Papers

We study the Shimura curves $X_D^*(1)$ of discriminant $D$ for $D=14, 15$ relating specific weight $4$ modular forms on these Shimura curves to the associated Heun functions for the quadrilateral group. We also discuss how the Picard-Fuchs equation of the associated family of abelian surfaces has solutions coming from modular forms on $X^*_{D}(1)$. We determine the rational exceptional sets of the associated Heun functions, and the algebraic values attained by the Heun function at these points. For example, $${\rm He}\left( 81, \frac{1}{2}; \frac{1}{3}, \frac{1}{6}, \frac{1}{2}, \frac{1}{2};-\frac{729}{112}\right)= \left( \frac{2^2 \cdot 3^3 \cdot 5^3}{7^5} \right)^{\frac{1}{6}} $$.

We study sparse regression codes (SPARCs) for lossy compression under simple greedy encoding rules, including both correlation-based and distance-based methods. We generalize the SPARC construction, and consider the class of additive orthogonal regression codes, of which standard SPARCs are a special case. For this class of codes, we derive nonasymptotic bounds on the squared-error distortion by tracking the evolution of the encoding residual across stages. Our results highlight the role of power allocation in controlling the distortion, allowing us to optimize the allocation based on the parameters of the code. The optimized allocation improves the finite-length compression performance of SPARCs, and our bounds provide distortion guarantees for lower complexity variants of SPARCs, like signed SPARCs and $K$-sparse SPARCs.

For a real $f\in \mathbb{N}^\mathbb{N}$, it is in general not the case that every set which is both $\Sigma^1_1(f)$ and $\Pi^1_1(f)$ is the $f$-section of a $\Delta^1_1$ set. However, there are reals $f$ for which this, in fact, does happen; we say that such a real strongly relativizes $\Delta^1_1$. In this paper, we will prove that the reals which strongly relativize $\Delta^1_1$ are exactly the hyperlow reals, i.e., the $f\in \mathbb{N}^\mathbb{N}$ with $\omega_1^f=\omega_1^{\text{ck}}$. We will also study reals that strongly relativize the classes $\Delta^0_\alpha$ for $1\leq \alpha<\omega_1^{\text{ck}}$. We characterize the reals that strongly relativize $\Delta^0_1$ subsets of $\mathbb{N}$ as the reals which are computably dominated. For $1\leq \alpha<\omega_1^{\text{ck}}$, we show that there are continuum many reals which strongly relativize $\Delta^0_\alpha$ subsets of $\mathbb{N}$. We will also find non-trivial examples of reals which strongly relativize $\Delta^0_{\alpha}$ ($1\leq \alpha<\omega_1^{\text{ck}}$) subsets of Baire space.

Let $M$ be a $\mathrm{II}_1$ factor, $N$ a tracial von Neumann algebra, and $\Phi: M \rightarrow N$ a subtracial completely positive map. For an irreducible $\mathrm{II}_1$ subfactor $P \subseteq M$, we characterize when $\Phi$ exhibits a flattening property under conjugation by unitaries in $P$. To be specific, we show that the failure of a Pimsner-Popa type inequality for $E_P \circ \Phi^* \circ \Phi$ is the precise obstruction, equivalently characterized by left weak mixing of a naturally associated $P$-$N$ bimodule. As an application, we obtain an asymptotic orthogonalization result generalizing a result of Popa.

The tournament isomorphism problem is one of the two fundamental bottlenecks to designing better algorithms for the graph isomorphism problem. Though the problem has been investigated for more than five decades, compared to graphs, there are only very few results on the isomorphism problem of tournaments. For most classes of tournaments neither hardness nor polynomial-time solvability is known. Tournaments of bounded VC dimension are such a class for which no results are available, even though the VC dimension is arguably one of the most robust and central notions of combinatorial tameness. Resolving an open problem of Neuen and Grohe, we show that the isomorphism problem for tournaments of VC dimension $d$ can be decided in time $n^{O(d\log d)}$. Consequently, automorphism groups of tournaments of bounded VC dimension can be computed in polynomial time. To this end, we develop a new method to isomorphism-invariantly decompose tournaments. To facilitate recursion, we introduce the notion of a patched tournament and analyze bounded VC dimension in patched tournaments. We design a recursive algorithm that balances the size of the decomposed pieces against their number and makes use of the structure of near twins. In an orthogonal direction, it is known that a hereditary class of tournaments has unbounded VC dimension if and only if it contains all 2-colorable tournaments. As a second result, we show that also this class does not form an obstruction towards polynomial-time isomorphism testing and indeed show that isomorphism of tournaments of bounded chromatic number is polynomial-time decidable.

