Papers

We review and study the notion of Higgs Grassmannians, which are schemes parametrizing the Higgs subbundles of a given Higgs bundle over a smooth variety. We write their equations as closed subschemes of the usual Grassmann bundles and investigate their geometry. Often the Higgs Grassmannians generically have 0-dimensional fibers over the base variety, thus implying that Higgs subbundles are "scarce." We characterize the structure of the Higgs Grassmannians by analyzing the local Jordan type of the Higgs field. A refined analysis of the rank 2 case is also provided in terms of the discriminant of the characteristic polynomial. We apply our characterizations to the Simpson system of a smooth variety to provide a streamlined proof of its semistability, and we establish a structural relationship between the rank 1 Higgs Grassmannian and the spectral cover of the Higgs bundle. Finally, we introduce the schemes of flags of Higgs subbundles of a given Higgs bundle, and the Quot schemes parametrizing Higgs quotients; we conclude with some examples.

In this paper, we analyze the solvability of the discrete nonlinear Schrödinger equation \begin{equation*} i\beta(\Delta_t+\nabla_t)\phi(t,k) +\gamma |\phi(t,k)|^2\phi(t,k) +\varepsilon \Delta_k^2\phi(t,k-1) = g(t,\phi(t,k)), \end{equation*} where $\Delta_t$ and $\Delta_k$ denote the standard forward difference operators in the variables $t$ and $k$, respectively, $\nabla_t$ denotes the standard backward difference operator in $t$, and \begin{equation*} \Delta_k^2\phi(t,k-1) = \phi(t,k+1)-2\phi(t,k)+\phi(t,k-1) \end{equation*} is the discrete Laplacian operator in the spatial variable $k$. Throughout, we will assume the parameters $\beta$ and $\varepsilon$ are positive real numbers, the parameter $\gamma$ is a nonzero real number, and the potential function $g:\mathbb{Z}\times\mathbb{C}\to \mathbb{C}$ is continuous.

We prove that smooth quartic threefolds are symplectically irrational: they cannot be related to projective space by a sequence of symplectic blow-ups, blow-downs, and deformations. This recovers the classical irrationality theorem of Iskovskikh-Manin. The obstruction is constructed from big quantum cohomology, using the multiplicities of eigenvalues of quantum multiplication by the Euler vector field. To prove its invariance, we establish a decomposition theorem for quantum cohomology under symplectic blow-ups, following the work of Iritani.

In the Bohr model of the hydrogen atom, the energy levels are a negative constant divided by the square of the level number. It is well known that special pairs of transitions exist that have the same energy difference, and a systematic treatment of this is given in the paper by Do and Phan (arXiv:2010.08338). In this paper we describe a simple method (using equal norms of Gaussian integers, and the Brahmagupta–Fibonacci identity) for finding all such pairs of transitions. We also analyze cascades of equal-frequency transitions, and use a theorem due to Fermat to show that cascades with more than three levels are not possible. We conclude the paper by going beyond Bohr's 1913 model, and analyzing some Diophantine equations related to the nonrelativistic Schrödinger Equation for hydrogen, some of which are similar to the Diophantine equations studied in the main part of the paper.

Purpose: This work aims to develop an image reconstruction algorithm for wide-angle digital breast tomosynthesis (DBT) that has improved depth resolution and in-plane contrast while reducing non-uniformity artifacts. Approach: The image reconstruction algorithm is an extension of our prior work on sparsity-regularized iterative image reconstruction. The algorithm is performed in two stages as explained in a prior work. The first stage consists of a low-resolution reconstruction that exploits sparsity for quantitative accuracy. In this work, this first stage is augmented with a formulation that includes the estimation of a "background" image, which absorbs low-frequency artifacts that cause image non-uniformity. Results: The new algorithm is demonstrated on three patient cases for which the data are acquired on a wide-angle DBT system. Conclusion: The results on the shown cases indicate that the algorithm design goals have been met, but additional empirical results and task-based assessment are needed to strengthen this conclusion.

We study the connection between the Mersenne numbers $M(n) = 2^n-1$ and the dynamics of the angle-doubling map. Within this framework, we develop an algorithm to compute divisors of Mersenne numbers without explicitly evaluating $M(n)$. Determining whether $M(n)$ is prime for a prime $n$ (and knowing if there are infinitely many of them), is a central problem, traditionally addressed with the help of the Lucas-Lehmer test. We provide an alternative approach based on dynamical methods. As an application, we prove that $M(2{,}199{,}023{,}254{,}451)$ (with approximately $6.6 \times 10^{11}$ digits) is composite by exhibiting a non-trivial divisor.

