Papers

The Delannoy numbers $d(n,k)$ count lattice paths from $(0,0)$ to $(n-k,k)$ using steps $(1,0),(0,1)$ and $(1,1)$. This paper introduces a graded poset $\mathcal{D}_n$ on the Delannoy paths ending on the line $x+y=n$, whose rank-generating function is the Delannoy polynomial $d_n(x)=\sum_{k=0}^n d(n,k)x^k$. We prove that $\\mathcal{D}_n$ is a self-dual lattice, which is call the Delannoy lattice. By establishing an explicit symmetric Boolean decomposition, we show that $\mathcal{D}_n$ is a symmetric Boolean order, thereby recovering the $\gamma$-positivity of $d_n(x)$. Such a decomposition is refined to a symmetric chain decomposition with the chain cover property, and is applied to determine all maximum antichains. We also investigate other combinatorial aspects of $\mathcal{D}_n$, including supersolvability, the Möbius number, characteristic polynomials, and zeta polynomials.

We study the homology of percolation clusters on disk-shaped patches of the regular $\{3,q\}$ triangle tilings for $q\ge7$. Varying $q$ gives a one-parameter family of hyperbolic tilings, from mildly curved ($q=7$) to deeply hyperbolic ($q=20$), and this paper measures and derives loop content as a function of curvature. A single growth-rate parameter $\lambda(q)$, the asymptotic ratio of successive ring sizes, governs every result. Total persistent $H_1$ of the bond-percolation complex, per vertex, is measured across $q=7,\dots,20$ and found to be a nearly linear function of $1/\lambda(q)$. An exact identity reduces total loop persistence to the weight of the lattice's minimum spanning tree minus a boundary correction. The boundary correction follows from the ring recursion, and the spanning-tree weight is derived to first order in $1/\lambda(q)$ by attaching rings one at a time to a contracted interior, with a correction for loops that close through the ring outside. The resulting formula has no free parameters, gives the large-curvature intercept in closed form as $5/4-2\pi/(3\sqrt3)$, and agrees with the measurements to within the size of the next-order term.

For a (quasi)projective variety $X$ and an automorphism $f$ of $X$, we define moduli spaces of twisted endomorphisms of sheaves on $X$. If the automorphism $f$ has finite order, we construct a sheaf of noncommutative algebras $\mathcal{A}_0(f)$ and show that sheaves with a twisted endomorphism can be described as $\mathcal{A}_0(f)$-modules. We compute the topological Euler characteristics of Hilbert schemes of points of the algebra $\mathcal{A}_0(f)$, generalizing the answer for the case where $f$ is the identity automorphism.

Let $p>5$ be a prime. We prove a conjecture of Z.-W. Sun asserting that $ \sum_{k=1}^{p-1}\frac{1}{k^4\binom{2k}{k}} \equiv \frac{H_{p-1}}{p^3}-\frac{7}{45}pB_{p-5}\pmod{p^2}.$ As a consequence, we obtain $ \sum_{k=1}^{p-1} \frac{H_{k-1}^{(2)}}{k^2\binom{2k}{k}} \equiv \frac{H_{p-1}}{3p^3} +\frac{26}{135}pB_{p-5} \pmod{p^2},$ an equivalent conjecture of K. Hessami Pilehrood and T. Hessami Pilehrood. Here $H_n^{(m)}$ denotes the generalized harmonic numbers of order $m$, with $H_n=H_n^{(1)}$, and $B_n$ denotes the $n$th Bernoulli number. The proof combines $p$-adic congruences involving harmonic numbers, finite-sum identities, and complex residue calculations via the residue theorem.

We investigate the least squares linear regression problem with random partial Discrete Fourier Transform (DFT) matrices, providing a rigorous analysis of the model's generalization error. By leveraging tools from random matrix theory, we derive exact non-asymptotic bounds for the risk of the Moore-Penrose estimator, which hold for finite-dimensional problems and reveal the precise dependence on key parameters such as the sample size, dimension, and noise variance. Then we obtain a characterization of the double descent phenomenon in the linear regression context, demonstrating how the risk evolves when the number of parameters $p$ and the number of samples $n$ tend to infinity, with $p/n$ fixed. The analysis relies on applications of the Stieltjes transform for random Fourier matrices, enabling a precise description of the spectral properties of these matrices and their impact on regression performance. To validate our theoretical findings, we present several numerical examples that illustrate the double descent curves. These simulations align closely with our derived bounds, confirming their predictive power in both under-parameterized and over-parameterized regimes.

