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ZETA FUNCTIONS ENUMERATING SUBFORMS OF QUADRATIC FORMS

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arXiv 2024年

In this paper, we introduce and study the Dirichlet series enumerating (proper) equivalence classes of full rank subforms/sublattices of a given quadratic form/lattice, focusing on the positive definite binary case. We obtain formulas linking this Dirichlet series with Dirichlet series counting ideal classes of the imaginary quadratic field associated with the quadratic form. Utilizing the result, we provide explicit formulas of the Dirichlet series for several lattices, including square lattice and hexagonal lattice. Moreover, we investigate some analytic properties of this Dirichlet *** Codes 11E12, 11E16, 11H06, 11M41 Copyright © 2024, The Authors. All rights reserved.

关键词： Equivalence classes

Visualising Feature Learning in Deep Neural Networks by Diagonalizing the Forward Feature Map

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arXiv 2024年

作者： Nam, Yoonsoo Mingard, Chris Lee, Seok Hyeong Hayou, Soufiane Louis, Ard Department of Theoretical Physics University of Oxford United Kingdom Department of Theoretical Physics Department of Chemistry University of Oxford United Kingdom Center for Quantum Structures in Modules and Spaces Seoul National University Korea Republic of Simons Institute UC Berkeley United States

Deep neural networks (DNNs) exhibit a remarkable ability to automatically learn data representations, finding appropriate features without human input. Here we present a method for analysing feature learning by decomposing DNNs into 1) a forward feature-map Φ: χ → p that maps the input dataspace χ to the p post-activations of the penultimate layer, and 2) a final linear layer that classifies the data. We diagonalize Φ with respect to the gradient descent operator and track feature learning by measuring how the eigenfunctions and eigenvalues of Φ change during training. Across many popular architectures and classification datasets, we find that DNNs converge, after just a few epochs, to a minimal feature (MF) regime dominated by a number of eigenfunctions equal to the number of classes. This behaviour resembles the neural collapse phenomenon studied at longer training times. For other DNN-data combinations, such as a fully connected network on CIFAR10, we find an extended feature (EF) regime where significantly more features are used. Optimal generalisation performance upon hyperparameter tuning typically coincides with the MF regime, but we also find examples of poor performance within the MF regime. Finally, we recast the phenomenon of neural collapse into a kernel picture which can be extended to broader tasks such as regression. Copyright © 2024, The Authors. All rights reserved.

关键词： Deep neural networks

Twisted equivalences in spectral algebraic geometry

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arXiv 2021年

作者： Chough, Chang-Yeon Center for Quantum Structures in Modules and Spaces Seoul National University Seoul08826 Korea Republic of

We study twisted derived equivalences for schemes in the setting of spectral algebraic geometry. To this end, we introduce the notion of a twisted equivalence and show that a twisted equivalence for perfect spectral algebraic stacks admitting a quasi-finite presentation supplies an equivalence between the stacks, which compensate for the failure of twisted derived equivalences for non-affine schemes to provide an isomorphism of the schemes. In the case of (not necessarily connective) commutative ring spectra, we also prove a spectral analogue of Rickard's theorem, which shows that a derived equivalence of associative rings induces an isomorphism between their *** Codes 18G80, 14A20, 16U70, 16E40, 14F08 Copyright © 2021, The Authors. All rights reserved.

关键词： Algebra

UNIVERSALITY LIFTING FROM A GENERAL BASE FIELD

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arXiv 2024年

作者： Kala, Vítězslav Kim, Daejun Lee, Seok Hyeong Charles University Faculty of Mathematics and Physics Department of Algebra Sokolovská 83 Praha 818600 Czech Republic Department of Mathematics Education Korea University Seoul02841 Korea Republic of Center for Quantum Structures in Modules and Spaces Seoul National University Seoul08826 Korea Republic of

Given a totally real number field F, we show that there are only finitely many totally real extensions of K of a fixed degree that admit a universal quadratic form defined over F. We further obtain several explicit classification results in the case of relative quadratic *** Codes 11E12, 11E20, 11R04, 11R80, 11H06 Copyright © 2024, The Authors. All rights reserved.

关键词： Number theory

REGULAR SCHUR LABELED SKEW SHAPE POSETS AND THEIR 0-HECKE modules

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arXiv 2023年

作者： Kim, Young-Hun Lee, So-Yeon Oh, Young-Tak Center for Quantum structures in Modules and Spaces Seoul National University Seoul08826 Korea Republic of Department of Mathematics Sogang University Seoul04107 Korea Republic of Department of Mathematics Institute for Mathematical and Data Sciences Sogang University Seoul04107 Korea Republic of

Assuming Stanley’s P -partitions conjecture holds, the regular Schur labeled skew shape posets are precisely the finite posets P with underlying set {1, 2, . . ., |P |} such that the P -partition generating function is symmetric and the set of linear extensions of P, denoted ΣL(P), is a left weak Bruhat interval in the symmetric group S|P |. We describe the permutations in ΣL(P) in terms of reading words of standard Young tableaux when P is a regular Schur labeled skew shape poset, and classify ΣL(P)’s up to descent-preserving isomorphism as P ranges over regular Schur labeled skew shape posets. The results obtained are then applied to classify the 0-Hecke modules MP associated with regular Schur labeled skew shape posets P up to isomorphism. Then we characterize regular Schur labeled skew shape posets as the finite posets P whose linear extensions form a dual plactic-closed subset of S|P |. Using this characterization, we construct distinguished filtrations of MP with respect to the Schur basis when P is a regular Schur labeled skew shape poset. Further issues concerned with the classification and decomposition of the 0-Hecke modules MP are also *** Codes 20C08, 06A07, 05E10, 05E05 Copyright © 2023, The Authors. All rights reserved.