The Ellis semigroup of a topological dynamical system contains algebraic, topological and dynamical information. It is invariant under conjugacy. Despite this wealth of structure, two non-conjugate dynamical systems can have the same Ellis semigroup. We identify a class of minimal dynamical systems inside which this cannot happen, that is, for which the Ellis semigroup is a complete conjugacy invariant.

We propose a minimal dynamical model of adaptive softmax routing for a two-expert Mixture-of-Experts (MoE) layer. The model is obtained as a mean-field limit of a discrete reinforcement rule: the selected expert receives a small score increment, while all scores undergo regularizing decay. In the symmetric case the limiting system has a supercritical pitchfork bifurcation: for weak feedback there is a unique stable balanced state, whereas above a critical feedback strength two stable asymmetric states appear. When an external asymmetry is added, the pitchfork unfolds into a pair of fold bifurcations forming a cusp in the control-parameter plane. We derive exact parametric equations for the bifurcation set and the local normal form of the cusp catastrophe. Numerical experiments connect this picture to empirical expert load, a small trainable MoE model, hard top-1 PyTorch routing, and a small classification experiment on digits. The results provide a controlled low-dimensional mechanism for abrupt transitions to load imbalance in adaptive MoE routers.

This paper develops new aspects of the interplay between shifted symplectic geometry and classical Poisson geometry, focusing on lagrangian morphisms into 2-shifted symplectic groups. We establish a Lie-type correspondence between such morphisms and Dirac structures in transitive Courant algebroids given by the product of an exact Courant algebroid and a quadratic Lie algebra. As a key application, we identify the global objects integrating quasi-Poisson manifolds, which we call multiplicative D-valued moment maps; this extends the integration of Poisson manifolds to symplectic groupoids and the lifting of Poisson actions to multiplicative hamiltonian actions. We devise systematic constructions of quasi-symplectic groupoids via fibred products of 2-shifted lagrangians, extending classical reduction procedures. This places known constructions, such as the integrations of Poisson homogeneous spaces and Poisson quotients, into a broader, conceptual framework, while yielding new examples.

We construct a $1$-bounded completely multiplicative function $f$ whose logarithmically-averaged partial sums satisfy $$ \limsup_{x \rightarrow \infty} \frac{\left|\sum_{n \leq x} \frac{f(n)}{n}\right|}{1+\exp\left(\sum_{p \leq x} \frac{\text{Re}(f(p))}{p}\right)} = \infty. $$ This disproves a conjecture of Goldmakher from 2009.

Let $R$ be a local or positively graded ring with a regular presentation $R \cong Q/I$ where $I$ is a monomial ideal generated by $n$ elements on a regular sequence. In Briggs-Grifo-Pollitz (2025), the authors classify the cohomological support varieties $\mathcal{V}_R(R)$ for $n \leqslant 5$. In this paper we extend their results to classify the varieties that can occur as $\mathcal{V}_R(R)$ for $n=6$. Moreover, we provide two families of rings, one realizing cohomological support varieties of unbounded codimension, the other realizing an unbounded number of components. Finally, we answer a question of Gintz (2026) about the varieties that occur as $\mathcal{V}_R(R)$ where $I$ is given by the edge ideal of a cycle.

Model merging combines fine-tuned checkpoints into a single multi-task model without retraining. Existing methods - such as task arithmetic, model soups, TIES, and DARE - are computationally efficient and empirically successful, but rely on heuristic design choices and lack formal optimality guarantees. We show that merging can be formulated as a convex quadratic programme over residual updates, yielding weights that minimise a squared-output calibration objective using calibration inputs and fine-tuned model outputs, and subsuming existing methods as special cases. Our framework yields a closed-form diagnostic - the fraction of residual energy captured by a chosen basis - that predicts downstream merge quality using only the calibration set. Empirically, the QP matches or outperforms existing methods in the single-layer setting, and we characterise when the optimal basis provides significant gains over the cheaper diagonal QP. We extend to multi-layer merging via a sequential layer-wise algorithm and demonstrate consistent gains across language and vision benchmarks.