Let $G$ be an isotropic reductive group and $X$ be a projective $G$-homogeneous variety. Using results from Bialynicki-Birula, Hesselink and Iversen, Brosnan showed that if $G$ is of inner type, the motive of $X$ can be expressed as a direct sum of Tate twists of motives of projective homogeneous varieties for the anisotropic kernel of $G$. We provide a SageMath implementation of this decomposition, based on a depth-first search of the Cayley graph of the Weyl group of $G$, ensuring the complexity scales with the size of the motive rather than the full Weyl group. As applications, we provide new motivic decompositions for some exceptional groups and show how to extend Karpenko's decompositions for classical groups to characteristic $2$.

We develop an arbitrary-degree geometry-conforming finite element (GC-FE) framework for two-dimensional elliptic boundary value and interface problems on curved domains. Using the Frenet–Serret transformation, curved-boundary and interface-fitted segments are represented exactly, while polynomials in Frenet coordinates generate generally nonpolynomial local shape functions in physical coordinates. For interface-unfitted meshes, GC-FE spaces on curved-boundary elements are coupled with geometry-conforming immersed finite element (GC-IFE) spaces on interface-cut elements, with standard polynomial spaces used elsewhere. We establish optimal approximation, inverse, and trace estimates for the GC-FE spaces. For fitted meshes, we prove well-posedness and optimal error estimates in energy and $L^2$ norms for a symmetric interior penalty discontinuous Galerkin discretization. By retaining the prescribed curves exactly, the method avoids the geometric variational crime associated with curved-geometry approximation and requires no corresponding geometric consistency estimates. Numerical experiments confirm the predicted rates, show global accuracy comparable to nodal isoparametric finite elements and smaller true-interface trace errors in the reported tests, and demonstrate the coupled GC-FE-GC-IFE method on interface-unfitted meshes.

We show that an inductive AH-system with diagonal connecting maps describes an action of the canonical AF-groupoid on the spectrum of the canonical C$^*$-diagonal, and that the canonical groupoid model is given by the transformation groupoid associated to this action. This divides the groupoid structure into two distinct aspects: the acting AF-groupoid (which is well-studied) and the unit space on which it acts. We describe the unit space by enriching the Bratteli diagram with extra topological information, and apply these descriptions to examples of interest, including Villadsen algebras of the first kind.

We introduce and study a Fisher information metric \(g^F_\tau\) associated to the conjugate heat kernel of a Ricci flow \((M^n,g_t)\). This tensor measures, at a fixed scale, how the pointed heat-kernel measure changes when the base point is moved. We prove that \(g^F_\tau\) is monotone in scale and satisfies \(g^F_\tau\le g_t\). We relate its trace to the pointed Nash entropy and prove a matrix square identity for the Fisher defect \(g_t-g^F_\tau.\) This identity gives a rigidity theorem for the equality case; on closed connected flows one has the strict inequalities \(0<g^F_\tau<g_t\) at every positive scale, while in the complete case equality forces a Euclidean splitting. We also develop several consequences of this point of view. These include a sharp reverse Poincaré inequality for the heat semigroup, a contraction formula for \(\varphi\)-divergences along conjugate heat flow, and a canonical construction of heat-kernel splitting maps from large eigenvalues of the Fisher endomorphism. As applications, we relate pointed Nash entropy close to \(0\) to small Fisher deficit at comparable scales, and we obtain a codimension-one Fisher-metric criterion for applying Bamler's \(\varepsilon\)-regularity theorem.

