Abstract We study families of singular holomorphic foliations on smooth complex algebraic varieties whose total intersection defines a foliation of unexpectedly low codimension.
Abstract To connect conformal field theories (CFTs) to probabilistic lattice models, recent works of Hongler et al. and Adame‐Carrillo have introduced a novel definition of local fields of the lattice models. Local fields in this picture are probabilistically concrete: they are built from random variables in the model. The key insight is that discrete complex analysis ideas allow to equip the space of local fields with the main structure of a CFT: a representation of the Virasoro algebra. In this article, for the first time, we fully analyze the structure of the space of local fields of a lattice model as a representation, and use this to establish a one‐to‐one correspondence between the local fields of a lattice model and those of a CFT. The CFT we consider is probabilistically realized in terms of the gradient of the Gaussian free field (GFF). Its space of local fields is just a bosonic Fock space for two chiral symmetry algebras. The corresponding lattice model is the discrete GFF. Our first main result is that the space of local fields of polynomials in the gradient of the discrete GFF is isomorphic to the Fock space. Notably, local fields in this setup make sense with both Dirichlet and Neumann boundary conditions. Our second main result is that with the appropriate renormalization, correlation functions of local fields of the discrete GFF converge in the scaling limit to the correlation functions of the CFT. The renormalization needed is, conceptually correctly, according to the eigenvalue of the Virasoro generator on the local field.
Let be a ‐cofinite vertex operator algebra, not necessarily rational or self‐dual. In this paper, we establish various versions of the sewing‐factorization (SF) theorems for conformal blocks associated to grading‐restricted generalized modules of (where ). In addition to the versions announced in the Introduction of the first part of this series of papers by Gui and Zhang, we prove the following coend version of the SF theorem. Let be a compact Riemann surface with incoming and outgoing marked points, and let be another compact Riemann surface with incoming and outgoing marked points. Assign and to the incoming marked points of and , respectively. For each , assign and its contragredient to the outgoing marked points of and , respectively. Denote the corresponding spaces of conformal blocks by and . Let be the ‐pointed surface obtained by sewing along their outgoing marked points. Then, the sewing of conformal blocks — proved to be convergent in the second part of this series of papers by Gui and Zhang — yields an isomorphism of vector spaces We also discuss the relationship between conformal blocks and the modular functors defined using Lyubashenko's coend/construction.
In 1873, James C. Maxwell conjectured that the electric field generated by point charges in generic position has at most isolated zeroes. The first (nonoptimal) upper bound was only obtained in 2007 by Gabrielov, Novikov, and Shapiro, who also posed two additional interesting conjectures. In this article, we give the best upper bound known to date on the number of zeroes of the electric field, and construct a counterexample to Conjecture 1.8 by Gabrielov, Novikov, and Shapiro that the number of equilibria cannot exceed those of the distance function defined by the unit point charges. Finally, we note that it is quite possible that Maxwell's quadratic upper bound is not tight, so it is prudent to find lower bounds. Hence, we also explore examples and construct configurations of charges achieving the highest ratios of the number of electric field zeroes by point charges found to this day.
We introduce a new functional equation in birational geometry, whose solutions can be used to construct two-dimensional topological quantum field theories (2d TQFTs), infinite-dimensional in many interesting examples. The solutions of the equation give rise to a hierarchy of graph potentials, which, in the simplest setup, are Laurent polynomials associated with colored trivalent graphs, in such a way that the birational type of a graph potential depends only on the homotopy type of the graph. We give an efficient computational method to compute the partition functions of these TQFTs. We elaborate on the key example, related to mirror symmetric description of -character varieties, and propose how graph potentials, used to define these 2d TQFTs, can also eventually be used as a foundation for the dual construction of 4d Donaldson-Floer-Witten theories. This paper is the first in a series, and we give a survey of the applications of graph potentials in the other parts. (A similar formalism and examples were introduced independently by Kontsevich-Odesskii under the name of multiplication kernels; we hypothesize that the connection between the two can be understood via 3d mirror symmetry.)
We prove an asymptotic formula for the second moment of the Rankin-Selberg central -values , where is a fixed cuspidal representation of that is tempered and unramified at every place, while varies over a family of automorphic representations of ordered by their (archimedean or non-archimedean) conductors. As another application of our method, we prove existence of infinitely many cuspidal representations of such that and do not vanish simultaneously where and are cuspidal representations of that are unramified and tempered at every place and have trivial central characters.
