Papers

We study the sample complexity of regularized optimal transport for general convex regularizations including the Kullback–Leibler divergence and $L^p$ penalties. Our main results are non-asymptotic bias and variance bounds for the empirical cost, with explicit dependence on the regularization parameter and on the intrinsic dimension of the marginals. Our approach simultaneously improves, unifies, and extends existing finite-sample bounds. In particular, we improve the state of the art for entropic optimal transport, and we obtain the first fully quantitative results for $L^p$ regularization with $1<p<\infty$. For the quadratic transport cost, we deduce that quadratically regularized optimal transport (i.e., $L^2$ regularization) estimates the unregularized optimal transport cost at rate $n^{-2/(d+4)}$, the fastest non-asymptotic rate currently available for any estimator based on regularized optimal transport.

The increasing number of orbital and sub-orbital launches makes it necessary to investigate the environmental impacts of launch vehicles and incorporate eco-design considerations into their development. In response, the European Space Agency has promoted Life Cycle Assessment (LCA) as a standardization methodology to mitigate environmental impacts of present and future space missions. This need is further amplified in the NewSpace, where numerous configurations and innovative technologies are explored, reinforcing the importance of integrating environmental considerations. At early design stages, launch vehicle architecture can be formalized through a multi-physics optimization problem based on Multidisciplinary Design Analysis and Optimization (MDAO) methods, where disciplines such as propulsion, aerodynamics, structure, and trajectory are coupled to obtain trade-offs among candidate configurations. This paper proposes a methodology to integrate an LCA discipline within an MDAO framework for launch vehicle design. The approach relies on parametric life-cycle inventories depending on design and coupling variables, covering component and propellant production as well as transport to the launch site. Launch emissions are evaluated from optimized trajectory profiles and characterized in terms of climate change impact. The methodology is illustrated on a representative expendable launch vehicle, where multi-objective optimizations assess trade-offs between performance and environmental indicators. Results highlight antagonistic behaviors among environmental impact categories, emphasizing the importance of carefully defining environmental objectives in eco-design studies. The generic nature of the methodology lays the foundation for integrating LCA into early-stage launch vehicle design, enabling exploration of trade-offs between performance, cost, and environmental considerations.

The purpose of this paper is twofold. In the first part, we revisit the description of the $\infty$-category of Borel complete equivariant spectra for a finite group given by Mathew-Naumann-Noel, introduce a version with coefficients, and then consider Borel equivariance for profinite groups. Here we identify two generally differing notions: levelwise Borel completeness and the hypercompletion thereof. In the second part, we study variants of smooth Artin motives, which are subcategories of the $\infty$-categories of effective Nisnevich and étale Voevodsky motives over a base scheme $S$ that are controlled by the étale fundamental group $\pi_1^{\mathrm{\'et}}(S)$. In the Nisnevich case, we extend a theorem of Voevodsky and identify smooth Artin motives with modules over the Bredon cohomology spectrum for the profinite group $\pi_1^{\mathrm{\'et}}(S)$. In the étale case, we show that the difference between our two notions of profinite Borel completeness is precisely the difference between étale sheaves and hypersheaves on finite étale schemes.

Given two conformal asymptotically hyperbolic metrics which are either both simple or both negatively curved, we show that if their (marked) renormalized boundary distances coincide for some choices of conformal representatives in their conformal infinities, then the two metrics are equal.

We study the Riesz $2$-energy of point configurations on the two-dimensional sphere arising from the Diamond ensemble, a construction of well-distributed points introduced by Beltrán and Etayo in 2020. For this family of point sets, we derive an explicit formula for the expected $2$-energy, valid for general choices of the parameters. Using this formula, we analyze a specific realization of the Diamond ensemble and obtain the asymptotic expansion of its expected $2$-energy. As a consequence, we establish a new upper bound for the minimal Riesz $2$-energy on the sphere, improving upon all previously known upper bounds. In particular, our result yields, for the first time, a negative coefficient in the quadratic term of the asymptotic expansion, bringing the upper bound significantly closer to the conjectured constant.

