iTHEMS数学セミナー
100 イベント
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セミナー
Topological data analysis from a practical and mathematical perspective
2020年7月15日(水) 16:00 - 18:10
池 祐一 (株式会社富士通研究所 人工知能研究所 研究員)
1. Topological data analysis and its applications In this talk, I will explain some methods in topological data analysis (TDA) and their applications. First I recall persistent homology, which is a central tool to analyze the "shape" of a point cloud set. Then I show several applications to material science and time-series analysis. I also talk about our collaborative research with Inria on noise-robust persistent homology and an automated vectorization method of persistence diagrams. 2. Persistence-like distance on sheaf category and displacement energy In this talk, I will talk about relation among sheaf theory, persistence modules, and symplectic geometry. We introduce a persistence-like distance on Tamarkin sheaf category and prove a stability result with respect to Hamiltonian deformation of sheaves. Based on this result, we propose a new sheaf-theoretic method to give a lower bound of the displacement energy of compact subsets of a cotangent bundle. This is a joint work with Tomohiro Asano.
会場: via Zoom
イベント公式言語: 英語
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セミナー
Universal Error Bound for Constrained Quantum Dynamics
2020年6月24日(水) 16:00 - 18:10
濱崎 立資 (理化学研究所 数理創造プログラム (iTHEMS) 上級研究員 / 理化学研究所 開拓研究本部 (CPR) 濱崎非平衡量子統計力学理研白眉研究チーム 理研白眉研究チームリーダー)
In quantum mechanics, the existence of large energy gaps allows us to trace out the degrees of freedom of irrelevant energy scale. Consequently, we can treat a system within a constrained subspace obtained by the projection of the total Hilbert space. While this statement has widely been used to approximate quantum dynamics in various contexts, a general and quantitative justification stays lacking. In this talk, we show a universal and rigorous error bound for such a constrained-dynamics approximation in generic gapped quantum systems [1,2]. This universal bound is a linear function of time that only involves the energy gap and coupling strength, provided that the latter is much smaller than the former. If time allows, I will briefly talk about generalizations of our result to e.g., quantum many-body systems and open quantum systems.
会場: via Zoom
イベント公式言語: 英語
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セミナー
Information geometry of operator scaling
2020年6月17日(水) 16:00 - 18:10
相馬 輔 (東京大学 大学院情報理工学系研究科 助教)
Matrix scaling is a classical problem with a wide range of applications. It is known that the Sinkhorn algorithm for matrix scaling is interpreted as alternating e-projections from the viewpoint of classical information geometry. Recently, a generalization of matrix scaling to completely positive maps called operator scaling has been found to appear in various fields of mathematics and computer science, and the Sinkhorn algorithm has been extended to operator scaling. In this study, the operator Sinkhorn algorithm is studied from the viewpoint of quantum information geometry through the Choi representation of completely positive maps. The operator Sinkhorn algorithm is shown to coincide with alternating e-projections with respect to the symmetric logarithmic derivative metric, which is a Riemannian metric on the space of quantum states relevant to quantum estimation theory. This talk is based on joint work with Takeru Matsuda.
会場: via Zoom
イベント公式言語: 英語
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セミナー
Knotted 2-spheres in the 4-space and Yang-Mills gauge theory
2020年5月27日(水) 16:00 - 18:10
谷口 正樹 (理化学研究所 数理創造プログラム (iTHEMS) 基礎科学特別研究員)
The classification problem of knots is one of the central topics in a study of topology. In the first part, we review classical knot theory and theory of 2-dimensional knots in the 4-dimensional space. In the second part, we focus on a problem considered in differential topology. In the studies of differential topology, people are interested in the difference between continuous and smooth. As the main result of this talk, we introduce a theorem that tells us the difference between continuous and smooth 2-dimensional knots. The proof uses Yang-Mills gauge theory for 4-manifolds obtained by the surgery of 2-knots.