In this paper, we consider the following Caffarelli-Kohn-Nirenberg (CKN) inequality where , with is the optimal constant and with Based on the ideas of Deng, Sun, and Wei (Duke Math. J 174 (2025)) and Frank and Peteranderl (Calc. Var. 63 (2024)), we develop a suitable strategy to derive the following sharp stability of the critical points at infinity of the above CKN inequality in the degenerate case with (the Felli-Schneider curve): let and be a nonnegative function such that then provided that is sufficiently small, where with being the dual space of , with being the unique extremal function of the above CKN inequality which is positive and radial up to dilations and scalar multiplications and . The above stability is sharp in the sense that the power of the right-hand side cannot be improved any more. The significant finding in our result is that in the degenerate case, the power of the optimal stability is an absolute constant (independent of and ) which is quite different from the nondegenerate case considered in Deng, Sun, and Wei (Duke Math. J) and Wei and Wu (Math. Ann. 384 (2022), 1509-1546). We also believe that our proof-based strategy might be useful in studying many other degenerate problems.
We investigate symmetry and quantitative approximate symmetry for an overdetermined problem related to the fractional torsion equation in a regular open, bounded set . Specifically, we show that if has positive reach and the non-local normal derivative introduced in Dipierro, Ros-Oton and Valdinoci [Rev. Mat. Iberoam. 33 (2017), no. 2, 377-416] is constant on an external surface parallel and sufficiently close to , then must be a ball. Remarkably, this conclusion remains valid under the sole assumption that is convex. Moreover, we analyze the quantitative stability of this result under two distinct sets of assumptions on . Finally, we extend our analysis to a broader class of overdetermined Dirichlet problems involving the fractional Laplacian.
In this article we show that if is a log canonical compact K & auml;hler threefold pair such that is nef of numerical dimension , then is semi-ample. This result combined with our previous work shows that the log abundance holds for log canonical compact K & auml;hler threefold pairs.
The celebrated Haglund-Haiman-Loehr (HHL) formula provides an explicit monomial expansion of the Macdonald polynomials. In 1994, Butler introduced a refinement I-lambda,I- mu[X; q, t]=(T lambda H mu similar to[X; q, t] - T mu H lambda similar to[X; q, t])/(T-lambda-T-mu) of the Macdonald polynomial and conjectured its Schur positivity. According to the Science Fiction conjecture by Bergeron and Garsia, this refinement represents the "intersection" of Macdonald polynomials. In this work, we introduce a novel combinatorial tool, the column exchange rule, which enables us to derive a positive monomial expansion for Butler's symmetric function I-lambda,I- mu, thereby refining the HHL formula. Additionally, we prove Butler's conjecture on the Schur positivity of I-lambda,I- mu in specific cases.
This paper investigates boundary-layer solutions of the singular Keller-Segel system (proposed in Keller and Segel [J. Theor. Biol. 30 (1971), 377-380]) in multi-dimensional domains, which describes cells' chemotactic movement toward the concentration gradient of the nutrient they consume, subject to a zero-flux boundary condition for the cell density and a Dirichlet boundary condition for the nutrient. The steady-state problem of the system reduces to a scalar nonlocal Dirichlet elliptic problem with a singularity. By analyzing this nonlocal problem, we establish the existence of a unique steady-state solution which forms a boundary spike-layer profile as the nutrient diffusion coefficient . For radially symmetric domains, we derive explicit expansions for the boundary-layer steepness and thickness in terms of the domain radius (for small ), which quantifies the influence of radius on the profile and thickness. Additionally, we prove the nonlinear exponential stability of this boundary-layer steady state in radially symmetric domains. The key challenge in our analysis is the emergence of a singularity for small in both stationary and time-dependent problems. To address this, we reduce the nonlocal steady-state problem to a local one and conduct a refined analysis via the barrier method and Fermi coordinates, yielding sharp estimates for the local steady-state solution near the boundary. This approach enables us to determine the asymptotic profile of the nonlocal problem's solution as , accurately capturing and properly resolving the singularity to establish our main results. For the time-dependent problem in radially symmetric domains, we employ a variable transformation to eliminate the singularity, ultimately proving the nonlinear stability of the unique steady-state solution. Our analysis leverages the equation governing the radial mass distribution function relative to the steady state, along with delicate time-weighted energy estimates.