A family of subsets of $[n]$ is $r$-fork-free if none of its members is strictly contained in $r$ other distinct members. For each fixed integer $r\ge2$, we prove that the maximum size of such a family is \[ \binom{n}{\lfloor n/2\rfloor} \left(1+\frac{2(r-1)}{n}+o(n^{-1})\right). \] This determines the second-order term and matches the upper bound of De Bonis and Katona. Our lower bound comes from a construction on two adjacent middle levels, using a finite ordered collection of disjoint coordinate blocks and a condition on sums modulo $n$. For each prescribed error, the blocks are fixed before $n$ tends to infinity, so the construction works for every sufficiently large $n$.

At Hermite rank two, the constant in the weighted law of the iterated logarithm under long-range dependence is represented as the largest eigenvalue of a suitably normalized positive integral operator on the unit interval. The same eigenvalue determines the exponential-moment threshold and the logarithmic right-tail rate of the weighted second-chaos limit, which in the unweighted case is the Rosenblatt law. The weighted third spectral moment is evaluated in closed form. Convergent two-sided spectral enclosures are obtained, and in the weight-concentration limit the leading eigenvalue is characterized by a scalar equation with a uniform geometric remainder. At the unweighted memory boundary, the constant decays like the square root of the distance to criticality. Its leading coefficient is determined to $25$ decimal places with certified full-operator residual bounds. After division by the square-root boundary factor, the weight-concentration and memory limits commute. Their common coefficient differs from the unweighted boundary coefficient and lies in the certified interval $(1.370323114331,\,1.370323114332)$. The joint asymptotic formula is established with a uniform two-parameter remainder bound.

We consider a fixed finite number of independent two-dimensional discrete Gaussian free fields on a box of side $n$, with zero boundary values, conditioned so that the vector-valued field stays outside a fixed ball on a macroscopic interior domain. We prove that the field, divided by $\log n$, converges in distribution in $L^2$ to twice the equilibrium potential of the domain multiplied by a uniform random unit vector. We also identify the exact capacity cost of the conditioning event and obtain alignment on a deterministic density-one set of sites, with no restriction on their mutual distance. The proof combines the norm repulsion theorem of Korzhenkova and Sepúlveda with Gaussian large deviations and the equality case in the capacity variational problem.

Reweighting a probability law by a positive numéraire and pushing forward the payoff-to-numéraire ratio yields the change-of-numéraire reweighting: the swap-rate law under the annuity measure, the numéraire-inversion involution of martingale optimal transport, and the population form of self-normalized importance sampling. We characterize exactly when it is Wasserstein-stable. Along weakly convergent inputs with uniformly integrable numéraires and a strictly positive limiting numéraire mean, convergence of the reweighted laws is equivalent to a uniform integrability condition, computed under the inputs, on one explicit family in the payoff and numéraire. At first order the numéraire cancels, the sole obstruction being payoff mass where the numéraire vanishes; absent such mass, prices and first moments pass to the limit under a uniformly integrable payoff alone, with no assumption on the reciprocal numéraire. Two-atom examples with bounded inputs attain the threshold exactly. Applications characterize a standing moment assumption in the transport literature and delimit which annuity-measure statistics survive.

Let $p_c(\mathcal F)$, $q(\mathcal F)$, and $q_f(\mathcal F)$ denote the threshold, expectation threshold, and fractional expectation threshold of a family $\mathcal F$ of nonempty subsets of a finite set, respectively. We prove that there is an absolute constant $C>0$ such that, if $\mathcal F$ has VC dimension at most $d\ge1$, then $p_c(\mathcal F)\le Cq(\mathcal F)\log(d+1)$. More generally, for every $0<\varepsilon\le1/2$, a binomial random set of density $\min\{1,Cq(\mathcal F)\log((d+1)/\varepsilon)\}$ contains a member of $\mathcal F$ with probability at least $1-\varepsilon$. Consequently, $q_f(\mathcal F)\le Cq(\mathcal F)\log(d+1)$, verifying Talagrand's integral–fractional conjecture for families of any fixed VC dimension. We also prove that if a $k$-spread probability measure has support of VC dimension at most $d$, then a binomial random set of density $\min\{1,(C/k)\log((d+1)/\varepsilon)\}$ contains a member of its support with probability at least $1-\varepsilon$. In both random-containment results, the factor \(\log((d+1)/\varepsilon)\) is optimal up to absolute constants. As an application of the spread theorem, we prove that every $n$-uniform family of VC dimension at most $d$ with more than $(C p^{-1}\log((d+1)/\varepsilon))^n$ members contains a $(p,\varepsilon)$-robust sunflower. In particular, every such family with more than $(Cr\log(d+1))^n$ members contains an $r$-sunflower, improving the recent bound $(Crd)^n$ of Ge, Wang, Xu, and Zhao.

