检索结果-内蒙古大学图书馆

FLOER THEORY FOR THE VARIATION OPERATOR OF AN ISOLATED SINGULARITY

学校读者我要写书评

暂无评论

arXiv 2023年

作者： Bae, Hanwool Cho, Cheol-Hyun Choa, Dongwook Jeong, Wonbo 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 Seoul Korea Republic of Korea Institute for Advanced Studies Dongdaemoon-gu Seoul02455 Korea Republic of Department of Mathematical Sciences Research Institute in Mathematics Seoul National University Gwanakgu Seoul Korea Republic of

The variation operator in singularity theory maps relative homology cycles to compact cycles in the Milnor fiber using the monodromy. We construct its symplectic analogue for an isolated singularity. We define a new Floer cohomology, called monodromy Lagrangian Floer cohomology, which provides categorifications of the standard theorems on the variation operator and the Seifert form. The key ingredients are a special class Γ in the symplectic cohomology of the inverse of the monodromy and its closed-open images. For isolated plane curve singularities whose A'Campo divide has depth zero, we find an exceptional collection consisting of non-compact Lagrangians in the Milnor fiber corresponding to a distinguished collection of vanishing cycles under the variation *** Codes 53D37, 53D40, 32S50, 32S55 Copyright © 2023, The Authors. All rights reserved.

关键词： Topology

Position: Solve Layerwise Linear Models First to Understand Neural Dynamical Phenomena (Neural Collapse, Emergence, Lazy/Rich Regime, and Grokking)

学校读者我要写书评

暂无评论

arXiv 2025年

作者： Nam, Yoonsoo Lee, Seok Hyeong Dominé, Clémentine Park, Yea Chan London, Charles Choi, Wonyl Göring, Niclas Lee, Seungjai Department of Theoretical Physics University of Oxford Oxfordshire United Kingdom Center for Quantum Structures in Modules and Spaces Seoul National University Seoul Korea Republic of Gatsby Computational Neuroscience Unit University College London London United Kingdom Center for AI and Natural Science Korea Institute For Advanced Study Seoul Korea Republic of Department of Computer Science University of Oxford Oxfordshire United Kingdom Department of Computer Science Boston University MA United States Department of Mathematics Incheon National University Incheon Korea Republic of

In physics, complex systems are often simplified into minimal, solvable models that retain only the core principles. In machine learning, layerwise linear models (e.g., linear neural networks) act as simplified representations of neural network dynamics. These models follow the dynamical feedback principle, which describes how layers mutually govern and amplify each other’s evolution. This principle extends beyond the simplified models, successfully explaining a wide range of dynamical phenomena in deep neural networks, including neural collapse, emergence, lazy and rich regimes, and grokking. In this position paper, we call for the use of layerwise linear models retaining the core principles of neural dynamical phenomena to accelerate the science of deep learning. © 2025, CC BY.

关键词： Neural network models

Applications of the theory of floer to symmetric spaces

学校读者我要写书评

暂无评论

arXiv 2021年

作者： Bae, Hanwool Chow, Chi Hong Leung, Naichung Conan Center for Quantum Structures in Modules and Spaces Seoul National University Seoul Korea Republic of The Institute of Mathematical Sciences Department of Mathematics The Chinese University of Hong Kong Shatin Hong Kong

We quantize the problem considered by Bott-Samelson who applied Morse theory to any compact symmetric space G/K and the associated real flag manifold G/B which is a real locus of a complex partial flag variety Gℂ/Pσ. We prove that the Pontryagin ring H-∗(Ω(G/K)) of the based loop space Ω(G/K) is isomorphic to the Floer cohomology ring HF∗(G/B,G/B) after localization. When G/K is a Lie group, this is a conjecture of Peterson, proved combinatorially by Lam-Shimozono, in the context of quantum cohomologies of complex flag varieties. Our approach is geometric in nature: we construct a Lagrangian correspondence from T ∗(G/K) to Gℂ/Pσ which geometrically composes with a cotangent fiber to G/B, and compute the linear part of the associated Ma'u-Wehrheim-Woodward's A∞homomorphism from a Floer model of Ω(G/K) to CF∗(G/B,G/B). The crux is to make use of the geometry of G/K to construct specific perturbation data which enables us to reduce the computations to the case when G/K is a *** Codes 57R58, 57T15 © 2021, CC BY.