We prove several results on the number of solutions to the asymptotic problem in . Firstly, we discuss criteria that ensure uniqueness. Given a Jordan curve in the asymptotic boundary of , we show that uniqueness of the minimal surfaces with asymptotic boundary is equivalent to uniqueness in the smaller class of stable minimal disks. Then we show that if a quasicircle (or more generally, a Jordan curve of finite width) is the asymptotic boundary of a minimal surface with principal curvatures less than or equal to 1 in absolute value, then uniqueness holds. In the direction of non-uniqueness, we construct an example of a quasicircle that is the asymptotic boundary of uncountably many pairwise distinct stable minimal disks.
Let be an affine diagonal iterated function system (IFS) on . Suppose that for each , there exists so that , and that for each , the IFS on the real line is exponentially separated. Under these assumptions, we show that the Hausdorff dimension of the attractor of is equal to , where is the affinity dimension. This follows from a result regarding self-affine measures, which says that, under the additional assumption that the linear parts of the maps in are all contained in a one-dimensional subgroup, the dimension of an associated self-affine measure is equal to the minimum of its Lyapunov dimension and . Most of the proof is dedicated to an entropy increase result for convolutions of with general measures of nonnegligible entropy, where entropy is measured with respect to nonconformal partitions corresponding to the Lyapunov exponents of . It turns out that with respect to these partitions, the entropy across scales of repeated self-convolutions of behaves quite differently compared to the conformal case. The analysis of this nonconformal multiscale entropy is the main ingredient of the proof, and is also the main novelty of this paper.
We prove that every poset with bounded cliquewidth and with sufficiently large dimension contains the standard example of dimension k $k$ as a subposet. This applies in particular to posets whose cover graphs have bounded treewidth, as the cliquewidth of a poset is bounded in terms of the treewidth of the cover graph. For the latter posets, we prove a stronger statement: Every such poset with sufficiently large dimension contains the Kelly example of dimension k $k$ as a subposet. Using this result, we obtain a full characterization of the minor-closed graph classes C $\mathcal {C}$ such that posets with cover graphs in C $\mathcal {C}$ have bounded dimension: They are exactly the classes excluding the cover graph of some Kelly example. Finally, we consider a variant of poset dimension called Boolean dimension, and we prove that posets with bounded cliquewidth have bounded Boolean dimension. The proofs rely on Colcombet's deterministic version of Simon's factorization theorem, which is a fundamental tool in formal language and automata theory, and which we believe deserves a wider recognition in structural and algorithmic graph theory.
The principal aim of this paper is to establish a purity theorem for Mahler functions that is reminiscent of famous purity theorems for G $G$ -functions by D. Chudnovsky and G. Chudnovsky and for E $E$ -functions (and, more generally, for holonomic arithmetic Gevrey series) by Y. André. Our approach is based on a preliminary study of independent interest of the nature of the solutions of Mahler equations. Roughly speaking, we prove a reduction result for Mahler systems, implying that any Mahler equation admits a complete basis of solutions formed of what we call generalized Mahler series. These are sums involving Puiseux series, Hahn series of a very special type, and solutions of inhomogeneous equations of order 1 with constant coefficients. In the light of B. Adamczewski, J. P. Bell, and D. Smertnig's recent height gap theorem, we introduce a natural filtration on the set of generalized Mahler series according to the arithmetic growth of the coefficients of the Puiseux series involved in their decomposition. This filtration has five pieces. Our purity theorem states that the membership of a generalized Mahler series to one of the three largest pieces of this filtration propagates to any other generalized Mahler series solution of its minimal Mahler equation. We also show that this statement does not extend to the smallest two pieces.
