arXiv subjects

Anton Alekseev

Publications and source records attributed to Anton Alekseev.

At least 19 recordsLinked to original sources

arXiv2026

Batalin-Vilkovisky structures on moduli spaces of flat connections

Let $Σ$ be a compact oriented 2-manifold (possibly with boundary), and let $\mathcal G_Σ$ be the linear span of free homotopy classes of closed oriented curves on $Σ$ equipped with the Goldman Lie bracket $[\cdot, \cdot]_\mathrm{Goldman}$ defined in terms of intersections of curves. A theorem of Goldman gives rise to a Lie homomorphism $Φ^\mathrm{even}$ from $(\mathcal G_Σ, [\cdot, \cdot]_\text{Goldman})$ to functions on the moduli space of flat connections $\mathcal{M}_Σ(G)$ for $G=U(N), GL(N)$, equipped with the Atiyah-Bott Poisson bracket. The space $\mathcal{G}_Σ$ also carries the Turaev Lie cobracket $δ_\mathrm{Turaev}$ defined in terms of self-intersections of curves. In this paper, we address the following natural question: which geometric structure on moduli spaces of flat connections corresponds to the Turaev cobracket? We give a constructive answer to this question in the following context: for $G$ a Lie supergroup with an odd invariant scalar product on its Lie superalgebra, and for nonempty $\partialΣ$, we show that the moduli space of flat connections $\mathcal{M}_Σ(G)$ carries a natural Batalin-Vilkovisky (BV) structure, given by an explicit combinatorial Fock-Rosly formula. Furthermore, for the queer Lie supergroup $G=Q(N)$, we define a BV-morphism $Φ^\mathrm{odd}\colon \wedge \mathcal{G}_Σ \to \mathrm{Fun}(\mathcal{M}_Σ(Q(N)))$ which replaces the Goldman map, and which captures the information both on the Goldman bracket and on the Turaev cobracket. The map $Φ^\mathrm{odd}$ is constructed using the "odd trace" function on $Q(N)$.

math.QA↗

arXiv2026

Group valued moment maps for even and odd simple $G$-modules

Let $G$ be a complex simple Lie group, and $\mathfrak{g}$ its Lie algebra. It is well known that a finite-dimensional $G$-module $V$ carrying a nondegenerate invariant bilinear form gives rise to a Hamiltonian Poisson space with a quadratic moment map $μ$. We show that under condition $\mathrm{Hom}_{\mathfrak{g}}({\textstyle{\bigwedge}}^3 V, S^3V)=0$ this space can be viewed as a quasi-Poisson space with the same bivector, and with the group valued moment map $Φ= \exp \circ μ$. Furthermore, we show that by modifying the bivector by the standard $r$-matrix for $\mathfrak{g}$ one obtains a space with a Poisson action of the Poisson-Lie group~$G$, and with the moment map in the sense of Lu taking values in the dual Poisson-Lie group~$G^\ast$.

math.RT↗

arXiv2026

The kernel of formal polylogarithms

Polylogarithmic functions (polylogs) in $n$ variables can be viewed as elements of $(U\mathfrak{p}_{m})^*$, the dual of the universal enveloping algebra of the Lie algebra $\mathfrak{p}_{m}$ of infinitesimal spherical pure braids with $m=n+3$ strands. Polylogs with $m=4,5$ are used in the theory relating double shuffle relations and Drinfeld associators \cite{furusho_double_2011}. We give explicit formulas for elements of $(U\mathfrak{p}_{m})^*$ representing polylogs, and compute the left ideal $J_{m} \subset U\mathfrak{p}_{m}$ given by their joint kernel. We introduce Lie subalgebras $\mathfrak{k}_{m}=\mathfrak{p}_{m} \cap J_{m}$, and we compute them for $m=4, 5$.