会場: via Zoom
イベント公式言語: 英語
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セミナー
How many electrons can atoms bind?
2020年5月13日(水) 16:00 - 18:10
後藤 ゆきみ (理化学研究所 数理創造プログラム (iTHEMS) 基礎科学特別研究員)
In this talk, I will introduce the mathematical studies on the ionization problem. Some experimental & numerical evidences say that any doubly charged atomic ion X^{2-} is not stable. This 'fact' is called the ionization conjecture in mathematical physics literatures. My hope is to illustrates the interplay between mathematical and physical ideas. The talk is directed towards researchers on various aspects of quantum mechanics. In the first part, we will discuss the many-body aspects of quantum mechanics and introduce some basic notions. The second part will deal with the mathematical results in some approximation theories.
会場: via Zoom
イベント公式言語: 英語
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セミナー
From Eigenvalues to Resonances
2020年5月1日(金) 16:00 - 18:10
三上 渓太 (理化学研究所 数理創造プログラム (iTHEMS) 研究員)
Resonance is one of the most studied object in mathematical study of Schrödinger operators. One possible reason is that resonance is appeared in many other fields like arithmetic, physics, and topography. This series of talks target both mathematicians and researchers in other fields. The goal of the talk is to introduce the study of resonances for two body Schrödinger operators. In the first part, we briefly review spectral theory and how we use it in the study of Schrödinger operators. The aim of this part is to introduce the audience some basic notions used in the study of Schrödinger operators. In the second part, we give brief introduction of resonances and its application to both mathematicians and researchers in other fields. We start from mathematical definition of resonances to its applications in the other fields.
会場: via zoom
イベント公式言語: 英語
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セミナー
Index of the Wilson-Dirac operator revisited: a discrete version of Dirac operator on a finite lattice
2020年2月25日(火) 16:00 - 18:10
古田 幹雄 (東京大学 教授)
The Wilson-Dirac operator is a discrete version of Dirac operator defined on regular lattices. When the discrete version is a fine approximation of the Dirac operator on a Z/2-graded Clifford module on a torus, it is known that (1) an integer-valued index is defined for the Wilson-Dirac operator, and (2) the index is equal to the Atiyah-Singer index of the Dirac operator on the torus. These have been well established up to around 2000. The strategy of all the previous works is to make use of the discrete version of the heat kernel for Neuberger's overlap Dirac operator. Therefore the strategy cannot be generalized to mod 2 index nor family version of index. In this talk I would like to explain a new approach to the index of Wilson-Dirac operator which can be immediately generalized to these various cases. Joint work with H. Fukaya, S. Matsuo, T. Onogi, S. Yamaguchi and M. Yamashita.
会場: セミナー室 (160号室)
イベント公式言語: 英語
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セミナー
Solved and open problems regarding the neighborhood grid data structure
2020年2月7日(金) 16:00 - 18:10
スクロツキ マーティン (理化学研究所 数理創造プログラム (iTHEMS) 客員研究員 / Fellow, German Academic Scholarship Foundation, Germany)
2020年2月7日 16:00-17:00 17:10-18:10
会場: セミナー室 (160号室)
イベント公式言語: 英語
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セミナー
Semiclassical methods in mathematical quantum mechanics
2020年1月23日(木) 16:00 - 18:10
中村 周 (学習院大学 教授)
Plan of the seminar: we separate each talk into two. In the first 60 minutes the speaker gives an introductory talk for non-mathematicians. After a short break, the second 60 minutes is spent for a bit more detailed talk for mathematicians (working in other areas). We welcome you joining both parts of the seminar or only the first/second half. Talk 1: Semiclassical analysis, microlocal analysis and scattering theory. I plan to talk about overview on the semiclassical analysis and related topics, especially its intrinsic relationship with microlocal analysis and (microlocal) scattering theory. Roughly speaking, the microlocal analysis is an application of semiclassical idea to the analysis of singularities, and its analogue in momentum space is the microlocal scattering theory. We discuss basic notions of these, and mention several recent results. Talk 2: Microlocal structure of the scattering matrix with long-range perturbations. As an example of topics discussed in Talk 1, we discuss recent results on the scattering matrix with long-range perturbations. In particular, we show that the scattering matrix is expressed as a Fourier integral operator, and in some cases we can decide its spectral properties. Our approach is fairly geometric and abstract, and thus applies not only to usual Schrödinger operators but also to higher order operators and discrete Schrödinger operators.