How many Slater determinants does an accurate variational description of interacting fermions require? Exact expansions in a finite basis need combinatorially many, and state-of-the-art fermionic neural quantum states stack growing numbers of them. We prove that, in the norms that govern variational calculations, at most two are needed, independently of the number of particles and of the target accuracy. A single universal Slater determinant-specified in advance, independent of both the system and the state-multiplied by a smooth bosonic wave function approximates any fermionic wave function in up to three spatial dimensions in the first-order Sobolev norm, which controls the variational energy. Reaching the second-order Sobolev norm-for Coulomb interactions, the domain of the Hamiltonian, which bounds the variance of the local energy at the core of variational Monte Carlo-requires at most one additional fixed determinant, and only in three dimensions. Antisymmetry therefore costs at most two universal determinants and no expressiveness: generalized Slater-Jastrow neural quantum states are variationally complete.

Let $P$ be a real polynomial of degree at most $m$ on $\mathbb{R}^d$. We construct finitely many correlated Gaussian moments that determine $P$ up to an orthogonal change of variables. The number of copies and the rational covariance matrices are specified in terms of $d,m$. On coefficient balls, the induced map on orthogonal orbits has a Hölder continuous inverse with an explicit exponent. For $m\geq3$, the normalised local stationary-phase expansion at the critical branch through the origin equals the Gaussian moment series. Its coefficients recover the same finite moment vector, without assumptions on the other critical points. If the leading homogeneous part has an isolated complex critical point, we compute the twisted de Rham ranks and construct formal lattices on each fixed coupling support.

Differentiable solver layers embed constrained optimization into end-to-end learning systems, but fixed-depth unrolling must trade off solution quality, feasibility, and computational budget. We propose Residual-Controlled Douglas–Rachford Splitting (RCDRS), a differentiable solver layer for conic linear programs. RCDRS treats an unrolled solver as a feedback-controlled dynamical system, where a causal controller adapts the relaxation and objective-drive parameters while preserving the projection-splitting structure of Douglas–Rachford splitting. Theoretically, we show that each fixed admissible block remains an averaged relaxed DRS operator and admits finite-step fixed-point residual bounds. We further analyze safeguarded time-varying rollouts as summable perturbations of a limiting averaged operator, and recover terminal primal-dual diagnostics from the final splitting state. Experiments on mixed-cone benchmarks and engineering applications show that RCDRS improves solution quality, feasibility and downstream decision performance. The code is available at https://anonymous.4open.science/r/RC-DRS-180C/.

Let $F:\mathbb{R}^3\to\mathbb{R}$ be a polynomial that is irreducible over $\mathbb{R}$ with $\text{deg} F\geq2$. We prove that, for any finite $X\subset Z(F)$ that does not concentrate on affine lines, \[ E(X)=\#\{(a,b,c,d)\in X^4: a+b=c+d\}\ll_{\text{deg} F,\,\epsilon}(\# X)^{2+\epsilon}. \] In particular, this answers a question of Bourgain and Demeter concerning finite subsets of the unit sphere.

An autonomous robot efficiently exploring an unknown environment, such as looking for water sources on Mars, faces two simultaneous demands: building an accurate information map while quickly finding the regions of greatest value, and paying for every meter of travel and the cost of every measurement it takes. Classical information-seeking and reward-seeking criteria address only one of these objectives at a time. Here, we propose Expected Free Energy (EFE), the principled action-selection objective from active inference, as a unifying criterion for budgeted robotic informative path planning. Maintaining a Gaussian-process belief over the information field, our agent plans continuous trajectories that minimize expected free energy under hard path-length constraints. The results from multiple realizations show that EFE-based planning yields accurate posterior maps and locates the highest-value regions simultaneously, outperforming information-theoretic baselines under the same settings. In robotic exploration, these unified, easy-to-tune principled information-gathering strategies facilitate autonomous deployment while enforcing efficiency and resource constraints.

We propose a measure of lottery complexity that combines probabilistic dispersion with dissimilarity between outcomes. Taking degenerate lotteries, which yield a single outcome with certainty, as the simplest alternatives, we measure complexity by proximity to this class. Specifically, for each degenerate lottery, we compute the expected distance between its certain outcome and the lottery's outcomes, and take the minimum across all degenerate lotteries. We provide an axiomatic characterization establishing uniqueness up to positive rescaling, study how complexity changes under outcome compression, and characterize maximally complex lotteries. Finally, we introduce Complexity-Adjusted Expected Utility preferences and characterize adherence to stochastic dominance and strong risk aversion.

