arXiv subjects

Yuya Mizuno

Publications and source records attributed to Yuya Mizuno.

At least 19 recordsLinked to original sources

arXiv2025

Fans and polytopes in tilting theory III: Classification of convex $g$-fans of rank 3

The $g$-fan $\Sigma(A)$ of a finite dimensional algebra $A$ is a non-singular fan in its real Grothendieck group, defined by tilting theory. If the union ${\rm P}(A)$ of the simplices associated with the cones of $\Sigma(A)$ is convex, we call $A$ $g$-convex. In this case, the $g$-polytope ${\rm P}(A)$ of $A$ is a reflexive polytope. Thus, in each dimension, there are only finitely many isomorphism classes of fans that can be realized as $g$-fans of $g$-convex algebras. An important problem is to classify such fans for a fixed dimension $d$. In this paper, we give a complete answer for the case $d=3$: we prove that there are precisely 61 convex $g$-fans of dimension 3 up to isomorphism. Our method is based on the decomposition of fans into the $2^3$ orthants in the real Grothendieck group of $A$, together with a detailed analysis of possible sequences of $g$-vectors arising from iterated mutations.

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arXiv2024

Multi-Subject Image Synthesis as a Generative Prior for Single-Subject PET Image Reconstruction

Large high-quality medical image datasets are difficult to acquire but necessary for many deep learning applications. For positron emission tomography (PET), reconstructed image quality is limited by inherent Poisson noise. We propose a novel method for synthesising diverse and realistic pseudo-PET images with improved signal-to-noise ratio. We also show how our pseudo-PET images may be exploited as a generative prior for single-subject PET image reconstruction. Firstly, we perform deep-learned deformable registration of multi-subject magnetic resonance (MR) images paired to multi-subject PET images. We then use the anatomically-learned deformation fields to transform multiple PET images to the same reference space, before averaging random subsets of the transformed multi-subject data to form a large number of varying pseudo-PET images. We observe that using MR information for registration imbues the resulting pseudo-PET images with improved anatomical detail compared to the originals. We consider applications to PET image reconstruction, by generating pseudo-PET images in the same space as the intended single-subject reconstruction and using them as training data for a diffusion model-based reconstruction method. We show visual improvement and reduced background noise in our 2D reconstructions as compared to OSEM, MAP-EM and an existing state-of-the-art diffusion model-based approach. Our method shows the potential for utilising highly subject-specific prior information within a generative reconstruction framework. Future work may compare the benefits of our approach to explicitly MR-guided reconstruction methodologies.

physics.med-ph↗

arXiv2024

Generative-Model-Based Fully 3D PET Image Reconstruction by Conditional Diffusion Sampling

Score-based generative models (SGMs) have recently shown promising results for image reconstruction on simulated positron emission tomography (PET) datasets. In this work we have developed and implemented practical methodology for 3D image reconstruction with SGMs, and perform (to our knowledge) the first SGM-based reconstruction of real fully 3D PET data. We train an SGM on full-count reference brain images, and extend methodology to allow SGM-based reconstructions at very low counts (1% of original, to simulate low-dose or short-duration scanning). We then perform reconstructions for multiple independent realisations of 1% count data, allowing us to analyse the bias and variance characteristics of the method. We sample from the learned posterior distribution of the generative algorithm to calculate uncertainty images for our reconstructions. We evaluate the method's performance on real full- and low-count PET data and compare with conventional OSEM and MAP-EM baselines, showing that our SGM-based low-count reconstructions match full-dose reconstructions more closely and in a bias-variance trade-off comparison, our SGM-reconstructed images have lower variance than existing baselines. Future work will compare to supervised deep-learned methods, with other avenues for investigation including how data conditioning affects the SGM's posterior distribution and the algorithm's performance with different tracers.

physics.med-ph↗

arXiv2024

Likelihood-Scheduled Score-Based Generative Modeling for Fully 3D PET Image Reconstruction

