数学前沿

This article revisits earlier work by the last author together with Kay Magaard. We correct mistakes in results about the Sylow 2- and 3-structure of groups that act with fixity 3 and one result about possible components, we improve some results and we add details.

We study optimal control of non-linear graphon dynamics in which the interaction kernel itself is the control. Motivated by opinion formation in social networks, we ask how changing who influences whom can steer a population toward a desired outcome. We develop a framework using interaction kernels with bounded variation in the receiver label, providing compactness while accommodating finite networks. We prove existence of an optimal pair and show that finite-agent approximations recover the continuum optimal value under successive refinement of the network and initial data, with subsequential convergence of suitably chosen discrete optimizers. Numerical experiments use neural networks to approximate optimal controls for bounded-confidence and graphon Kuramoto dynamics, illustrating how changes in interactions can prevent fragmentation and promote synchronization.

Given a finite point set satisfying condition $\mathcal{A}$, the subset selection problem asks, how large of a subset satisfying condition $\mathcal{B}$ can be extracted? In this paper, we make progress on three instances of subset selection problems in planar point sets. Let $n,s\in\mathbb{N}$ with $n\geq s$, and let $P\subseteq\mathbb{R}^2$ be a set of $n$ points, where at most $s$ points lie on the same line. Firstly, we select a general position subset of $P$. This problem was proposed by Erd\H{o}s under the regime when $s$ is a constant. For $s$ being non-constant, we give new lower and upper bounds on the maximum size of such a subset. In particular, we show that in the worst case such a set can have size at most $O(n^{5/6+o(1)}/\sqrt{s})$ when $3\leq s\leq n^{1/3}$ and $O(n/s)$ when $n^{1/3}\leq s\leq n$. Secondly, we select a monotone general position subset of $P$, that is, a subset in general position where the points are ordered from left to right and their $y$-coordinates are either non-decreasing or non-increasing. We present bounds on the maximum size of such a subset. In particular, when $s=\Omega(\sqrt{n})$, our upper and lower bounds differ at most by a logarithmic factor. Lastly, we select a subset of $P$ with pairwise distinct slopes. This problem was initially studied by Erd\H{o}s, Graham, Ruzsa, and Taylor on the grid. We show that for $s=O(\sqrt{n})$ such a subset of size $\Omega((n/\log{s})^{1/3})$ can always be found in $P$. When $s=\Theta(\sqrt{n})$, this matches a lower bound given by Zhang on the grid. As for the upper bound, we show that in the worst case such a subset has size at most $O(\sqrt{n})$ for $2\leq s\leq n^{3/8}$ and $O((n/s)^{4/5})$ for $n^{3/8}\leq s=O(\sqrt{n})$. The proofs use a wide range of tools such as incidence geometry, probabilistic methods, the hypergraph container method, and additive combinatorics.

We observe that the $2$-norm distance $d_{U,2}$ between the unitary orbits of normal elements in a $\mathrm{II}_1$ factor $\mathcal{M}$ is equal to the $2$-Wasserstein distance between the spectral measures induced by the trace $\tau_\mathcal{M}$. Using classification and optimal transport theory, we deduce an analogous $2$-norm equation for normal operators $x$ and $y$ in simple, separable, unital, nuclear, $\mathcal{Z}$-stable $\mathrm{C}^*$-algebras that are either monotracial, or real rank zero with finitely many extremal traces, provided that $\sigma(x)=\sigma(y)$ is convex. Consequently, $d_{U,2}$ equips the set of approximate unitary equivalence classes of contractive normal elements of $\mathcal{M}$ with the structure of a compact length space. The same is true of the set of equivalence classes of embeddings into the Jiang-Su algebra $\mathcal{Z}$ of classifiable tracial $2$-Wasserstein spaces over compact, convex planar domains.

Modelling qualitative uncertainty in formal argumentation is essential both for practical applications and theoretical understanding. Yet, most existing work focuses on abstract models of argumentation under uncertainty. Following a recent trend in the literature, we tackle the open question of studying plausible instantiations of these abstract models. To do so, we root the uncertainty of arguments in their components in structured models: rules and premises. Our main conceptual contribution is to provide an intuitive notion of comparative expressivity suitable for both abstract and structured formalisms. Roughly, one formalism for arguing with qualitative uncertainty is at least as expressive as another if the former can generate the same set of abstract argumentation frameworks (completions) as the latter. On the technical side, we prove negative and positive results comparing the expressivity of abstract and structured models of argumentation with uncertainty. These results concern incomplete abstract argumentation frameworks and their extension with dependencies on the abstract side, and ASPIC$^+$, assumption-based argumentation (ABA) and deductive argumentation on the structured side.

