iTHEMS Math Seminar
97 events
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Seminar
Matrix estimation via singular value shrinkage
June 21 (Wed) at 15:30 - 16:30, 2023
Takeru Matsuda (Unit Leader, Statistical Mathematics Collaboration Unit, RIKEN Center for Brain Science (CBS))
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.
Venue: Hybrid Format (3F #359 and Zoom), Main Research Building
Event Official Language: English
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Seminar
Around homogeneous spaces of complex semisimple quantum groups
June 7 (Wed) at 14:00 - 16:30, 2023
Kan Kitamura (Ph.D. Student, Graduate School of Mathematical Sciences, The University of Tokyo)
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.
Venue: Seminar Room #359 (Main Venue) / via Zoom
Event Official Language: English
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Seminar
Hydrodynamic limit and the fluctuating hydrodynamics for large-scale interacting systems
May 24 (Wed) at 14:00 - 16:30, 2023
Kohei Hayashi (Visiting Researcher, RIKEN Interdisciplinary Theoretical and Mathematical Sciences Program (iTHEMS))
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.
Venue: Seminar Room #359 (Main Venue) / via Zoom
Event Official Language: English
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Seminar
Generalized AKS scheme of integrability via vertex algebra
May 9 (Tue) at 16:15 - 17:15, 2023
Wenda Fang (Ph.D. Student, Research Institute for Mathematical Sciences (RIMS), Kyoto University)
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.
Venue: Seminar Room #359
Event Official Language: English
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Seminar
On the Beem-Nair conjecture
May 9 (Tue) at 15:00 - 16:00, 2023
Syun Furihata (Ph.D. Student, Research Institute for Mathematical Sciences (RIMS), Kyoto University)
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.
Venue: Seminar Room #359
Event Official Language: English
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Seminar
Quantum modularity of quantum invariants and related techniques
April 11 (Tue) at 13:00 - 15:30, 2023
Yuya Murakami (JSPS Research Fellow PD, Faculty of Mathematics, Kyushu University)
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.
Venue: Seminar Room #359 (Main Venue) / via Zoom
Event Official Language: English
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Seminar
Frobenius algebras associated with the α-induction for equivariantly braided tensor categories
April 10 (Mon) at 14:00 - 16:30, 2023
Mizuki Oikawa (Ph.D. Student / JSPS Research Fellow DC, Graduate School of Mathematical Sciences, The University of Tokyo)
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.
Venue: Seminar Room #359 / via Zoom
Event Official Language: English
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Seminar
Coherent sheaves, quivers, and quantum groups
February 17 (Fri) at 14:00 - 16:00, 2023
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.
Venue: Hybrid Format (Common Room 246-248 and Zoom)
Event Official Language: English
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Seminar
Quantum groups and cohomology theories
February 15 (Wed) at 14:00 - 16:00, 2023
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.
Venue: Hybrid Format (Common Room 246-248 and Zoom)
Event Official Language: English
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Seminar
Math and Physics of Seiberg-Witten theory
January 20 (Fri) at 16:00 - 18:10, 2023
Nobuo Iida (JSPS Research Fellow PD, School of Science, Tokyo Institute of Technology)
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.
Venue: Hybrid Format (Common Room 246-248 and Zoom)
Event Official Language: English
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Seminar
CM minimization and special K-stability
December 16 (Fri) at 14:00 - 16:30, 2022
Masafumi Hattori (Ph.D. Student, Department of Mathematics, Graduate School of Science, Kyoto University)
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.
Venue: via Zoom
Event Official Language: English
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Seminar
Tropical methods in Enumerative Geometry and Mirror Symmetry
November 25 (Fri) at 14:00 - 16:00, 2022
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.
Venue: Hybrid Format (Common Room 246-248 and Zoom)
Event Official Language: English
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Seminar
Mathematics of Post-Quantum Cryptography
November 18 (Fri) at 14:00 - 16:30, 2022
Yusuke Aikawa (Researcher, Information Technology R&D Center, Mitsubishi Electric Corporation)
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.
Venue: Hybrid Format (Common Room 246-248 and Zoom)
Event Official Language: English
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Seminar
Arithmetic dynamics on algebraic varieties
November 11 (Fri) at 14:00 - 16:30, 2022
Yosuke Matsuzawa (Associate Professor, Department of Mathematics, Graduate School of Science, Osaka Metropolitan University)
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.
Venue: Hybrid Format (Common Room 246-248 and Zoom)
Event Official Language: English
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Seminar
Measuring diversity: species similarity
October 28 (Fri) at 16:00 - 17:00, 2022
Tom Leinster (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.
Venue: via Zoom
Event Official Language: English
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Seminar
Measuring diversity: the axiomatic approach
October 21 (Fri) at 16:00 - 17:00, 2022
Tom Leinster (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.
Venue: via Zoom
Event Official Language: English
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Seminar
Product Replacement Algorithm, Semidefinite Programming, and Operator Algebras
August 2 (Tue) at 16:00 - 17:00, 2022
Narutaka Ozawa (Professor, Research Institute for Mathematical Sciences (RIMS), Kyoto University)
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.
Venue: Hybrid Format (Common Room 246-248 and Zoom)
Event Official Language: English
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Seminar
Introduction to instanton knot homology
July 25 (Mon) at 16:00 - 18:00, 2022
Hayato Imori (Ph.D. Student, Division of Mathematics and Mathematical Sciences, Graduate School of Science, Kyoto University)
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.
Venue: Hybrid Format (Common Room 246-248 and Zoom)
Event Official Language: English
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Seminar
Seiberg-Witten Floer homotopy
July 15 (Fri) at 14:00 - 16:30, 2022
Hokuto Konno (Assistant Professor, Graduate School of Mathematical Sciences, The University of Tokyo)
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).
Venue: Hybrid Format (Common Room 246-248 and Zoom)
Event Official Language: English
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Seminar
Algebraic geometry in mixed characteristic
June 10 (Fri) at 14:00 - 16:30, 2022
Shou Yoshikawa (Special Postdoctoral Researcher, RIKEN Interdisciplinary Theoretical and Mathematical Sciences Program (iTHEMS))
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.
Venue: Hybrid Format (Common Room 246-248 and Zoom)
Event Official Language: English
97 events