Let Φ : = { ( x 1 , … , x d ) → ( r i , 1 x 1 + a i , 1 , … , r i , d x d + a i , d ) } i ∈ Λ $\Phi:=\lbrace (x_{1},\ldots,x_{d})\rightarrow (r_{i,1}x_{1}+a_{i,1},\ldots,r_{i,d}x_{d}+a_{i,d})\rbrace _{i\in \Lambda }$ be an affine diagonal iterated function system (IFS) on R d $\mathbb {R}^{d}$ . Suppose that for each 1 ⩽ j 1 < j 2 ⩽ d $1\leqslant j_{1}<j_{2}\leqslant d$ , there exists i ∈ Λ $i\in \Lambda$ so that | r i , j 1 | ≠ | r i , j 2 | $|r_{i,j_{1}}|\ne |r_{i,j_{2}}|$ , and that for each 1 ⩽ j ⩽ d $1\leqslant j\leqslant d$ , the IFS t → r i , j t + a i , j i ∈ Λ $\left\lbrace t\rightarrow r_{i,j}t+a_{i,j}\right\rbrace _{i\in \Lambda }$ on the real line is exponentially separated. Under these assumptions, we show that the Hausdorff dimension of the attractor of Φ $\Phi$ is equal to min dim A Φ , d $\min \left\lbrace \dim _{A}\Phi,d\right\rbrace $ , where dim A Φ $\dim _{A}\Phi$ is the affinity dimension. This follows from a result regarding self-affine measures, which says that, under the additional assumption that the linear parts of the maps in Φ $\Phi$ are all contained in a one-dimensional subgroup, the dimension of an associated self-affine measure μ $\mu$ is equal to the minimum of its Lyapunov dimension and d $d$ . Most of the proof is dedicated to an entropy increase result for convolutions of μ $\mu$ with general measures θ $\theta$ of nonnegligible entropy, where entropy is measured with respect to nonconformal partitions corresponding to the Lyapunov exponents of μ $\mu$ . It turns out that with respect to these partitions, the entropy across scales of repeated self-convolutions of θ $\theta$ behaves quite differently compared to the conformal case. The analysis of this nonconformal multiscale entropy is the main ingredient of the proof, and is also the main novelty of this paper.
The existence of acyclic complete matchings on the face poset of a regular CW complex implies that the underlying topological space of the CW complex is contractible by discrete Morse theory. In this paper, we construct explicitly acyclic complete matchings on any nontrivial Bruhat interval [ v , w ] $[v,w]$ based on any reflection order on the Coxeter group W $W$ . We then apply this combinatorial result to regular CW complexes arising from the theory of total positivity. As an application, we show that the totally nonnegative Springer fibers are contractible. This verifies a conjecture of Lusztig. As another application, we show that the totally nonnegative fibers of the natural projection from full flag varieties to partial flag varieties are contractible. This leads to a much simplified proof of the regularity property on totally nonnegative partial flag varieties compared to the proofs by Galashin et al. and Bao and He.
The principal aim of this paper is to establish a purity theorem for Mahler functions that is reminiscent of famous purity theorems for -functions by D. Chudnovsky and G. Chudnovsky and for -functions (and, more generally, for holonomic arithmetic Gevrey series) by Y. Andr & eacute;. Our approach is based on a preliminary study of independent interest of the nature of the solutions of Mahler equations. Roughly speaking, we prove a reduction result for Mahler systems, implying that any Mahler equation admits a complete basis of solutions formed of what we call generalized Mahler series. These are sums involving Puiseux series, Hahn series of a very special type, and solutions of inhomogeneous equations of order 1 with constant coefficients. In the light of B. Adamczewski, J. P. Bell, and D. Smertnig's recent height gap theorem, we introduce a natural filtration on the set of generalized Mahler series according to the arithmetic growth of the coefficients of the Puiseux series involved in their decomposition. This filtration has five pieces. Our purity theorem states that the membership of a generalized Mahler series to one of the three largest pieces of this filtration propagates to any other generalized Mahler series solution of its minimal Mahler equation. We also show that this statement does not extend to the smallest two pieces.
Let be a hyperk & auml;hler manifold, and let be a nef and big divisor on . We show that the fixed part of the linear system is reduced, and as a consequence, that is mobile. If has dimension 4, we also show that if the fixed part of is not empty, the mobile part induces a (rational) Lagrangian fibration.
While several instances of shifted convolution problems for GL(3) x GL(2) have been solved, the case where one factor is the classical divisor function and one factor is a GL(3) Fourier coefficient has remained open. We solve this case in the present paper. The proof involves two intertwined applications of different types of delta symbol methods. As an application we establish an asymptotic formula for central values of L-functions for a GL(3) automorphic form twisted by Dirichlet characters to moduli q < Q.