We introduce Controlled Stochastic Activity Networks (Controlled SANs), extending classical SANs with explicit control actions and policies for stochastic discrete-event and real-time systems. The framework combines nondeterminism, probabilistic branching, and stochastic timing through controlled automata, controlled probabilistic automata, and controlled stochastic automata. At the nondeterministic level, we distinguish structurally available controls from behaviorally admissible transitions and establish a constructive modeling-power result: every decidable, effectively finitely branching controlled automaton with finite activity and control sets and a finite nonempty initial-state set admits a computable, well-behaved controlled activity network realization whose reachable restriction is effectively isomorphic to the original. We define memoryless, finite-memory, stack-augmented, and tape-augmented policies, investigate their language-theoretic expressiveness, and develop behavioral equivalences including bisimulation and stochastic isomorphism. At the probabilistic level, contingent activity-indexed control assignments and an activity-generation law yield a discrete-time Markov decision process (DTMDP) representation; conversely, every conventional DTMDP admits a single-activity representation. Under Markovian assumptions, controlled stochastic automata admit continuous-time Markov decision process (CTMDP) projections, and conventional CTMDPs admit single-activity representations. Randomized-policy correspondences follow the adopted product-form convention. These results provide a semantic basis for verification, reward optimization, policy synthesis, and potential applications to reinforcement learning, safety-critical control, and agentic AI.

The Sinai billiard map $T$ on the two-torus, i.e., the periodic Lorentz gas, is a discontinuous map. We provide two separate conditions for a finite horizon billiard to admit equilibrium states. The first one -- namely \emph{negligible singularities} -- only concerns the billiard table and implies upper semi-continuity for the metric pressure map of $T$. The condition of negligible singularities is shown to be generic (dense $G_\delta$) among billiard tables with $C^{3+\alpha}$ boundaries, in the $C^{3+\alpha}$ topology, and implies the sparse recurrence condition for the null potential. Compactness of the set of $T$-invariant measures is proved. For finite horizon Sinai billiards, we provide a bound on the defect of upper semi-continuity of the metric pressure map. From it, a condition of existence of equilibrium states only involving potentials is deduced. In both situations, for the potentials $g \equiv 0$ and $g = -h_{\rm top}(\Phi^1) \tau$, we recover and extend the recent results of the existence of measures of maximal entropy (MME) for both the map $T$ and the flow $\Phi^t$, due to Baladi and Demers for $T$, jointly with Carrand for $\Phi^t$. Upper semi-continuity of the metric entropy is shown. Assuming negligible singularities, equidistribution of the periodic orbits with respect to the unique MME is proved. A bound on the topological tail entropy is given.

We consider the left translation action of an element $a$ on a finite-volume semisimple homogeneous space $G/\Gamma$, where $\operatorname{Ad}(a)$ is diagonalizable over $\mathbb{R}$. Assuming exponential mixing, we prove a sub-Gaussian large-deviation estimate for compactly supported smooth test functions. As an application, we prove effective uniqueness of the measure of maximal entropy. We also obtain quantitative pointwise equidistribution for Haar-almost every point on each full expanding horosphere.

Sierksma's conjecture asserts that every affine map $a\colon\Delta_{(d+1)(r-1)}\longrightarrow\mathbb{R}^d$ has at least $((r-1)!)^d$ unordered Tverberg partitions into $r$ parts. Until now, lower bounds growing with $d$ were known only when $r$ is a prime power. As one of our main results we prove that for every integer $r\geq3$, every decomposition $r=kq+s$ into $q=p^\alpha$ a prime power, $k\geq1$ and $s\geq0$, and every affine map $a$, the number $\operatorname{Tv}(a)$ of unordered Tverberg partitions satisfies $$\operatorname{Tv}(a)\ \geq\ \frac{q}{r!}\,\Big(\frac{q}{\alpha+1}\Big)^{\big\lceil\frac{(r-1-2s)(d+1)}{2}\big\rceil}.$$

We study a variational discretization in time of Aubry--Mather theory for Tonelli Lagrangians on the flat torus. We define discrete Ma\~n\'e, Aubry and Mather sets, establish their compactness and inclusion properties, and investigate their convergence as the time step tends to zero. We prove upper-limit results for the Ma\~n\'e sets and, when the continuous Aubry set has finitely many static classes, for the Aubry sets. We obtain full Kuratowski and Hausdorff convergence of the Aubry sets when this set is a single equilibrium or periodic orbit, and, under additional assumptions, a finite union of equilibria. Every weak limit of discrete Mather measures is a continuous Mather measure. For equilibrium Aubry sets, we give sufficient conditions for exact preservation of Mather sets and measure families at small time steps. A mechanical example illustrates these results and exhibits exact preservation of the Aubry set as well.