Medical image reconstruction with pre-trained score-based generative models (SGMs) has advantages over other existing state-of-the-art deep-learned reconstruction methods, including improved resilience to different scanner setups and advanced image distribution modeling. SGM-based reconstruction has recently been applied to simulated positron emission tomography (PET) datasets, showing improved contrast recovery for out-of-distribution lesions relative to the state-of-the-art. However, existing methods for SGM-based reconstruction from PET data suffer from slow reconstruction, burdensome hyperparameter tuning and slice inconsistency effects (in 3D). In this work, we propose a practical methodology for fully 3D reconstruction that accelerates reconstruction and reduces the number of critical hyperparameters by matching the likelihood of an SGM's reverse diffusion process to a current iterate of the maximum-likelihood expectation maximization algorithm. Using the example of low-count reconstruction from simulated [$^{18}$F]DPA-714 datasets, we show our methodology can match or improve on the NRMSE and SSIM of existing state-of-the-art SGM-based PET reconstruction while reducing reconstruction time and the need for hyperparameter tuning. We evaluate our methodology against state-of-the-art supervised and conventional reconstruction algorithms. Finally, we demonstrate a first-ever implementation of SGM-based reconstruction for real 3D PET data, specifically [$^{18}$F]DPA-714 data, where we integrate perpendicular pre-trained SGMs to eliminate slice inconsistency issues.

physics.med-ph↗

arXiv2024

$\tau$-tilting theory and silting theory of skew group algebra extensions

Let $\Lambda$ be a finite dimensional algebra with an action by a finite group $G$ and $A:= \Lambda *G$ the skew group algebra. One of our main results asserts that the canonical restriction-induction adjoint pair of the skew group algebra extension $\Lambda \subset A$ induces a poset isomorphism between the poset of $G$-stable support $\tau$-tilting modules over $\Lambda$ and that of $(\!\!\!\mod G)$-stable support $\tau$-tilting modules over $A$. We also establish a similar poset isomorphism of posets of appropriate classes of silting complexes over $\Lambda$ and $A$. These two results generalize and unify preceding results by Huang-Zhang, Breaz-Marcus-Modoi and the second and the third authors. Moreover, we give a practical condition under which $\tau$-tilting finiteness and silting discreteness of $\Lambda$ are inherited to those of $A$. As applications we study $\tau$-tilting theory and silting theory of the (generalized) preprojective algebras and the folded mesh algebras. Among other things, we determine the posets of support $\tau$-tilting modules and of silting complexes over preprojective algebra $\Pi(\Bbb{L}_{n})$ of type $\Bbb{L}_{n}$.

math.RT↗

arXiv2024

Derived preprojective algebras and spherical twist functors

We study silting objects over derived preprojective algebras of acyclic quivers by giving a direct relationship between silting objects, spherical twist functors and mutations. Especially, for a Dynkin quiver, we establish a bijection between the elements of the braid group and the set of isomorphism classes of basic silting objects over the derived preprojective algebra.

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arXiv2024

Fans and polytopes in tilting theory I: Foundations

For a finite dimensional algebra $A$ over a field $k$, the 2-term silting complexes of $A$ gives a simplicial complex $Δ(A)$ called the $g$-simplicial complex. We give tilting theoretic interpretations of the $h$-vectors and Dehn-Sommerville equations of $Δ(A)$. Using $g$-vectors of 2-term silting complexes, $Δ(A)$ gives a nonsingular fan $Σ(A)$ in the real Grothendieck group $K_0(\mathsf{proj} A)_{\mathbb{R}}$ called the $g$-fan. We give several basic properties of $Σ(A)$ including sign-coherence, sign decomposition, idempotent reductions, Jasso reductions, pairwise positivity and a connection with Newton polytopes of $A$-modules. Moreover, $Σ(A)$ gives a (possibly infinite and non-convex) polytope $P(A)$ in $K_0(\mathsf{proj} A)_{\mathbb{R}}$ called the $g$-polytope of $A$. We call $A$ $g$-convex if $P(A)$ is convex. In this case, we show that it is a reflexive polytope, and that the dual polytope is given by the 2-term simple minded collections of $A$. There are precisely 7 convex $g$-polyogons up to isomorphism. We give a classification of algebras whose $g$-polytopes are smooth Fano. We study $g$-fans and $g$-polytopes of two important classes of algebras. We show that the $g$-fan of a classical or generalized preprojective algebra is given by the Coxeter fan. It is $g$-convex if and only if it is of type $A$ or $B$, and in this case, its $g$-polytope is the dual polytope of the short root polytope. Moreover we classify Brauer graph algebras which are $g$-convex, and describe their $g$-polytopes as the root polytopes of type $A$ or $C$.

