## Seminar information archive

Seminar information archive ～08/19｜Today's seminar 08/20 | Future seminars 08/21～

### 2014/11/04

#### Tuesday Seminar on Topology

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

The coarse geometry of Teichmuller space. (ENGLISH)

**Brian Bowditch**(University of Warwick)The coarse geometry of Teichmuller space. (ENGLISH)

[ Abstract ]

We describe some results regarding the coarse geometry of the

Teichmuller space

of a compact surface. In particular, we describe when the Teichmuller

space admits quasi-isometric embeddings of euclidean spaces and

half-spaces.

We also give some partial results regarding the quasi-isometric rigidity

of Teichmuller space. These results are based on the fact that Teichmuller

space admits a ternary operation, natural up to bounded distance

which endows it with the structure of a coarse median space.

We describe some results regarding the coarse geometry of the

Teichmuller space

of a compact surface. In particular, we describe when the Teichmuller

space admits quasi-isometric embeddings of euclidean spaces and

half-spaces.

We also give some partial results regarding the quasi-isometric rigidity

of Teichmuller space. These results are based on the fact that Teichmuller

space admits a ternary operation, natural up to bounded distance

which endows it with the structure of a coarse median space.

#### Seminar on Probability and Statistics

16:30-17:40 Room #052 (Graduate School of Math. Sci. Bldg.)

Conditions for consistency of a log-likelihood-based information criterion in normal multivariate linear regression models under the violation of normality assumption

**YANAGIHARA, Hirokazu**(Graduate School of Science, Hiroshima University)Conditions for consistency of a log-likelihood-based information criterion in normal multivariate linear regression models under the violation of normality assumption

### 2014/10/29

#### Operator Algebra Seminars

16:30-18:00 Room #122 (Graduate School of Math. Sci. Bldg.)

The classification of easy quantum groups (ENGLISH)

**Sven Raum**(RIMS, Kyoto Univ.)The classification of easy quantum groups (ENGLISH)

#### Lie Groups and Representation Theory

16:30-18:00 Room #118 (Graduate School of Math. Sci. Bldg.)

Harmonic analysis on reductive p-adic symmetric spaces. (ENGLISH)

**Patrick Delorme**(UER Scientifique de Luminy Universite d'Aix-Marseille II )Harmonic analysis on reductive p-adic symmetric spaces. (ENGLISH)

[ Abstract ]

In this lecture we will review the Plancherel formula that

we got by looking to neighborhoods at infinity of the

symmetric spaces and then using the method of Sakellaridis-Venkatesh

for spherical varieties for a split group. For us the group

is not necessarily split. We will try to show what questions

are raised by this work for real spherical varieties.

We will present in the last part a joint work with Pascale

Harinck and Yiannis Sakellaridis in which we prove Paley-Wiener

theorems for symmetric spaces.

In this lecture we will review the Plancherel formula that

we got by looking to neighborhoods at infinity of the

symmetric spaces and then using the method of Sakellaridis-Venkatesh

for spherical varieties for a split group. For us the group

is not necessarily split. We will try to show what questions

are raised by this work for real spherical varieties.

We will present in the last part a joint work with Pascale

Harinck and Yiannis Sakellaridis in which we prove Paley-Wiener

theorems for symmetric spaces.

#### Classical Analysis

16:00-17:00 Room #117 (Graduate School of Math. Sci. Bldg.)

Whittaker functions and Barnes-Type Lemmas (ENGLISH)

**Eric Stade**(University of Colorado Boulder)Whittaker functions and Barnes-Type Lemmas (ENGLISH)

[ Abstract ]

In the theory of automorphic forms on GL(n,R), which concerns harmonic analysis and representation theory of this group, certain special functions known as GL(n,R) Whittaker functions play an important role. These Whittaker functions are generalizations of classical Whittaker (or, more specifically, Bessel) functions.

Mellin transforms of products of GL(n,R) Whittaker functions may be expressed as certain Barnes type integrals, or equivalently, as hypergeometric series of unit argument. The general theory of automorphic forms predicts that these Mellin transforms reduce, in certain cases, to products of gamma functions. That this does in fact occur amounts to a whole family of generalizations of the so-called Barnes' Lemma and Barnes' Second Lemma, from the theory of hypergeometric series. We will explore these generalizations in this talk.