关键词： Lie groups

ON CATEGORICAL ENTROPY FROM THE VIEWPOINT OF SYMPLECTIC TOPOLOGY

学校读者我要写书评

暂无评论

arXiv 2022年

作者： Bae, Hanwool Choa, Dongwook Jeong, Wonbo Karabas, Dogancan Lee, Sangjin Center for Quantum Structures in Modules And Spaces Seoul National University Seoul Korea Republic of Department of Mathematics Korea Institute for Advanced Study Seoul02455 Korea Republic of Department of Mathematical Sciences Research Institute in Mathematics Seoul National University Gwanak-Gu Seoul Korea Republic of The University of Tokyo Institutes for Advanced Study The university of Tokyo Chiba Kashiwa277-8583 Japan Pohang37673 Korea Republic of

In this paper, motivated by symplectic topology, we explore categorical entropy and present two main results. The first result establishes a relation between categorical entropies of functors on a category and its localisation. Additionally, it demonstrates analogies between the notions of topological and categorical entropy. This result is then applied to symplectic topology, where we provide a method for calculating the categorical entropy of a functor on a (partially) wrapped Fukaya category, assuming that the functor is induced by a compactly supported symplectic automorphism. For the second main result of the paper, we observe the existence of natural examples of symplectic manifolds whose Fukaya categories satisfy a type of Floer-theoretic duality. Motivated by this observation, we prove that categorical entropy can be computed from the morphism spaces under the assumption of duality. The formula is similar to a result of [DHKK14], which is proven for the case of smooth and proper *** Codes 53D37, 18G80, 37B40 Copyright © 2022, The Authors. All rights reserved.

关键词： Entropy

THE PROJECTIVE COVER OF TABLEAU-CYCLIC INDECOMPOSABLE Hn(0)-modules

学校读者我要写书评

暂无评论

arXiv 2020年

作者： Choi, Seung-Il Kim, Young-Hun Nam, Sun-Young 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 Research Institute for Basic Science Sogang University Seoul04107 Korea Republic of

Let α be a composition of n and σ a permutation in S(α). This paper concerns the projective covers of Hn(0)-modules Vα, Xα, and Sσ/αwhose images under the quasisymmetric characteristic are the dual immaculate quasisymmetric function, the extended Schur function, and the quasisymmetric Schur function when σ is the identity, respectively. First, we show that the projective cover of Vα is the projective indecomposable module Pα due to Norton, and Xα and the φ-twist of the canonical submodule Sσ/β,C of Sσ/β for (β, σ)'s satisfying suitable conditions appear as homomorphic images of Vα. Second, we introduce a combinatorial model for the φ-twist of Sσ/α and derive a series of surjections starting from Pα to the φ-twist of Sidα,C. Finally, we construct the projective cover of every indecomposable direct summand Sσ/α,E of Sσ/α. As a byproduct, we give a characterization of triples (σ, α, E) such that the projective cover of Sσ/α,E is *** Codes 20C08, 05E05, 05E10 Copyright © 2020, The Authors. All rights reserved.

关键词：

A COMPUTATION of the RING STRUCTURE in WRAPPED FLOER HOMOLOGY

学校读者我要写书评

暂无评论

arXiv 2019年

作者： Bae, Hanwool Kwon, Myeonggi Center for Quantum Structures in Modules and Spaces Seoul National University Seoul Korea Republic of Department of Mathematics Education Sunchon National University Sunchon57922 Korea Republic of

We give an explicit computation of the ring structure in wrapped Floer homology of a class of real Lagrangians inAk-type Milnor fibers. In theAk-type plumbing description, those Lagrangians correspond to the cotangent fibers or the diagonal Lagrangians. The main ingredient of the computation is to apply a version of the Seidel representation. For a technical reason, we first carry out computations in v-shaped wrapped Floer homology, and this in turn gives the desired ring structure via the Viterbo transfer map. Copyright © 2019, The Authors. All rights reserved.

关键词：