iTHEMS数学セミナー
85 イベント
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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
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セミナー
Introduction to instanton knot homology
2022年7月25日(月) 16:00 - 18:00
井森 隼人 (京都大学 大学院理学研究科 数学・数理解析専攻 博士課程)
Floer theory is an infinite-dimensional version of Morse theory and has provided powerful invariants in the study of low-dimensional topology. In the context of Yang-Mills gauge theory, some versions of Floer homology groups for knots have been developed. These knot invariants are called instanton knot homology groups and are strongly related to representations of the fundamental group of the knot complement. In this talk, the speaker introduces basic constructions of instanton knot homology groups and recent developments related to the equivariant version of instanton knot homology theory.
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
Seiberg-Witten Floer homotopy
2022年7月15日(金) 14:00 - 16:30
今野 北斗 (東京大学 大学院数理科学研究科 助教)
I will survey a mathematical object called the Seiberg-Witten Floer homotopy type introduced by Manolescu. This is a machinery that extracts interesting aspects of 3- and 4-dimensional manifolds through the Seiberg-Witten equations. This framework assigns a 3-manifold to a "space" (more precisely, the stable homotopy type of a space), and this space contains rich information that is strong enough to recover the monopole Floer homology of the 3-manifold, which is known already as a strong invariant. I shall sketch how this theory is constructed along Manolescu's original work, and introduce major applications. If time permits, I will also explain recent developments of Seiberg-Witten Floer homotopy theory. If you are not familiar with the mathematical formulation of TQFT and categorification, I recommended you to watch Dr. Sano's recent talk in advance (see related links).
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
Algebraic geometry in mixed characteristic
2022年6月10日(金) 14:00 - 16:30
吉川 翔 (数理創造プログラム 基礎科学特別研究員)
In algebraic geometry, we study the geometry of algebraic varieties, which are sets defined by algebraic equations. There are two types of algebraic varieties, they are varieties over characteristic zero and varieties over positive characteristic. Algebraic geometry in characteristic zero is similar to analytic geometry, so it is related to many other subjects. In this talk, I will introduce the notion of algebraic geometry in positive characteristic and relationships between positive characteristic and characteristic zero. In order to study it, we consider families consisting of varieties over characteristic zero and varieties over positive characteristic, called mixed characteristic.
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
A mathematical formulation of two-dimensional conformal field theory
2022年5月23日(月) 14:00 - 16:30
森脇 湧登 (数理創造プログラム 基礎科学特別研究員)
The mathematical construction of non-trivial quantum field theory in four dimensions, known as the "Yang-Mills existence and mass gap problem", is a very important issue in mathematical sciences. There are many examples of rigorous quantum field theories in two dimensions, although the four dimensions have not yet been solved. In particular, two-dimensional conformal field theory, which is a quantum field theory with conformal symmetry, has good properties and can be formulated mathematically using algebraic structures formed by "products of a field and a field" (operator product expansion). In this talk, this algebraic formulation (full vertex algebra) will be explained. Various construction methods and concrete examples (construction using codes, construction from quantum groups, and construction by deformation) will then be discussed. All the talk here is mathematical, but I will try to speak in a way that is motivated by physics as much as possible throughout the talk. I hope to receive various comments from the viewpoints of other fields.
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
Khovanov homology theory - an introduction to categorification
2022年5月13日(金) 14:00 - 16:30
佐野 岳人 (数理創造プログラム 基礎科学特別研究員)
Jones polynomial is a knot invariant discovered by V. F. R. Jones in 1984. Not only that it is a useful mathematical tool, the discovery led to opening up a new research area, quantum topology, which connects quantum mechanics and low-dimensional topology. In 2000, M. Khovanov introduced a “categorification of the Jones polynomial”, which is now called Khovanov homology, and made categorification one of the fundamental concept in knot theory. Now what does categorification mean, and what is it good for? In this talk, assuming that many of the audience are not familiar with abstract category theory, I will start from easy examples of categories and categorifications, for example categorification of natural numbers, and explain why they are something natural to think of. In the latter part, I will briefly explain the construction of Khovanov homology, and introduce several related topics.