This talk will not require any specific knowledge of automorphic forms.

In the theory of automorphic forms on GL(n,R), which concerns harmonic analysis and representation theory of this group, certain special functions known as GL(n,R) Whittaker functions play an important role. These Whittaker functions are generalizations of classical Whittaker (or, more specifically, Bessel) functions.

Mellin transforms of products of GL(n,R) Whittaker functions may be expressed as certain Barnes type integrals, or equivalently, as hypergeometric series of unit argument. The general theory of automorphic forms predicts that these Mellin transforms reduce, in certain cases, to products of gamma functions. That this does in fact occur amounts to a whole family of generalizations of the so-called Barnes' Lemma and Barnes' Second Lemma, from the theory of hypergeometric series. We will explore these generalizations in this talk.

This talk will not require any specific knowledge of automorphic forms.

### 2014/10/28

#### Number Theory Seminar

16:40-18:50 Room #002 (Graduate School of Math. Sci. Bldg.)

A p-adic Labesse-Langlands transfer (English)

Plectic cohomology (English)

**Judith Ludwig**(Imperial college) 16:40-17:40A p-adic Labesse-Langlands transfer (English)

[ Abstract ]

Let B be a definite quaternion algebra over the rationals, G the algebraic group defined by the units in B and H the subgroup of G of norm one elements. Then the classical transfer of automorphic representations from G to H is well understood thanks to Labesse and Langlands, who proved formulas for the multiplicity of irreducible admissible representations of H(adeles) in the discrete automorphic spectrum.

The goal of this talk is to prove a p-adic version of this transfer. By this we mean an extension of the classical transfer to p-adic families of automorphic forms as parametrized by certain rigid analytic spaces called eigenvarieties. We will prove the p-adic transfer by constructing a morphism between eigenvarieties, which agrees with the classical transfer on points corresponding to classical automorphic representations.

Let B be a definite quaternion algebra over the rationals, G the algebraic group defined by the units in B and H the subgroup of G of norm one elements. Then the classical transfer of automorphic representations from G to H is well understood thanks to Labesse and Langlands, who proved formulas for the multiplicity of irreducible admissible representations of H(adeles) in the discrete automorphic spectrum.

The goal of this talk is to prove a p-adic version of this transfer. By this we mean an extension of the classical transfer to p-adic families of automorphic forms as parametrized by certain rigid analytic spaces called eigenvarieties. We will prove the p-adic transfer by constructing a morphism between eigenvarieties, which agrees with the classical transfer on points corresponding to classical automorphic representations.

**Jan Nekovar**(Université Paris 6) 17:50-18:50Plectic cohomology (English)

### 2014/10/27

#### Algebraic Geometry Seminar

14:50-16:20 Room #122 (Graduate School of Math. Sci. Bldg.)

On projective varieties with very large canonical volume (ENGLISH)

**Meng Chen**(Fudan University)On projective varieties with very large canonical volume (ENGLISH)

[ Abstract ]

For any positive integer n>0, a theorem of Hacon-McKernan, Takayama and Tsuji says that there is a constant c(n) so that the m-canonical map is birational onto its image for all smooth projective n-folds and all m>=c(n). We are interested in the following problem "P(n)": is there a constant M(n) so that, for all smooth projective n-fold X with Vol(X)>M(n), the m-canonical map of X is birational for all m>=c(n-1). The answer to “P_n" is positive due to Bombieri when $n=2$ and to Todorov when $n=3$. The aim of this talk is to introduce my joint work with Zhi Jiang from Universite Paris-Sud. We give a positive answer in dimensions 4 and 5.

For any positive integer n>0, a theorem of Hacon-McKernan, Takayama and Tsuji says that there is a constant c(n) so that the m-canonical map is birational onto its image for all smooth projective n-folds and all m>=c(n). We are interested in the following problem "P(n)": is there a constant M(n) so that, for all smooth projective n-fold X with Vol(X)>M(n), the m-canonical map of X is birational for all m>=c(n-1). The answer to “P_n" is positive due to Bombieri when $n=2$ and to Todorov when $n=3$. The aim of this talk is to introduce my joint work with Zhi Jiang from Universite Paris-Sud. We give a positive answer in dimensions 4 and 5.

