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談話会/Colloquium

Title

Cluster integrable systems

Date

2019年7月10日(水) 16:30〜17:30    (16:00より1階ロビーでtea)

Place

京都大学数理解析研究所 (RIMS) 110号室
(Rm110, Research Institute for Mathematical Sciences, Kyoto University)

Speaker

Michael Gekhtman 氏 (京大・数理研 & University of Notre Dame)

Abstract

 Combinatorial structures embedded into a definition of cluster algebras proved instrumental in reimagining many important integrable models and helped to discover new instances of complete integrability. The talk will provide an overview of an interaction between theories of cluster algebras and integrable systems with examples ranging from dilogarithm identities to pentagram maps and their generalizations to discrete Toda-like systems that ``live'' on double Bruhat cells.

Comment

Title

Homotopy theory of $A_n$-spaces in Lie groups

Date

2019年6月26日(水) 16:30〜17:30    (16:00より105談話室でtea)

Place

京都大学大学院理学研究科3号館110講演室
(Rm110, Building No.3, Faculty of Science, Kyoto University)

Speaker

蔦谷 充伸 (Mitsunobu Tsutaya)氏 (九大・数理)

Abstract

An $A_n$-space is a topological space equipped with a continuous unital binary operation satisfying certain higher homotopy associativity conditions depending on $n=1, 2, \ldots, \infty$. Similarly, we can also define several versions of higher homotopy commutativities of $A_n$-spaces. Lie groups are basic examples of $A_\infty$-spaces. The speaker has been working on problems in homotopy theory of $A_n$-spaces in Lie groups.
In this talk, we will review the basics of higher homotopy associativity and commutativity and the results in Lie groups especially related to higher homotopy commutativity.
[pdf]

Comment

Title

Construction of symplectic field theory and smoothness of Kuranishi structure

Date

2019年6月19日(水) 16:30〜17:30    (16:00より1階ロビーでtea)

Place

京都大学数理解析研究所 (RIMS) 110号室
(Rm110, Research Institute for Mathematical Sciences, Kyoto University)

Speaker

石川 卓 (Suguru Ishikawa)氏 (京大・数理研)

Abstract

 Symplectic field theory (SFT) is a generalization of Gromov-Witten invariant and Floer homology for contact maniflods and symplectic cobordisms between them. It was introduced by Eliashberg, Givental and Hofer around 2000, and its algebraic structure was well studied by them. However, for a long time, it was a difficult problem to construct SFT by counting pseudoholomorphic curves. Recently, I succeeded in its construction by using Kuranishi theory, a theory developed by Fukaya and Ono for the construction of Gromov-Witten inavriant and Floer homology for general symplectic manifolds. In this talk, I explain about this work. Especially, I will talk about smoothness of Kuranishi structure.

Comment

Title

Nonbacktracking spectrum of random matrices

Date

2019年6月12日(水) 16:30〜17:30    (16:00より105談話室でtea)

Place

京都大学大学院理学研究科3号館127大会議室
(Rm127, Building No.3, Faculty of Science, Kyoto University)
[お部屋が変更になりました]

Speaker

Charles Bordenave 氏 (CNRS Marseille)

Abstract

The nonbacktracking operator has been introduced in the 80's by Sunada and Hashimoto in the context of the Ihara zeta function on graphs. In 2013, Krzakala et al. have used this matrix for the design of an algorithm to detect communities in social networks. In recent years, this nonbacktracking matrix has been promoted as a powerful tool to analyse the interplay between geometry and spectrum of a graph. In this talk, we will introduce this matrix and give some recents results on the spectrum of random graphs or random matrices which rely on the use of the nonbacktracking matrix.

Comment

Title

Vanishing of open Jacobi diagrams with odd legs

Date

2019年6月5日(水) 16:30〜17:30    (16:00より1階ロビーでtea)

Place

京都大学数理解析研究所 (RIMS) 110号室
(Rm110, Research Institute for Mathematical Sciences, Kyoto University)

Speaker

石川 勝巳 (Katsumi Ishikawa)氏 (京大・数理研)

Abstract

 有向結び目のKontsevich不変量は一連の量子不変量や有限型不変量を統括する極めて強力な不変量である一方で、その計算は難しく、基本的な性質ですらわかっていないことも多いが、これは値をとるJacobi図の空間の複雑さに因るところも大きい。例えば、Kontsevich不変量が結び目の可逆性を判定できないという予想は奇数個の1価頂点をもつ開Jacobi図が開Jacobi図の空間に於いて0となるという予想に翻訳されるが、このような単純に思える問題ですら未だに解決されていないのである。
 本講演ではKontsevich不変量と開Jacobi図の空間について簡単に復習した後、開Jacobi図の空間の持つ幾つかの興味深い性質を紹介し、それらにより上記の予想を7-ループ以下の開Jacobi図に対して肯定的に解決する。これはMoskovich-大槻による3-ループの場合の結果の拡張となっている。

Comment

Title

On stability of blow-up solutions of the Burgers vortex type for the Navier-Stokes equations with a linear strain

Date

2019年5月29日(水) 16:30〜17:30    (16:00より105談話室でtea)

Place

京都大学大学院理学研究科3号館110講演室
(Rm110, Building No.3, Faculty of Science, Kyoto University)

Speaker

前川 泰則 (Yasunori Maekawa)氏 (京大・理)

Abstract

 We discuss the three-dimensional Navier-Stokes equations in the presence of the axisymmetric linear strain, where the strain rate depends on time in a specific manner. It is known that the system admits solutions which blow up in finite time and whose profiles are in a backward self-similar form of the familiar Burgers vortices. In this talk it is shown that the existing stability theory of the Burgers vortex leads to the stability of these blow-up solutions as well. The secondar y blow-up is also observed when the strain rate is relatively weak. Joint work with Christophe Prange (Universite de Bordeaux) and Hideyuki Miura (Tokyo Institute of Technology).