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
Recurrence theorems for topological Markov chains
2022年4月22日(金) 17:00 - 19:00
Cédric Ho Thanh (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)
Recurrence theorems place conditions under which probabilistic systems, specifically Markov chains, are expected to visit certain states infinitely often. For example, a printer with its many moving parts and the random requests it receives, may be described as a probabilistic system, and recurrence of the "ready to print" state is desirable. Recurrence theorems in the case of finite Markov chains are widely known. In this talk, we are interested in generalization to the infinitary setting. As it turns out, some care has to be put in the definition of infinite Markov chains. Rather than simply infinite, the introduct topological Markov chains, and show how standard constructions can be naturally extended to thisframework: path spaces, cylinder sets, as well as the semantic of LTL and PCTL. With all these tools in hand, we finally state our recurrence theorems. This is work in progress in collaboration with Natsuki Urabe and Ichiro Hasuo. This seminar is hold in a hybrid style. If you want attend the seminar onsite, please contact to Keita Mikami.
会場: コモンルーム 246-248号室 とZoomのハイブリッド開催 (メイン会場)
イベント公式言語: 英語
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セミナー
Explore the possibility to control hurricanes
2022年3月18日(金) 16:00 - 18:00
Lin Li (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)
Hurricanes, also known as tropical cyclones and typhoons, are the biggest and the most devastating storms on Earth. In this seminar, I will talk about the possibility to control hurricanes with existing human capability. Energetically speaking, controlling hurricanes is a very challenging task due to a large gap: hurricanes are gigantic heat engines with a power of around 1014 Watt, while the most powerful manmade engines have the power of only 108 Watt. This six-order-magnitude gap is the major obstacle toward using existing engines to control hurricanes. To fill in this gap, we propose to utilize the chaotic nature of hurricanes, namely, the sensitivity of a chaotic system to its initial condition, to control hurricanes. In this presentation, I will first review the basics of hurricanes and existing chaos control methods, and then present my thoughts on hurricane control and preliminary results I acquired since joining Prediction Science Laboratory. Future directions on using reinforcement learning to control hurricanes will also be discussed. Since it is a very challenging task, I welcome any discussions, questions, and comments. I hope we can make the hurricane-risk-free future come earlier.
会場: via Zoom
イベント公式言語: 英語
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セミナー
Extracting rules from trained machine learning models with applications in Bioinformatics
2022年3月11日(金) 16:00 - 18:00
Pengyu Liu (情報統合本部 (R-IH) 医療データ数理推論チーム 特別研究員)
Recently, Machine learning methods have achieved great success in various areas. However, some machine learning-based models are not explainable (e.g., Artificial Neural Networks), which may affect the massive applications in medical fields. In this talk, we first introduce two approaches that extract rules from trained neural networks. The first one leads to an algorithm that extracts rules in the form of Boolean functions. The second one extracts probabilistic rules representing relations between inputs and the output. We demonstrate the effectiveness of these two approaches by computational experiments. Then we consider applying an explainable machine learning model to predict human Dicer cleavage sites. Human Dicer is an enzyme that cleaves pre-miRNAs into miRNAs. We develop an accurate and explainable predictor for the human Dicer cleavage site -- ReCGBM. Computational experiments show that ReCGBM achieves the best performance compared with several existing methods. Further, we find that features close to the center of pre-miRNA are more important for the prediction.
会場: via Zoom
イベント公式言語: 英語
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セミナー
Introduction to stability conditions 2
2022年3月9日(水) 16:00 - 17:30
小関 直記 (Postdoctoral Research Associate, School of Mathematics, University of Edinburgh, UK)
In 2002, Bridgeland defined the notion of stability conditions on a triangulated category, motivated by string theory and mirror symmetry. Since then, Bridgeland stability conditions have been found very useful not only in Mathematical Physics, but also in various areas of Pure Mathematics. In the first part, I will review basic background and open problems in the theory of Bridgeland stability conditions. In the second part, I will explain recent developments of the theory, especially its applications to algebraic geometry.
会場: via Zoom
イベント公式言語: 英語
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セミナー
Introduction to stability conditions 1
2022年3月2日(水) 16:00 - 17:30
小関 直記 (Postdoctoral Research Associate, School of Mathematics, University of Edinburgh, UK)
In 2002, Bridgeland defined the notion of stability conditions on a triangulated category, motivated by string theory and mirror symmetry. Since then, Bridgeland stability conditions have been found very useful not only in Mathematical Physics, but also in various areas of Pure Mathematics. In the first part, I will review basic background and open problems in the theory of Bridgeland stability conditions. In the second part, I will explain recent developments of the theory, especially its applications to algebraic geometry.
