AI for Science

For a traveling salesperson problem (TSP) of n cities, we present a compact quantum encoding based on a time-register representation of tours. A candidate route is represented as a sequence of n-1 city labels over discrete time steps, with one fixed start city and the remaining cities encoded in binary registers. We describe three ingredients of the construction: uniform route generation over the route register, a reversible validity oracle, and a phase oracle that encodes the total tour cost. The validity oracle checks both that the non-start city labels form a permutation and, for incomplete graphs, that every directed edge used by the route exists. The cost oracle then accumulates the start-edge, intermediate-transition, and return-edge costs into a tour-dependent phase for valid routes. This yields a coherent superposition of candidate routes with feasibility and tour-length information embedded directly in the quantum state. The complete construction uses O(n log n) qubits, while a naive implementation has worst-case elementary-gate complexity O(n^3 log n). The encoding is compatible with amplitude amplification or spectral filtering techniques such as the quantum singular value transform (QSVT) or Grover's algorithm. However, due to the exponentially small fraction of valid tours, the overall complexity remains exponential even when combined with amplitude amplification.

Let $\g$ be a simple complex Lie algebra of type $G_2$, $F_4$, or $E_8$, and let $G$ be the unique connected simply connected complex Lie group with $\mathrm{Lie}(G)=\g$ and compact real form $K$. We prove a triangular decomposition theorem for the lower crystal lattice $\OAztG$ of the quantized function algebra $\OtG$, establishing that $\OAztG=A_0\text{-alg}<\RAzp \cup \RAzm>.$ This extends the triangular decomposition recently obtained for types $A_n, B_n, C_n, D_n, E_6$, and $E_7$ in~\cite{DDPa} to all simple complex Lie algebras. As a consequence, we obtain: (i) the inclusion $\OAztG\subseteq\OAztK$ conjectured by Matassa-Yuncken and (ii) the crystal limit $\CpKo$ is a compact quantum semigroup with a unique bi-invariant (Haar) state.

We introduce \textit{Prior-Informed Flow Matching (PIFM)}, a conditional flow model for graph reconstruction. Reconstructing graphs from partial observations remains a key challenge; classical embedding methods often lack global consistency, while modern generative models struggle to incorporate structural priors. PIFM bridges this gap by integrating embedding-based priors with continuous-time flow matching. Grounded in a permutation equivariant version of the distortion-perception theory, our method first uses a prior, such as GraphSAGE or node2vec, to form an informed initial estimate of the adjacency matrix based on local information. It then applies rectified flow matching to refine this estimate, transporting it toward the true distribution of clean graphs and learning a global coupling. Experiments on different datasets demonstrate that PIFM consistently enhances classical embeddings, outperforming them and state-of-the-art generative baselines in reconstruction accuracy.

In this paper we build a bijection between the meet-irreducible elements of the lattice of the compositions of $n$ with parts in $[1,p]$ equipped with the dominance order, and the edges of the $(n,p)$-Turán graph. Using this bijection, we then compute asymptotically the average value of some statistics on those meet-irreducible compositions.

We prove that, if $G$ is a finite almost simple group and $H$ is a maximal subgroup of $G$, then the $10$th term of the derived series of $H$ is perfect. The same is true if $G$ is perfect and $H$ is core-free. The constant $10$ is best possible.

We provide a structural analysis for McCarthy algebras, the variety generated by the three-element algebra defining the logic of McCarthy (the non-commutative version of Kleene three-valued logics). Our analysis will be conducted in a very general algebraic setting by introducing McCarthy algebras as a subvariety of unital bands (idempotent monoids) equipped with an involutive (unary) operation $'$ satisfying $x''\approx x$; herein referred to as i-ubands. Prominent (commutative) subvarieties of i-ubands include Boolean algebras, ortholattices, Kleene algebras, and involutive bisemilattices, hence i-ubands provides an algebraic common ground for several non-classical logics. Our main contributions consist in providing for McCarthy algebras: reduced and equivalent axiomatizations; a semilattice decomposition theorem; and representations as certain decorated posets from which the algebraic structure can be uniquely determined.