关键词： Algebra

RATIONAL HOMOLOGY DISK SMOOTHINGS AND LEFSCHETZ FIBRATIONS

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arXiv 2022年

作者： Choi, Hakho School of Mathematics Korea Institute for Advanced Study 85 Hoegiro Dongdaemungu Seoul02455 Korea Republic of Center for Quantum Structures in Modules and Spaces Seoul National University Seoul08226 Korea Republic of

In this article, we generalize the results discussed in [EMV11] by introducing a genus to generic fibers of Lefschetz fibrations. That is, we give families of relations in the mapping class groups of genus-1 surfaces with boundaries that represent rational homology disk smoothings of weighted homogeneous surface singularities whose resolution graphs are 3-legged with a bad central vertex. © 2022, CC BY.

关键词：

WEAK BRUHAT INTERVAL modules FOR GENOMIC SCHUR FUNCTIONS

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arXiv 2022年

作者： Kim, Young-Hun Yoo, Semin Center for quantum structures in modules and spaces Seoul National University Seoul08826 Korea Republic of Discrete Mathematics Group Institute for Basic Science 55 Expo-ro Yuseong-gu Daejeon34126 Korea Republic of

Let λ be a partition of a positive integer n. The genomic Schur function Uλ was introduced by Pechenik–Yong in the context of the K-theory of Grassmannians. Recently, Pechenik provided a positive combinatorial formula for the fundamental quasisymmetric expansion of Uλ in terms of increasing gapless tableaux. In this paper, for each 1 ≤ m ≤ n, we construct an Hm(0)-module Gλ;m whose image under the quasisymmetric characteristic is the mth degree homogeneous component of Uλ by defining an Hm(0)-action on increasing gapless tableaux. We provide a method to assign a permutation to each increasing gapless tableau, and use this assignment to decompose Gλ;m into a direct sum of weak Bruhat interval modules. Furthermore, we determine the projective cover of each summand of the direct sum *** Codes 20C08, 05E10, 05E05, 14M15 Copyright © 2022, The Authors. All rights reserved.

关键词： Combinatorial mathematics

CLUSTER CATEGORIES FROM FUKAYA CATEGORIES

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arXiv 2022年

作者： Bae, Hanwool Jeong, Wonbo Kim, Jongmyeong Center for Quantum Structures in Modules and Spaces Seoul National University Seoul08826 Korea Republic of Department of Mathematical Sciences Research Institute in Mathematics Seoul National University Seoul08826 Korea Republic of Pohang37673 Korea Republic of

We show that the derived wrapped Fukaya category (Equation presented). the derived compact Fukaya category (Equation presented). and the cocore disks LQ of the plumbing space XQd+1 form a Calabi-Yau triple. As a consequence, the quotient category (Equation presented). becomes the cluster category associated to Q. One of its properties is a Calabi-Yau structure. Also it is known that this quotient category is quasi-equivalent to the Rabinowitz Fukaya category due to the work of Ganatra-Gao-Venkatesh. We compute the morphism space of LQ in (Equation presented). using the Calabi-Yau structure, which is isomorphic to the Rabinowitz Floer cohomology of LQMSC Codes 53D37, 16E45, 18G80 Copyright © 2022, The Authors. All rights reserved.

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A TOPOLOGICAL CHARACTERIZATION OF SYMPLECTIC FILLINGS OF SEIFERT 3-MANIFOLDS

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arXiv 2022年

作者： Choi, Hakho Park, Jongil Jeon, JaeKwan Center for Quantum Structures in Modules and Spaces Seoul National University Seoul08826 Korea Republic of Department of Mathematical Sciences Seoul National University Seoul08826 Korea Republic of Department of Mathematics Chungnam National University Daejeon34134 Korea Republic of

In this paper, we investigate a surgical interpretation for minimal symplectic fillings of a given Seifert 3-manifold equipped with a canonical contact structure. Consequently, we determine a necessary and sufficient condition for a minimal symplectic filling of a Seifert 3-manifold satisfying certain conditions to be obtained by a sequence of rational blowdown surgery from the minimal resolution of the corresponding weighted homogeneous surface singularity. Furthermore, as an application, we prove that every minimal symplectic filling of a large family of Seifert 3-manifolds with a canonical contact structure is in fact realized as a Milnor fiber of the corresponding weighted homogeneous surface singularity in the Appendix. Copyright © 2022, The Authors. All rights reserved.

关键词： Filling