## Colloquium

Title

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Cluster integrable systems
**

Date

2019.7.10 (Wed) 16:30-17:30 (16:00- tea)

Place

Rm110, Research Institute for Mathematical Sciences, Kyoto University

Speaker

Michael Gekhtman (RIMS, Kyoto University & 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.

Title

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Homotopy theory of $A_n$-spaces in Lie groups
**

Date

2019.6.26 (Wed) 16:30-17:30 (16:00- tea)

Place

Rm110, Building No.3, Faculty of Science, Kyoto University

Speaker

Mitsunobu Tsutaya (Kyushu University)

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]

Title

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Construction of symplectic field theory and smoothness of Kuranishi structure
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Date

2019.6.19 (Wed) 16:30-17:30 (16:00- tea)

Place

Rm110, Research Institute for Mathematical Sciences, Kyoto University

Speaker

Suguru Ishikawa (RIMS, Kyoto University)

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.

Title

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Nonbacktracking spectrum of random matrices
**

Date

2019.6.12 (Wed) 16:30-17:30 (16:00- tea)

Place

[changed] Rm127, Building No.3, Faculty of Science, Kyoto University

Speaker

Charles Bordenave (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.

Title

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Vanishing of open Jacobi diagrams with odd legs
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Date

2019.6.5 (Wed) 16:30-17:30 (16:00- tea)

Place

Rm110, Research Institute for Mathematical Sciences, Kyoto University

Speaker

Katsumi Ishikawa (RIMS, Kyoto University)

Title

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

Date

2019.5.29 (Wed) 16:30-17:30 (16:00- tea)

Place

Rm110, Building No.3, Faculty of Science, Kyoto University

Speaker

Yasunori Maekawa (Kyoto University)

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).

Title

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

Date

2019.5.22 (Wed) 16:30-17:30 (16:00- tea)

Place

Rm110, Research Institute for Mathematical Sciences, Kyoto University

Speaker

Xing Cheng (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.

Title

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Scaling limits of random walks on random graphs in critical regimes
**

Date

2019.5.15 (Wed) 15:00-16:00

Place

Rm420, Research Institute for Mathematical Sciences, Kyoto University

Speaker

David Croydon (RIMS, Kyoto University)

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.

Title

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Cluster structures on strata of flag manifolds
**

Date

2019.5.15 (Wed) 16:30-17:30

Place

Rm420, Research Institute for Mathematical Sciences, Kyoto University

Speaker

Bernard Leclerc (RIMS, Kyoto University & 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 $G/P$.

Title

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

Date

2019.5.8 (Wed) 16:30-17:30 (16:00- tea)

Place

Rm110, Building No.3, Faculty of Science, Kyoto University

Speaker

Kenji Matsuki (Kyoto University & Purdue University)

Abstract

Title

**
Structure and randomness in II$_1$ factors
**

Date

2019.4.24 (Wed) 16:30-17:30 (16:00- tea)

Place

Rm110, Building No.3, Faculty of Science, Kyoto University

Speaker

Sorin Popa (Kyoto University & UCLA)

Abstract

Title

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Anomalous diffusions and fractional order differential equations
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Date

2019.4.17 (Wed) 16:30-17:30 (16:00- tea)

Place

Rm110, Research Institute for Mathematical Sciences, Kyoto University

Speaker

Zhen-Qing Chen (Kyoto University & 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.

Title

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Schubert calculus and quantum integrability
**

Date

2019.4.10 (Wed) 16:30-17:30 (16:00- tea)

Place

Rm110, Building No.3, Faculty of Science, Kyoto University

Speaker

Paul Zinn-Justin (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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