会場: 研究本館 435-437号室
イベント公式言語: 英語
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セミナー
Multiple Zeta Values: Interrelation of Series and Integrals
2019年12月17日(火) 16:00 - 18:10
山本 修司 (慶應義塾大学 准教授)
Plan of the seminar: we separate each talk into two. In the first 60 minutes the speaker gives an introductory talk for non-mathematicians. After a short break, the second 60 minutes is spent for a bit more detailed talk for mathematicians (working in other areas). We welcome you joining both parts of the seminar or only the first/second half. Abstract: This is an introduction to multiple zeta values (MZVs). Although the study of MZVs is related to various areas of mathematics, we will concentrate on the algebraic structures of MZVs themselves. The key point is that MZVs have two kinds of representations: nested series and iterated integrals. We present how these two representations yield rich algebraic relations among MZVs.
会場: セミナー室 (160号室)
イベント公式言語: 英語
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セミナー
Noncommutative crepant resolutions and some higher dimensional flops
2019年12月4日(水) 16:00 - 18:10
原 和平 (早稲田大学 理工学術院 日本学術振興会特別研究員)
Plan of the seminar: we separate each talk into two. In the first 60 minutes the speaker gives an introductory talk for non-mathematicians. After a short break, the second 60 minutes is spent for a bit more detailed talk for mathematicians (working in other areas). We welcome you joining both parts of the seminar or only the first/second half. Abstract: We will talk about the theory noncommutative resolution of singularities. Noncommutative resolution is a noncommutative analog of usual (geometric) resolution of singularities, and allows us to generalise the idea of McKay correspondence to a large class of singularities. In the first part of the talk, we discuss the classical McKay correspondence, the definition of noncommutative crepant resolution, and some known results in lower dimensions. In the second half, we will discuss some concrete examples of noncommutative crepant resolutions in higher dimensions.
会場: セミナー室 (160号室)
イベント公式言語: 英語
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セミナー
Some topics in projective geometry of algebraic varieties
2019年11月8日(金) 16:00 - 18:10
伊藤 敦 (名古屋大学 大学院多元数理科学研究科 助教)
Plan of the seminar: we separate each talk into two. In the first 60 minutes the speaker gives an introductory talk for non-mathematicians. After a short break, the second 60 minutes is spent for a bit more detailed talk for mathematicians (working in other areas). We welcome you joining both parts of the seminar or only the first/second half. Abstract: We talk about Gauss maps and projective dual varieties, which are classical objects in projective geometry of algebraic varieties. In particular, we explain Gauss maps in positive characteristic and projective dual varieties of toric varieties in characteristic 0.
会場: セミナー室 (160号室)
イベント公式言語: 英語
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セミナー
Atiyah-Hirzebruch spectral sequence in the band theory
2019年10月24日(木) 16:00 - 18:10
塩崎 謙 (京都大学 基礎物理学研究所 助教)
Plan of the seminar: we separate each talk into two. In the first 60 minutes the speaker gives an introductory talk for non-mathematicians. After a short break, the second 60 minutes is spent for a bit more detailed talk for mathematicians (working in other areas). We welcome you joining both parts of the seminar or only the first/second half. Abstract: The topological nature of the band theory in crystalline systems can be well described by the topological K-theory over the Brillouin zone torus. In the first part of my talk, I will present the band-theory understanding of the grading of the K-group, and how the exactness axiom and the Mayer-Vietoris sequence are naturally understood. In the second part, I discuss how to compute the differentials of the Atiyah-Hirzebruch spectral sequence associated with a cell decomposition.