We introduce a new structural framework for constructing multivariate Alexander quandles based on a groupoid ${\cal G}$, composed of a disjoint union of delooping groupoids of deck groups. Compared to standard Alexander modules over group rings ${\mathbb{Z}}[{\mathbb{Z}}]$, the multivariate version involves modules over groupoid rings ${\mathbb{Z}}[{\cal G}]$. A manifestly categorical description is given in terms of ${\mathbb{Z}}$-Algebroids, whose functorial module theory yields an equivalent definition of multivariate Alexander modules. Using the category of elements construction, we define multivariate Alexander quandles from this groupoid framework. Three classes of multivariate quandle operations follow. In contrast to earlier approaches, we abstract the algebraic structure from explicit link-dependence. As a consequence, we have a structural framework for: (i) enumerating new classes of Alexander quandles, and (ii) finding new coloring invariants of links. We show that maps between ${\mathbb{Z}}$-modules, facilitated via multivariate quandle operations, realize a quiver presentation of quandles, which makes manifest the oidification of Alexander modules. Oidified Alexander modules, based on groupoids, provide a constructive framework for composing univariate quandles to obtain new multivariate ones. Finally, we comment on the possibility of new link invariants coming from groupoid-based formulations, including the fundamental groupoid.

We show that any convex co-compact hyperbolic surface with exponent of convergence of Poincaré series $\delta \in (\frac25,\frac{14}{27})$ has an essential spectral gap of size $\beta=\tfrac78(\tfrac12-\delta)+\tfrac{1}{32}\delta-\epsilon$ for any $\epsilon>0$. In particular, for $\delta=\frac12$ this becomes $\beta=\tfrac{1}{64}-\epsilon$. We show existence of the gap by proving a new Fractal Uncertainty Principle, using a two-ends Furstenberg theorem of O'Regan-Wu-Yi [arXiv:2607.08461].

We show that the cohomology rings of toric Richardson varieties in the Grassmannian are finite truncations of the ring of quasisymmetric functions. We exhibit an affine paving of each such variety whose cell closures give rise to the basis of fundamental quasisymmetric functions. We similarly interpret the homology of these varieties in terms of the ring of noncommutative symmetric functions and show that the expansion of the homological class of any torus-invariant subvariety into the affine paving basis agrees with the expansion of a corresponding generalized noncommutative ribbon function into the ribbon basis. By taking the direct limit of all toric Richardson varieties, we obtain an ind-variety equipped with a weak $H$-group structure whose cohomology is the Hopf algebra of quasisymmetric functions. We conjecture that it is isomorphic to a similar $H$-group constructed by Baker–Richter. As a byproduct, we deduce that the $f$-vectors of shard polytopes are log-concave, making the first progress on a question of Ferroni–Schröter for matroid base polytopes.

We describe a family of convex uniformly continuous functions $E^\lambda_F$, $\lambda>0$, on the set of states of a bipartite quantum system (consisting of finite-dimensional or infinite-dimensional subsystems), which monotonically increase and converge pointwise to the Entanglement of Formation (EoF) as $\lambda\to0$. These functions are "nonselective" entanglement monotones defined in a way close to the construction of the Moreau-Yosida regularization (the Moreau envelope) of a convex function on a convex set used in the modern convex analysis. So, we call the functions $E^\lambda_F$ the Moreau-Yosida approximations of the EoF and describe their equivalent definitions and basic properties. The semicontinuity bounds for the EoF (presented in [Lob.J.Math., 46(6), 2632-2658]) allow us to obtain easily computable upper bounds on the difference $E_F(\rho)-E^\lambda_F(\rho)$ for a given state $\rho$. These bounds give easily computable bounds on the rate of uniform convergence of the function $E^\lambda_F$ to the EoF as $\lambda\to0^+$ on the sets of states with bounded rank/energy of one of the marginal states. We also discuss sufficient conditions for the coincidence of $E_F(\rho)$ and $E^\lambda_F(\rho)$ at a given state $\rho$ for all $\lambda$ small enough and consider several classes of states for which such coincidence takes place. The conjectured selective LOCC-monotonicity of the functions $E^\lambda_F$ and a possible way to prove it are briefly discussed. Secondary parts of the article are written with the help of ChatGPT-5.6.