A proof of optimal-order $H^1$-norm error estimates is given for $A$-stable backward difference full discretizations (of order 1 and 2) of Willmore flow for closed two-dimensional surfaces. The numerical method discretizes a coupled system of evolution equations by evolving surface finite elements of polynomial degree at least two in space and backward difference method of order 1 or 2 in time. The convergence analysis is based on a stability analysis, based on energy estimates exploiting the anti-symmetric structure of the second-order system, in combination with Dahlquist's $G$-stability and the multiplier techniques of Nevanlinna and Odeh, with a new upper bound in the spirit of Dahlquist. Numerical experiments illustrate and complement the theoretical results.

In connection to the Brown-Erdős-Sós conjecture, we give a short local averaging proof of a density theorem for linear uniform hypergraphs. Let $r \ge 3$, $k \ge 3$, and suppose that $n \ge (r-2)(k-2)+1$. If $H$ is a linear $r$-uniform hypergraph on $n$ vertices and \[|E(H)| \geq \frac{k-2}{r^2((r-2)(k-2)+1)}n^2 + \frac{n}{r},\] then $H$ contains $k$ edges spanning at most $(r-2)k+3$ vertices. In the standard linear-density normalization, this gives the asymptotic density threshold $c \geq \frac{r-1}{r} \cdot \frac{k-2}{(r-2)(k-2)+1} + o(1)$. In particular, this yields a simple proof of the large-uniformity form of the Brown-Erdős-Sós theorem, due to Keevash and Long. In the case of triple systems, our bound becomes $c \geq \frac{2(k-2)}{3(k-1)} + o(1)$, improving upon a bound of $\frac{4}{5}$ due to Santos and Tyomkyn.

Higher-order tangent bundles possess a rich tower of fibrations, suggesting the existence of geometric structures compatible with their iterated bundle structure. In this paper, we introduce the notion of a connection tower on a higher-order tangent bundle and study the geometric structures induced by such towers. In particular, we show that connection towers determine natural multiconnections, adapted splittings of the tangent bundle, and canonical vector bundle structures on higher-order tangent bundles. We then construct a specific connection tower induced by the Levi-Civita connection of a Riemannian manifold. This construction extends the classical Dombrowski connection map on the tangent bundle and leads naturally to a family of higher-order Sasaki metrics. We study the associated lifts of vector fields and derive explicit Lie bracket formulas for these lifts, together with structural identities for the induced multiconnection. Finally, we determine the Levi-Civita connection of the higher-order Sasaki metrics and derive explicit geodesic equations on the second- and third-order tangent bundles. We also obtain characterization results relating geodesics of the higher-order Sasaki metrics to geodesics on the base manifold.

We propose a fast deterministic scheme for the space-homogeneous Boltzmann equation that exploits the low-rank structure of the velocity distribution. This paper consists of two independent contributions. The first is a lifting-projection (LP) scheme, inspired by the approach in the recent theoretical breakthroughs \cite{guillen2025landau, imbert2026monotonicity, guillen2025landau2} on the well-posedness of the Landau and Boltzmann equations. In particular, the approach lifts the nonlinear 3D Boltzmann equation to the 6D linear Kac master equation, advanced over a single time step, and projected back to its marginal in 3D. The second contribution is a low-rank tensor method for evaluating the collision operator, in which the lifted solution is represented in tensor train (TT) format and computed via a TT cross approximation algorithm with interpolation, complemented by a TT-friendly conservation correction that enforces conservation of mass, momentum, and energy. When the solution is low-rank in velocity, the method scales linearly in $n$ when cubic interpolation is used (and quadratic in $n$ when spectral interpolation is used), where $n$ is the number of grid points in each velocity direction. Therefore, our methods offer significant computational savings over existing deterministic solvers in such cases. Numerical experiments on 2D and 3D benchmarks, including the BKW exact solution and anisotropic initial data, confirm the computational scaling, the expected order of accuracy and verify the effectiveness of the conservation correction.