We generalize Milnor link invariants to surface-links in 4-space, possibly with boundary. To this end, we introduce the notion of cut-diagram, which is a 2-dimensional analogue of Gauss diagrams. To each cut-diagram, we associate a group extending the fundamental group of the exterior of a surface-link, and we extract Milnor-type invariants from its successive nilpotent quotients. We show that this yields concordance invariants for surface-links, and that some even are link-homotopy invariants. We give several concrete applications, including realization and classification results. The theory of cut-diagrams is further investigated, heading towards a combinatorial approach to surfaces in 4-space.

We provide efficient algorithms to compute convergents of quadratic irrationals. We show that for square roots, in settings where Galois' refinement of Lagrange's theorem holds, certain decimations of the sequence of convergents are signed Chebyshev sequences, which can be also be generated by a Householder method.

Given $m \ge 2$ discrete probability distributions over $n$ states each, the minimum-entropy coupling is the minimum-entropy joint distribution whose marginals are the same as the input distributions. Computing the minimum-entropy coupling is NP-hard, but there has been significant progress in designing approximation algorithms; prior to this work, the best known polynomial-time algorithms attain guarantees of the form $H(\operatorname{ALG}) \le H(\operatorname{OPT}) + c$, where $c \approx 0.53$ for $m=2$, and $c \approx 1.22$ for general $m$ [CKQGK '23]. A main open question is whether this task is APX-hard, or whether there exists a polynomial-time approximation scheme (PTAS). In this work, we design an algorithm that produces a coupling with entropy $H(\operatorname{ALG}) \le H(\operatorname{OPT}) + \varepsilon$ in running time $n^{O(\operatorname{poly}(1/\varepsilon) \cdot \operatorname{exp}(m) )}$: showing a PTAS exists for constant $m$.

We investigate the problem of finding the minimum number of pieces necessary for dividing a three-dimensional sphere or a ball and reassembling it to form $n$ congruent copies of the original object, generalising a known result by Raphael Robinson.

Can we determine someone's geographic location solely from the sounds they hear? Are acoustic signals enough to localize within a country, state, or even city? In this work, we tackle the challenge of global-scale audio geolocation, with a particular focus on wildlife and natural sounds. We posit that bioacoustic signals contain informative geolocation cues because of well-defined geographic ranges of species. To test this hypothesis, we benchmark image geolocation and soundscape mapping methods, design oracles and species-centric baselines, and propose a hybrid approach that combines species range prediction with retrieval-based geolocation. We further ask whether geolocation improves with species-diverse recordings and spatiotemporal aggregation across neighboring samples. Finally, we extend our study to multimodal geolocation with case studies from movies that combine both audio and visual content. Our results highlight the potential of incorporating bioacoustic signals into geospatial tasks, motivating future work on species recognition and audio geolocation.

In this paper we study the spectral analysis of Bochner-Kodaira Laplacians on an Abelian variety, complex projective space $\mathbb{P}^{n}$ and a Grassmannian with a holomorphic line bundle. By imitating the method of creation and annihilation operators in physics, we convert those eigensections (of the \textquotedblleft higher energy" level) into holomorphic sections (of the \textquotedblleft lowest energy" level). This enables us to endow these spectral bundles, which are defined over the dual Abelian variety, with natural holomorphic structure. Using this conversion expressed in a concrete way, all the higher eigensections are explicitly expressible using holomorphic sections formed by theta functions. Moreover, we give an explicit formula for the dimension of the space of higher-level eigensections on $\mathbb{P}^{n}$ through vanishing theorems and the Hirzebruch-Riemann-Roch theorem. These give a theoretical study related to some problems newly discussed by string theorists using numerical analysis. Some partial results on Grassmannians are proved and some directions for future research are indicated.