会場: セミナー室 (160号室)
イベント公式言語: 英語
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セミナー
力学系の平均次元と情報理論
2019年10月21日(月) - 23日(水)
塚本 真輝 (九州大学 教授)
Oct.21 15:30-16:30, 16:40-17:40, Okochi Hall Oct.22 13:30-14:30, room #435-437, Main Research Building Oct.23 13:30-14:30, room #435-437, Main Research Building
会場: 大河内記念ホール / 研究本館 435-437号室
イベント公式言語: 日本語
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セミナー
Complex analysis on a neighborhood of a complex submanifold and its applications
2019年7月30日(火) 16:00 - 18:10
小池 貴之 (大阪市立大学 大学院理学研究科 数学教室 講師)
Plan of the seminar: we separate each talk into two. In the first 60 minutes the speaker gives an introductory talk for non-mathematicians. After a short break, the second 60 minutes is spent for a bit more detailed talk for mathematicians (working in other areas). We welcome you joining both parts of the seminar or only the first/second half. Abstract: We explain our recent study on the complex analytic structure of a small tubular neighborhood of a complex submanifold, which is based on T. Ueda's classification theory. We also explain how to apply them to: (i) a study on (non-) existence of a smooth Hermitian metric on a nef line bundle over a projective manifold with semi-positive curvature, and (ii) a study on non-projective and non-Kummer K3 surfaces.
会場: セミナー室 (160号室)
イベント公式言語: 英語
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セミナー
Introduction to Schroedinger Operators
2019年7月12日(金) 16:00 - 18:10
三上 渓太 (理化学研究所 数理創造プログラム (iTHEMS) 研究員)
Plan of the seminar: we separate each talk into two. In the first 60 minutes the speaker gives an introductory talk for non-mathematicians. After a short break, the second 60 minutes is spent for a bit more detailed talk for mathematicians (working in other areas). We welcome you joining both parts of the seminar or only the first/second half. Abstract: In this seminar, I will talk about mathematical study of Schroedinger operators (or Schroedinger equation). Part 1: I will talk about what mathematicians do to find a solution to Schroedinger equation. The goal of the first part is to be able to check the existence of solutions of Schroedinger equations in terms of decay/growth rate of potentials. Part 2: I will talk about what can we say about solutions to Schroedinger equation constructed in the first part. Especially, the relationship to the corresponding classical mechanic is introduced.
会場: セミナー室 (160号室)
イベント公式言語: 英語
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セミナー
Asymmetric metric and coarse geometry
2019年6月20日(木) 16:00 - 18:10
児玉 大樹 (理化学研究所 数理創造プログラム (iTHEMS) 客員研究員 / 東北大学 材料科学高等研究所 (AIMR) 助教)
Plan of the seminar: we separate each talk into two. In the first 60 minutes the speaker gives an introductory talk for non-mathematicians. After a short break, the second 60 minutes is spent for a bit more detailed talk for mathematicians (working in other areas). We welcome you joining both parts of the seminar or only the first/second half. Abstract: For most of mathematicians, metric is symmetric. However, we can define asymmetric metric without any difficulty. "Coarse" is a notion to describe some large scale viewpoint. For example, the set of real numbers is coarse equivalent to the set of integers (with respect to standard metric). I will discuss asymmetric metric space in "coarse" sense. Part 1: I will define metric space and asymmetric metric space. I will also explain a notion of coarse equivalence. Part 2: I will discuss what kind of asymmetric metric space is not coarse equivalent to (symmetric) metric space. I also would like to give other generalizations of metric.