Spectral optimal partition and segregation problems are deeply connected with harmonic maps, eigenfunctions, and the fine structure of nodal sets for linear elliptic equations. In this paper, we show that replacing the spectral energy by torsional rigidity leads to a genuinely different theory. The resulting optimal configurations are governed locally not by harmonic equations, but by torsion-type energies and unstable free boundary problems, thereby creating a natural bridge between optimal partition theory and the analysis of sublinear free boundary phenomena. We prove existence of optimal torsional partitions and segregated torsional configurations, together with optimal Lipschitz regularity of the associated nonlinear eigenfunctions. We establish a strong unique continuation principle, characterize the admissible vanishing orders and the corresponding blow-up profiles, derive sharp Hausdorff dimension estimates for the nodal set and its singular subset, and prove $C^{1,\alpha}$-regularity of the regular part of the free boundary. The proofs combine variational arguments with Almgren-type and Weiss-type monotonicity formulae adapted to the intrinsically sublinear torsional regime, blow-up analysis, and tools from geometric measure theory.

In this work, the performance of different spectral deferred corrections (SDC) methods applied to initial value problems for differential-algebraic equations (DAEs) of index one is analyzed. The SDC method solves a series of correction equations, and after each iteration, the numerical solution is corrected by adding the obtained approximation of the error. The formulation of the SDC method allows for a parallelization across the method to obtain small-scale parallelism, for which a number of processes equal to the number of collocation nodes can be used. Recently, an analytical approach to finding optimal diagonal coefficients for parallel SDC was proposed. So far, no analysis with the new coefficients for parallel SDC applied to DAEs was done. We demonstrate that parallel SDC methods solve DAE problems with high accuracy faster than the associated sequential SDC schemes, obtaining speedup in a small-scale parallel regime.

We study the prime geodesic theorem in congruence classes of $\mathrm{SL}_2(\mathbb{Z})$. We generalize previous work of Luo and Sarnak and of Soundararajan and Young, and prove that the geodesic analogue of the Chebotarev density theorem holds with exponent $25/36 + \varepsilon$. In particular, we deduce that the prime geodesic theorem holds with exponent $25/36 + \varepsilon$ for any congruence subgroup of $\mathrm{SL}_2(\mathbb{Z})$.

We equip quotients of Real bordism with the structure of a ring involution, an important source of examples being the truncated Real Brown-Peterson spectra. Motivated by this, we orient Lubin-Tate theory by higher truncated Brown-Peterson spectra, which is a key input for Meier-Shi-Zeng's transchromatic isomorphism theorem. We use these orientations to characterize the higher truncated Brown-Peterson spectra that are equivalent to a form of Lubin-Tate theory after chromatic localization.

We establish a homological algebra framework for proving polynomiality of higher homological Dehn functions of groups. As an application, we show a combination theorem for polynomial Dehn functions, which is reminiscent of a theorem of Brown for finiteness properties.

This paper introduced a Hierarchical Generative Density Model (HGDM) for the long-term propagation of orbital fragmentation clouds. Validation against high-fidelity Monte Carlo simulations showed that the proposed surrogate accurately reproduces the dominant multidimensional structures of propagated clouds while consistently outperforming classical band-formation approximations based on independent angular variables. Accurate cloud reconstructions were obtained using only a few hundred to a few thousand propagated fragments, yielding reductions exceeding two orders of magnitude in computational cost and three orders of magnitude in storage requirements; future work will investigate its application to large-scale debris-environment evolution and collision-cascade simulations associated with the Kessler syndrome.

Recently, a generalization of the Cesàro operator to several variables was introduced as a tuple acting on the Drury-Arveson space \cite{P}. We determine the norm of these Cesàro-type operators and utilize it along with this Cesàro tuple definition to generalize the classical Rhaly operator and recover some basic facts about this operator when applied to the generalization.

The theory of Latin squares has a long history. While the objects themselves appeared earlier, the study of their general mathematical theory dates back to Euler in the 18th century. Latin squares can be interpreted as 3-dimensional permutation hypermatrices, and alternating sign matrices often arise as a natural generalisation of permutation matrices. In 2018, Brualdi and Dahl introduced a generalisation of classical Latin squares using alternating sign hypermatrices. Inspired by their definition, we develop the theory of Italian squares, a related generalisation of Latin squares, together with a notion of orthogonality that resolves an inconsistency in the definition of Brualdi and Dahl. Building on classical questions from Latin square theory, we obtain results including upper bounds on the maximal size of a pairwise orthogonal set, conditions for the existence of an orthogonal mate, infinite families of orthogonal pairs, and transversals. As part of our exploration of alternating sign matrices, we also prove a Frobenius-König type result for a class of $(0,\pm1)$-matrices.