We initiate a `Peterson program' that seeks to extend Dale Peterson's theory for the quantum cohomology of flag varieties to arbitrary Schubert varieties, and that incorporates a mirror-symmetric approach. In this first paper we introduce a generalisation of the (localised) Peterson variety associated to an arbitrary Schubert variety $\check X_{w,P}$ inside a partial flag variety $\check G/\check P$. This generalisation is possibly a non-reduced affine scheme that lies in the Langlands dual full flag variety $G/B$. Additionally, we construct a Lie-theoretic `superpotential' associated to $\check X_{w,P}$, generalising earlier ones for the flag varieties $\check G/\check P$ from [Rie08], and we show that its relative critical point locus recovers the generalised Peterson variety. We make the key conjecture that for smooth Fano Schubert varieties the coordinate ring of the generalised Peterson variety recovers the quantum cohomology ring of $\check X_{w,P}$ localised at the quantum parameters. For smooth Fano Schubert varieties $X_{w,B}$ in $\check G/\check B$ we furthermore map our generalised Peterson variety to the cotangent bundle $\mathcal T^*(T)$ of the maximal torus $T$ of $G$ and construct a partial compactification that we conjecture models the full quantum cohomology ring of $\check X_{w,B}$, in analogy with the Givental-Kim presentation of $QH^*(\check G/\check B)$ via the degenerate leaf of the Kostant Toda lattice of $\check G$. These conjectures are verified for all smooth Schubert divisors in the complete type $A$ flag variety, and we show that our superpotential agrees with that introduced for Grassmannian Schubert varieties by Rietsch and Williams, after a suitable isomorphism of the domains.

This article develops a finite-horizon multiple-stopping framework and applies it to three American-style contracts on a geometric random walk in a Cox–Ross–Rubinstein market: an American put, a Russian option, and a floating-strike geometric-average Asian put. The general problem is represented by recursively defined Snell envelopes, with unused exercise rights encoded by a cemetery time; this yields an ordered optimal exercise vector without requiring all rights to be exercised. For the American put, a median representation of successive marginal values yields diminishing marginal values, nested exercise regions, and monotone exercise thresholds without relying on convexity of the marginal value. After suitable state reductions, analogous marginal-value arguments give threshold-type optimal exercise rules for the Russian and geometric-average Asian options. Independent random maturity is also incorporated, and its effect on the corresponding stopping regions is identified.

In this paper, we derive a closed-form for the proximity operator of the weighted squared $\ell_{2,\infty}$ norm. Moreover, we derive a closed-form for the proximity operator of a weakly convex version of this function. The proof involves using known results for compute the proximity operator of a supremum function, which requires finding a solution of an auxiliary problem. We find the explicit solution of this auxiliary problem by finding the KKT multipliers and the critical point associated to the first-order optimality conditions. Additionally, we present applications to denoising problems and weighted max-min dispersion problems.

We describe an algorithm for computing the zeta function of a proper, smooth curve over a finite field $k$ of characteristic $p$, when the curve is given together with some auxiliary data, including a lift $C$ to the valuation ring in a finite extension of $\Q_p$. The algorithm is denominator-free if the ramification is at most $p$. Our method computes the matrix of the action of a semilinear Frobenius on the first de Rham cohomology group of the curve by means of Poincaré duality, using cup products that can be computed from local expansions of a globally defined lift of Frobenius. Its complexity is softly cubic in the field degree for (general) smooth, planar curve, for which we work out our general estimates in more detail. We make explicit how to compute a suitable basis of the first de Rham cohomology group of $C$, base on 1-forms with `locally integrable polar parts', in both the general case and when the curve is smooth planar. We show the crystalline Frobenius preserves the first de Rham cohomology group of $C$ if the ramification is at most $p$, improving upon known results. In an appendix we prove a well-known formula for the cup product, and a formula by Serre, on the first de Rham cohomology group for a curve in characteristic zero, for which no reference seems to exist.

We study the moiré pattern formed by the intersection of a planar Sierpiński gasket and a copy rotated about its centre. At every resonant angle, we prove that the intersection is of finite type and compute its Hausdorff and Minkowski dimensions from an explicit finite matrix; for Lebesgue-almost every angle, we bound its upper Minkowski dimension by $\overline{\dim}_{\mathrm B}(\mathcal S_\theta)\le 2\dim_{\mathrm H}(S)-2$. Finally, we conjecture that both dimensions equal this bound at every non-resonant angle. Sections 2 and 3 are formalised in Lean.

