# iTHEMS数学セミナー

## From Yang-Mills theory to enumerative geometry on Calabi-Yau 4-folds

2021年8月6日16:00 - 18:10

ヤーロン・ツァオ (数理創造プログラム 研究員)

Yang-Mills theory was studied from mathematical perspectives in the 1970s by Atiyah and his collaborators (notably Drinfeld, Hitchin, Singer). Subsequent breakthroughs were made on dimensions 3 and 4 by Floer and Donaldson (based on deep analytic results obtained by Uhlenbeck and Taubes) in the 1980s. In 1996, Donaldson and Thomas proposed to study Yang-Mills theories on dimensions bigger than 4. In higher dimensions, the analytic method is limited and algebro-geometric method is heavily used instead. This powerful tool usually enables us to compute partition functions and lead to amazing links to other invariants in enumerative geometry, e.g. Gromov-Witten and Gopakumar-Vafa invariants. In this talk, I will review some of these inspiring stories and discuss how my works on Calabi-Yau 4-folds fit into them.

会場: via Zoom

イベント公式言語: 英語

## An introduction to modular functions, conformal field theories, and moonshine phenomena

2021年7月2日16:00 - 18:10

及川 瑞稀 (数理創造プログラム 大学院生リサーチ・アソシエイト / 数理創造プログラム 研修生 / 東京大学 大学院数理科学研究科 博士課程)

Moonshine phenomena are certain mysterious connections between modular functions and finite groups. The first example is the celebrated monstrous moonshine, which connects the J-invariant and the Monster group. Surprisingly, this relationship can be well understood in terms of chiral conformal field theory. In this talk, I would like to explain what is chiral conformal field theory and how it gives moonshine phenomena. In the first part of the talk, the notion of modular function will be introduced and the precise statement of the monstrous moonshine will be given. Then the monstrous moonshine will be explained in terms of vertex operator algebra, a mathematical model of chiral conformal field theory. In the second part of the talk, we focus on the question: what is chiral conformal field theory mathematically? In addition to vertex operator algebras, other mathematical models of chiral conformal field theory, namely conformal nets and Segal conformal field theories, will be introduced. Recent progress on the relationship among these three models, including the Carpi--Kawahigashi--Longo--Weiner correspondence and the geometric realization of conformal nets will also be reviewed.

会場: via Zoom

イベント公式言語: 英語

## Stable eigenvalues of compact anti-de Sitter 3-manifolds

2021年6月18日16:00 - 18:10

甘中 一輝 (数理創造プログラム 基礎科学特別研究員)

Geometric objects that have been investigated in detail so far, such as closed Riemann surfaces, are sometimes locally homogeneous. Loosely speaking, their infinitesimal behavior is the same at each point. In this talk, I would like to explain the idea of investigating such objects using the Lie group theory.In the first part of the talk, I will recall the notions of Lie group actions and their quotient spaces with examples, and then explain the definitions of locally homogeneous spaces and their deformations (Teichmüller spaces). In the second part of the talk, I will consider anti-de Sitter manifolds as a special case, i.e., Lorentzian manifolds of negative constant curvature. As in the Riemannian case, a differential operator called the Laplacian (or the Klein-Gordon operator) is defined on Lorentzian manifolds. Unlike the Riemannian case, it is no longer an elliptic differential operator but a hyperbolic differential operator. In its spectral analysis, new phenomena different from those in the Riemannian case have been discovered in recent years, following pioneering works by Toshiyuki Kobayashi and Fanny Kassel. I would like to explain stable eigenvalues of the hyperbolic Laplacian of anti-de Sitter 3-manifolds with recent progress.

会場: via Zoom

イベント公式言語: 英語

## Loewner's theorem for maps on operator domains / The structure of maps on the space of all quantum pure states that preserve a fixed quantum angle

2021年5月24日16:00 - 18:10

森 迪也 (数理創造プログラム 基礎科学特別研究員)

This talk is divided into two independent topics. In the first part of my talk we consider the order structure of hermitian matrices. Given two matrix domains (open connected sets of n-by-n hermitian matrices), what is the general form of order isomorphisms between them? I will explain that there is a complete correspondence between the class of order isomorphisms and that of biholomorphic mappings. In the second part we consider the metric structure of the space P(H) of all quantum pure states (= the projective space of a complex Hilbert space H). Wigner's theorem asserts that every surjective isometry of P(H) onto itself is implemented by a unitary or an antiunitary operator. Uhlhorn generalized Wigner's theorem by showing that every bijective transformation of P(H) that preserves orthogonality is implemented by a unitary or an antiunitary operator. We consider some variants of Uhlhorn's result. The first part is joint work with P. Semrl (Univ. of Ljubljana), and the second part with G.P. Geher (Univ. of Reading). Only basic linear algebra is assumed in both parts.