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

On the minimality of canonically attached singular Hermitian metrics on certain nef line bundles (JAPANESE)

**Takayuki Koike**(University of Tokyo)On the minimality of canonically attached singular Hermitian metrics on certain nef line bundles (JAPANESE)

[ Abstract ]

We apply Ueda theory to a study of singular Hermitian metrics of a (strictly) nef line bundle $L$. Especially we study minimal singular metrics of $L$, metrics of $L$ with the mildest singularities among singular Hermitian metrics of $L$ whose local weights are plurisubharmonic. In some situations, we determine a minimal singular metric of $L$. As an application, we give new examples of (strictly) nef line bundles which admit no smooth Hermitian metric with semi-positive curvature.

We apply Ueda theory to a study of singular Hermitian metrics of a (strictly) nef line bundle $L$. Especially we study minimal singular metrics of $L$, metrics of $L$ with the mildest singularities among singular Hermitian metrics of $L$ whose local weights are plurisubharmonic. In some situations, we determine a minimal singular metric of $L$. As an application, we give new examples of (strictly) nef line bundles which admit no smooth Hermitian metric with semi-positive curvature.

### 2014/10/25

#### Harmonic Analysis Komaba Seminar

13:30-16:30 Room #128 (Graduate School of Math. Sci. Bldg.)

Approximation in Banach space by linear positive operators (JAPANESE)

Local ill-posedness of the Euler equations in a critical Besov space (JAPANESE)

**Yoshihiro Sawano**(Tokyo Metropolitan University) 13:30-14:30Approximation in Banach space by linear positive operators (JAPANESE)

[ Abstract ]

We obtain a sufficient condition for the

convergence of positive linear operators in Banach

function spaces on Rn and derive a Korovkin type

theorem for these spaces. Also, we generalized

this result via statistical sense. This is a joint

work with Professor Arash Ghorbanalizadeh.

We obtain a sufficient condition for the

convergence of positive linear operators in Banach

function spaces on Rn and derive a Korovkin type

theorem for these spaces. Also, we generalized

this result via statistical sense. This is a joint

work with Professor Arash Ghorbanalizadeh.

**Tsuyoshi Yoneda**(Tokyo Institute of Technology) 15:00-16:30Local ill-posedness of the Euler equations in a critical Besov space (JAPANESE)

### 2014/10/22

#### Operator Algebra Seminars

16:30-18:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Free independence in ultraproduct von Neumann algebras and applications (ENGLISH)

**Yusuke Isono**(Kyoto Univ.)Free independence in ultraproduct von Neumann algebras and applications (ENGLISH)

#### Mathematical Biology Seminar

14:50-16:20 Room #122 (Graduate School of Math. Sci. Bldg.)

Fast reaction limit of a system with different reaction terms (JAPANESE)

**Harunori Monobe**(Meiji Institute for Advanced Study of Mathematical Sciences)Fast reaction limit of a system with different reaction terms (JAPANESE)

### 2014/10/21

#### Tuesday Seminar on Topology

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Vanishing theorems for p-local homology of Coxeter groups and their alternating subgroups (JAPANESE)

**Toshiyuki Akita**(Hokkaido University)Vanishing theorems for p-local homology of Coxeter groups and their alternating subgroups (JAPANESE)

[ Abstract ]

Given a prime number $p$, we estimate vanishing ranges of $p$-local homology groups of Coxeter groups (of possibly infinite order) and alternating subgroups of finite reflection groups. Our results generalize those by Nakaoka for symmetric groups and Kleshchev-Nakano and Burichenko for alternating groups. The key ingredient is the equivariant homology of Coxeter complexes.

Given a prime number $p$, we estimate vanishing ranges of $p$-local homology groups of Coxeter groups (of possibly infinite order) and alternating subgroups of finite reflection groups. Our results generalize those by Nakaoka for symmetric groups and Kleshchev-Nakano and Burichenko for alternating groups. The key ingredient is the equivariant homology of Coxeter complexes.

