iTHEMS数学セミナー
80 イベント
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セミナー
Classification of Meromorphic Spin 2-dimensional Conformal Field Theories of Central Charge 24
2023年9月19日(火) 15:00 - 16:30
Möller Sven (Group Leader, Department of Mathematics, University of Hamburg, Germany)
We classify the self-dual (or holomorphic) vertex operator superalgebras (SVOAs) of central charge 24, or in physics parlance the purely left-moving, spin 2-dimensional conformal field theories with just one primary field. There are exactly 969 such SVOAs under suitable regularity assumptions and the assumption that the shorter moonshine module VB^# is the unique self-dual SVOA of central charge 23.5 whose weight-1/2 and weight-1 spaces vanish. Additionally, there might be self-dual SVOAs arising as "fake copies" of VB^# tensored with a free fermion F. We construct and classify the self-dual SVOAs by determining the 2-neighbourhood graph of the self-dual (purely bosonic) VOAs of central charge 24 and also by realising them as simple-current extensions of a dual pair containing a certain maximal lattice VOA. We show that all SVOAs besides VB^# x F and potential fake copies thereof stem from elements of the Conway group Co_0, the automorphism group of the Leech lattice. By splitting off free fermions F, if possible, we obtain the classification for all central charges less than or equal to 24. This is based on joint work with Gerald Höhn (arXiv:2303.17190)
会場: セミナー室 (359号室)
イベント公式言語: 英語
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セミナー
Quasi-BPS categories
2023年9月13日(水) 10:00 - 11:30
戸田 幸伸 (東京大学 カブリ数物連携宇宙研究機構 (Kavli IPMU) 教授)
In this talk, I will explain the notion of "Quasi-BPS category". This is the (yet to be defined) category which categorifies BPS invariants on Calabi-Yau 3-folds, and plays an important role in categorical wall-crossing in Donaldson-Thomas theory. I will explain the motivation of quasi-BPS categories, give definition in the case of symmetric quivers with potential (a local model of CY 3-folds), and their properties. If time permits, I will explain quasi-BPS categories for local K3 surfaces and their relation to derived categories of hyperkahler manifolds. This is a joint work in progress with Tudor Padurariu.
会場: セミナー室 (359号室)
イベント公式言語: 英語
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セミナー
Introduction to braid groups
2023年7月5日(水) 14:00 - 16:30
根上 春 (千葉大学 大学院融合理工学府数学情報科学専攻 博士課程)
Part 1 (14:00-15:00): Introduction to braid groups Braid groups are groups that are defined by figures formed by the entanglement of n strings. Besides this geometric realization, it is a very interesting field where algebra and analysis intersect. In the first half of this seminar, aimed mainly at those unfamiliar with braid groups, we will introduce three aspects of braid groups and review the history of the research. In particular, in the area of its relation to analysis, the relationship between KZ equations and braid groups will be introduced. Part 2 (15:30-16:30): Representations of braid groups and the relationship between monodromy representations of KZ equations In the second half of the talk, after a brief introduction to representation theory, we will introduce the Katz-Long-Moody construction, a method of constructing infinite series of representations of the semi-direct product of braid group and free group. We will also show that its special case is isomorphic to multiplicative middle convolution, a method for constructing monodromy representations of KZ equations. Lastly, we will also discuss the connection between representations of braid groups and knot invariants. The talk includes joint work with Kazuki Hiroe.
会場: セミナー室 (359号室) / via Zoom
イベント公式言語: 英語
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セミナー
Matrix estimation via singular value shrinkage
2023年6月21日(水) 15:30 - 16:30
松田 孟留 (理化学研究所 脳神経科学研究センター (CBS) 統計数理連携ユニット ユニットリーダー)
In this talk, I will introduce recent studies on shrinkage estimation of matrices. First, we develop a superharmonic prior for matrices that shrinks singular values, which can be viewed as a natural generalization of Stein’s prior. This prior is motivated from the Efron–Morris estimator, which is an extension of the James–Stein estimator to matrices. The generalized Bayes estimator with respect to this prior is minimax and dominates MLE under the Frobenius loss. In particular, since it shrinks to the space of low-rank matrices, it attains large risk reduction when the unknown matrix is close to low-rank (e.g. reduced-rank regression). Next, we construct a theory of shrinkage estimation under the “matrix quadratic loss”, which is a matrix-valued loss function suitable for matrix estimation. A notion of “matrix superharmonicity” for matrix-variate functions is introduced and the generalized Bayes estimator with respect to a matrix superharmonic prior is shown to be minimax under the matrix quadratic loss. The matrix-variate improper t-priors are matrix superharmonic and this class includes the above generalization of Stein’s prior. Applications include matrix completion and nonparametric estimation.