We prove that $N$ points in $\mathbb{R}^3$ determine at least $N^{2/3-o(1)}$ distinct distances.

Let $(a_n)_{n\geqslant 0}$ be a sequence of integers. Its dual sequence $(a_n^*)_{n\geqslant 0}$ is defined by \begin{equation*} a_n^* := \sum_{k=0}^{n} \binom{n}{k}(-1)^k a_k. \end{equation*} Let $p>3$ be a prime. In this paper we mainly investigate congruences modulo $p^2$ involving central binomial coefficients and $p$-adic dual sequences. For example, we prove that for any sequence $(a_k)_{k\ge0}$ of $p$-adic integers, \begin{align*} \sum^{(p-1)/2}_{k=0}\binom{2k}{k}^2\frac{a_{2k}}{16^k}\equiv\left( \frac{-1}{p}\right) \sum_{k=0}^{p-1}\frac{\mathcal{P}_{k}}{16 ^{k}}a_{k}^*\pmod{p^2}, \end{align*} where $(\mathcal{P}_n)_{n\ge0}$ are the Catalan–Larcombe–French numbers given by \begin{equation*} \mathcal{P}_0=1,\quad \mathcal{P}_1=8, \quad n^2 \mathcal{P}_n = 8(3n^2-3n+1)\mathcal{P}_{n-1}-128(n-1)^2\mathcal{P}_{n-2} \quad (n\ge2). \end{equation*} We also establish a new formula for $\sum_{k=0}^{(p-1)/2}\binom{2k}{k}a_{2k}^*/4^k \pmod{p^2}$ and as a consequence we confirm some conjectures of Z.-W. Sun \cite{Sun2014CANT} on the generalized central trinomial coefficients $T_{2k}(b,c)$, i.e., the coefficient of $x^{2k}$ in $(x^2+bx+c)^{2k}$, where $b,c$ are integers.

We initiate the study of four point functions in the symplectic bi-Grassmannian framework for four dimensional conformal field theories. We derive factorization formulae analogous to those for scattering amplitudes and bootstrap several examples of AdS$_5$ exchange correlation functions involving scalars, photons, fermions, gluons and gravitons. These correlators are rational functions of the minors of the Grassmannian matrices in contrast to their momentum space counterparts which are much more complicated. We then develop the $GL(2,\mathbb{R})$ twistor space formalism, deriving the Penrose transform and the twistorial Grassmannian where we find remarkably simple expressions. We also find an elegant and natural extension to $\mathcal{N}=1$ supersymmetry, deriving the supersymmetric Penrose transform and the super-twistor Grassmannian. Performing a half-Fourier transform from twistor space, we derive the supersymmetric symplectic bi-Grassmannian corresponding to spinor-helicity variables. We test our formalism in the context of AdS$_5$ $\mathcal{N}=1$ super-Yang Mills theory where we impose consistent factorization and input the bootstrapped four-fermion correlator to obtain those involving gluons, scalars and fermions. Finally, we discuss the application to supergravity theories.

The paper focuses on information theoretic converse bounds for the alignment of $m$ correlated graphs and for the detection of correlation among $m$ graphs. A simple idea for $m\geq 3$ is that if the alignment of $m-1$ of the graphs is revealed as extra information (by a genie for example) then it is still necessary to produce the alignment between the one remaining graph and the others, i.e. the last matching must be accomplished. For both Gaussian and Erdos-Renyi models, the last-matching problem is equivalent to one with two observed graphs, providing a path to extend converse bounds for $m=2$ to larger $m$. While the method is rather obvious for alignment, we show that the method can also be used to derive converse bounds for weak detection of correlation.

