iTHEMS Math Seminar
87 events

Introduction to operator algebras
May 17 (Fri) at 15:00  17:00, 2024
Kan Kitamura (Special Postdoctoral Researcher, iTHEMS)
I will give a quick introduction to operator algebras. Operator algebras in this talk consist of linear operators over some Hilbert space. Their study was initiated by Murray and von Neumann, motivated partially by the mathematical foundation of quantum mechanics. Starting from the definitions of a few basic notions, I will explain that commutative operator algebras can be interpreted as spaces. On the other hand, simple operator algebras (i.e., those without nontrivial ideals) form a class of operator algebras opposite to commutative ones and have attracted many operator algebraists. I will try to introduce several examples of simple operator algebras, some of which appear in mathematical physics. If time permits, I will also give recent results on ideals in C*algebras. People with any scientific background are welcome.
Venue: via Zoom / Seminar Room #359
Event Official Language: English

Knot Theory in Doubly Periodic Tangles and Applications
January 19 (Fri) at 15:00  16:30, 2024
Sonia Mahmoudi (Assistant Professor, Mathematical Science Group, Advanced Institute for Materials Research (AIMR), Tohoku University)
Doubly periodic entangled structures offer an interesting framework for modeling and investigating diverse materials and physical phenomena, from micro to large scales. Specifically, a doubly periodic tangle (DP tangle) is characterized as an embedding of an infinite number of curves in the thickened plane, derived as the lift of a link in the thickened torus to the universal cover. DP tangles play a crucial role in scientific research, particularly in fields such as materials science, molecular chemistry, and biology. Despite their widespread applications, a universally accepted mathematical description of DP tangles is currently lacking. One of the key challenges arises from the infinite possibilities in choosing a periodic cell (referred to as a motif) for a DP tangle, taking into account various periodic boundary conditions. In this presentation, we conduct a comprehensive examination of the concept of topological equivalence of DP tangles, offering insights into potential classifications and applications in the process.
Venue: Hybrid Format (3F #359 and Zoom), Main Research Building
Event Official Language: English

Tropical geometry and period integrals
December 13 (Wed) at 14:00  16:30, 2023
Yuto Yamamoto (Special Postdoctoral Researcher, iTHEMS)
Tropical geometry is a field of mathematics that naturally emerges when considering the limits of spaces with respect to some parameters. One of the motivations to study tropical geometry is to describe the behaviors of the spaces under the limit. In this math seminar, starting with a brief introduction to tropical geometry, we discuss its application to computation of period integrals, which are one of the most fundamental quantities of complex manifolds. The goal is to compute asymtptotics of period integrals for complex hypersurfaces in toric varieties using tropical geometry, and observe that the Riemann zeta values (or the gamma classes) appear in the result of the computation. The first half of the talk will be a brief introduction to tropical geometry for nonexperts including those who are working outside mathematics, and everyone will be welcome.
Venue: Hybrid Format (3F #359 and Zoom), Main Research Building
Event Official Language: English

Seminar
Introduction and prospects of topological recursion
November 17 (Fri) at 15:00  17:00, 2023
Osuga Kento (JSPS Research Fellow PD, Graduate School of Mathematical Sciences, The University of Tokyo)
Topological recursion is a universal recursive formalism that connects many branches in mathematical physics, such as enumerative geometry, algebraic geometry, integrable hierarchy, matrix models, 2d gravity, and more. In the first half of this talk, I will give a pedagogical overview of topological recursion and present simple examples from which we learn how topological recursion works. Then in the second half, I will present some ongoing research projects as well as a few future directions in topological recursion.
Venue: Seminar Room #359
Event Official Language: English

Seminar
Geometry of special nilpotent orbits
November 15 (Wed) at 14:00  15:30, 2023
Baohua Fu (Professor, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, China)
Special nilpotent orbits play a key role in representation theory, but their geometry is little understood. I'll first report a joint work with Yongbin Ruan and Yaoxiong Wen proposing a mirror symmetry conjecture for special nilpotent orbits and then a joint work with Daniel Juteau, Paul Levy and Eric Sommers on the proof of sliced version of Lusztig's conjecture on special pieces.
Venue: via Zoom
Event Official Language: English

Seminar
A cluster algebra structure in the quantum cohomology ring of a quiver variety
October 24 (Tue) at 10:00  11:30, 2023
Yingchun Zhang (Postdoctoral Researcher, Institute for Advanced Study in Mathematics, Zhejiang University, China)
The GromovWitten theory of a quiver variety is expected to be preserved by quiver mutation according to Seiberg duality, which has been proved to be true for Atype and starshaped quivers. Cluster algebra can be constructed for a given quiver via quiver mutation. The two subjects GromovWitten and cluster algebra seem to differ a lot. Howerver, when we move to the quantum cohomology ring of a quiver variety, BeniniParkZhao’s work “indicates” that there should be a cluster algebra structure in the quantum cohomology ring of the quiver variety. In this talk, I will introduce our recent work about the construction of such a cluster algebra structure in the quantum cohomology of a quiver variety. In particular, we will give a proof of the construction for Atype cluster algebra in quantum cohomology of a flag variety. This is a joint work with Weiqiang He.
Venue: Seminar Room #359
Event Official Language: English