会場: 研究本館 3階 359号室とZoomのハイブリッド開催
イベント公式言語: 英語
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セミナー
Around homogeneous spaces of complex semisimple quantum groups
2023年6月7日(水) 14:00 - 16:30
北村 侃 (東京大学 大学院数理科学研究科 博士課程)
Murray and von Neumann initiated the study of operator algebras motivated by the mathematical foundations of quantum physics. Operator algebras give good language to treat quantum symmetries, such as quantum groups. In this talk, I would like to give an overview of this topic first. Then, I discuss the q-deformations of complex semisimple Lie groups. From an operator algebraic viewpoint, we can treat them as "locally compact" quantum groups. Especially, I will focus on its homogenous spaces coming from discrete quantum subgroups with a motivation toward the quantum analog of lattices. Unlike the classical setting, we can obtain a complete classification of its discrete quantum subgroups.
会場: セミナー室 (359号室) (メイン会場) / via Zoom
イベント公式言語: 英語
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セミナー
Hydrodynamic limit and the fluctuating hydrodynamics for large-scale interacting systems
2023年5月24日(水) 14:00 - 16:30
林 晃平 (数理創造プログラム 訪問研究員)
In these decades, a great deal of works has been devoted to understand macroscopic phenomena, such as diffusion, aggregation or pattern formation, from the viewpoint of microscopic systems. Hydrodynamic limit, or fluctuating hydrodynamics, is a fundamental framework to explain the macroscopic behavior of physical quantities in mathematically rigorous ways from a system of the vast numbers of microscopic agents under random interactions, which system is called the large-scale interacting system. In this framework, our central aim is to derive partial differential equations (PDEs) which describe time evolution of some macroscopic quantities, starting from the large-scale interacting systems; hydrodynamic limit is a procedure to derive deterministic PDEs with help of the law of large numbers, whereas stochastic PDEs are derived under the scale of the central limit theorem by fluctuating hydrodynamics. In this talk, I would like to explain basic concepts of hydrodynamic limit and fluctuating hydrodynamics, through some simple models. In the first part, I will give a concise exposition on Markov processes as preliminaries and then state some results on scaling limits of simple exclusion processes as a pedagogical example. In the second part, I will talk about recent progress on universality which appears in fluctuating hydrodynamics. Especially, I would like to talk about the universality of the Kardar-Parisi-Zhang equation, and its mathematical background.
会場: セミナー室 (359号室) (メイン会場) / via Zoom
イベント公式言語: 英語
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セミナー
Generalized AKS scheme of integrability via vertex algebra
2023年5月9日(火) 16:15 - 17:15
Wenda Fang (京都大学 数理解析研究所 (RIMS) 博士課程)
In this talk, we define and study the classical R-matrix for vertex Lie algebra, based on which we propose to construct a new vertex Lie algebra. As an application, using the classical R-matrix we defined, we give a new scheme to construct infinite-dimensional (Liouville) integrable systems via the Feigin-Frenkel center. This seminar is on-site only.
会場: セミナー室 (359号室)
イベント公式言語: 英語
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セミナー
On the Beem-Nair conjecture
2023年5月9日(火) 15:00 - 16:00
降旗 駿 (京都大学 数理解析研究所 (RIMS) 博士課程)
Given a simple Lie group G, we have an open immersion (constructed by Beem and Nair) from the Kostant-Toda lattice associated to G into the universal centralizer of G. They expected that a free field realization of the chiral universal centralizer of G at the critical level will be obtained by the chiralization of this immersion. In this talk, we will verify this conjecture is true by constructing an embedding from the chiral universal centralizer into an appropriate vertex operator algebra at any level. This seminar is on-site only.
会場: セミナー室 (359号室)
イベント公式言語: 英語
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セミナー
Quantum modularity of quantum invariants and related techniques
2023年4月11日(火) 13:00 - 15:30
村上 友哉 (九州大学 数理学研究院 日本学術振興会 特別研究員 PD)
In this talk, I will present my recent work[1] and related research. In the first half of the talk, I will provide an overview of the concept of quantum modularity of quantum invariants, and briefly discuss my main result. In the second half, I will provide a more detailed explanation of my main result and the proof.
会場: セミナー室 (359号室) (メイン会場) / via Zoom
イベント公式言語: 英語
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セミナー
Frobenius algebras associated with the α-induction for equivariantly braided tensor categories
2023年4月10日(月) 14:00 - 16:30
及川 瑞稀 (東京大学 大学院数理科学研究科 博士課程／日本学術振興会 特別研究員 DC)
In this talk, I would like to introduce my work https://arxiv.org/abs/2303.11845. In the first half of the talk, I will give an introduction of tensor categories. In the latter half, I will explain about my construction of some tensor categories and Frobenius algebras.