We introduce the almost isoclinic region in the oriented Lagrangian Grassmannian ${\rm Lag}^+(n)$ of $\mathbb{C}^n$, an intrinsic higher-dimensional analog of a natural convex region in ${\rm Lag}^+(2) \simeq \mathbb S^1\times \mathbb S^2$. A Lagrangian submanifold is called almost isoclinic if its Gauss map takes values in this region, extending the graphical condition that the characteristic angles of the tangent plane remain uniformly close. We construct a canonical positive function $\Lambda$ on this region and prove that $\log \Lambda $ is concave with respect to the invariant Grassmannian metric. This property yields subharmonicity and monotonicity formulas for minimal Lagrangians and Lagrangian mean curvature flow. As applications, we prove rigidity and Bernstein-type results, including that a complete connected almost isoclinic minimal Lagrangian with a positive lower bound for $\Lambda$ must be a Lagrangian $n$-plane.

Given a pair of (round) 2-dimensional spheres, one of which has radius three times that of the other, the Lie algebra of local infinitesimal symmetries for the distribution corresponding to rolling the spheres along each other (without letting them slip or twist) happens to be isomorphic to the split real form of the exceptional simple Lie algebra of rank 2. These exceptional local symmetries appear only for this specific 1:3 ratio of radii, however, and while there are several proofs of this result, a straightforward kinematic explanation for the seemingly miraculous appearance of an exceptional simple Lie group in this situation has long been desired. In this paper, we provide such an explanation, relating the ratio of radii to the intersections of a pair of one-parameter subgroups that can be seen from the rolling sphere perspective. The approach does not require the split-octonions, as we construct the exceptional simple Lie group directly by Tanaka prolongation to aid in visualizing the underlying geometry.

For a complete Riemannian manifold with nonnegative Ricci curvature, we prove two sharp volume growth order estimates, thereby resolve a conjecture of Gromov in 1986. There first is that a uniform deficit in the volume of unit balls, an analog of positive macroscopic scalar curvature, forces codimension one volume growth, and the second one is that a uniformly positive scalar curvature lower bound forces codimension two growth known as the codimension two volume growth conjecture.

We generalize intersection numbers for combinatorial designs to designs in finite meet-semilattices satisfying suitable regularity conditions. While designs in regular semilattices go back to Delsarte, our regularity assumptions are weaker than his and need not give rise to an association scheme. In this framework, we extend Mendelsohn's equations, prove a generalized Singleton bound with Steiner systems as equality cases, and determine the block intersection distribution at any block of a Steiner system. In particular, this distribution is independent of the chosen block. Specializing to several classical semilattice families, our results recover a number of well-known distributions in coding and design theory. In the Hamming and the $q$-Hamming (or bilinear forms) schemes, they give the local distance distributions of MDS and MRD codes, respectively. In the Johnson and $q$-Johnson (or Graßmann) schemes, they reproduce the block intersection distribution of classical and $q$-analog Steiner systems, equivalently the distance distribution of diameter-perfect constant-weight codes and diameter-perfect constant-dimension subspace codes. For the $q$-Johnson schemes, to the best of our knowledge, this result is new. As a further illustration, we apply our theory to designs of perfect matchings. Our approach provides a unified treatment of these cases in the strongest form known in the literature, determining the distribution relative to each individual block or codeword, without averaging and without linearity or additivity assumptions. Moreover, it identifies the natural double-counting objects underlying these distributions, leading to formulas in the regularity parameters of the semilattice and avoiding the more cumbersome expressions that arise in eigenvalue-based approaches via the ambient association scheme.

It is established that the reduced Hartree-Fock grand potential at positive temperature converges in the semiclassical limit to a Thomas-Fermi grand potential. In particular, a density-potential dual representation of the limiting functional is proposed for a general class of fermionic entropy functions. The corresponding dual problem has no duality gap and admits a maximizer. The resulting grand potential converges to the established zero-temperature grand potential as the temperature tends to zero and, for the Fermi-Dirac entropy, agrees with the standard positive-temperature Thomas-Fermi free-energy functional. The proof also establishes a semiclassical trace formula that is uniform over locally precompact families of potentials subject to a common confinement condition.

