これまでのセミナー (2018年度)

第1回（2018年4月25日(水) 16:40--18:10）

講師：奥山和美氏 [信州大学]

題目： Resurgence analysis of 2d Yang-Mills theory on a torus

概要： We study the large $N$ 't Hooft expansion of the partition function of 2d Yang-Mills theory on a torus. We compute the $1/N$ genus expansion of both the chiral and the full partition function of 2d Yang-Mills using the recursion relation found by Kaneko and Zagier. Then we study the large order behavior of this genus expansion, from which we extract the non-perturbative correction using the resurgence relation. It turns out that the genus expansion is not Borel summable and the coefficient of 1-instanton correction, the so-called Stokes parameter, is pure imaginary. We find that the non-perturbative correction obtained from the resurgence is reproduced from a certain analytic continuation of the grand partition function of a system of non-relativistic fermions on a circle. Our analytic continuation is different from that considered in [hep-th/0504221].

第２回（2018年５月２３日（水）16:40--18:10)

講師：坂井哲氏（北海道大学）

場所：立教大学池袋キャンパス 4号館4階4405号室

題目：Hyperscaling for oriented percolation in 1+1 space-time dimensions

概要：Consider nearest-neighbor oriented percolation in d + 1 space–time dimensions. Let $\rho$, $\eta$, $\nu$ be the critical exponents for the survival probability up to time t,the expected number of vertices at time t connected from the space–time origin, and the gyration radius of those vertices, respectively. We prove that the hyperscaling inequality $d\nu \ge \eta + 2\rho$, which holds for all $d \ge 1$ and is a strict inequality above the upper-critical dimension 4, becomes an equality for $d = 1$, i.e., $\nu = \eta + 2\rho$, provided existence of at least two among those exponents. The key to the proof is the recent result on the critical box-crossing property by Duminil-Copin et al (to appear in Probab. Theory Relat. Fields).

第３回（2018年６月６日（水） 16:40-18:10)

講師：沙川貴大氏（東京大学）

場所：立教大学池袋キャンパス 4号館4階4405号室

Title: Eigenstate thermalization and the second law in many-body quantum systems

Abstract: The origin of macroscopic irreversibility under microscopic reversible dynamics is a fundamental problem ever since the seminal thought by Boltzmann. In recent years, this problem has attracted renewed attentions, in light of modern statistical mechanics and experimental progress in ultracold atoms. In particular, it has been recognized that the eigenstate thermalization hypothesis (ETH) plays a crucial role in understanding the mechanism of thermalization in isolated quantum systems. In this talk, starting from a brief review of this topic, I will present our recent results. First, I will talk about a rigorous proof of a weak version of the ETH [1] and systematic numerical verification of its strong version [2]. Next, I will talk about the second law for pure quantum states, where the heat bath is initially in a single energy eigenstate [1]. Our rigorous proof of the second law is based on the ETH and the Lieb-Robinson bound.

[1] E. Iyoda, K. Kaneko, T. Sagawa, Phys. Rev. Lett. 119, 100601 (2017).

[2] T. Yoshizawa, E. Iyoda, T. Sagawa, Phys. Rev. Lett. 120, 200604 (2018).

第4回（2018年7月18日（水） 16:40-18:10)

講師：中野史彦氏（学習院大学）

場所：立教大学池袋キャンパス 4号館4階4405号室

Title: Density of states and level statistics for 1d Schroedinger operators

Abstract: We consider the 1d Schroedinger operator with random potential decaying of order α.

The results include :

(1) the fluctuation of density of states with different behavior depending on α,

(2) the level statistics asymptotically obeys clock, Sineβ, and Poisson processes for super-critical, critical, and sub-critical cases, respectively.

(3) if the time permits, we discuss some recent results on the eigenfunction statistics.

Joint work with S. Kotani and T. K. Duy.