会場: via Zoom

イベント公式言語: 英語

## Geometry of canonical metrics on Kähler manifolds

2021年5月14日16:00 - 18:10

井上 瑛二 (数理創造プログラム 基礎科学特別研究員)

The aim of this talk is to report recent trends in Kähler geometry. Kähler geometry consists of two aspects: the one is algebraic geometry and the other is metric geometry.The first one hour is an introduction for non-mathematicians. I begin with a simple example of algebraic variety from ancient Greek, which I believe is the simplest example illustrating motivation for compact complex manifolds. On the other hand, I explain the first motivation for canonical metrics in Kähler geometry via Riemann’s uniformization theorem.The last one hour is an introduction to recent trends in Kähler geometry, especially Kähler-Einstein metrics. The existence of Kähler-Einstein metrics turns out to be related to geometry of degenerations of space, which is so called Yau-Tian-Donaldson conjecture. I explain various aspects of this topic. We encounter deep studies in metric geometry, birational geometry and non-archimedean geometry. I finally explain recent breakthrough on Kähler-Ricci flow.The goal of this talk is the starting point of my study. I briefly explain my study if time permits.

会場: via Zoom

イベント公式言語: 英語

## Alternative tsunami observing and forecasting systems

2021年4月22日16:00 - 18:10

イヤン・ムリア (理化学研究所 開拓研究本部 (CPR) 三好予測科学研究室 研究員)

Dedicated tsunami observing systems are mostly expensive and are often not sustainable. Therefore, alternative approaches should be implemented to overcome the issues. We introduced innovative ways to observe tsunamis using existing instrumentation available on unconventional platforms such as commercial vessels and airplanes. Our study demonstrated that the accuracy of the proposed observing systems is adequate for detecting large tsunamis offshore. The use of such systems is expected to provide more cost-effective and sustainable observations for the future. Additionally, we also developed a tsunami forecasting system based on machine learning to improve or complement the conventional methods that typically require considerable computational resources. On the contrary, the main appealing feature of the machine learning is the computational speed that would be suitable for a real-time prediction of tsunami inundation or flooding. We found that the application of machine learning can significantly improve the computing time without sacrificing the accuracy compared to the conventional methods.

会場: via Zoom

イベント公式言語: 英語

## Long-time behavior of moving solids in a fluid and the kinetic theory of gases

2021年4月7日16:00 - 18:10

小池 開 (京都大学 工学研究科 特別研究員)

Understanding dynamics of solids in a fluid is a fundamental problem in fluid dynamics. Due to the growing interest in engineering in out-of-equilibrium situations, moving boundary problems for kinetic equations such as the Boltzmann equation have become an active area of research. In the first part of the talk, I shall explain recent, especially mathematical, developments in this field. Then in the second part, I'd like to explain my results concerning the long-time behavior of a point particle moving in a 1D viscous compressible fluid. These results aim to give some explanation of related numerical simulations for a BGK model of the Boltzmann equation.

会場: via Zoom

イベント公式言語: 英語

## The Green-Tao theorem for number fields

2021年3月22日16:00 - 18:10

甲斐 亘 (東北大学 理学部数学科 助教)

5, 11, 17, 23, 29 are prime numbers which form an arithmetic progression of length 5. A famous theorem of Ben Green and Terence Tao in 2008 says there are arbitrarily long arithmetic progressions of prime numbers. Algebraic number theorists are also interested in more general numbers like square roots of integers. Recently, Mimura, Munemasa, Seki, Yoshino and I have established a generalization of the Green-Tao theorem in such a direction. In the first 50 minutes of my talk, I would like to explain some background and technology behind the Green-Tao theorem. In the second half after a break, I explain the concept of number fields to formulate our generalization of their result. I will also discuss how one of the new difficulties, which I call the norm vs length conflict, is handled by a technique called Geometry of Numbers. *Please contact Keita Mikami or Hiroyasu Miyazaki's mailing address to get access to the Zoom meeting room.