会場: セミナー室 (359号室) / via Zoom
イベント公式言語: 英語
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セミナー
Coherent sheaves, quivers, and quantum groups
2023年2月17日(金) 14:00 - 16:00
Gufang Zhao (Senior Lecturer, University of Melbourne, Australia)
This talk aims to illustrate symmetries in geometry. The first half surveys a few examples of parametrizing coherent sheaves on a variety and how quantum groups control the symmetry of parametrization space. The second half aims to illustrate some special cases when the variety is a local toric 3-Calabi-Yau.
会場: コモンルーム 246-248号室 とZoomのハイブリッド開催
イベント公式言語: 英語
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セミナー
Quantum groups and cohomology theories
2023年2月15日(水) 14:00 - 16:00
Yaping Yang (Senior Lecturer, University of Melbourne, Australia)
In the first half of my talk, I will review quantum groups at roots of unity and their representation theory. In the second half, I will explain a construction of new quantum groups using cohomology theories from topology. The construction uses the so-called cohomological Hall algebra associated to a quiver and an oriented cohomology theory. In examples, we obtain the Yangian, quantum loop algebra and elliptic quantum group, when the cohomology theories are the cohomology, K-theory, and elliptic cohomology respectively.
会場: コモンルーム 246-248号室 とZoomのハイブリッド開催
イベント公式言語: 英語
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セミナー
Math and Physics of Seiberg-Witten theory
2023年1月20日(金) 16:00 - 18:10
飯田 暢生 (東京工業大学 理学院 日本学術振興会 特別研究員 PD)
Math and physics have developed through interactions with each other. For example, classical mechanics and calculous were born together. Einstein's theory of gravitation is written in the language of pseudo-Riemann geometry. Since the late 20th century, physicists centering on Edward Witten have revolutionized modern geometry. Seiberg-Witten theory is one of such breakthroughs, for both mathematicians and physicists. In physics it is regarded as a theory describing strong coupling (i.e. low energy) behavior of some supersymmetric gauge theories. It showes confinement (by a mechanism similar to superconductivity) and electric magnetic duality. Even though this story has not been mathematically justified yet, it is regarded as an important trigger of developments in understanding non perturbative aspects of quantum field theory and string theory, and stimulates broad fields of physics and math. In math, Seiberg-Witten theory is regarded as a fundamental tool to study 3 and 4-dimensional geometry. This is based on a PDE called Seiberg-Witten equation, which originates from the "electric magnetic dual description" of monopoles, but people can use it as a tool to study geometry without knowing such a physical origin. In this talk, developments of Seiberg-Witten theory from both viewpoints will be reviewed and if the time permits, works in math by the speaker and collaborators will be discussed. The speaker thinks it is unusual for a mathematician to talk about something that has not been mathematically justified yet, but hopes this talk will lead to new interactions between math and physics.
会場: コモンルーム 246-248号室 とZoomのハイブリッド開催
イベント公式言語: 英語
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セミナー
CM minimization and special K-stability
2022年12月16日(金) 14:00 - 16:30
服部 真史 (京都大学 大学院理学研究科 数学教室 博士課程)
Odaka proposed a conjecture predicting that the degrees of CM line bundles for families with fixed general fibers are strictly minimized if the special fibers are K-stable. This conjecture is called CM minimization and a quantitative strengthening of the conjecture of separatedness of moduli spaces of K-stable varieties (K-moduli). This conjecture was already shown for K-ample (Wang-Xu), Calabi-Yau (Odaka) and Fano varieties (Blum-Xu). In this talk, we introduce a new class, special K-stable varieties, and settle CM minimization for them, which is a generalization of the above results. In addition, we would like to explain an important application of this, construction of moduli spaces of uniformly adiabatically K-stable klt trivial fibrations over curves as a separated Deligne-Mumford stack in a joint work with Kenta Hashizume to appear. This is based on arXiv:2211.03108.
会場: via Zoom
イベント公式言語: 英語
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セミナー
Tropical methods in Enumerative Geometry and Mirror Symmetry
2022年11月25日(金) 14:00 - 16:00
Michel Van Garrel (Assistant Professor, School of Mathematics, University of Birmingham, UK)
Abstract for the 1st hour: Enumerative Geometry has been a feature of mathematics from its beginnings, just think about the number of lines in the plane passing through 2 points. I will take you on a history of the subject and its relationship to other areas of mathematics and physics. Abstract for the 2nd hour: Many problems in mathematics are solved by taking a limit and solving the limiting problem. Tropical geometry is a key technique that allows us to do this systematically. I will talk about the following problem. Take the complex projective plane S and an elliptic curve E in S. Count algebraic maps from the affine line into the complement S \ E. This counting problem is solved via tropical geometry as I will describe in this talk.