Interactions between Algebraic Topology and Representation Theory by Toric Code
October 2 (Mon)  4 (Wed), 2023
Minkyu Kim (Research Fellow, School of Mathematics, Korea Institute for Advanced Study (KIAS), Republic of Korea)
Toric code is an error correction code designed by Kitaev in late 1990’s, which contributes to the birth of topological quantum computation. The goal of these lectures is to introduce toric code and related mathematics. We will explain an interaction between lowdimensional topology and representation of Drinfeld double. Especially, we will encode several operations (e.g. braidings) on representations into topology and geometry on surfaces. If time allows, we will give an overview of how toric code arises from chain complexes, which will be the prequel of our talk at TokyoSeoul Conference on Oct 6. These lectures will be fundamental and concrete. We hope that the audience are familiar with basic concepts of finite groups and Hopf algebras. These lectures will be held from Oct 2 to Oct 4, each from 13:30 to 15:00, for a total of 3 lectures. Oct 2 (mon) Introduction to toric code. Oct 3 (tue) Introduction to nonabelian toric code. Oct 4 (wed) Further studies on toric code.
Venue: via Zoom / Seminar Room #359
Event Official Language: English

Seminar
Classification of Meromorphic Spin 2dimensional Conformal Field Theories of Central Charge 24
September 19 (Tue) at 15:00  16:30, 2023
Möller Sven (Group Leader, Department of Mathematics, University of Hamburg, Germany)
We classify the selfdual (or holomorphic) vertex operator superalgebras (SVOAs) of central charge 24, or in physics parlance the purely leftmoving, spin 2dimensional 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 selfdual SVOA of central charge 23.5 whose weight1/2 and weight1 spaces vanish. Additionally, there might be selfdual SVOAs arising as "fake copies" of VB^# tensored with a free fermion F. We construct and classify the selfdual SVOAs by determining the 2neighbourhood graph of the selfdual (purely bosonic) VOAs of central charge 24 and also by realising them as simplecurrent 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)
Venue: Seminar Room #359
Event Official Language: English

Seminar
QuasiBPS categories
September 13 (Wed) at 10:00  11:30, 2023
Yukinobu Toda (Professor, Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU), The University of Tokyo)
In this talk, I will explain the notion of "QuasiBPS category". This is the (yet to be defined) category which categorifies BPS invariants on CalabiYau 3folds, and plays an important role in categorical wallcrossing in DonaldsonThomas theory. I will explain the motivation of quasiBPS categories, give definition in the case of symmetric quivers with potential (a local model of CY 3folds), and their properties. If time permits, I will explain quasiBPS categories for local K3 surfaces and their relation to derived categories of hyperkahler manifolds. This is a joint work in progress with Tudor Padurariu.
Venue: Seminar Room #359
Event Official Language: English

Introduction to braid groups
July 5 (Wed) at 14:00  16:30, 2023
Haru Negami (Ph.D. Student, Graduate School of Science and Engineering, Chiba University)
Part 1 (14:0015:00): 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 (15:3016:30): 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 KatzLongMoody construction, a method of constructing infinite series of representations of the semidirect 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.
Venue: Seminar Room #359 / via Zoom
Event Official Language: English

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 lowrank matrices, it attains large risk reduction when the unknown matrix is close to lowrank (e.g. reducedrank regression). Next, we construct a theory of shrinkage estimation under the “matrix quadratic loss”, which is a matrixvalued loss function suitable for matrix estimation. A notion of “matrix superharmonicity” for matrixvariate 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 matrixvariate improper tpriors 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

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 qdeformations 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

Hydrodynamic limit and the fluctuating hydrodynamics for largescale interacting systems
May 24 (Wed) at 14:00  16:30, 2023
Kohei Hayashi (Visiting Researcher, 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 largescale 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 largescale 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 KardarParisiZhang equation, and its mathematical background.
Venue: Seminar Room #359 (Main Venue) / via Zoom
Event Official Language: English

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 Rmatrix for vertex Lie algebra, based on which we propose to construct a new vertex Lie algebra. As an application, using the classical Rmatrix we defined, we give a new scheme to construct infinitedimensional (Liouville) integrable systems via the FeiginFrenkel center. This seminar is onsite only.
Venue: Seminar Room #359
Event Official Language: English

Seminar
On the BeemNair 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 KostantToda 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 onsite only.
Venue: Seminar Room #359
Event Official Language: English

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

Frobenius algebras associated with the αinduction 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

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 3CalabiYau.
Venue: Hybrid Format (Common Room 246248 and Zoom)
Event Official Language: English

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 socalled 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, Ktheory, and elliptic cohomology respectively.
Venue: Hybrid Format (Common Room 246248 and Zoom)
Event Official Language: English

Seminar
Math and Physics of SeibergWitten 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 pseudoRiemann geometry. Since the late 20th century, physicists centering on Edward Witten have revolutionized modern geometry. SeibergWitten 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, SeibergWitten theory is regarded as a fundamental tool to study 3 and 4dimensional geometry. This is based on a PDE called SeibergWitten 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 SeibergWitten 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 246248 and Zoom)
Event Official Language: English
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