We establish the acoustic and Stokes-Fourier hydrodynamic limits for the Boltzmann-Fermi-Dirac (BFD) equation on the torus of dimension 2 or more. For each limit, we consider fluctuations of appropriately scaled solutions of the BFD equation around a constant background Fermi-Dirac distribution. By adapting the weak solution framework developed by Bardos, Golse and Levermore for the classical Boltzmann equation to handle the cubic nonlinearity of the BFD collision operator, we show that these fluctuations converge weakly in time to a unique limit governed by the respective hydrodynamic equation.

We prove global well-posedness and scattering for the three-dimensional focusing energy-critical nonlinear Schrödinger equation with initial data below the ground-state energy and gradient thresholds. This extends the seminal result of Kenig and Merle (Invent. Math., 2006) to general data in $\dot H^1(\mathbb R^3)$. Classical concentration compactness and the exclusion of rapid frequency cascades reduce the problem to the quasi-soliton scenario. The main ingredient is a frequency-localized interaction Morawetz estimate based on an explicit convex weight whose time-dependent radius is determined implicitly by the mass density of the high-frequency component. The ground-state gap controls the focusing contribution with constants independent of the mass of this component. The radius is chosen to retain quartic spacetime control, while the continuity equation expresses the moving-radius contribution as a nonnegative square, up to frequency-truncation errors. We absorb these errors by closing the Morawetz estimate together with an auxiliary spacetime bound for the nonlinearity. The resulting quartic bound excludes the quasi-soliton scenario and completes the proof.

This paper investigates the numerical resolution of the Beurling-LASSO (BLASSO), a convex optimization framework that promotes sparsity in the space of measures. We consider its application to the estimation of Gaussian mixture models (GMMs) with an unknown number of components and unknown diagonal covariance matrices. Our approach combines the Conic Particle Gradient Descent (CPGD) principle with Riemannian gradient descent, to account for the underlying Fisher-Rao geometry of Gaussian distributions. Our contributions are twofold. First, we provide theoretical guarantees for the convergence of our algorithm. In particular, we establish exponential local convergence under a non-degeneracy condition on the solution and relate this assumption to a separation condition on the underlying statistical target. Second, we address practical implementation aspects of CPGD and present numerical experiments illustrating its performance. On the test cases considered, these experiments suggest that CPGD is more robust to overspecification of the number of components than the EM algorithm. We also investigate the impact of component separation on recovery accuracy.

We prove nearly quadratic lower bounds for randomized algorithms for linear optimization and uniform sampling over convex bodies in the membership oracle model. For linear optimization, this matches the known nearly quadratic upper bound up to a polylog factor in the dimension. For uniform sampling, this improves on the previous linear lower bound. Our construction also implies the same lower bound for volume estimation.

We study the deterministic oracle complexity of finding approximate solutions to composite monotone inclusion problems, formed by the sum of a smooth single-valued monotone operator and a maximally monotone set-valued operator, under the tangent-residual criterion. We introduce the Anchored Extra-Proximal (AEP) framework, which combines an anchored extrapolation step with an inexact anchored proximal update satisfying a relative-error condition. The framework recovers the composite Fast Extragradient method in the first-order setting and yields natural second- and higher-order extensions by replacing the operator in the implicit update with its Taylor approximation at the extrapolated point. For every $p\geq 2$, assuming that the $(p-1)$th derivative of the single-valued operator is Lipschitz continuous, we combine this construction with a bisection line search to obtain a $p$th-order method that finds a point with tangent residual at most $\varepsilon$ in $\widetilde{O}(\varepsilon^{-2/(3p-1)})$ oracle calls. This improves all prior upper bounds for $p$th-order methods: in particular, it improves the previous best-known $\widetilde{O}(\varepsilon^{-1/p})$ tangent-residual complexity as well as the classical $O(\varepsilon^{-2/(p+1)})$ bound of higher-order hybrid proximal extragradient methods under the weaker duality-gap criterion. We complement this result with a worst-case lower bound of $\Omega(\varepsilon^{-2/(3p-1)})$ for every deterministic algorithm in the $p$th-order oracle model, without restricting the algorithm to tensor steps or any other prescribed update structure. Thus, the proposed method attains the optimal dependence on $\varepsilon$, up to logarithmic factors, for all $p\geq2$.