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arXiv2024

Dimensions of $\tau$-tilting modules over path algebras and preprojective algebras of Dynkin type

In this paper, we introduce a new generating function called $d$-polynomial for the dimensions of $\tau$-tilting modules over a given finite dimensional algebra. Firstly, we study basic properties of $d$-polynomials and show that it can be realized as a certain sum of the $f$-polynomials of the simplicial complexes arising from $\tau$-rigid pairs. Secondly, we give explicit formulas of $d$-polynomials for preprojective algebras and path algebras of Dynkin quivers by using a close relation with $W$-Eulerian polynomials and $W$-Narayana polynomials. Thirdly, we consider the ordinary and exponential generating functions defined from $d$-polynomials and give closed-form expressions in the case of preprojective algebras and path algebras of Dynkin type $\mathbb{A}$.

math.RT↗

arXiv2023

Fans and polytopes in tilting theory II: $g$-fans of rank 2

The $g$-fan of a finite dimensional algebra is a fan in its real Grothendieck group defined by tilting theory. We give a classification of complete $g$-fans of rank 2. More explicitly, our first main result asserts that every complete sign-coherent fan of rank 2 is a $g$-fan of some finite dimensional algebra. Our proof is based on three fundamental results, Gluing Theorem, Rotation Theorem and Subdivision Theorem, which realize basic operations on fans in the level of finite dimensional algebras. For each of 16 convex sign-coherent fans $Σ$ of rank 2, our second main result gives a characterization of algebras $A$ of rank 2 satisfying $Σ(A)=Σ$. As a by-product of our method, we prove that for each positive integer $N$, there exists a finite dimensional algebra $A$ of rank 2 such that the Hasse quiver of the poset of 2-term silting complexes of $A$ has precisely $N$ connected components.

math.RT↗

arXiv2022

Shard theory for $g$-fans

For a finite dimensional algebra $A$, the notion of $g$-fan $\Sigma(A)$ is defined from two-term silting complexes of $A$ in the real Grothendieck group $K_0(\mathsf{proj} A)_{\mathbb{R}}$. In this paper, we discuss the theory of shards to $\Sigma(A)$, which was originally defined for a hyperplane arrangement. We establish a correspondence between the set of join-irreducible elements of the poset of torsion classes of $\mathrm{mod} A$ and the set of shards of $\Sigma(A)$ for $g$-finite algebra $A$. Moreover, we show that the semistable region of a brick of $\mathrm{mod} A$ is exactly given by a shard. We also give a poset isomorphism of shard intersections and wide subcategories of $\mathrm{mod} A$.

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arXiv2022

Arc diagrams and 2-term simple-minded collections of preprojective algebras of type $A$

We study an explicit description of semibricks and 2-term simple-minded collections over preprojective algebras of type $A$ via arc diagrams. We provide a bijection between the set of noncrossoing arc diagrams (resp. the set of double arc diagrams), which is in bijective correspondence with elements of the symmetric group, and the set of semibricks (resp. the set of 2-term simple-minded collections) over the algebra. Moreover we define a mutation and a partial order on the set of double arc diagrams. In particular, we obtain a poset isomorphism between the symmetric group and the set of 2-term simple-minded collections. As an application of our results, we study semibricks of some quotient algebras of the preprojective algebras of type $A$ and we reprove some important results shown by the other authors.