会場: via Zoom

イベント公式言語: 英語

## Scattering theory for half-line Schrödinger operators: analytic and topological results

2020年12月7日16:00 - 18:10

井上 秀樹 (名古屋大学)

Levinson’s theorem is a surprising result in quantum scattering theory, which relates the number of bound states and the scattering part of the underlying quantum system. For the last about ten years, it has been proved for several models that once recast in an operator algebraic framework this relation can be understood as an index theorem for the Møller wave operators. Resulting index theorems are called topological version of Levinson’s theorem or shortly topological Levinson’s theorem. In this talk, we first review the background and the framework of our investigation. New analytical and topological results are provided for Schrödinger operators on the half-line. This talk is based on my Ph.D thesis.

会場: via Zoom

イベント公式言語: 英語

## Flat and spherical surface approximations

2020年11月30日16:00 - 17:30

マーティン・スクロツキ (数理創造プログラム 訪問研究員 / German Academic Scholarship Foundation Fellow)

State-of-the-art acquisition devices produce surface representations of increasingly high resolution. While these detailed representations are important for production, they are problematic e.g. when exchanging drafts via the internet or when a quick rendering for comparison is necessary. In the first part of the talk, I will present results and further research questions from a paper I recently co-authored on 'Variational Shape Approximation'. This approach aims at linearizing the input surface and representing it via a set of localized planar segments. In the second part of the talk, I will present some ongoing research on surface representations via balls. This work started with constructions from spherical neodym magnets and provided a set of mathematical questions. These investigations are joint work with FU Berlin and OIST.

会場: via Zoom

イベント公式言語: 英語

## Representations of fundamental groups and 3-manifold topology

2020年11月16日16:00 - 18:10

北山 貴裕 (東京大学 大学院数理科学研究科 准教授)

In 3-dimensional topology the great progress during the last two decades revealed that various properties of 3-manifolds are well understood from their fundamental groups. I will give an introduction to the study of splittings of 3-manifolds along surfaces, with an emphasis on an application of group representations. A fundamental and difficult problem in general is to find surfaces essentially embedded in a given 3-manifold. I will explain how such surfaces are detected by deformations of representations of the fundamental group, and what information of detected surfaces is described in terms of topological invariants derived from representations.

会場: via Zoom

イベント公式言語: 英語

## Efficient probabilistic assessment of building performance: sequential Monte Carlo and decomposition methods

2020年11月13日16:00 - 18:10

ティエンフォン・ホウ (数理創造プログラム 特別研究員 / 理化学研究所 開拓研究本部 (CPR) 三好予測科学研究室 特別研究員 / 理化学研究所 計算科学研究センター (R-CCS) データ同化研究チーム 特別研究員)

The use of numerical simulations for complex systems is common. However, significant uncertainties may exist for many of the involved variables, and in order to ensure the reliability of our simulation results and the safety of such complex systems, a stochastic approach providing statistics of the probability distribution of the results is of crucial importance. However, when a highly accurate result is required, the conventional Monte Carlo based probabilistic methodology inherently requires many repetitions of the deterministic analysis and in cases where that deterministic simulation is (relatively) time consuming, such probabilistic assessment can easily become computationally intractable. Hence, to reduce the computational expense of such probabilistic assessments as much as possible, the targets of this seminar are twofold: (1), to exploit an efficient sampling strategy to minimize the number of needed simulations of Monte Carlo based probabilistic analysis; (2), to investigate a surrogate model to reduce the computational expense of single deterministic simulation. This seminar contains two parts and will be accompanied by a set of illustrative building physical case studies (analysis of the heat and moisture transfer through building components). The first part of this seminar focusses on the use of quasi-Monte Carlo based probabilistic assessment for building performance, since it has the potential to outperform the standard Monte Carlo method. More specifically, the quasi-Monte Carlo sampling strategies and related error estimation techniques will be introduced in detail. In addition, questions on under which conditions the quasi-Monte Carlo can outperform the standard Monte Carlo method will be answered by a set of analyses. The second part of this seminar targets the investigation of using model order reduction methods for optimizing the deterministic simulation, given that it generally allows a (large) reduction of the simulation time without losing the dynamic behavior of the conventional models (such as the transient finite element analysis). Particularly, the fundamental concepts of one common model order reduction method – proper orthogonal decomposition (POD) will be provided, and its potential use for simulating (building physical) problems with different levels of non-linearity and complexity will be illustrated.