In this work, we focus on deriving the absolute moments of the general ized hyperbolic (GH) distribution via its mean-variance mixture structure. Conditioning on the GIG mixing variable reduces the problem to Gaussian moment calculations and known GIG power moments, avoiding direct integration with Bessel functions. The variance gamma (VG), normal inverse Gaussian (NIG), hyperbolic, Normal, and Student t distributions emerge as natural special or limiting cases of the generalized hyperbolic family. This approach offers a substantially simpler and more transparent derivation, facilitating both implementation and verification.

We prove a Fuglede-type quantitative isoperimetric inequality for the Pansu sphere, the conjectured isoperimetric set in the sub-Riemannian Heisenberg group $\mathbb{H}^1$. We show that, among suitable volume-preserving normal perturbations of the Pansu sphere that are small in a weighted $C^2$-norm, the difference between the sub-Riemannian perimeters of the perturbed and unperturbed regions is bounded below by a positive constant times the square of their $L^1$-distance. The proof is based on a Riemannian approximation argument with uniform estimates.

We bound the error term in the counting function of cubic extensions of $\mathbb{Q}$ by $\mathcal{O}_\epsilon(X^{\theta+\epsilon})$, with $\theta < 2/3$, improving upon results due to Bhargava, Taniguchi, and Thorne. The proof proceeds by relating a certain discriminant-reducing sieve, due to Bhargava, Shankar, and Tsimerman, to averages of Artin characters associated with the Artin $L$-function $\zeta_K(s)/\zeta(s)$, with $K$ a cubic field. Conditional on the Generalised Riemann Hypothesis (GRH) for Dedekind zeta functions $\zeta_K(s)$, we obtain further savings and show that the error term is $\mathcal{O}(X^{2/3-1/51+\epsilon})$. Moreover, for the analogous smooth counting problem, we obtain a bound $\mathcal{O}(X^{1/2+\epsilon})$ for the error term, conditionally improving results of Shankar, S\"odergren, and Templier. Our arguments can also handle finitely many splitting conditions. The methods we use to study cubic fields are natural extensions of methods which are used to study the one-level density associated with the above family of Artin $L$-functions. In this direction, we improve the admissible support $[-\sigma,\sigma]$, from the previously known admissible value $\sigma < 2/5$, to $\sigma < 1$, conditional on the GRH. This also proves that at least $75\%$ of these $L$-functions are non-vanishing at the central point, conditionally improving upon results of Shankar, S\"odergren and Templier. Finally, one may study similar questions over a rational function field $\mathbb{F}_q(T)$, with $q$ coprime to $2$ and $3$. Here, we obtain a proportion of non-vanishing of at least $75\%$ for the analogous family of $L$-functions. Furthermore, we obtain a bound $\mathcal{O}\left(X^{1/2+\epsilon}\right)$ for the error term in the counting function of cubic function fields, improving previous results of the author. Over function fields, our results are unconditional.

We compute the local and global Denef--Loeser topological zeta functions of the determinant on the space of symmetric matrices. It verifies the strong monodromy conjecture of the symmetric determinant function. The computation relates to the geometry of complete quadrics and wonderful embeddings.

Focusing on reaction-diffusion models with an interface region dividing the domain, we study the stability of systems as the permeability parameter approaches zero or infinity under Kedem-Katchalsky boundary conditions imposed on the membrane or interface region. These boundary conditions establish that the flux across the interface manifold is proportional to the density jump across that region, with the constant of proportionality defined as the permeability parameter. We analyze both limiting behaviors: when permeability vanishes and when infinite flux is permitted. This analysis is carried out for both the well-known symmetric case, where permeability is identical in both directions across the membrane, and the asymmetric case, where it depends on the direction of the flux. For the asymmetric case, we obtain novel and surprising results that reveal a richer variety of behaviors in this class of interface problems and that differ from the scalar models.

We study the calibration of continuous-time martingale models to observed derivative prices via a dynamic martingale optimal transport problem that penalizes deviations from the volatility of a prescribed reference model. Using $q_x$-Bass martingales, we approximate this problem through a sequence of discrete-time problems and prove that the continuous-time interpolations of their optimizers are asymptotically optimal. Building on the theory of $q_x$-Bass martingales, we develop numerical methods for the discrete-time problems that do not require full marginal distributions at calibration maturities.