Data consisting of a graph with a function mapping into $\mathbb{R}^d$ arise in many data applications, encompassing structures such as Reeb graphs, geometric graphs, and knot embeddings. As such, the ability to compare and cluster such objects is required in a data analysis pipeline, leading to a need for distances between them. In this work, we study the interleaving distance on discretization of these objects, called mapper graphs when $d=1$, where functor representations of the data can be compared by finding pairs of natural transformations between them. However, in many cases, computation of the interleaving distance is NP-hard. For this reason, we take inspiration from recent work by Robinson to find quality measures for families of maps that do not rise to the level of a natural transformation, called assignments. We then endow the functor images with the extra structure of a metric space and define a loss function which measures how far an assignment is from making the required diagrams of an interleaving commute. Finally we show that the computation of the loss function is polynomial with a given assignment. We believe this idea is both powerful and translatable, with the potential to provide approximations and bounds on interleavings in a broad array of contexts.

Andréka and Maddux classified the relation algebras with at most 3 atoms, and in particular they showed that all of them are representable. Hirsch and Cristiani showed that the network satisfaction problem (NSP) for each of these algebras is in P or NP-hard. The literature contains many results on representations of relation algebras; in particular, some relation algebras with four atoms are not representable. We extend the result of Cristiani and Hirsch to relation algebras with at most 4 atoms: the NSP is always either in P or NP-hard. To this end, we construct universal, fully universal, or even normal representations for these algebras, whenever possible.

We conduct a computability-theoretic study of Ramsey-like theorems of the form "Every coloring of the edges of an infinite clique admits an infinite sub-clique avoiding some pattern", with a particular focus on transitive patterns. As it turns out, the patterns corresponding to separable permutations play an important role in the computational features of the statement. We prove that the avoidance of any separable permutation is equivalent to the existence of an infinite homogeneous set in standard models, while this property fails for any other pattern. For this, we develop a novel argument for relativized diagonal non-computation.

We prove that the $τ$-tilting graph of any quasi-silted algebra is connected and has the reachable-in-face property. Our approach utilizes $τ$-reduction and wall and chamber structures. In particular, we observe a sufficient condition on the wall and chamber structure under which the connectivity of $τ$-tilting graphs is preserved under taking quotients of algebras. As an immediate consequence, the connectivity of $τ$-tilting graphs is also established for several new classes of algebras.

Motivated by the discrete logarithmic Minkowski problem we study for a given matrix $U\in\mathbb{R}^{n\times m}$ its cone-volume set $C_{\tt cv}(U)$ consisting of all the cone-volume vectors of polytopes $P(U,b)=\{ x\in\mathbb{R}^n : U^\intercal x\leq b\}$, $b\in\mathbb{R}^n_{\geq 0}$. We will show that $C_{\tt cv}(U)$ is a path-connected semialgebraic set which extends former results in the planar case or for particular polytopes. Moreover, we define a subspace concentration polytope $P_{\tt scc}(U)$ which represents geometrically the subspace concentration conditions for a finite discrete Borel measure on the sphere. This is up to a scaling the basis matroid polytope of $U$, and these two sets, $P_{\tt scc}(U)$ and $C_{\tt cv}(U)$, also offer a new geometric point of view to the discrete logarithmic Minkowski problem.

We find identities involving differential operators in the generic artinian reduction of the Stanley-Reisner ring of a simplicial sphere in any positive characteristic. These identities generalize the characteristic 2 identities used by Papadakis and Petrotou to give a proof of the algebraic g-conjecture. We show that these identities are a shadow of an identity on the degree map, and we use them to prove the anisotropy of certain forms on the generic artinian reduction of the Stanley--Reisner ring and to prove weak Lefschetz results.

