Japanese / English

Research Institute for Science & Technology, Tokyo University of Science
Research Project "Geometry and Analyses of Various Natural Phenomena"

Workshop "Discrete geometry"

Date: March 2 (Sat), 2024

HyFlex: Face to Face and Zoom

Place: Tokyo University of Science, Kagurazaka Campus, Building No. 11, Room 11-1

Invited Speakers (confirmed)

Program

Title & Abstract

• Toru Kajigaya (Tokyo University of Science) "Variational problems on finite graphs"
  Based on geometric variational problems, we explore "good" realizations for mappings from finite graphs to a Riemannian manifold, which realize configurations of graph in a space, such as triangulations of surfaces or networks in a higher dimensional space. Realizations obtained as critical points of length or energy functionals can be regarded as an extension of closed geodesics, and various studies using differential geometric approaches have already investigated, e.g. the existence, stability, standard realizations on surfaces, and explicit or numerical constructions.
  In this talk, I will focus on our recent results regarding the stability and the standard realizations, and introduce related research and problems.


• Atsushi Kasue (Kanazawa University) "The Kuramochi compactification of a weighted infinite graph and Mosco convergence "
  We consider a weighted infinite graph and the Kuramochi compactification of the graph associated to the Hilbert space consisting of functions with finite Dirichlet sum.　A closed form on the Hilbert space satisfying the Beuring - Deny conditions which lies between the Dirichlet and the Neumann forms is taken up. We discuss Mosco convergence on the forms induced on finite subgraphs or the Kuramochi boundary endowed with harmonic measure.


• Hisashi Naito (Nagoya University) "Discrete geometric analysis and material sciences "
  Discrete geometric analysis is an analysis using geometric properties of graphs. In the physical sciences, graph theory is also widely used in discussions such as the molecular orbital method and the tight binding approximation of crystal structures. In addition, the theory of the standard realization of topological crystals by Kotani-Sunada, which describes crystal structures based on graph theory, is one successful example. The standard realization of topological crystals by Kotani-Sunada, which gives a description of crystal structures using graph theory and the variational principle, has attracted attention as a new approach to the study of crystal structures.
  
  In this talk, we discuss the geometry of trivalent discrete surfaces as models of sp^2 carbon structures such as fullerenes and carbon nanotubes, construction of negatively curved sp^2 carbon structures using standard realizations of crystal lattices, and their application to material sciences.


• Koji Hashimoto (Kyoto University) "Unification of Symmetries Inside Neural Networks: Transformer, Feedforward and Neural ODE"
  Understanding the inner workings of neural net- works, including transformers, remains one of the most challenging puzzles in machine learning. This study introduces a novel approach by applying the principles of gauge symmetries, a key concept in physics, to neural network architectures. By regarding model functions as physical observables, we find that parametric redundancies of various machine learning models can be inter- preted as gauge symmetries. We mathematically formulate the parametric redundancies in neural ODEs, and find that their gauge symmetries are given by spacetime diffeomorphisms, which play a fundamental role in Einstein’s theory of gravity. Viewing neural ODEs as a continuum version of feedforward neural networks, we show that the parametric redundancies in feedforward neural networks are indeed lifted to diffeomorphisms in neural ODEs. We further extend our analysis to transformer models, finding natural correspondences with neural ODEs and their gauge symmetries. The concept of gauge symmetries sheds light on the complex behavior of deep learning models through physics and provides us with a unifying perspective for analyzing various ma- chine learning architectures. This talk is based on the collaboration with Yuji Hirono and Akiyoshi Sannai, https://arxiv.org/abs/2402.02362


• Masashi Yasumoto (Tokushima University) "Expanding discrete differential geometry of surfaces"
  Discrete differential geometry concerns about establishing a theory of discrete geometric objects that preserves properties of differential geometric objects, and various discrete theories have been developed for different purposes. This time we focus on discrete surface theory based on differential geometry and integrable systems, in particular discrete surface theory in Minkowski space. Discrete surfaces in Minkowski space possess similar properties of discrete surfaces in the Euclidean space. On the other hand, because interesting properties that the Euclidean objects do not have appear, they are more interesting objects than the previous ones.
  
  In this talk we first introduce a theory of discrete zero mean curvature surfaces in Minkowski space, their properties, and their constructions. As an application, we will report our ongoing project on constructions of discrete minimal surfaces in Euclidean space. If time permits, we would like to introduce some interesting facts that our discrete surfaces appear in the study of architectural geometry, which was discovered by Mueller and Pottmann.

Supports

• Research Institute for Science & Technology, Tokyo University of Science
• Grants-in-Aid for Scientific Research (C), No. 23K03100 (Makiko Sumi Tanaka)

Organizers

• Makiko Sumi Tanaka (Department of Mathematics, Faculty of Science and Technology, Chair)
• Naoyuki Koike (Department of Mathematics, Faculty of Science)
• Susumu Hirose (Department of Mathematics, Faculty of Science and Technology)
• Toru Kajigaya (Department of Mathematics, Faculty of Science)
• Kurando Baba (Department of Mathematics, Faculty of Science and Technology)