Let $F\in S_{2k}^{new}(M,\chi^2)$ be a Hecke newform, and $D$ a fundamental discriminant with $(-1)\chi(-1)D>0$. Via the Shimura Correspondence, we get associated modular forms of half-integral weight, and Waldspurger showed that for such $g=\sum c_nq^n \in S_{k+1/2}(4M',\chi;F)$, if $c_{|D|}(g)\neq 0$, then $|c_{|D|}(g)|^2 \propto |D|^{k-1/2}L(F\otimes \overline{\chi}\otimes \chi_D,k)$. Kohnen refined this in the case that $f\in S_{2k}^{new}(N)$ for $N$ odd and squarefree, making explicit $|c_{|D|}(g)|^2$ for $g \in S^+_{k+1/2}(4N;f)$. However, $c_{|D|}(g)=0$ unless $\left(\frac{D}{p}\right)=w_p(f)$ for all $p|N$. In this paper, we produce more generally $g \in S^+_{k+1/2}(4\ \mathrm{lcm}(N,N_1^2),\chi;f\otimes \chi)$, where $\chi$ is a character of odd squarefree conductor $N_1$, using a family of generalised Shimura/Shintani maps $\Psi_\chi^D,{\Psi_\chi^D}^*$ due to Kojima and Tokuno. Extending their work (and Kohnen's), we compute its Petersson norm, and give an explicit formula for all the $|c_m(g)|^2$. The key technical ingredient is an explicit set of representatives for $L^0_{M_1,M}(M_1^2D^2)/\Gamma_0(M)$ of binary quadratic forms. We also use a new multiplicity-one result based on work of Kohnen, Ueda, and Waldspurger. In the case $N_1|N$, we reconstruct forms previously considered by Baruch-Mao and Prasanna, which change Kohnen's condition to $\left(\frac{D}{p}\right)=-w_p(f)$ for all $p|N_1$, thus relating twisted $L$-values to squares of Fourier coefficients for all classes of discriminants.

Leader-follower gradient iterations, including bilevel and two-time-scale methods, can exhibit large finite-time excursions despite asymptotic stability. For cascaded iterations with block matrix J = [[A,0],[C,D]], symmetric diagonal blocks of norm at most gamma < 1, and coupling norm at most c, we prove an exact, dimension-free minimax formula: at every horizon t, the worst-case amplification equals the spectral norm of [[gamma^t,0],[c t gamma^(t-1),gamma^t]], attained by an aligned two-dimensional quadratic. Consequently, the worst transient is on the order of 1 + c/(1-gamma), and eigenvalues alone cannot predict finite-time gain. Under martingale noise, the aligned system has stationary follower variance sigma^2 c^2 (1+gamma^2)/(1-gamma^2)^3, establishing cubic inverse-gap scaling. For quadratic followers, the coupling is linked to follower sensitivity R by C = (I-D)R. When the leader is isotropic and the follower map is positive semidefinite, amplification is at most 1 plus the operator norm of R, independently of the spectral gaps. A look-ahead Newton update eliminates follower memory and is minimax-optimal among linear updates that track the leader exactly. Experiments validate the bounds and show that accelerated or inexact follower solvers can exceed the sensitivity bound when their effective maps have negative eigenvalues. Code and experiment scripts: https://github.com/Axym-Labs/transient-amplification

For a directed graph with a chosen root, the arborescence polynomial sums the products of edge weights over all spanning trees directed toward that root. In a finite graph cover, each lifted edge inherits the variable of its base edge. At corresponding roots, the ratio of the cover's arborescence polynomial to the base polynomial is known to be an integer polynomial. We prove that every coefficient of this ratio is nonnegative, resolving the positivity conjecture of Pylyavskyy. The key to the proof is to allow fibers of unequal size and replace the permutations defining a cover by arbitrary maps. For these more general systems, we prove a stronger coefficient-positivity theorem for a determinant derived from the graph Laplacian. The enlarged class is closed under two reductions: contraction of an edge over a base loop and elimination of a base vertex. Both yield nonnegative recurrences for the relevant coefficients in terms of smaller systems, allowing an inductive proof. We also pose the problem of finding a direct combinatorial interpretation of the coefficients.

We study the initial boundary value problem for the compressible Navier--Stokes--Poisson equations without damping in the three-dimensional half-space and in smooth exterior domains. The velocity satisfies the no-slip boundary condition, and the electric potential satisfies the homogeneous Neumann boundary condition. For small perturbations of a constant state in $H^3$, with a small initial electric field in $L^2$, we establish global existence and uniqueness of strong solutions. The uniform estimates combine density dissipation, tangential and normal derivative estimates, and a modified energy expressed through the Helmholtz projection. In exterior domains, boundary normal coordinates and localized elliptic estimates control the geometric terms. We also obtain algebraic decay in the full energy norm. In the half-space, finite tangential negative Sobolev norms of order $-s$, with $\frac12\leq s<1$, yield the rate $(1+t)^{-s/2}$. In exterior domains, integrable initial density and velocity perturbations yield the same form of estimate for $0<s<\frac14$. The exterior decay proof uses whole-space negative-order estimates for localized variables and the time integrability of the dissipation.