math.QA↗

arXiv2025

The tetrahedral Horn problem and asymptotics of U(n) 6j symbols

Horn's problem is concerned with characterizing the eigenvalues $(a,b,c)$ of Hermitian matrices $(A,B,C)$ satisfying the constraint $A+B=C$ and forming the edges of a triangle in the space of Hermitian matrices. It has deep connections to tensor product invariants, Littlewood-Richardson coefficients, geometric invariant theory and the intersection theory of Schubert varieties. This paper concerns the tetrahedral Horn problem which aims to characterize the tuples of eigenvalues $(a,b,c,d,e,f)$ of Hermitian matrices $(A,B,C,D,E,F)$ forming the edges of a tetrahedron, and thus satisfying the constraints $A+B=C$, $B+D=F$, $D+C=E$ and $A+F=E$. Here we derive new inequalities satisfied by the Schur-polynomials of such eigenvalues and, using eigenvalue estimation techniques from quantum information theory, prove their satisfaction up to degree $k$ implies the existence of approximate solutions with error $O(\ln k / k)$. Moreover, the existence of these tetrahedra is related to the semiclassical asymptotics of the $6j$-symbols for the unitary group $U(n)$, which are maps between multiplicity spaces that encode the associativity relation for tensor products of irreducible representations. Using our techniques, we prove the asymptotics of norms of these $6j$-symbols are either inverse-polynomial or exponential depending on whether there exists such tetrahedra of Hermitian matrices.

math.RT↗

arXiv2025

Human-Annotated NER Dataset for the Kyrgyz Language

We introduce KyrgyzNER, the first manually annotated named entity recognition dataset for the Kyrgyz language. Comprising 1,499 news articles from the 24.KG news portal, the dataset contains 10,900 sentences and 39,075 entity mentions across 27 named entity classes. We show our annotation scheme, discuss the challenges encountered in the annotation process, and present the descriptive statistics. We also evaluate several named entity recognition models, including traditional sequence labeling approaches based on conditional random fields and state-of-the-art multilingual transformer-based models fine-tuned on our dataset. While all models show difficulties with rare entity categories, models such as the multilingual RoBERTa variant pretrained on a large corpus across many languages achieve a promising balance between precision and recall. These findings emphasize both the challenges and opportunities of using multilingual pretrained models for processing languages with limited resources. Although the multilingual RoBERTa model performed best, other multilingual models yielded comparable results. This suggests that future work exploring more granular annotation schemes may offer deeper insights for Kyrgyz language processing pipelines evaluation.

cs.CL↗

arXiv2025

LAMeD: LLM-generated Annotations for Memory Leak Detection

Static analysis tools are widely used to detect software bugs and vulnerabilities but often struggle with scalability and efficiency in complex codebases. Traditional approaches rely on manually crafted annotations -- labeling functions as sources or sinks -- to track data flows, e.g., ensuring that allocated memory is eventually freed, and code analysis tools such as CodeQL, Infer, or Cooddy can use function specifications, but manual annotation is laborious and error-prone, especially for large or third-party libraries. We present LAMeD (LLM-generated Annotations for Memory leak Detection), a novel approach that leverages large language models (LLMs) to automatically generate function-specific annotations. When integrated with analyzers such as Cooddy, LAMeD significantly improves memory leak detection and reduces path explosion. We also suggest directions for extending LAMeD to broader code analysis.

