is the unique invariant maximizing measure. The function is the uniform limit of a recursively defined sequence of trigonometric polynomials with rational coefficients; every parameter of the recursion is given by an explicit formula, every step is exact, and the rate of convergence is explicit. For expanding circle endomorphisms, ergodic toral endomorphisms, and the one-sided full shift, the construction can moreover be arranged so that the function is given by a Fourier series that converges uniformly in the classical frequency order, each of whose coefficients is rational and computable exactly by a finite procedure. We also show that no continuous function whose unique maximizing measure is Haar measure has absolutely summable Fourier coefficients, so that no such function lies in the Wiener algebra. For expanding maps of the circle, the Fourier coefficients of a continuous function with a unique maximizing measure of full support admit no closed form, in the sense that they satisfy no linear recurrence with polynomial coefficients; this follows from a result of independent interest, that a continuous function on the circle whose Fourier coefficients satisfy such a recurrence is Hölder continuous.

A Banach space $X$ has the SHAI property if, for every non-zero Banach space $Y$, every surjective algebra homomorphism from the algebra $\mathcal{L}(X)$ of bounded linear operators on $X$ onto $\mathcal{L}(Y)$ is injective. In this work, we prove that the Bourgain-Rosenthal-Schechtman spaces have the SHAI property, thereby answering a question posed by Johnson, Phillips, and Schechtman in 2022.

Motivated by refinements of Kunz's theorem involving the growth of Betti numbers, we study Betti and Bass numbers associated to the relative Frobenius. Under appropriate hypotheses on a local ring homomorphism $\varphi \colon R \to S$, we obtain bounds on the growth of the Betti and Bass numbers of relative Frobenius pushforwards of homologically finite complexes in terms of the corresponding invariants of the residue field of the closed fiber $S/\mathfrak m S$. These results extend previous work on Betti numbers to coefficients and provide a dual perspective via Bass numbers, contributing to the study of how Frobenius actions encode information about ring homomorphisms.

Cluster varieties contain the union of cluster tori and the points not in this union are called $\textit{deep points}$, and the locus of these points is called the $\textit{deep locus}$. In arXiv:2402.16970, a description of this locus is conjectured for locally acyclic cluster algebras, in particular, stating that this should be the stabilizer locus of the cluster automorphism group. We resolve this conjecture for the case of cluster algebras arising from surfaces, introduced in arXiv:math/0608367, and the remaining finite type cases as categorized in arXiv:math/0208229. In particular, we show locally acyclic surface type cluster varieties have no deep points not contained in the stabilizer locus, and finite type cluster algebras also have no deep points not contained in the stabilizer locus, validating the mysterious points conjecture in these cases.

Consider a free product $(M,\varphi)= (M_1,\varphi_1)* (M_2,\varphi_2)$ of non-trivial von Neumann algebras. Using amalgamated free product techniques, we establish irreducibility and weak spectral gap results for free product subalgebras and the centralizer subalgebra of the free product state. In particular, it is shown that, under a mild dimension constraint, the inclusion of the diffuse summand of these subalgebras into the corresponding corner of $M^\omega$ is irreducible for any free ultrafilter $\omega\in \beta\mathbb{N}\setminus \mathbb{N}$. We also obtain analogous irreducibility results for graph products of von Neumann algebras.

We give a self-contained analysis of the symmetric Lanczos algorithm without reorthogonalization when computations are performed inexactly. We show the computed Lanczos vectors are well-behaved locally. From this we (i) prove Paige's loss-of-orthogonality identity, and (ii) construct a Greenbaum-type backward stability result, exhibiting a nearby problem on which exact Lanczos produces the computed tridiagonal matrix. This allows us to derive several facts about the computed Ritz values at the original eigenvalues. Our analysis simplifies the analyses of Paige and Greenbaum, at the cost of hiding polynomial factors in the iteration count.

The recent work of Sarkar and Zhang (2025) introduced Positive Tail Dependence Under the Null (PTDN) and developed Generalized Shifted Benjamini-Hochberg (BH) procedures for two-sided Gaussian $z$- and $t$-testing under known covariance structures. This paper develops further consequences of that framework. First, we derive explicit dependence-adaptive lower and upper bounds for the FDR of the original BH procedure in terms of the conditional variance parameters $\tau_i=1-R_i^2$, where $R_i^2$ is the squared multiple correlation between the $i$th statistic and the remaining coordinates. These bounds recover the exact BH FDR under independence and provide finite-sample, covariance-specific information complementary to generic bounds. We also identify conditions under which the coordinate-specific calibration of shifted BH can provide a rejection advantage over the original BH procedure. Second, we consider the practically important setting in which the covariance matrix is unknown but an independent Wishart estimator is available. Using simultaneous lower confidence bounds for the $\tau_i$'s, we construct a confidence-bound shifted BH procedure and establish finite-sample FDR control. To our knowledge, this is the first shifted-BH-type procedure with a finite-sample guarantee for two-sided Gaussian mean testing under a completely unknown covariance matrix estimated independently. Numerical studies illustrate the behavior of the covariance-adaptive bounds, the potential advantage of shifted BH over BH, and the performance of confidence-bound shifting under unknown covariance.

We prove that every prime Fano threefold of genus $\geq 7$ in positive characteristic is a linear section of a Mukai variety.

Let $X$ be a prime Fano threefold of genus $g\geq 8$ in positive characteristic. For every nontrivial decomposition $g =rs$, we prove that there exists a unique Mukai bundle on $X$ of type $(r, s)$. To this end, we show that every smooth hyperplane section of $X$ is Brill-Noether general.

In this paper we prove the equality of the dimensions IndX and dimX for any stratifiable spaces. As a consequence, we obtain the equality of the dimensions IndX, indX and dimX for stratifiable spaces with a countable network. It has been proven that every stratifiable space is an S-space (possesses an S-network).

