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
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, 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
Alexander Braverman, Michael Finkelberg, Hiraku Nakajima
Summary: This article deduces the Kazhdan-Lusztig conjecture on the multiplicities of simple modules over a simple complex Lie algebra in Verma modules in category O using the equivariant geometric Satake correspondence and the analysis of torus fixed points in zastava spaces. Similar speculations are made for affine Lie algebras and W-algebras.
JOURNAL FUR DIE REINE UND ANGEWANDTE MATHEMATIK
(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
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
Mathematics
Emily Clader, Felix Janda, Yongbin Ruan, Yang Zhou
Summary: This paper introduces a technique for proving wall-crossing formulas in the gauged linear sigma model, without assuming factorization properties of the virtual class. Applying this technique to the gauged linear sigma model associated to a complete intersection in weighted projective space, a uniform proof of the wall-crossing formula in both the geometric and the Landau-Ginzburg phase is obtained.
DUKE MATHEMATICAL JOURNAL
(2021)
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, Applied
Ahmed J. Zerouali
Summary: The article introduces a method of considering a certain class of moduli spaces on bordered surfaces from a quasi-Hamiltonian perspective, by parameterizing flat G-connections to describe twisted local systems, constructing a Duistermaat-Heckman measure that is invariant under twisted conjugation, and characterizing it using a localization formula for its Fourier coefficients.
JOURNAL OF GEOMETRY AND PHYSICS
(2021)
Article
Mathematics
M. Bertola, D. Korotkin
Summary: By using the embedding of the moduli space of generalized GL(n) Hitchin's spectral covers into the moduli space of meromorphic Abelian differentials, variational formulae for the period matrix, canonical bidifferential, prime form, and Bergman tau function were studied, leading to generalize the residue formulae from Donagi-Markman's formula. The computation of second derivatives of the period matrix reproduces the formula derived in [2] using the framework of topological recursion.
INTERNATIONAL MATHEMATICS RESEARCH NOTICES
(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
Andrea Bianchi
Summary: This paper investigates the moduli space of Riemann surfaces with ordered and directed marked points. It shows a homotopy equivalence between the moduli space and a component of the simplicial Hurwitz space associated with a partially multiplicative quandle. Furthermore, it provides a new proof of the Mumford conjecture on the stable rational cohomology of moduli spaces of Riemann surfaces and presents a combinatorial model for the infinite loop space of Hurwitz flavor.
ADVANCES IN MATHEMATICS
(2023)
Article
Mathematics
Hans Franzen, Markus Reineke, Silvia Sabatini
Summary: We present a large class of Fano varieties by studying line bundles on quiver moduli spaces and their global sections, and provide several examples including moduli spaces of point configurations, Kronecker moduli, and toric quiver moduli.
CANADIAN MATHEMATICAL BULLETIN-BULLETIN CANADIEN DE MATHEMATIQUES
(2021)
Article
Mathematics
Andrei Negut
Summary: This article proves a conjecture about the equivariant K-theory of affine Laumon spaces and establishes a connection between the quantum affine algebra and the quantum toroidal algebra by reinterpreting the action of the latter on the K-theory in terms of the shuffle algebra.
ANNALES SCIENTIFIQUES DE L ECOLE NORMALE SUPERIEURE
(2022)
Article
Mathematics
Yao Yuan
Summary: The study focuses on Le Potier's strange duality conjecture on P2, specifically looking at the strange duality map SDc,d. By utilizing tools in quiver representation theory, it is shown that SDc,d is an isomorphism when r = n or r = n - 1 or d <= 3, and in general SDc,d is injective for any n >= r > 0 and d > 0.
ADVANCES IN MATHEMATICS
(2021)