cs.SE↗

arXiv2025

Multiple Horn problems for planar networks and invertible matrices

The multiplicative multiple Horn problem is asking to determine possible singular values of the combinations $AB, BC$ and $ABC$ for a triple of invertible matrices $A,B,C$ with given singular values. There are similar problems for eigenvalues of sums of Hermitian matrices (the additive problem), and for maximal weights of multi-paths in concatenations of planar networks (the tropical problem). For the planar network multiple Horn problem, we establish necessary conditions, and we conjecture that for large enough networks they are also sufficient. These conditions are given by the trace equalities and rhombus inequalities (familiar from the hive description of the classical Horn problem), and by the new set of tetrahedron equalities. Furthermore, if one imposes Gelfand-Zeitlin conditions on weights of planar networks, tetrahedron equalities turn into the octahedron recurrence from the theory of crystals. We give a geometric interpretation of our results in terms of positive varieties with potential. In this approach, rhombus inequalities follow from the inequality $Φ^t \leqslant 0$ for the tropicalized potential, and tetrahedron equalities are obtained as tropicalization of certain Plücker relations. For the multiplicative problem, we introduce a scaling parameter $s$, and we show that for $s$ large enough (corresponding to exponentially large/small singular values) the Duistermaat-Heckman measure associated to the multiplicative problem concentrates in a small neighborhood of the octahedron recurrence locus.

math.RT↗

arXiv2024

Syntactic Transfer to Kyrgyz Using the Treebank Translation Method

The Kyrgyz language, as a low-resource language, requires significant effort to create high-quality syntactic corpora. This study proposes an approach to simplify the development process of a syntactic corpus for Kyrgyz. We present a tool for transferring syntactic annotations from Turkish to Kyrgyz based on a treebank translation method. The effectiveness of the proposed tool was evaluated using the TueCL treebank. The results demonstrate that this approach achieves higher syntactic annotation accuracy compared to a monolingual model trained on the Kyrgyz KTMU treebank. Additionally, the study introduces a method for assessing the complexity of manual annotation for the resulting syntactic trees, contributing to further optimization of the annotation process.

cs.CL↗

arXiv2024

$\nabla^2$DFT: A Universal Quantum Chemistry Dataset of Drug-Like Molecules and a Benchmark for Neural Network Potentials

Methods of computational quantum chemistry provide accurate approximations of molecular properties crucial for computer-aided drug discovery and other areas of chemical science. However, high computational complexity limits the scalability of their applications. Neural network potentials (NNPs) are a promising alternative to quantum chemistry methods, but they require large and diverse datasets for training. This work presents a new dataset and benchmark called $\nabla^2$DFT that is based on the nablaDFT. It contains twice as much molecular structures, three times more conformations, new data types and tasks, and state-of-the-art models. The dataset includes energies, forces, 17 molecular properties, Hamiltonian and overlap matrices, and a wavefunction object. All calculations were performed at the DFT level ($ω$B97X-D/def2-SVP) for each conformation. Moreover, $\nabla^2$DFT is the first dataset that contains relaxation trajectories for a substantial number of drug-like molecules. We also introduce a novel benchmark for evaluating NNPs in molecular property prediction, Hamiltonian prediction, and conformational optimization tasks. Finally, we propose an extendable framework for training NNPs and implement 10 models within it.

physics.chem-ph↗

arXiv2024

HJ-Ky-0.1: an Evaluation Dataset for Kyrgyz Word Embeddings

One of the key tasks in modern applied computational linguistics is constructing word vector representations (word embeddings), which are widely used to address natural language processing tasks such as sentiment analysis, information extraction, and more. To choose an appropriate method for generating these word embeddings, quality assessment techniques are often necessary. A standard approach involves calculating distances between vectors for words with expert-assessed 'similarity'. This work introduces the first 'silver standard' dataset for such tasks in the Kyrgyz language, alongside training corresponding models and validating the dataset's suitability through quality evaluation metrics.