会場: コモンルーム 246-248号室 とZoomのハイブリッド開催
イベント公式言語: 英語
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セミナー
Mathematics of Post-Quantum Cryptography
2022年11月18日(金) 14:00 - 16:30
相川 勇輔 (三菱電機株式会社 情報技術総合研究所 研究員)
Cryptography keeps our everyday information communications secure. Cryptography based on key sharing have been used mainly for military purposes since ancient times in human history, but with the advent of the Internet, cryptography that does not require key sharing has become necessary. In 1976, Diffie and Hellman proposed the concept of public key cryptography, which does not require key sharing among communicators. Since then, research on public key cryptography has progressed, involving not only computer science but also mathematics, and has become an essential technology for the society we live in. The security of public key cryptography is supported by computational hardness of problems derived from mathematics. For example, the integer factoring problem is a basis for the security of RSA cryptography, and the discrete logarithm problem is for elliptic curve cryptography. However, in 1994, Shor proposed an efficient quantum algorithm that solves these problems. This means that emergence of large-scale quantum computers will break RSA and elliptic curve cryptography we use today. For this reason, research on next-generation cryptography, so-called Post-Quantum Cryptography (PQC for short), is currently underway to prepare for a future in which quantum computers will emerge. In this talk, without assuming any knowledge of cryptography, I will give a brief overview of cryptography and the progress of PQC. The first half of the talk will mainly outline the relationship between mathematics and cryptography, while the second half will discuss isogeny-based cryptography, one of the promising PQC, with our recent results.
会場: コモンルーム 246-248号室 とZoomのハイブリッド開催
イベント公式言語: 英語
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セミナー
Arithmetic dynamics on algebraic varieties
2022年11月11日(金) 14:00 - 16:30
松澤 陽介 (大阪公立大学 大学院理学研究科 数学専攻 准教授)
The study of self-maps of algebraic varieties is a relatively new and active area in mathematics. Such a self-map can be considered as a discrete dynamical system, and we can study the asymptotic properties of such systems from various points of views, including number theoretic viewpoint. I will introduce several problems in arithmetic dynamics and some of my results in this area.
会場: コモンルーム 246-248号室 とZoomのハイブリッド開催
イベント公式言語: 英語
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セミナー
Measuring diversity: species similarity
2022年10月28日(金) 16:00 - 17:00
トム・レンスター (Professor, University of Edinburgh, UK)
Traditional measures of the diversity of an ecological community depend only on how abundant the species are, not the similarities or differences between them. To better reflect biological reality, species similarity should be incorporated. Mathematically, this corresponds to moving from probability distributions on sets to probability distributions on metric spaces. I will explain how to do this and how it can change ecological judgements. Finally, I will describe a surprising theorem on maximum diversity (joint with Meckes and Roff), which reveals close connections between maximum diversity and invariants of geometric measure.
会場: via Zoom
イベント公式言語: 英語
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セミナー
Measuring diversity: the axiomatic approach
2022年10月21日(金) 16:00 - 17:00
トム・レンスター (Professor, University of Edinburgh, UK)
Ecologists have been debating the best way to measure diversity for more than 50 years. The concept of diversity is relevant not only in ecology, but also in other fields such as genetics and economics, as well as being closely related to entropy. The question of how best to quantify diversity has surprising mathematical depth. I will argue that the best approach is axiomatic: to enable us to reason logically about diversity, the measures we use must satisfy certain mathematical conditions, and those conditions dramatically limit the choice of measures. This point will be illustrated with a theorem: using a simple model of ecosystems, the only diversity measures that behave logically are the Hill numbers, which are very closely related to the Rényi entropies of information theory.
会場: via Zoom
イベント公式言語: 英語
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セミナー
Product Replacement Algorithm, Semidefinite Programming, and Operator Algebras
2022年8月2日(火) 16:00 - 17:00
小澤 登高 (京都大学 数理解析研究所 (RIMS) 教授)
Suppose you are given a large finite set G and want to estimate the size |G| or see how a typical element x in G looks like. In this talk, G will be a finite group generated by g_1,...,g_d. The "Product" Replacement Algorithm" is a popular algorithm for random sampling in the group G. The PRA shows outstanding performance in practice, but the theoretical explanation has remained mysterious. I will talk how an infinite-dimensional topological-algebraic analysis (operator algebra theory) connects this problem to a convex (semidefinite) optimization problem that can be rigorously solved by computer. This talk is intended for a general audience.
会場: コモンルーム 246-248号室 とZoomのハイブリッド開催
イベント公式言語: 英語
80 イベント