A family of sets is $s$-intersecting if every pair of its sets has at least $s$ elements in common. It is an $s$-star if all its members have some $s$ elements in common. A family of sets is called $s$-EKR if all its $s$-intersecting subfamilies have size at most that of some $s$-star. For example, the classic 1961 Erdős-Ko-Rado theorem states essentially that the family of $r$-sized subsets of $\{1,2,\ldots,n\}$ is $s$-EKR when $n$ is a large enough function of $r$ and $s$, and the 1967 Hilton-Milner theorem provides the near-star structure of the largest non-star intersecting family of such sets. Two important conjectures along these lines followed: by Chvátal in 1974, that every subset-closed family of sets is 1-EKR, and by Holroyd and Talbot in 2005, that, for every graph, the family of all its $r$-sized independent sets is 1-EKR when every maximal independent set has size at least $2r$. In this paper we present similar 1-EKR results for families of length-$r$ paths in graphs, specifically for sun graphs, which are cycles with pendant edges attached in a uniform way, and theta graphs, which are collections of pairwise internally disjoint paths sharing the same two endpoints. We also prove $s$-EKR results for such paths in suns, and give a Hilton-Milner type result for them as well. A set is a transversal of a family of sets if it intersects each member of the family, and the transversal number of the family is the size of its smallest transversal. For example, stars have transversal number 1, and the Hilton-Milner family has transversal number 2. We conclude the paper with some transversal results involving what we call triangular families, including a few results for projective planes.

The best-worst method is a well-known distance based multi-criteria decision-making method used for computing the weights of decision criteria. This article provides a comprehensive analytical examination of the taxicab distance based model of the method, with the objectives of investigating the uniqueness of these solutions, and performing a rigorous consistency analysis. To achieve this, an optimal modification based optimization problem, equivalent to the original one, is first formulated. This reformulated problem is then solved analytically, and the optimal weight sets are derived from its solutions. Contrary to the prevailing understanding derived from numerical experiments with the taxicab model, our analytical framework proves that the model can, in fact, lead to multiple optimal weight sets, and we formally establish the conditions for this occurrence. A mixed-integer linear programming model is then employed to compute the consistency index. A decision-maker-aided selection strategy is also proposed for addressing non-uniqueness of optimal weight sets. In addition, threshold values of the consistency ratio to assess the admissibility of given preferences are also established. This framework provides a solid mathematical foundation that enhances the understanding of the model and eliminates the requirement for optimization software. By significantly improving the model's computational efficiency, it enables implementation in large-scale, dynamic real-world applications such as electricity market bidding strategies and portfolio rebalancing under market volatility. The effectiveness of the proposed framework is demonstrated through numerical examples, and its practical applicability is illustrated via a smartphone selection problem.

The paper establishes results following two interconnected directions. 1. Let $G$ be a Roelcke precompact closed subgroup of the group $\mathrm{Sym}(ω)$ of permutations of the natural numbers. Let $\mathrm{Aut}(G)$ denote the group of continuous automorphisms of $G$. Then $\mathrm{Inn}(G)$ is closed in $\mathrm{Aut}(G)$, where $\mathrm{Aut}(G)$ carries the topology of pointwise convergence for its (faithful) action on the cosets of open subgroups. Under the stronger hypothesis that~$G$ is oligomorphic, $\+ N_G/G$ is profinite, where $\+ N_G$ denotes the normaliser of~$G$ in $\mathrm{Sym}(ω)$, and the topological group $\mathrm{Out}(G)= \mathrm{Aut}(G)/\mathrm{Inn}(G)$ is totally disconnected, locally compact. 2a. We provide a general method to show smoothness of the isomorphism relation for appropriate Borel classes of oligomorphic groups. We apply it to two such classes: the oligomorphic groups with no algebraicity, and the oligomorphic groups with finitely many {essential} subgroups up to conjugacy. 2b. Using this method we also show that if $G$ is in such a Borel class, then $\mathrm{Aut}(G)$ is topologically isomorphic to an oligomorphic group, and $\mathrm{Out}(G)$ is profinite.