cs.CL↗

arXiv2024

KyrgyzNLP: Challenges, Progress, and Future

Large language models (LLMs) have excelled in numerous benchmarks, advancing AI applications in both linguistic and non-linguistic tasks. However, this has primarily benefited well-resourced languages, leaving less-resourced ones (LRLs) at a disadvantage. In this paper, we highlight the current state of the NLP field in the specific LRL: kyrgyz tili. Human evaluation, including annotated datasets created by native speakers, remains an irreplaceable component of reliable NLP performance, especially for LRLs where automatic evaluations can fall short. In recent assessments of the resources for Turkic languages, Kyrgyz is labeled with the status 'Scraping By', a severely under-resourced language spoken by millions. This is concerning given the growing importance of the language, not only in Kyrgyzstan but also among diaspora communities where it holds no official status. We review prior efforts in the field, noting that many of the publicly available resources have only recently been developed, with few exceptions beyond dictionaries (the processed data used for the analysis is presented at https://kyrgyznlp.github.io/). While recent papers have made some headway, much more remains to be done. Despite interest and support from both business and government sectors in the Kyrgyz Republic, the situation for Kyrgyz language resources remains challenging. We stress the importance of community-driven efforts to build these resources, ensuring the future advancement sustainability. We then share our view of the most pressing challenges in Kyrgyz NLP. Finally, we propose a roadmap for future development in terms of research topics and language resources.

cs.CL↗

arXiv2024

Neural Click Models for Recommender Systems

We develop and evaluate neural architectures to model the user behavior in recommender systems (RS) inspired by click models for Web search but going beyond standard click models. Proposed architectures include recurrent networks, Transformer-based models that alleviate the quadratic complexity of self-attention, adversarial and hierarchical architectures. Our models outperform baselines on the ContentWise and RL4RS datasets and can be used in RS simulators to model user response for RS evaluation and pretraining.

cs.IR↗

arXiv2024

Poisson brackets and coaction maps of regularized holonomies of the KZ equation

We derive explicit closed formulas for the Kirillov-Kostant-Souriau (KKS) coaction maps of open path regularized holonomies of the Knizhnik-Zamolodchikov (KZ) equation, and the corresponding Poisson brackets for the Lie algebra ${\rm gl}(N, \mathbb{C})$. Our main technical tool is a certain projection of the generalized pentagon equation of \cite{AFR2024}.

math.QA↗

arXiv2024

WKB asymptotics of Stokes matrices, spectral curves and rhombus inequalities

We consider an $n\times n$ system of ODEs on $\mathbb{P}^1$ with a simple pole $A$ at $z=0$ and a double pole $u={\rm diag}(u_1, \dots, u_n)$ at $z=\infty$. This is the simplest situation in which the monodromy data of the system are described by upper and lower triangular Stokes matrices $S_\pm$, and we impose reality conditions which imply $S_-=S_+^\dagger$. We study leading WKB exponents of Stokes matrices in parametrizations given by generalized minors and by spectral coordinates, and we show that for $u$ on the caterpillar line (which corresponds to the limit $(u_{j+1}-u_j)/(u_j - u_{j-1}) \to \infty$ for $j=2, \cdots, n-1$), the real parts of these exponents are given by periods of certain cycles on the degenerate spectral curve $Γ(u_{\rm cat}(t), A)$. These cycles admit unique deformations for $u$ near the caterpillar line. Using the spectral network theory, we give for $n=2$, and $n=3$ exact WKB predictions for asymptotics of generalized minors in terms of periods of these cycles. Boalch's theorem from Poisson geometry implies that real parts of leading WKB exponents satisfy the rhombus (or interlacing) inequalities. We show that these inequalities are in correspondence with finite webs of the canonical foliation on the root curve $Γ^r(u, A)$, and that they follow from the positivity of the corresponding periods. We conjecture that a similar mechanism applies for $n>3$. We also outline the relation of the spectral coordinates with the cluster structures considered by Goncharov-Shen, and with ${\mathcal N}=2$ supersymmetric quantum field theories in dimension four associated with some simple quivers.