会場: via Zoom
イベント公式言語: 英語
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セミナー
How to understand Earth science system using data science
2022年2月25日(金) 16:00 - 18:00
Kaman Kong (理化学研究所 計算科学研究センター (R-CCS) 複合系気候科学研究チーム 特別研究員)
Hi everyone, my name is Kaman Kong. After I graduated from Nagoya University last April, I joined the computational climate science research team, R-CCS at Kobe. Although I have still not yet had the important results now, I would like to share my idea and future plan here. In this talk, different from the previous seminar, I would like to highlight how to use data science approaches to understand our Earth system science. In the first 60 minutes, I would like to share my research experiences in ecosystems, dust outbreaks, and atmospheric sciences and try to discuss their limitation in my study. After a 10-minute break, the 30 minutes will be spent discussing the potential methodology to overcome these limitations and new opportunities and challenges in Earth system science. (Part 1) In the first 60 minutes, I would like to talk about the relationships among ecosystems, dust outbreaks, and atmospheric conditions. I used the models of dust and ecosystem to explore seasonal variations of threshold wind speed, an index of soil susceptibility to dust outbreak, and its relations with land surface conditions, such as plant growth and soil moisture and temperature changes, in the Mongolian grasslands. On the other side, I am improving the weather forecast model to accurately predict dust emission and discuss its effects on the Earth system. Meanwhile, I am integrating the dust model into the ecosystem model. During this period, I realized there are many uncertainties of simulation. (Part 2) In the second 30 minutes, I will explain these limitations as I mentioned before and try to discuss how to solve these problems. For example, using deep learning to identify the green and brown plants separately for discussing their different effect on the dust model. And, used data assimilation (e.g., EnKF and Bayesian calibration) to improve the simulated performance of land surface parameters (e.g., soil moisture and vegetation).
会場: via Zoom
イベント公式言語: 英語
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セミナー
Recent progress on dualities in W-superalgebras
2022年1月28日(金) 16:00 - 18:00
中塚 成徳 (東京大学 カブリ数物連携宇宙研究機構 (Kavli IPMU) 東京大学特別研究員)
Vertex superalgebras are algebras which describe the chiral part of two dimensional superconformal field theory. A rich and fundamental class is provided by the affine vertex superalgebras associated with simple Lie superalgebras and the W-superalgebras obtained from them by cohomology parametrized by nilpotent orbits. Historically, the W-algebras associated with simple Lie algebras and principal nilpotent orbit have been studied intensively and are well-known to play an essential role in the quantum geometric Langlands program. In particular, they enjoy a duality, called the Feigin-Frenkel duality, which is a chiral analogue of the isomorphism between centers of the enveloping algebras of simple Lie algebras in Langlands duality. Recently, physicists found a suitable generalization for other types of nilpotent orbits from study on four dimensional supersymmetric gauge theory. In this talk, I will report the recent progress on our understanding of dualities in W-superalgebras since then in terms of several aspects: algebras, modules, and fusion rules.
会場: via Zoom
イベント公式言語: 英語
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セミナー
The Ohsawa-Takegoshi $L^2$ extension theorem and variations of Bergman kernels
2022年1月14日(金) 16:00 - 18:00
細野 元気 (東北大学 大学院理学研究科 数学専攻)
In complex analysis and geometry, $L^2$ methods are very important and widely used. Recent studies show that the $L^2$ theory and the variational theory are closely related. In particular, the (optimal) $L^2$ extension theorem can be proved by subharmonicity of variations of Bergman kernels and vice versa. In this talk, I will explain the background, results, and key ideas of the proof. *Please contact Keita Mikami mailing address to get access to the Zoom meeting room.
会場: via Zoom
イベント公式言語: 英語
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セミナー
Generalized Bernoulli process and computation of proportional areas for Venn diagram
2021年12月8日(水) 16:00 - 18:00
入谷 亮介 (数理創造プログラム 研究員)
*For detailed information about the seminar, please refer to the email.
会場: via Zoom
イベント公式言語: 英語
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セミナー
The Conley index of topological dynamical systems
2021年12月3日(金) 16:00 - 18:00
森田 陽介 (京都大学 大学院理学研究科 助教)
The study of topological dynamical systems, i.e. continuous self-homeomorphisms (or continuous flows) on topological spaces, is important in both pure mathematics and applications. To each isolated invariant subset of a topological dynamical system, we can assign an invariant called the Conley index, which is (roughly speaking) a based space that describes the dynamics around the isolated invariant subset. It is used not only in the study of topological dynamical systems themselves but also in Manolescu’s construction of the Seiberg-Witten-Floer homotopy type (a spectrum-valued (3+1)-dimensional TQFT). In this talk, I am planning to explain a new construction of Conley indices, which is entirely non-homotopical and uses only basic general topology. *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
85 イベント