Let $n$ and $p$ denote the numbers of vertices and labels, respectively, in an undirected edge-labeled graph. Previous work showed that, under the Exponential Time Hypothesis (ETH), there is no deterministic algorithm with running time \[ (np)^{o\left(\frac{\log n}{(\log\log n)^2}\right)}. \] In this paper, we give a deterministic reduction that strengthens this conditional running-time lower bound to \[ (np)^{o(\log n)}\operatorname{poly}(|E|). \] The lower bound holds even for simple edge-labeled graphs. Since our reduction is deterministic, the same lower bound applies to bounded-error randomized algorithms under the randomized Exponential Time Hypothesis. On the resulting hard family, $p=n^{O(1)}$. Thus, under rETH, the lower bound matches the exponent order of the known randomized quasi-polynomial exact upper bound up to constant factors.

Stochastic network rewiring processes, in which edges dynamically rewire based on fixed node attributes, are widely used in applications ranging from social dynamics to neuroscience and form an important component of adaptive network modelling. In this paper, we identify low-dimensional collective variables (CVs) that capture the essential macroscopic behavior of such time-evolving networks and enable reduced-order descriptions of their dynamics. To this end, we apply the data-driven transition manifold approach to homophily-driven rewiring models, in which edges preferentially connect nodes with similar attributes. For two representative models, we find that the optimal CV is a consensus measure quantifying the fraction of edges whose incident nodes differ by less than a certain threshold. Building on the learned CV, we construct reduced macroscopic models using a data-driven approach based on sparse regression and through an analytical derivation using graphons. The latter yields a closed-form evolution equation for the consensus measure and analytically validates the identified CV.

Nahm sums and Rogers-Ramanujan type identities have attracted considerable attention in recent years. In this paper, we provide analytic proofs of five conjectural identities on modular rank four Nahm sums that were proposed by Cao and Wang. Moreover, we reveal that the conjectures of Shi-Wang and Cao-Wang are closely related.

Let $P$ be a sub-Markov matrix on a finite set $S$, representing the transition probabilities of a Markov chain on \(S\) absorbed at a cemetery point $\partial\notin S$. We consider a reinforced process \((X_n,\mu_n)\) defined as follows: $(X_n)$ behaves like a chain with kernel $P$ until it dies, and when it dies at time $n$, it is instantaneously “resurrected” at a point sampled according to its weighted past occupation measure $$ \mu_n = \frac{1}{W_n} \left( w_0\mu_0+\sum_{k=1}^n w_k\delta_{X_k} \right), \qquad W_n=\sum_{k=0}^n w_k, $$ where the positive weights $w_k$ satisfy certain technical assumptions, a typical example being given by $w_k = k^q$, with $q\geq -1$. When $P$ is irreducible, the behaviour of $(\mu_n)$ is well understood \cite{AFP}, \cite{bansaye2022non}: it converges almost surely toward the unique quasi-stationary distribution (QSD) of $P$. The purpose of this paper is to investigate the general situation where $P$ is not irreducible. Under generic assumptions on $P$, there are finitely many QSDs. We prove that the asymptotic selection depends on the summability of the inverse cumulative weights $1/W_n$. If $$ \sum_{n\geq 0}\frac1{W_n}=\infty, $$ then $(\mu_n)$ almost surely converges toward the QSD associated with the largest Perron value. If instead $$ \sum_{n\geq0}\frac1{W_n}<\infty, $$ then each QSD is selected with positive probability. In particular, for polynomial weights $w_0=1$ and $w_k=k^q$, $k\geq1$, this gives almost sure selection of the QSD with largest Perron value for $-1\leq q\leq 0$, whereas each quasi-stationary distribution is selected with positive probability for $q>0$.