math-ph↗

arXiv2024

Generalized Pentagon Equations

Drinfeld defined the Knizhinik--Zamolodchikov (KZ) associator $Φ_{\rm KZ}$ by considering the regularized holonomy of the KZ connection along the {\em droit chemin} $[0,1]$. The KZ associator is a group-like element of the free associative algebra with two generators, and it satisfies the pentagon equation. In this paper, we consider paths on $\mathbb{C}\backslash \{ z_1, \dots, z_n\}$ which start and end at tangential base points. These paths are not necessarily straight, and they may have a finite number of transversal self-intersections. We show that the regularized holonomy $H$ of the KZ connection associated to such a path satisfies a generalization of Drinfeld's pentagon equation. In this equation, we encounter $H$, $Φ_{\rm KZ}$, and new factors associated to self-intersections, to tangential base points, and to the rotation number of the path.

math.QA↗

arXiv2024

Symplectic geometry of Teichmüller spaces for surfaces with ideal boundary

A hyperbolic 0-metric on a surface with boundary is a hyperbolic metric on its interior, exhibiting the boundary behavior of the standard metric on the Poincaré disk. Consider the infinite-dimensional Teichmüller spaces of hyperbolic 0-metrics on oriented surfaces with boundary, up to diffeomorphisms fixing the boundary and homotopic to the identity. We show that these spaces have natural symplectic structures, depending only on the choice of an invariant metric on sl(2,R). We prove that these Teichmüller spaces are Hamiltonian Virasoro spaces for the action of the universal cover of the group of diffeomorphisms of the boundary. We give an explicit formula for the Hill potential on the boundary defining the moment map. Furthermore, using Fenchel-Nielsen parameters we prove a Wolpert formula for the symplectic form, leading to global Darboux coordinates on the Teichmüller space.

math.DG↗

arXiv2023

The Goldman-Turaev Lie bialgebra and the Kashiwara-Vergne problem in higher genera

For a compact oriented surface $Σ$ of genus $g$ with $n+1$ boundary components, the space $\mathfrak{g}(Σ)$ spanned by free homotopy classes of loops in $Σ$ carries the structure of a Lie bialgebra equipped with a natural decreasing filtration, whose structure morphisms are called the Goldman bracket and the (framed) Turaev cobracket. We address the following Goldman-Turaev (GT) formality problem: construct a Lie bialgebra homomorphism $θ$ from $\mathfrak{g}(Σ)$ to its associated graded ${\rm gr}\, \mathfrak{g}(Σ)$ such that ${\rm gr} \, θ= {\rm id}$. In order to solve it, we define a family of higher genus Kashiwara-Vergne (KV) problems for an element $F\in {\rm Aut}(L)$, where $L$ is a free Lie algebra. In the case of $g=0$ and $n=2$, it is the classical KV problem from Lie theory. For $g>0$, these KV problems are new. We show that an element $F$ induces a GT formality map if and only if it is a solution of the KV problem. A crucial step in solving the higher genus KV problem is to construct solutions for the case of $g=1$ and $n=1$ in terms of certain elliptic associators following Enriquez. By solving the KV problem, we establish the GT formality for every $g$ and $n$, with the exception of some framings for $g=1$ in which case the GT formality actually does not hold. Furthermore, we introduce pro-unipotent groups ${\rm KV}$ and ${\rm KRV}$ which act on the space of solutions of the KV problem freely and transitively. There are injective maps ${\rm GT}_1\to {\rm KV}, {\rm GRT}_1\to {\rm KRV}$ from Grothendieck-Teichmüller groups. As an application, we show that the Johnson obstruction given by the Turaev cobracket coincides with the one given by the Enomoto-Satoh trace. As part of our study, we prove a uniqueness theorem for non-commutative divergence cocycles on the group algebra of a free group which is of independent value.

math.GT↗

arXiv2023

Benchmarking Multilabel Topic Classification in the Kyrgyz Language

Kyrgyz is a very underrepresented language in terms of modern natural language processing resources. In this work, we present a new public benchmark for topic classification in Kyrgyz, introducing a dataset based on collected and annotated data from the news site 24.KG and presenting several baseline models for news classification in the multilabel setting. We train and evaluate both classical statistical and neural models, reporting the scores, discussing the results, and proposing directions for future work.

cs.CL↗