Suppose $μ$ and $ν$ are probability measures on $\mathbb{R}$ satisfying $μ\leq_{cx} ν$. Let $a$ and $b$ be convex functions on $\mathbb{R}$ with $a \geq b \geq 0$. We are interested in finding $$\sup_{\mathbf{M}} \sup_τ \mathbb{E}^{\mathbf{M}} \left[ a(X) I_{ \{ τ= 1 \} } + b(Y) I_{ \{ τ= 2 \} } \right] $$ where the first supremum is taken over consistent models $\mathbf{M}$ (i.e., filtered probability spaces $(Ω, \mathbf{F}, \mathbb{F}, \mathbb{P})$ such that $Z=(z,Z_1,Z_2)=(\int_{\mathbb{R}} x μ(dx) = \int_{\mathbb{R}} y ν(dy), X, Y)$ is a $(\mathbb{F},\mathbb{P})$ martingale, where $X$ has law $μ$ and $Y$ has law $ν$ under $\mathbb{P}$) and $τ$ in the second supremum is a $(\mathbb{F},\mathbb{P})$-stopping time taking values in $\{1,2\}$. Our contributions are first to characterise and simplify the dual problem, and second to completely solve the problem under some structural assumptions on the measures $μ$ and $ν$ (namely that $μ$ and $ν$ are absolutely continuous probability measures that satisfy the Dispersion Assumption). A key finding is that the canonical set-up in which the filtration is that generated by $Z$ is not rich enough to define an optimal model and additional randomisation is required. This holds even though the marginal laws $μ$ and $ν$ are atom-free. The problem has an interpretation of finding the robust, or model-free, no-arbitrage bound on the price of a Bermudan option with two possible exercise dates, given the prices of co-maturing European options.

Piecewise-Deterministic Markov Processes (PDMPs) hold significant promise for sampling from complex probability distributions. However, their practical implementation is hindered by the need to compute model-specific bounds. Conversely, while Hamiltonian Monte Carlo (HMC) offers a generally efficient approach to sampling, its inability to adaptively tune step sizes impedes its performance when sampling complex distributions like funnels. To address these limitations, we introduce three innovative concepts: (a) a Metropolis-adjusted approximation for PDMP simulation that eliminates the need for explicit bounds without compromising the invariant measure, (b) an adaptive step size mechanism compatible with the Metropolis correction, and (c) a No U-Turn Sampler (NUTS)-inspired scheme for dynamically selecting path lengths in PDMPs. These three ideas can be seamlessly integrated into a single, `doubly-adaptive' PDMP sampler with favourable robustness and efficiency properties.

In our series of papers, we prove that smooth Fano threefolds in positive characteristic lift to the ring of Witt vectors. Moreover, we show that they satisfy Akizuki-Nakano vanishing, $E_1$-degeneration of the Hodge to de Rham spectral sequence, and torsion-freeness of Crystalline cohomologies. In this paper, we establish these results for the case when $|-K_X|$ is very ample and the Picard group is generated by $ω_X$.

It is well-known that the Sobolev spaces $W^{k,p}(\mathbb R^d)$ are vector lattices with respect to the pointwise almost everywhere order if $k \in \{0,1\}$, but not if $k \ge 2$. In this note, we consider negative $k$ and show that the span of the positive cone in $W^{k,p}(\mathbb R^d)$ is a vector lattice in this case. We also prove a related abstract result: if $(T(t))_{t \in [0,\infty)}$ is a positive $C_0$-semigroup on a Banach lattice $X$ with order continuous norm, then the span of the cone $X_{-1,+}$ in the extrapolation space $X_{-1}$ is a vector lattice. This complements results obtained by Bátkai, Jacob, Wintermayr, and Voigt in the context of perturbation theory and provides additional context for the theory of infinite-dimensional positive systems.