For each $s>0$, we consider a random Dirichlet series $X(s)=\sum_{k\geq 1}k^{-1/2-s}a_k\eta_k$, where $\eta_1$, $\eta_2,\ldots$ are independent and identically distributed random variables with mean zero and finite positive variance, and $(a_k)_{k\geq 1}$ is a deterministic sequence of real numbers satisfying $\sum_{k\geq 1}k^{-1-2s}a_k^2<\infty$ for each $s>0$ and $\sum_{k\geq 1}k^{-1}a_k^2=\infty$. We investigate the almost-sure fluctuations of $X(s)$ as $s\to0+$. Under these minimal assumptions, we construct examples exhibiting several non-standard forms of the law of the iterated logarithm (LIL) along suitable sequences: the normalization and the upper and lower limit constants may differ from their classical counterparts. We also show that a regular growth condition of the form $\sum_{k\leq n}k^{-1}a_k^2\sim c(\log n)^\beta$, where $c,\beta>0$, is not by itself sufficient to ensure a standard LIL. Finally, under an additional counting condition controlling the frequency of indices at which the weights $a_k$ are comparatively large, we prove that $(2{\rm Var}\,[X(s)]\log\log({\rm Var}\,[X(s)]))^{-1/2}X(s)$ has the almost-sure cluster set $[-1,1]$ as $s\to 0+$. The latter result is applied to several coefficient sequences of number-theoretic origin.

We study \emph{$T$-Robinson spaces}, a tree-based generalization of Robinson spaces in which every path of a compatible tree induces a Robinson subspace. This framework extends the classical notion of Robinsonian representations from linear orderings to tree structures, allowing the modeling of hierarchical and branching data. We establish a complete combinatorial characterization of $T$-Robinson spaces by proving their equivalence with several graph- and hypergraph-theoretic properties. In particular, we show that a dissimilarity space is $T$-Robinson if and only if all its level graphs are dually chordal with a common compatible tree. Combined with the characterization of hypertrees established by Brucker~\cite{brucker2005hypertrees}, this yields the equivalent characterization in terms of the associated cluster, ball, and 2-ball hypergraphs being hypertrees. Building upon these structural results, we develop a recognition algorithm with complexity \(O(K n^{2})\), where \(K\) denotes the number of minimum spanning trees of the dissimilarity space, improving upon existing hypertree-based approaches whenever \(K\) remains moderate. We further introduce a quantitative measure of $T$-Robinson structure that evaluates the extent to which an arbitrary dissimilarity space admits a tree-like representation. Finally, we discuss applications to real-world datasets, illustrating how $T$-Robinson spaces provide an interpretable framework for analyzing and organizing relational data.

Following a hint of Semmes, we employ Stong's bijections between integer lattices to construct space-filling surfaces, which are higher-dimensional analogues of space-filling curves. For each $m\geq 2$ we build $\alpha$-Hölder continuous parameterizations $f:[0,1]^m\rightarrow[0,1]^{m+1}$ with sharp exponent $\alpha=m/(m+1)$. In particular, there exist $(2/3)$-Hölder continuous surjections from squares to cubes. This solves Arnold's problem 1988–5.

Richardson tableaux were introduced by Karp and Precup in their study of irreducible components of Springer fibers that are Richardson varieties. Guo gave an explicit bijection between Richardson tableaux and Motzkin paths through noncrossing involutions and the Robinson–Schensted (RS) algorithm. In this paper, we study Richardson tableaux of a fixed shape from the viewpoint of Motzkin paths. We introduce a shape algorithm, independent of the RS algorithm, that directly determines the shape of the corresponding Richardson tableau from a Motzkin path. Based on this algorithm, we construct a local bijection which keeps track of the major statistic and reproves combinatorially Karp and Precup's $q$-enumeration formula for Richardson tableaux of a given shape. We also prove in two ways a conjecture of Guo on the comajor generating function for Richardson tableaux with a prescribed number of odd columns.

We construct families of sign-changing solutions for the four-dimensional Brezis–Nirenberg problem \[ -\Delta u=u^3+\varepsilon u\quad\text{in }\Omega,\qquad u=0\quad\text{on }\partial\Omega, \] as $\varepsilon\to0^+$. A Lyapunov–Schmidt reduction shows that the location and relative scales of the bubbles are governed by a signed Green–Robin interaction matrix. We formulate an abstract existence criterion in terms of a simple positive eigenvalue admitting a positive eigenvector and a stable critical set. We then apply it to a positive–negative pair in a general domain and to several symmetric multi-peak configurations, including alternating regular polygons, orthogonal polygons, one central peak surrounded by peaks of the opposite sign, and aligned three-, four-, and five-peak patterns. For the two-peak solution we also prove that it has exactly two nodal domains and, under a natural balance condition and connectedness of the boundary, that the closure of its nodal set meets the boundary.

Inspired by quantum field-theoretic constructions of 4d $\mathcal{N}=1$ quiver gauge theories that flow to superconformal field theories (SCFTs), we construct 7d manifolds of $G_2$-holonomy, which geometrically engineer these quivers in M-theory. Field theoretically, the 4d quivers are obtained by flux torus compactifications of 6d $\mathcal{N}=(1,0)$ SCFTs. The 6d theory compactified on a circle gives rise to a 5d KK-theory, which has a geometric realization as M-theory on a non-compact elliptically fibered Calabi-Yau threefold. Following the field-theoretical prescription, these local geometries are fibered over a circle to realize (topological) $G_2$-holonomy manifolds. We carry this out concretely in the case of the rank 1 E-string theory and its 4d quivers and construct new families of $G_2$-holonomy spaces.