We analyze the decrease of simplex diameters under iterated refinement of spherical Delaunay complexes. Unlike in ordinary subdivision, the refined Delaunay complex need not be a subdivision of the previous one, so mesh contraction is not automatic. We derive explicit contraction bounds for several families of Steiner points, including Delaunay analogues of barycentric and edgewise subdivision. The proof reduces the problem to sharp covering estimates for Euclidean simplices. These estimates are obtained through a strengthening of Maurey's empirical method via pivotal sampling and a dimension-dependent version of the approximate Carathéodory theorem. Theoretical results and numerical experiments show that Delaunay refinements achieve stronger contraction than their subdivision counterparts.

In this paper, we improve the marked-vertex strategy introduced by Fu and Liang, and design the algorithm to compute all mutation paths and generator elements of cluster automorphism groups efficiently. As an application, we get generators of cluster automorphism groups of cluster algebras of finite mutation type of rank 4.

We resolve Erdős Problem 1061, the question whether the number \[ S(x)=\#\{(a,b)\in\mathbb{N}^2:a+b\le x, \ \sigma(a)+\sigma(b)=\sigma(a+b)\} \] of ordered solutions has a linear asymptotic $S(x)\sim cx$. In fact the opposite extreme holds at every fixed logarithmic scale: for every \(R>0\), \[ \lim_{x\to\infty}\frac{S(x)}{x(\log x)^R}=+\infty. \] The construction begins with three integers having the same abundancy index and reduces the divisor-sum identity to two equations in six primes. After a linear change of variables, these equations lie on a split quadric. A three-parameter rational ruling of the quadric supplies many affine systems of six linear forms. An exact lattice-index calculation, an elementary codimension-two parameter sieve, and Bienvenu's higher-dimensional Siegel–Walfisz theorem give prime points uniformly on these planes. Coprime multiplier amplification then yields the stated resolution.

Given a smooth deformation of a Lie subalgebra, we establish a necessary and sufficient condition for its smooth triviality and derive an analogous criterion for Lie ideals. We then give a direct proof of the Moser trick for foliations, which forms the basis for extending this result to general Lie subalgebroids.

We construct a local tri-Hamiltonian structure of the asymmetric (3,1)-type generalized Ablowitz-Ladik (gAL) hierarchy at the full-dispersive level and rigorously prove its validity using the supervariable technique. All central invariants of the corresponding bi-Hamiltonian structures are computed. In addition, we construct a Frobenius manifold M arising from the dispersionless limit of this hierarchy and show that the dispersionless limits of the first flows of the (3,1)-type gAL hierarchy belong to the Principal Hierarchy of M.

For a number field $K/\mathbb{Q}$, the minimal index is the least positive integer $m$ for which there exists a monogenic order with index $m$ in the maximal order. For any $n>2$ and $N>1$, we construct infinitely many number fields of degree $n$ with minimal index greater than $N$.

We classify rational elliptic and quasi-elliptic surfaces with global vector fields over arbitrary algebraically closed fields of characteristic $p \geq 0$ different from $2$. For every such surface, we determine the multiple and reducible fibers, the identity component of the automorphism scheme, and the moduli. As a corollary, we deduce that rational (quasi-)elliptic surfaces with global vector fields are Jacobian if $p \neq 3,5$ and we describe all counterexamples in small characteristics.

In this work, we develop a multi-fidelity strategy for kinetic models in epidemiology with uncertain contact dynamics. Assessing and controlling the population-level effects of contact dynamics requires the development of models for understanding observable effects of heterogeneous contact structures, whose formation depends on complex social phenomena. These can be captured taking into account high-dimensional uncertain quantities. The proposed approach combines high-fidelity kinetic solvers with a hierarchy of low-fidelity surrogates, including reduced macroscopic models and coarse kinetic descriptions, remaining applicable even in regimes where a macroscopic closure is unavailable. This hierarchical framework identifies representative parameter samples and reconstructs full solutions via projection-based techniques, enabling efficient uncertainty propagation while drastically reducing computational cost. Numerical experiments in high-dimensional stochastic settings demonstrate that accurate statistical estimates of epidemic observables can be obtained with significantly reduced computational costs compared to standard approaches.

The Beauville-Voisin conjecture is about the Chow ring of hyper-Kähler varieties. We consider a variant version taking place in the ring of algebraic cycles modulo algebraic equivalence. We prove this variant version is true in codimension at least 3 for Fano varieties of lines on cubic fourfolds, and for double EPW sextics.