We introduce a dynamical version $\operatorname{rc} (A, \alpha)$ of the radius of comparison $\operatorname{rc} (A)$ of a unital C*-algebra, based on the dynamical Cuntz semigroup. We also give an intrinsic ordered semigroup definition, which agrees with $\operatorname{rc} (A, \alpha)$ when $A$ is residually stably finite, and is lower semicontinuous for equivariant direct limits with injective unital maps. We construct actions $\alpha$ of $G = \mathbb{Z} / 2 \mathbb{Z}$ on simple unital AH~algebras for which $\operatorname{rc} (A, \alpha)$ lies strictly between $\operatorname{rc} (A)$ and $\operatorname{rc} (A) / \operatorname{card} (G)$, and actions $\alpha$ of a finite group $G$ on unital C*-algebras for which $\operatorname{rc} (A, \alpha)$ lies strictly between $\operatorname{rc} (A)$ and $\operatorname{rc} (C^* (G, A, \alpha))$. For a minimal action of a countable discrete group $G$ on a zero dimensional compact metrizable space $X$, we prove that $\operatorname{rc} (C (X), \alpha) = 0$ if and only if the action has dynamical comparison as defined by Kerr. For finite group actions on simple unital stably finite C*-algebras, assuming the weak tracial Rokhlin property, we get $\operatorname{rc} (A, \alpha) \leq \operatorname{rc} (A) / \operatorname{card} (G)$, and assuming weak tracial strict approximate innerness, we get $\operatorname{rc} (A, \alpha) = \operatorname{rc} (A)$.

The Anderson tight-binding model is a fundamental model of quantum transport and localization in disordered media. Solving a problem left open by Bourgain and Kenig, this paper proves Anderson localization near the bottom of the spectrum for the lattice Anderson model with Bernoulli potential, in any dimension $d\ge 2$. The proof uses the multiscale framework of Fröhlich-Spencer and Bourgain-Kenig, and the main new ingredient is a probabilistic discrete unique continuation principle (PDUC) for the discrete Schrödinger equation. This PDUC is established via a bootstrap argument and a key probabilistic lemma proved by adaptively revealing the random potential.

Let $\alpha>0$ be an irrational number of finite type and let $\beta\in\mathbb{R}$. Assuming the infinitude of Siegel zeros (respectively, sufficiently strong Siegel zeros), we show that for every sufficiently small $\varepsilon>0$, there exist infinitely many primes $p$ such that \begin{gather*} \|\alpha p+\beta\|\leq p^{-1/3+\varepsilon} \qquad\text{(respectively, }\|\alpha p+\beta\|\leq p^{-1/3-1/20}\text{)}. \end{gather*}

Let $f$ be a holomorphic automorphism of a compact Kähler manifold $X$ with simple cohomological action. We prove that the saddle periodic points of $f$ equidistribute with respect to the equilibrium measure. In particular, our result applies to every holomorphic automorphism of a compact Kähler surface with positive topological entropy.

We study the problem of determining upper bounds on the maximal binary rank, $b_d$, of a $0,1$ matrix with real rank $d$, for small constant $d$. We prove a general upper bound of $b_d \leq \left\lfloor \sqrt{d2^d + 2d}\right\rfloor -1$. We also develop a general dimension reduction technique of bounding $b_d$ using known upper bounds on $b_{d-1}$ and $b_{d-2}$, as well as prove interesting general structural results on extremal matrices with binary rank $b_d$. This allows us to get improved non-trivial upper bounds on $b_d$ for constant values of $d$, which imply tight bounds on $b_d$ for $3 \leq d \leq 5$. Our results also imply bounds on the equivalent problem of finding the minimum number of bicliques needed to partition the edges of a bipartite graph whose reduced adjacency matrix has real rank at most $d$.

For a graph $G$, let $\tau_{\mathrm{iso}}(G)$ denote the number of isomorphism classes of its spanning trees. For every fixed $d\ge3$ and all sufficiently large $n$, we prove that every connected $n$-vertex graph $G$ with $\delta(G)\ge d$ satisfies \[\tau_{\mathrm{iso}}(G)\ge \tau_{\mathrm{iso}}(K_{d,n-d})=A_dn^{d-1}+O_d(n^{d-2}),\] for an explicit constant $A_d>0$, and $K_{d,n-d}$ is the unique minimizer. This confirms a conjecture of Bitonti, Michel and Scott and extends it to every $d\ge3$. We also show that any such graph with $O(n^{d-1})$ spanning-tree types has all but a bounded number of vertices with the same $d$ neighbours.