会場: セミナー室 (160号室)
イベント公式言語: 英語
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セミナー
Certain invariants as dimension
2019年5月24日(金) 16:00 - 18:10
大内 元気 (理化学研究所 数理創造プログラム (iTHEMS) 基礎科学特別研究員)
Plan of the seminar: we separate each talk into two. In the first 60 minutes the speaker gives an introductory talk for non-mathematicians. After a short break, the second 60 minutes is spent for a bit more detailed talk for mathematicians (working in other areas). We welcome you joining both parts of the seminar or only the first/second half. Abstract: In this talk, I would like to talk about certain invariants that look like dimension. This talk has independent two parts. In part 1, I will talk about finite metric spaces. In 2013, Leinster introduced the notion of magnitude of finite metric spaces. It measures effective number of points in finite metric spaces. Considering magnitude and scale transformation, Leinster and Willerton defined dimension of finite metric space with scale. I will explain the definition of magnitude of finite metric spaces and see examples. In part 2, I will talk about derived categories of smooth projective varieties or finite dimensional algebras. In 2014, Dimitrov, Heiden, Katzarkov and Kontsevich introduced the notion of entropy of endofunctors of derived categories. It measures complexity of endofunctors under iteration. Serre functor is an autoequivalence of derived category, that describes Serre duality. Entropy of Serre functor looks like dimension of derived categories. I will talk about known results for entropy of Serre functors and some related topics.
会場: セミナー室 (160号室)
イベント公式言語: 英語
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セミナー
Introduction to Singularity Theory in Algebraic Geometry
2019年5月16日(木) 16:00 - 18:10
佐藤 謙太 (理化学研究所 数理創造プログラム (iTHEMS) 基礎科学特別研究員)
Plan of the seminar: we separate each talk into two. In the first 60 minutes the speaker gives an introductory talk for non-mathematicians. After a short break, the second 60 minutes is spent for a bit more detailed talk for mathematicians (working in other areas). We welcome you joining both parts of the seminar or only the first/second half. In this talk, I will explain for all scientists how singularities are studied in algebraic geometry. In algebraic geometry, we study algebraic varieties, which are figures defined as the zero sets of polynomial equations. To study an algebraic variety, we often expect that the variety is smooth, that is, the variety locally resembles Euclidian spaces. However, even if we start from smooth varieties, we sometimes encounter non-smooth varieties. This is one of the reasons why we need to study singularities. Part I: In the first one hour, I will explain how singularities are studied. I will introduce two invariants of singularities by which we can compare singularities numerically. One invariant is defined in terms of resolution of singularities and the other is defined in terms of positive characteristic methods. I also explain a surprising relation of these invariants. Part II: In the second one hour, I will explain how singularity theory is used to study smooth projective varieties. I will introduce Minimal Model Program and explain the relation with singularity theory.
会場: セミナー室 (160号室)
イベント公式言語: 英語
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セミナー
Gauge theory and symmetries of 4-dimensional spaces
2019年4月26日(金) 16:00 - 18:10
今野 北斗 (理化学研究所 数理創造プログラム (iTHEMS) 基礎科学特別研究員)
Plan of the seminar: we separate each talk into two. In the first 60 minutes the speaker gives an introductory talk for non-mathematicians. After a short break, the second 60 minutes is spent for a bit more detailed talk for mathematicians (working in other areas). We welcome you joining both parts of the seminar or only the first/second half. Although the term "gauge theory" is usually used in physical contexts, in the early 1980's, mathematicians found that gauge theory has many striking applications to purely mathematical problems. Most of typical applications are related to topology of 4-dimensional spaces. As a recent development in this direction, I used gauge theory to study "the shape of the space of all symmetris of a 4-dimensional space". In the first one hour, I will explain a notion of mathematical spaces, called manifolds, and try to describe the idea: how mathematicians make use of gauge theory to study the topology of a 4-dimensional manifold. In the second one hour, I will explain what the space of symmetries of a manifold means, and which type of theorems about the space of symmetries can be obtained using gauge theory.
会場: セミナー室 (160号室)
イベント公式言語: 英語
100 イベント