Comment

Title

Long time behavior of the solutions of the mass-critical nonlinear Klein-Gordon equations

Date

2019年5月22日(水) 16:30〜17:30    (16:00より1階ロビーでtea)

Place

京都大学数理解析研究所 (RIMS) 110号室
(Rm110, Research Institute for Mathematical Sciences, Kyoto University)

Speaker

Xing Cheng 氏 (Hohai University)

Abstract

 In this talk, we will give the scattering of the mass-critical nonlinear Klein-Gordon equations both in the defocusing and focusing case. We establish the linear profile decomposition, then by using the solution of the mass-critical nonlinear Schrodinger equation to approximate the large scale profile, we can prove the scattering result by the concentration-compactness/rigidity method developed by C. E. Kenig and F. Merle.

Comment

大談話会

Title

Scaling limits of random walks on random graphs in critical regimes

Date

2019年5月15日(水) 15:00〜16:00

Place

京都大学数理解析研究所 (RIMS) 420 号室
(Rm420, Research Institute for Mathematical Sciences, Kyoto University)

Speaker

David Croydon 氏 (京大・数理研)

Abstract

 In describing properties of disordered media, physicists have long been interested in the behaviour of random walks on random graphs that arise in statistical mechanics, such as percolation clusters and various models of random trees. Random walks on random graphs are also of interest to computer scientists in studies of complex networks. In 'critical' regimes, many of the canonical models exhibit large-scale fractal behaviour, which mean it is often a challenge to describe their geometric properties, let alone the associated random walks. However, in recent years, the deep connections between electrical networks and stochastic processes have been advanced so that tackling some of the key examples of random walks on random graphs is now within reach. In this talk, I will introduce some recent work in this direction, and describe some prospects for future developments.

Comment 16:00-16:30 110号室にて Tea Break

大談話会

Title

Cluster structures on strata of flag manifolds

Date

2019年5月15日(水) 16:30〜17:30

Place

京都大学数理解析研究所 (RIMS) 420 号室
(Rm420, Research Institute for Mathematical Sciences, Kyoto University)

Speaker

Bernard Leclerc 氏 (京大・数理研 & Université Caen Normandie)

Abstract

 Let $G$ be a simple algebraic group split over $R$, for instance $G = SL(n,R)$. Generalizing the classical notion of totally positive matrices, Lusztig introduced in the 1990's the subset of totally positive (resp. totally nonnegative) elements of $G$. In 1998 he extended this notion to the partial flag manifolds $G/P$, for example the Grassmannians.
 One combinatorial problem arising from this is to find optimal criteria for an element of $G$ (or $G/P$) to be totally positive (resp. totally nonnegative). In 2001, Fomin and Zelevinsky invented the notion of a cluster algebra, motivated in part by this combinatorial problem which they solved completely in the case of $G$.
 After reviewing this story, I will outline some recent progress in the case of $G/P$.

Comment 16:00-16:30 110号室にて Tea Break

Title

Problem of Resolution of Singularities: Past, Present, and Future

Date

2019年5月8日(水) 16:30〜17:30    (16:00より105談話室でtea)

Place

京都大学大学院理学研究科3号館110講演室
(Rm110, Building No.3, Faculty of Science, Kyoto University)

Speaker

Kenji Matsuki 氏 (京大 & Purdue University)

Abstract

[pdf]

Comment

Title

Structure and randomness in II$_1$ factors

Date

2019年4月24日(水) 16:30〜17:30    (16:00より105談話室でtea)

Place

京都大学大学院理学研究科3号館110講演室
(Rm110, Building No.3, Faculty of Science, Kyoto University)

Speaker

Sorin Popa 氏 (京大 & UCLA)

Abstract

[pdf]

Comment

Title

Anomalous diffusions and fractional order differential equations

Date

2019年4月17日(水) 16:30〜17:30    (16:00より1階ロビーでtea)

Place

京都大学数理解析研究所 (RIMS) 110号室
(Rm110, Research Institute for Mathematical Sciences, Kyoto University)

Speaker

Zhen-Qing Chen 氏 (京大 & University of Washington)

Abstract

 Anomalous diffusion phenomenon has been observed in many natural systems, from the signaling of biological cells, to the foraging behavior of animals, to the travel times of contaminants in groundwater. In this talk, I will first discuss the interplay between anomalous sub-diffusions and time-fractional differential equations, including how they arise naturally from limit theorems for random walks. I will then present some recent results in this area, in particular on the probabilistic representation to the solutions of time fractional equations with source terms.

Comment

Title

Schubert calculus and quantum integrability

Date

2019年4月10日(水) 16:30〜17:30    (16:00より105談話室でtea)

Place

京都大学大学院理学研究科3号館110講演室
(Rm110, Building No.3, Faculty of Science, Kyoto University)

Speaker

Paul Zinn-Justin 氏 (The University of Melbourne)

Abstract

 We report on recent progress in the field of Schubert calculus, a classical branch of enumerative geometry, due to its surprising connection to quantum integrable systems. We shall see how the latter provide many explicit combinatorial formulae (``puzzle rules'') for intersection numbers for partial flag varieties, and their generalizations (e.g. in equivariant K-theory). We shall also discuss the connection with the work of Okounkov et al on quantum integrable systems and the equivariant cohomology of Nakajima quiver varieties. This is joint work with A. Knutson (Cornell).

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