会場: via Zoom

イベント公式言語: 英語

## Mathematical aspects of quasi-Monte Carlo integration

2020年11月5日16:00 - 18:10

鈴木 航介 (広島大学 大学院先進理工系科学研究科 助教)

In this talk, I will introduce mathematical aspects of quasi-Monte Carlo (QMC) integration. We aim to approximate the integral of a function on the d-dimensional hypercube [0,1]^d. A useful approach is Monte-Carlo (MC) integration, which uses randomly chosen samples. A drawback of MC is the rate of convergence; the standard deviation of the estimator converges as 1/sqrt(n) asymptotically in n. To have a better rate of convergence as O(log^d N/N) or more, QMC uses deterministic, uniformly distributed points. In the first part, I will give an overview of QMC, such as star-discrepancy, Koksma-Hlawka inequality, and some explicit constructions as lattices and digital nets. In the second part, I will show that QMC using lattices and digital nets can achieve a higher rate of convergence for smooth integrands.

会場: via Zoom

イベント公式言語: 英語

## Math Seminars by Dr. Genki Ouchi and Dr. Kenta Sato

2020年9月24日16:00 - 18:10

大内 元気
(数理創造プログラム 基礎科学特別研究員)

佐藤 謙太
(数理創造プログラム 基礎科学特別研究員)

[Talk 1] (16:00 - 17:00) Dr. Genki Ouchi Automorphism groups of cubic fourfolds and K3 categories In this talk, I would like to talk about symmetries of algebraic varieties, especially cubic fourfolds and K3 surfaces. It is known that symmetries of cubic fourfolds and K3 surfaces are related to sporadic finite groups as Mathieu groups and Conway groups in both algebraic geometry and string theory. Relations between cubic fourfolds and K3 surfaces are studied in the context of derived categories, Hodge theory and so on. I would like to explain the direct relation among symmetries of cubic fourfolds and K3 surfaces via their derived categories. [Talk 2] (17:10 - 18:10) Dr. Kenta Sato An algebraic approach to the four color theorem The four color theorem states that, given any separation of a plane into contiguous regions, no more than four colors are required to color the regions. Although this theorem was already proved about 40 years ago, another proof without using a computer is not found still now. In this talk, I will introduce an algebraic approach to this theorem, which states that a conjecture about singularities of algebraic varieties implies the four color theorem. In particular, I would like to focus on the connection of three different fields in mathematics: graph theory, convex geometry and algebraic geometry.

会場: via Zoom

イベント公式言語: 英語

## Maximal Regularity and Partial Differential Equations

2020年9月8日16:00 - 18:10

古川 賢 (理化学研究所 開拓研究本部 (CPR) 三好予測科学研究室 特別研究員)

The theory of maximal regularity is a powerful tool to get solutions having the best regularity to linear partial differential equations (PDEs) of parabolic type. The theory is also applicable to show well-posedness of various non-linear PDEs. In the first part, We introduce the history of the development of the theory of maximal regularity and the way to apply non-linear PDEs. In the second part, We give some applications to PDEs, e. g. the primitive equations, the Navier-Stokes equations, and elliptic equations with dynamic boundary conditions.

会場: via Zoom

イベント公式言語: 英語

## Stability of ferromagnetism in many-electron systems

2020年7月31日16:00 - 18:10

宮尾 忠宏 (北海道大学 理学部数学科 准教授)

First part Title: Stability of ferromagnetism in many-electron systems Abstract: I construct a model-independent framework describing stabilities of ferromagnetism in strongly correlated electron systems. Within the new framework, I reinterpret the Marshall-Lieb-Mattis theorem and Lieb’s theorem; in addition, from the new perspective, I prove that Lieb’s theorem still holds true even if the electron-phonon and electron-photon interactions are taken into account. I also examine the NagaokaThouless theorem and its stability. These examples verify the effectiveness of the new viewpoint. Second part Title: Order preserving operator inequalities in many-electron systems Abstract: In this talk, I will introduce order preserving operator inequalities and explain how these inequalities are applied to the mathematical study of ferromagnetism. As examples of applications, Lieb's theorem of the Hubbard model and its stabilities will be discussed in terms of the inequalities.

会場: via Zoom

イベント公式言語: 英語

## Topological data analysis from a practical and mathematical perspective

2020年7月15日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

## Universal Error Bound for Constrained Quantum Dynamics

2020年6月24日16:00 - 18:10

濱崎 立資 (数理創造プログラム 上級研究員 / 理化学研究所 開拓研究本部 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

## Information geometry of operator scaling

2020年6月17日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

## Knotted 2-spheres in the 4-space and Yang-Mills gauge theory

2020年5月27日16:00 - 18:10

谷口 正樹 (数理創造プログラム 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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