Consider a sequence of random polynomials $P_n(z) = \prod_{k=1}^{n}(z - X_k)$, where $\{X_k\}_k$ are i.i.d. random variables distributed uniformly on the unit disc $\mathbb{D}$. Let $\Lambda_n = \{z \in \mathbb{C}: |P_n(z)| < 1\}$ be the lemniscate of $P_n$, and let $\mathscr{C}(\Lambda_n)$ be the number of connected components of $\Lambda_n$. In this paper, we prove that $\lim_{n\to\infty}\frac{\mathbb{E}[\mathscr{C}(\Lambda_n)]}{\sqrt{n}}= \gamma$, and identify the constant $\gamma$.

In our previous papers we repeatedly emphasized the special role in Quaternionic Analysis of the conformal group SU(2,2) and other real forms of its complexification SL(4,C). In particular, the natural product map of the left and right regular functions into a larger representation that contains the doubly regular functions as a subquotient is an intertwining operator. In this paper we show, however, that the spaces of regular and doubly regular functions do not "interact" - there is no invariant trilinear form on the tensor product of these representations. To construct a natural invariant trilinear form, we reduce the conformal group symmetry to the symplectic subgroup Sp(4,R). This suggests a new approach to the Quaternionic Analysis in general, and we make the first steps in this paper. It turns out that the spaces of regular and doubly regular functions are still irreducible after the restriction to the symplectic subgroup and have a composed structure arising from the metaplectic representation of the double cover of Sp(4,R) - the metaplectic group. This also leads us to consider the double covers of the quaternionic spaces and non-trivial pairings between them. Our study of Quaternionic Analysis based on the symplectic symmetry group culminates in the construction of the invariant trilinear forms on the products of spaces of doubly regular functions and certain counterparts of regular and quasi regular functions. Additional motivation for constructing invariant trilinear forms comes from their application to spinor representations of certain quaternionic algebras based on the doubly regular functions. The latter can be viewed as the space of solutions of the Maxwell equation, and their spinor representations are of a great importance to quantum field theory. These spinor representations will be the subject of a forthcoming paper.

The notion of an exterior differential system (on a manifold) has recently been extended to the setting of a Lie algebroid. Here, we further develop the theory and we present two versions of the Cartan-Kähler theorem in the case where the anchor map of the Lie algebroid is surjective. We give an illustrative example and, as a concrete application, we make use of our results in a specific case of the so-called invariant inverse problem of the calculus of variations.

We prove that if an $n$-vertex graph $G$ is non-extremal and $T$ is a bounded degree tree on $n$ vertices, then $T\subset G$ even when the minimum degree of $G$ is less than $n/2$ by a linear term. We avoid the use of the Regularity lemma, instead we apply a vertex decomposition theorem by the author, which does not require a tower-type lower bound for $n.$

We introduce a class of continuous Volterra processes, called Volterra clocks, and study their singular limit as the memory kernel collapses to a Dirac mass at zero. The dynamics are parametrised by a function $f$ acting as a nonlinear time-change, generalising the Volterra square-root process and recovering it when $f$ is affine. In the singular limit, the continuous Volterra clock converges weakly to a pure-jump process given by first passage times of a Brownian motion to curved boundaries, including affine and square-root boundaries when $f$ is, respectively, affine or quadratic. Outside the affine setting, characteristic function methods are no longer available, and we instead identify the limit directly from the dynamics. We do this through a topological framework adapted to the time-change structure which involves Skorokhod's $M_1$ topology and a decorated notion of convergence. Our analysis unifies several regimes of interest for general Volterra clocks, including large-time asymptotics, fast mean reversion, and hyper-roughness. In particular, this subsumes and extends existing results in the affine setting.

Accurate estimation and forecasting of energy consumption are important for power-system operation, planning, and demand-side management. In practice, however, complete and timely measurements may not always be available, and the observed data can be partial, noisy, or delayed. This motivates the use of learned forecasting models for predicting the evolving consumption state, together with data assimilation methods for sequential forecast correction. In this work, we study a high-dimensional data assimilation problem for real energy-consumption data. \modeltext{The forward prediction is supplied by a pretrained black-box spatio-temporal forecasting model, which is treated as the state propagator in the filtering procedure.} We employ the Ensemble Score Filter (EnSF) to assimilate partial and noisy observations and to correct the forecast trajectory over time. The EnSF uses score-based diffusion models to approximate filtering distributions and avoids retraining neural-network score models during assimilation by using a closed-form score representation and Monte Carlo approximation. Numerical experiments demonstrate that open-loop propagation of the learned forecasting model can become unreliable over long horizons, while EnSF-based correction substantially improves state estimation. Comparisons with the Ensemble Kalman Filter (EnKF) further show that EnSF provides stronger correction under the nonlinear observation setting considered in this work.