臨時セミナー (2018年9月19日（水）16時40分 – 18時10分)

講師：山根宏之氏（富山大学）

場所：立教大学池袋キャンパス 4号館4階4405号室

Title: Generalized quantum groups, quantum superalgebras and Coxeter groupoids

（一般化された量子群、量子超代数とコクセター亜群）

Abstract: Coxeter introduced Coxeter groups in 1934, and he classified the finite Coxeter groups in 1935. Those are classified into A_n, B_n = C_n, D_n, F₄, E₆, E₇, E₈, G₂, I_n, H₃ and H₄. Coxeter groups appear in many areas of Algebra and Geometry. One of the areas is the representation theory of Lie algebras. Perhaps from 1970's, many researchers have considered that they need a notion of `Coxeter groupoids' which can be applied for study of the representation theory of Lie superalgebras. Recently, Coxeter groupoids became necessary for study of Hopf algebras including `the generalized quantum groups'. Before achieving (1) below, virtually using Coxeter groupoids, Hiroyuki Yamane decided the defining relations of finite and affine Lie superalgebras of type A-G and their quantum superalgebras (1991, 1994, 1994, 1999, 2001). With collaborators, Hiroyuki Yamane achieved the following results.

(1) Axiomatic definition, Defining relations and Matsumoto-type theorem of Coxeter groupoids (2008).

(2) Shapovalov determinants of the generalized quantum groups (2010).

(3) Classification of the finite dimensional irreducible representations of the generalized quantum groups (2015).

(4) Universal R-matrices of the generalized quantum groups (2015).

(5) Harish-Chandra type theorem of the center of the generalized quantum groups (2018).

(6) Bruhat order of the Coxeter groupoids (2018).

(7) Lusztig A-form of the multi-parameter quantum groups (2018).

In the talk, in addition to these topics, I will also explain the contents of the paper: I. Heckenberger, F. Spill, A. Torrielli, H. Yamane, Drinfeld second realization of the quantum affine superalgebras of D⁽¹⁾(2,1;x) via the Weyl groupoid. RIMS Kokyuroku Bessatsu B8 (2008), 171-216.

臨時セミナー (2018年10月10日（水）16時40分 – 18時10分)

講師：庵原謙治氏（リヨン第一大学）

場所：立教大学池袋キャンパス 4号館4階4404号室

Title: Lattice Lie Algebras of Witt type and their Representations

（Witt 型の Lattice Lie 環とその表現）

Abstract:

Lattice Lie 環というのは、Z^N-graded (N>1 は整数)-gradedな Lie 環であって, 各 graded subspace の次元が 1次元のものを言う。

本講演では、

1. Lattice Lie 環の分類について、

2. 特に Witt 代数を部分代数として含むような Lie 環について、その実現及び表現について,

知られている結果を紹介する。

A Lattice Lie algebra is a Lie algebra graded over Z^N for some integer N>1 each of which graded component is 1-dimensional.

In this talk,

1. classification of Lattice Lie algebras, and

2. in particular, for those contaning a graded subalgebra isomrphic to the Witt algebra, their realization and representations,

will be explained.

臨時セミナー (2018年12月19日（水）17時00分—18時30分)

講師：竹村剛一氏 (中央大学)

場所：立教大学池袋キャンパス 4号館4階4405号室

Title: ホインの微分方程式（Heun's differential equation）

Abstract: 超幾何微分方程式は３点 0,1,∞ に確定特異点をもつリーマン球面上の線形二階微分方程式の標準形であり、数学と物理の双方でよく現れるものである。そして、ホインの微分方程式はリーマン球面上の４点に確定特異点をもつ線形二階微分方程式の標準形である。本講演では、ホインの微分方程式についていくつかの側面から解説する。関連する事柄として、アクセサリーパラメーター、パンルヴェ-ホイン対応、q 変形、多項式解、楕円関数による表示とその応用を挙げておく。

It is well known that the hypergeometric equation is a standard form of the linear second order differential equation with three regular singuarities 0,1 and infinity on the Riemann sphere, and it appears frequently in mathematics and physics. Heun's differential equation is a standard form of the linear second order differential equation with four regular singuarities on the Riemann sphere. I explain some aspects of Heun's differential equation which would be concerned with accessory parameter, Painleve-Heun correspondence, q-deformation, polynomial solutions, expression in terms of elliptic function and its application.