Article
Mathematics
Diego Corro, Jesus Nunez-Zimbron, Masoumeh Zarei
Summary: We present an equivariant classification for orientable, closed, four-dimensional Alexandrov spaces admitting an isometric torus action, extending the equivariant classification of closed four-dimensional manifolds with torus actions by Orlik and Raymond. In addition, we demonstrate that such Alexandrov spaces are equivariantly homeomorphic to 4-dimensional Riemannian orbifolds with isometric T-2-actions, and obtain a partial homeomorphism classification.
JOURNAL OF GEOMETRIC ANALYSIS
(2022)
Article
Mathematics
Markus Reineke, Thorsten Weist
Summary: By studying Gromov-Witten invariants of rational curves, we can identify and count the moduli space of point configurations using Euler characteristics. S. Fomin and G. Mikhalkin established a recurrence relation via tropicalization, which is applied in the moduli space using Donaldson-Thomas invariants.
INTERNATIONAL MATHEMATICS RESEARCH NOTICES
(2021)
Article
Mathematics, Applied
Rodrigo A. Von Flach, Marcos Jardim, Valeriano Lanza
Summary: The study established an isomorphism between the moduli space of framed flags of sheaves on the projective plane and stable representations of a certain quiver. The weaker claim regarding the existence of unobstructed points in the quiver moduli space replaces one of the previous claims. Additionally, the research extends some results regarding the maximal stability chamber and the perfect obstruction theory for the quiver moduli space from the cited paper.
JOURNAL OF GEOMETRY AND PHYSICS
(2021)
Article
Mathematics, Applied
Marco Armenta, Thomas Bruestle, Souheila Hassoun, Markus Reineke
Summary: Motivated by problems in the neural networks setting, this study focuses on the moduli spaces of double framed quiver representations and provides both a linear algebra description and a representation theoretic description of these moduli spaces. By defining a network category, it is proven that the output of a neural network depends only on the corresponding point in the moduli space. Finally, a different perspective on mapping neural networks with a specific activation function to a moduli space is presented using the symplectic reduction approach to quiver moduli.
LINEAR ALGEBRA AND ITS APPLICATIONS
(2022)
Article
Mathematics
Giosue Muratore
Summary: This article proves that some Gromov-Witten numbers associated with rational contact (Legendrian) curves in any contact complex projective space with arbitrary incidence conditions are enumerative, and uses the Bott formula on the Kontsevich space to find the exact value of these numbers. The numbers of rational contact curves of low degree in P3 and P5 are computed as examples, and the results are consistent with existing results.
MICHIGAN MATHEMATICAL JOURNAL
(2023)
Article
Mathematics
Alexander H. W. Schmitt
Summary: This article reviews a constructive proof of Gabriel's theorem on the representation finiteness of Dynkin quivers, specifically focusing on the case of an equioriented quiver of type D. It then delves into the analysis of polystable representations of such a quiver, provides a direct proof of the characterisation of its semistable representations, and explains how a theorem of Abeasis and Koike can be derived from this characterisation. Finally, it demonstrates how these results can be applied to the study of moduli spaces of quiver bundles.
LINEAR & MULTILINEAR ALGEBRA
(2022)
Article
Mathematics
Kyoung-Seog Lee, Kyeong-Dong Park
Summary: The study focuses on the moduli spaces of Ulrich bundles on the Fano 3-fold V-5 of Picard number 1, degree 5, and index 2. It is proven that the moduli space of stable Ulrich bundles on V-5 can be associated with a smooth open subset of the moduli space of stable quiver representations with a dimension vector (r, r) of the Kronecker quiver.
JOURNAL OF ALGEBRA
(2021)
Article
Mathematics
Qirui Li
Summary: This article provides an explicit formula for the arithmetic intersection number of complex multiplication (CM) cycles on Lubin-Tate spaces, and proves the formula by formulating the intersection number at the infinite level. The formula works for all cases, whether the extensions are the same or different, and whether they are ramified or unramified over F. Additionally, the article demonstrates the linear arithmetic fundamental lemma for GL(2)(F).
DUKE MATHEMATICAL JOURNAL
(2022)
Article
Mathematics
V. V. Cherepanov
Summary: This paper investigates the effective actions of the compact torus Tn-1 on smooth compact manifolds M-2n of even dimension with isolated fixed points. It is shown that under certain conditions, the orbit space of such an action forms a manifold with corners. In the case of Hamiltonian actions, the orbit space is homotopy equivalent to a certain sphere, with applications to regular Hessenberg varieties and isospectral manifolds of Hermitian matrices.
SBORNIK MATHEMATICS
(2021)
Article
Mathematics
Jaroslaw Buczynski, Jaroslaw A. Wisniewski, Andrzej Weber
Summary: We prove the correctness of the LeBrun-Salamon Conjecture in low dimensions, and specifically discuss the properties of contact Fano manifolds and positive quaternion-Kahler manifolds.
JOURNAL OF DIFFERENTIAL GEOMETRY
(2022)
Article
Mathematics
Tamas Hausel, Michael Lennox Wong, Dimitri Wyss
Summary: In this paper, the motivic class of the open de Rham space on certain moduli spaces is determined, using motivic Fourier transform. The result agrees with the purity conjecture and also identifies the open de Rham spaces with quiver varieties. Additionally, natural complete hyperkähler metrics are constructed on them, expected to be of type ALF in four-dimensional cases.
PROCEEDINGS OF THE LONDON MATHEMATICAL SOCIETY
(2023)
Article
Physics, Mathematical
Johannes Christensen, Stefaan Vaes
Summary: This paper investigates the topologically free action of a countable group on a compact metric space and its effect on the 1-cocycles and diagonal 1-parameter groups of automorphisms in the reduced crossed product C*-algebra. The study shows that the KMS spectrum, determined by the inverse temperatures, heavily depends on the nature of the acting group G.
COMMUNICATIONS IN MATHEMATICAL PHYSICS
(2022)
Article
Mathematics
F. Galaz-Garcia, M. Kerin, M. Radeschi
Summary: An upper bound has been obtained for the rank of a Torus acting on a smooth, closed (simply connected) rationally elliptic manifold. In the case of maximal rank, the manifolds admitting such actions are classified up to equivariant rational homotopy equivalence.
MATHEMATISCHE ZEITSCHRIFT
(2021)
Article
Mathematics
Zahra Shabani, Ali Barzanouni, Xinxing Wu
Summary: This article introduces new types of recurrent sets on noncompact metric spaces for finitely generated semigroups that are conjugacy invariant, and explores the basic properties of chain recurrent sets for semigroups. Additionally, the notion of weak shadowing property for finitely generated group actions on compact metric spaces is defined, and the equivalence of shadowing and weak shadowing properties for group actions on a generalized homogeneous space without isolated points is proved.
HACETTEPE JOURNAL OF MATHEMATICS AND STATISTICS
(2021)
Article
Mathematics, Applied
Eduardo Fierro Morales, Richard Urzua-Luz
Summary: We show that for Z(p)-actions by homeomorphisms on the three dimensional torus, the Lefschetz fixed point theorem is optimal. Specifically, we demonstrate the existence of a free Zr-action whose induced Z(p)-action on the first homology group is the same as the given action A, establishing the normal form for this type of actions.
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY
(2021)