We present the construction of stationary solutions of Bose-Einstein condensate dark matter (BECDM) around a point-like gravitational source representing a black hole. The problem is formulated for general axisymmetric configurations, and we focus on two cases: the ground-state core solution and the first nonzero winding number configuration corresponding to a line vortex solution. The stationary equations are solved using an imaginary-time approach, which enables the construction of families of solutions across a wide range of self-interaction and black hole masses. We analyze the impact of these parameters on the density distribution and on the stability properties of the solutions, assessing stability through the turning point criterion based on the enthalpy functional, which allows us to identify stable and unstable branches along each family of solutions. It has been shown in the past that spherical core solutions act as attractors in the collapse of BECDM around black holes in the non-interacting case ($g=0$), supporting their astrophysical relevance. In the present work, the existence of a maximum mass for configurations with attractive self-interaction ($g<0$) allows us to infer the parameter range in which such solutions may also arise in this regime. Building on this picture, we show that stable vortex solutions of BECDM can also exist in the presence of a black hole, whose stability properties suggest that these configurations may likewise be compatible with physically relevant formation scenarios.

We derive a scale-free bound on the density of the maximum of a centered Gaussian vector. The basic bound is non-uniform, depends logarithmically on the dimension, and allows any covariance matrix. When the largest marginal variance is separated from zero, it implies that the density of the maximum is uniformly controlled at all quantiles above ${2}/{3}$, which is sufficient for many hypothesis testing applications; it yields validity of Gaussian and bootstrap approximations for maxima of high-dimensional sums at test levels $\alpha < {1}/{3}$ without further restricting the covariance. Under these same conditions, the argument is extended to show that the maximum absolute value of a Gaussian vector has a uniformly bounded density on the real line. The method also produces new bounds on the variance of the maximum. We discuss implications for high-dimensional correlation testing, time-uniform sequential inference, and non-parametric confidence bands under latent, low-dimensional structure.

In this paper the structures of the generalised Euler-Lagrange equations and their associated conserved quantities are derived for one-dimensional Herglotz variational problems of order $n$. Their derivations use the framework of moving frames and invariant calculus of variations. The knowledge of these structures not only offers a geometric insight, it may provide a more efficient path for the determination of extremals. This is exemplified with a Herglotz problem invariant under the restricted Lorentz group $SO^+(1,2)$.

We study the existence of an optimizer for a quantitative isoperimetric ratio $\mathcal{Q}_*$ in $\mathbb{R}^3$ involving the Hausdorff asymmetry. We prove that $\mathcal{Q}_*$ attains its minimum over the class $\mathcal{A}$ of convex bodies of fixed volume.

Spherical caps play a crucial role in establishing a criterion for the existence of solutions to the Yamabe problem on a compact Riemannian manifold with boundary, similar to the role played by the standard sphere in the problem on a closed Riemannian manifold. This problem is expressed in terms of a nonlinear boundary-value problem, where both the nonlinearity and the boundary condition are critical in the Sobolev sense. This work focuses on the existence of multiple solutions to the Yamabe problem on spherical caps. We show that if the spherical cap is contained in a hemisphere of the standard $n$-sphere and $n = 5$ or $n \geq 7$, the Yamabe problem has infinitely many sign-changing solutions. Our approach takes advantage of symmetries and is based on a careful analysis of the loss of compactness of the variational problem.

We consider a special class of timelike surfaces in the four-dimensional Minkowski space which are one-parameter systems of meridians of rotational hypersurfaces with spacelike axis and call them meridian surfaces of hyperbolic type. We show that all timelike meridian surfaces of hyperbolic type are surfaces with flat normal connection and give the complete classification of those surfaces with constant Gauss curvature. We also classify all minimal timelike meridian surfaces of hyperbolic type and all timelike meridian surfaces of hyperbolic type with non-zero constant mean curvature (CMC-surfaces). We show that there are no timelike meridian surfaces of hyperbolic type with parallel mean curvature vector field other than CMC surfaces lying in a hyperplane. Finally, we describe the class of timelike meridian surfaces of hyperbolic type with parallel normalized mean curvature vector field, but non-parallel mean curvature vector.