Classical results of Cauchy and Dehn imply that the 1-skeleton of a convex simplicial polyhedron $P$ is rigid i.e. every continuous motion of the vertices of $P$ in $\mathbb R^3$ which preserves its edge lengths results in a polyhedron which is congruent to $P$. This result was extended to convex smplicial polytopes in $\mathbb R^d$ for all $d\geq 3$ by Whiteley, and to generic realisations of 1-skeletons of simplicial $(d-1)$-manifolds in $\mathbb R^{d}$ by Kalai for $d\geq 4$ and Fogelsanger for $d\geq 3$. We will generalise Kalai's result by showing that, for all $d\geq 4$ and any fixed $1\leq k\leq d-3$, every generic realisation of the $k$-skeleton of a simplicial $(d-1)$-manifold in $\mathbb R^{d}$ is volume rigid, i.e. every continuous motion of its vertices in $\mathbb R^d$ which preserves the volumes of its $k$-faces results in a congruent realisation. In addition, we conjecture that our result remains true for $k=d-2$ and verify this conjecture when $d=4,5,6$.

This text is a survey on symmetric matrices. It serves as a script for a module to be taught at university.

The drastic changes in the global economy, geopolitical conditions, and disruptions such as the COVID-19 pandemic have impacted the cost of living and quality of life. It is essential to comprehend the long-term implications of the cost of living and quality of life in major economies. A transparent and comprehensive living index must include multiple dimensions of living conditions. In this study, we present an approach to quantifying the quality of life through the Global Ease of Living Index that combines various socio-economic and infrastructural factors into a single composite score. Our index utilises economic indicators that define living standards, which could help in targeted interventions to improve specific areas. We present a machine learning framework to address missing data for certain economic indicators in specific countries. We then curate and update the data and use a dimensionality reduction approach (Principal Component Analysis and Factor Analysis) to create the Ease of Living Index for major economies since 1970. Our work significantly adds to the literature by offering a practical tool for policymakers to identify areas needing improvement, such as healthcare systems, employment opportunities, and public safety. Our approach with open data and code can be easily reproduced and applied to various contexts, providing transparency and accessibility for ongoing research and policy development in quality-of-life assessment.

Previous work of Chan--Church--Grochow and Baker--Wang shows that the set of spanning trees in a plane graph $G$ is naturally a torsor for the Jacobian group of $G$. Informally, this means that the set of spanning trees of $G$ naturally forms a group, except that there is no distinguished identity element. We generalize this fact to graphs embedded on orientable surfaces of arbitrary genus, which can be identified with ribbon graphs. In this generalization, the set of spanning trees of $G$ is replaced by the set of spanning quasi-trees of the ribbon graph, and the Jacobian group of $G$ is replaced by the Jacobian group of the associated regular orthogonal matroid $M$ (along with an associated regular representation of $M$). Our proof shows, more generally, that the family of "BBY torsors" constructed by Backman--Baker--Yuen and later generalized by Ding admit natural generalizations to (regular representations of) regular orthogonal matroids. In addition to shedding light on the role of planarity in the earlier work mentioned above, our results represent one of the first substantial applications of orthogonal matroids (also called "even delta-matroids" or "Lagrangian orthogonal matroids") to a natural combinatorial problem about graphs.

We prove that the minimal size $M(π_n)$ of a maximal matching in the permutahedron $π_n$ is asymptotically $n!/3$. On the one hand, we obtain a lower bound $M(π_n) \ge n! (n-1) / (3n-2)$ by considering $4$-cycles in the permutahedron. On the other hand, we obtain an asymptotical upper bound $M(π_n) \le n!(1/3+o(1))$ by multiple applications of Hall's theorem (similar to the approach of Forcade (1973) for the hypercube) and an exact upper bound $M(π_n) \le n!/3$ by an explicit construction. We also derive bounds on minimum maximal matchings in products of permutahedra.