Fano quiver moduli

We show that the Poincaré bundle gives a fully faithful embedding from the derived category of a curve of sufficiently high genus into the derived category of its moduli space of bundles of rank r with fixed determinant of degree 1. Moreover we show that a twist of the embedding, together with 2 exceptional line bundles, gives the start of a semi-orthogonal decomposition. This generalises results of Narasimhan and Fonarev-Kuznetsov, who embedded the derived category of a single copy of the curve, for rank 2.

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“…as the parameter space for representations of 𝑄 of dimension vector d. It comes with an action of the group (16)…”

Section: Quiver Modulimentioning

confidence: 99%

“…The data 𝑄 ′ , d ′ , and 𝜃 ′ satisfy Assumption 2.4 by Proposition 3.9. We may therefore apply [16,Lemma 3.3], and obtain the identification…”

Section: The Line Bundlementioning

confidence: 99%

Admissible subcategories in derived categories of moduli of vector bundles on curves

Belmans

Mukhopadhyay

2019

Advances in Mathematics

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“…We show that this data manifold can be mapped to the moduli space while carrying the feature maps induced by the data, and then it is related to notions appearing in manifold learning [11,23]. Our results, therefore, create a new bridge between the mathematical study of these moduli spaces [25][26][27] and the study of the training dynamics of neural networks inside these moduli spaces.…”

Section: Previous Workmentioning

confidence: 67%

“…We will provide an explicit map from the input space to the moduli space of a neural network with which the data manifold can be translated to the moduli space. This will allow the use of mathematical theory for quiver moduli spaces [25][26][27] to manifold learning, representation learning and the dynamics of neural network learning [11,23].…”

Section: The Moduli Space Of a Neural Networkmentioning

confidence: 99%

“…The subset ϕ(W, f )(M) generates a sub-manifold of the moduli space (as it is well known in topology [46]) that parametrizes all possible outputs that the neural network (W, f ) can produce from inputs on the data manifold M. This means that the geometry of the data manifold M has been translated into the moduli space d M k ( Q), and this implies that the mathematical knowledge [25][26][27] that we have of the geometry of the moduli spaces d M k ( Q) can be used to understand the dynamics of neural network training, due to the universality of the description of neural networks we have provided. Following Consequence 1, one would like to look for correlations between the dimension of the moduli space and properties of neural networks.…”

Section: Consequencementioning

confidence: 99%

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The Representation Theory of Neural Networks

Armenta

Jodoin

2021

Mathematics

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In this work, we show that neural networks can be represented via the mathematical theory of quiver representations. More specifically, we prove that a neural network is a quiver representation with activation functions, a mathematical object that we represent using a network quiver. Furthermore, we show that network quivers gently adapt to common neural network concepts such as fully connected layers, convolution operations, residual connections, batch normalization, pooling operations and even randomly wired neural networks. We show that this mathematical representation is by no means an approximation of what neural networks are as it exactly matches reality. This interpretation is algebraic and can be studied with algebraic methods. We also provide a quiver representation model to understand how a neural network creates representations from the data. We show that a neural network saves the data as quiver representations, and maps it to a geometrical space called the moduli space, which is given in terms of the underlying oriented graph of the network, i.e., its quiver. This results as a consequence of our defined objects and of understanding how the neural network computes a prediction in a combinatorial and algebraic way. Overall, representing neural networks through the quiver representation theory leads to 9 consequences and 4 inquiries for future research that we believe are of great interest to better understand what neural networks are and how they work.

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Reductive quotients of klt singularities

Braun,

Greb,

Langlois

et al. 2024

Invent. math.

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We prove that the quotient of a klt type singularity by a reductive group is of klt type in characteristic 0. In particular, given a klt variety $X$ X endowed with the action of a reductive group $G$ G and admitting a quasi-projective good quotient $X\rightarrow X/\!/G$ X → X / / G , we can find a boundary $B$ B on $X/\!/G$ X / / G so that the pair $(X/\!/G,B)$ ( X / / G , B ) is klt. This applies for example to GIT-quotients of klt varieties. Our main result has consequences for complex spaces obtained as quotients of Hamiltonian Kähler $G$ G -manifolds, for collapsings of homogeneous vector bundles as introduced by Kempf, and for good moduli spaces of smooth Artin stacks. In particular, it implies that the good moduli space parametrizing $n$ n -dimensional K-polystable smooth Fano varieties of volume $v$ v has klt type singularities. As a corresponding result regarding global geometry, we show that quotients of Mori Dream Spaces with klt Cox rings are Mori Dream Spaces with klt Cox ring. This in turn applies to show that projective GIT-quotients of varieties of Fano type are of Fano type; in particular, projective moduli spaces of semistable quiver representations are of Fano type.

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Fano quiver moduli

Abstract: We exhibit a large class of quiver moduli spaces, which are Fano varieties, by studying line bundles on quiver moduli and their global sections in general, and work out several classes of examples, comprising moduli spaces of point configurations, Kronecker moduli, and toric quiver moduli.

Cited by 6 publications

References 20 publications

Admissible subcategories in derived categories of moduli of vector bundles on curves

Admissible subcategories in derived categories of moduli of vector bundles on curves

The Representation Theory of Neural Networks

Reductive quotients of klt singularities

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