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arXiv2020

Torsion pairs for quivers and the Weyl groups

We give an interpretation of the map $π^c$ defined by Reading, which is a map from the elements of a Coxeter group to the $c$-sortable elements, in terms of the representation theory of preprojective algebras. Moreover, we study a close relationship between $c$-sortable elements and torsion pairs, and give an explicit description of the cofinite torsion classes in the context of the Coxeter group. As a consequence, we give a proof of some conjectures proposed by Oppermann, Reiten, and the second author.

math.RT↗

arXiv2020

Simplicial complexes and tilting theory for Brauer tree algebras

We study 2-term tilting complexes of Brauer tree algebras in terms of simplicial complexes. We show the symmetry and convexity of the simplicial complexes as lattice polytopes. Via a geometric interpretation of derived equivalences, we show that the $f$-vector of simplicial complexes of Brauer tree algebras only depends the number of the edges of the Brauer trees and hence it is a derived invariant. In particular, this result implies that the number of 2-term tilting complexes, which is in bijection with support $τ$-tilting modules, is a derived invariant. Moreover, we apply our result to the enumeration problem of Coxeter-biCatalan combinatorics.

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arXiv2019

Two-term tilting complexes for preprojective algebras of non-Dynkin type

In this paper, we study two-term tilting complexes for preprojective algebras of non-Dynkin type. We show that there exist two families of two-term tilting complexes, which are respectively parameterized by the elements of the corresponding Coxeter group. Moreover, we provide the complete classification in the case of affine type by showing that any two-term silting complex belongs one of them. For this purpose, we also discuss the Krull-Schmidt property for the homotopy category of finitely generated projective modules over a complete ring.

math.RT↗

arXiv2019

Derived Picard groups of preprojective algebras of Dynkin type

In this paper, we study two-sided tilting complexes of preprojective algebras of Dynkin type. We construct the most fundamental class of two-sided tilting complexes, which has a group structure by derived tensor products and induces a group of auto-equivalences of the derived category. We show that the group structure of the two-sided tilting complexes is isomorphic to the braid group of the corresponding folded graph. Moreover we show that these two-sided tilting complexes induce tilting mutation and any tilting complex is given as the derived tensor products of them. Using these results, we determine the derived Picard group of preprojective algebras for type $A$ and $D$.

math.RT↗

arXiv2018

Discreteness of silting objects and t-structures in triangulated categories

We introduce the notion of ST-pairs of triangulated subcategories, a prototypical example of which is the pair of the bound homotopy category and the bound derived category of a finite-dimensional algebra. For an ST-pair $(\C,\D)$, we construct an order-preserving map from silting objects in $\C$ to bounded $t$-structures on $\D$ and show that the map is bijective if and only if $\C$ is silting-discrete if and only if $\D$ is $t$-discrete. Based on a work of Qiu and Woolf, the above result is applied to show that if $\C$ is silting-discrete then the stability space of $\D$ is contractible. This is used to obtain the contractibility of the stability spaces of some Calabi--Yau triangulated categories associated to Dynkin quivers.

math.RT↗

arXiv2018

Classifying tilting complexes over preprojective algebras of Dynkin type

We study tilting complexes over preprojective algebras of Dynkin type. We classify all tilting complexes by giving a bijection between tilting complexes and the braid group of the corresponding folded graph. In particular, we determine the derived equivalence class of the algebra. For the results, we develop the theory of silting-discrete triangulated categories and give a criterion of silting-discreteness.

math.RT↗

arXiv2015

Higher APR tilting preserves $n$-representation infiniteness

We show that $m$-APR tilting preserves $n$-representation infiniteness for $1\leq m\leq n$. Moreover, we show that these tilting modules provide different tilting modules for the corresponding higher preprojective algebras, which is $(n+1)$-CY algebras. We also study the interplay of the two kinds of tilting modules.

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