Let $\mathbf{\eta}$ be a positive Radon measure on the space of continuous paths from R into a locally compact Polish space $Y$, and assume that $\mathbf{\eta}$ admits an invariant measure. We explicit conditions on $\mathbf{\eta}$ such that successive Markovianisation of $\mathbf{\eta}$ over any dense and countable set of times have all limit points (in the weak-star topology on Radon measures) satisfying the strong Markov property. We show that if Y is a locally compact Polish group, and $\mathbf{\eta}$ is left or right translation invariant, then $\mathbf{\eta}$ satisfies such conditions. Our proof uses the Zermalo-Fraenkel and Axiom of Dependant Choice axiomatisation of set theory, in which countable products of compacts are compact.

We settle the problem of finding the sharp constant in the log Sobolev inequality on the $n$-cycle for all $n\ge 4$, by showing that it is equal to half of the spectral gap. We deduce this result from an optimal cubic Sobolev inequality.

We investigate the impact on survival of a modification of the evolution of a sub-stochastic Markov chain that involves random relocations at previously visited states. Our central result is that such preferential relocations increase the persistence rate, meaning the survival probability decays more slowly than for the benchmark chain without relocations, and this improvement is strict under mild assumptions. We derive explicit lower bounds on the ratio of persistence rates when the relocation distribution is highly dispersed. The analysis relies on ergodic properties of so-called chains with complete connections, and a Feynman–Kac approach to estimate persistence.

In this paper, we investigate two subclasses of univalent functions associated with the exponential mapping $\varphi(z)=e^{\alpha z}, \ 0<\alpha\le1,$ defined via the subordination conditions $\frac{zf'(z)}{f(z)}\prec e^{\alpha z}\quad \text{and} \quad 1+\frac{zf''(z)}{f'(z)}\prec e^{\alpha z}.$ These classes provide a natural exponential analogue of several classical subclasses arising in geometric function theory. We obtain sharp coefficient estimates, logarithmic coefficient inequalities and sharp bounds for the associated Hankel determinants. In particular, explicit estimates are derived for $H_{2,1}(F_f/2)$ for functions belonging to the exponential subclasses introduced above of starlike and convex functions. Our results extend and unify several earlier works on exponential subclasses and highlight connections with logarithmic coefficients and determinant functionals.

A sequence of positive integers $(a_1,a_2,\ldots,a_k)$ is called $\ell$-additive if $a_1+a_2+\cdots+a_k=\ell a_1$ or $\ell a_k$. In this paper, we prove that for all $k\geq3$, if $n$ is sufficiently large, then every permutation of $\{1,2,\ldots,n\}$ has a 2-additive subsequence of length $k$. We also provide polynomial bounds for the smallest $n$ such that every permutation of $\{1,2,\ldots,n\}$ has a 2-additive subsequence of length $k$. When only monotone subsequences are considered, we show that $18$ is the smallest $n$ such that every permutation of $\{1,2,\ldots,n\}$ has a monotone 2-additive subsequence of length three. Strong bounds are obtained for the minimum number of $\ell$-additive subsequences of any length, as well as monotone $2$-additive subsequences of length three. Using techniques in arithmetic Ramsey theory, we also show similar results for products and inverse sums.

Let $\Gamma_{2n}^\omega(p)$ be the level-$p$ principal congruence subgroup of $\text{Sp}_{2n}(\mathbb{Z})$ for all prime $p$. Borel–Serre demonstrated that the cohomology of $\Gamma_{2n}^\omega(p)$ vanishes above degree $n^2$. We prove that $\text{H}^{n^2}(\Gamma_{2n}^\omega(p); \mathbb{Q})$ surjects onto the homology of the quotient of the symplectic Tits building for $\mathbb{Q}$ by $\Gamma_{2n}^\omega(p)$ and we compute the homology of this quotient. We conclude that $\text{H}^{n^2}(\Gamma_{2n}^\omega(p);\mathbb{Q})$ is nontrivial and provide a lower bound of its rank.