Kelly and Trotter conjectured that dim(P x Q) >= dim P + dim Q - 2 for all finite posets P and Q. We prove the conjecture when dim P = dim Q = 3. This also disproves Trotter's conjecture that, for every 1 <= m <= n, there exist finite posets P and Q with dim P = m, dim Q = n, and dim(P x Q) = n. We further prove that dim(C_k x P) = 4 for every finite poset P with dim P = 3 and every crown C_k with k >= 3. The proof uses the classification of 3-irreducible posets and graphs of critical pairs. For the six infinite noncrown families, we construct explicit non-3-colorable subgraphs. The ten fixed posets are handled by an exhaustive 3-coloring search.

Pavez-Signé [Combin. Probab. Comput. 33 (2024), 121–128] conjectured a Dirac-type condition for spanning $H$-subdivisions and asked whether the subdivision paths can additionally be required to have similar lengths. Lee [European J. Combin. 124 (2025), 104059] resolved the existence conjecture in the stronger setting of digraphs. We answer the length-control question in this stronger directed setting: for every $\varepsilon>0$, there exists a constant $C_0>0$ such that, for every digraph $H$ with $h$ arcs and no isolated vertices, every $n$-vertex digraph $D$ with $n\ge C_0h$ and $\delta^0(D)\ge(1/2+\varepsilon)n$ contains a spanning $H$-subdivision whose subdivision paths have lengths differing by at most one.

We classify smooth compact connected Lagrangian immersions $X$ in the closed unit ball of $\C^n$, $n\ge2$, satisfying $H+\varepsilon X^\perp=0$, $\varepsilon\in\{-1,0,1\}$, with Legendrian boundary on the unit sphere and constant contact angle on each connected component. We prove that the boundary has at most two connected components. When the boundary is connected, $X$ is a diffeomorphism onto an equatorial Lagrangian $n$-disk. When the boundary has two components, $X$ splits globally as $X(s,p)=\gamma(s)\psi(p)$, where $\psi$ is a compact minimal Legendrian immersion in the unit sphere and $\gamma$ is an Anciaux profile with a unique radial minimum. The two contact angles are supplementary. In complex dimension two, every non-disk solution is a finite cover of a Lagrangian catenoid segment for $\varepsilon=0$ or of a rotational Anciaux annulus for $\varepsilon=\pm1$. In higher complex dimensions, iterated Calabi suspensions produce families whose minimal Legendrian links have nontrivial topology.

The Quot scheme $\operatorname{Quot}^{q}(\mathcal{O}_{\mathbb{P}^n}^{r})$ parametrizes the quotients of the trivial vector bundle of rank $r$ on $n$-dimensional projective space that have Hilbert polynomial $q$ and are flat over a base scheme. We identify numerical conditions on the polynomial $q$ that completely determine when this Quot scheme is smooth and irreducible. Our approach also uncovers further geometric features of the projective scheme $\operatorname{Quot}^{q} (\mathcal{O}_{\mathbb{P}^n}^{r})$ including the smoothness of the lexicographic point.

Let X_K be a projective geometrically normal variety over the fraction field of a DVR. We show that if X_K admits a projective model whose special fibre has at worst nodes in codimension one, then the Albanese variety Alb_{X_K/K} has semi-abelian reduction over the same base. As an application, we extend the classical nonexistence theorem of Fontaine and Abrashkin for abelian schemes over Z to the singular setting. For instance, if a flat projective Z-scheme X has normal and geometrically connected fibres, then H^1(X_Q, O_{X_Q})=0.

Let $G$ be a connected solvable algebraic group acting rationally on a finite-dimensional vector space $U$. Using a $G$-stable complete flag, we formulate a successive-quotient procedure that decides whether $U$ contains an open $G$-orbit. When the procedure succeeds, it constructs an open-orbit vector of minimum support cardinality and determines the image of a generic stabilizer in the maximal torus quotient. We also give an infinitesimal version detecting open separable orbits. We apply the criterion to the action of $Aut_Q(M)$ on $Ext^1_Q(M,M)^*$ where $M$ is a multiplicity-free representation of a Dynkin quiver. Rigidity of the corresponding component of Lusztig's nilpotent variety is thereby reduced to a rank test together with an acyclicity condition on a graph of active extension coordinates; the connected components of the resulting forest determine the generic indecomposable decomposition. For equioriented type $A$ this yields an explicit algorithm for a family of multisegments encoded by incidence matrices, including nonregular examples with repeated beginnings or ends.