### 2014/10/20

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Kodaira dimension of modular variety of type IV (JAPANESE)

**Shouhei Ma**(Tokyo Institute of Technology)Kodaira dimension of modular variety of type IV (JAPANESE)

#### Numerical Analysis Seminar

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Finite element method with various types of penalty on domain/boundary (ENGLISH)

**Guanyu Zhou**(The University of Tokyo)Finite element method with various types of penalty on domain/boundary (ENGLISH)

[ Abstract ]

We are concerned with several penalty methods (on domain/boundary)

combining with finite element method to solve some partial differential equations. The penalty methods are very useful and widely applied to various problems. For example, to solve the Navier-Stokes equations in moving boundary domain, the finite element method requires to construct the boundary fitted mesh at every times step, which is very time-consuming. The fictitious domain method is proposed to tackle this problem. It is to reformulate the equation to a larger fixed domain, called the fictitious domain, to which we can take a uniform mesh independent on the original moving boundary. The reformulation is based on a penalty method on do- main. Some penalty methods are proposed to approximate the boundary conditions which are not easy to handle with general FEM, such as the slip boundary condition to Stokes/Navier-Stokes equations, the unilateral boundary condition of Signorini’s type to Stokes equations, and so on. It is known that the variational crimes occurs if the finite element spaces or the implementation methods are not chosen properly for slip boundary condition. By introducing a penalty term to the normal component of velocity on slip boundary, we can solve the equations in FEM easily. For the boundary of Signorini’s type, the variational form is an inequality, to which the FEM is not easy to applied. However, we can approximate the variational inequality by a variation equation with penalty term, which can be solve by FEM directly. In above, we introduced several penalty methods with finite element approximation. In this work, we investigate the well-posedness of those penalty method, and obtain the error estimates of penalty; moreover, we consider the penalty methods combining with finite element approximation and show the error estimates.

We are concerned with several penalty methods (on domain/boundary)

combining with finite element method to solve some partial differential equations. The penalty methods are very useful and widely applied to various problems. For example, to solve the Navier-Stokes equations in moving boundary domain, the finite element method requires to construct the boundary fitted mesh at every times step, which is very time-consuming. The fictitious domain method is proposed to tackle this problem. It is to reformulate the equation to a larger fixed domain, called the fictitious domain, to which we can take a uniform mesh independent on the original moving boundary. The reformulation is based on a penalty method on do- main. Some penalty methods are proposed to approximate the boundary conditions which are not easy to handle with general FEM, such as the slip boundary condition to Stokes/Navier-Stokes equations, the unilateral boundary condition of Signorini’s type to Stokes equations, and so on. It is known that the variational crimes occurs if the finite element spaces or the implementation methods are not chosen properly for slip boundary condition. By introducing a penalty term to the normal component of velocity on slip boundary, we can solve the equations in FEM easily. For the boundary of Signorini’s type, the variational form is an inequality, to which the FEM is not easy to applied. However, we can approximate the variational inequality by a variation equation with penalty term, which can be solve by FEM directly. In above, we introduced several penalty methods with finite element approximation. In this work, we investigate the well-posedness of those penalty method, and obtain the error estimates of penalty; moreover, we consider the penalty methods combining with finite element approximation and show the error estimates.

### 2014/10/17

#### Geometry Colloquium

10:00-11:30 Room #126 (Graduate School of Math. Sci. Bldg.)

A finite diameter theorem on RCD spaces (JAPANESE)

**Yu Kitabeppu**(Kyoto University)A finite diameter theorem on RCD spaces (JAPANESE)

[ Abstract ]

I will talk about a finite diameter theorem on RCD spaces of (possibly) infinite dimension. An RCD space is a generalization of a concept of a manifold with bounded Ricci curvature. Savar¥'e proves the "self-improving property" on RCD spaces via the Gamma calculus. Because of his work and Kuwada’s duality argument, we are able to get the L^{¥infty}-contraction of heat kernels. I will show the main result by combining the contraction property and a simple lemma.

I will talk about a finite diameter theorem on RCD spaces of (possibly) infinite dimension. An RCD space is a generalization of a concept of a manifold with bounded Ricci curvature. Savar¥'e proves the "self-improving property" on RCD spaces via the Gamma calculus. Because of his work and Kuwada’s duality argument, we are able to get the L^{¥infty}-contraction of heat kernels. I will show the main result by combining the contraction property and a simple lemma.

### 2014/10/15

#### Operator Algebra Seminars

16:30-18:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Classification of actions of compact abelian groups on subfactors with index less than 4 (ENGLISH)

**Koichi Shimada**(Univ. Tokyo)Classification of actions of compact abelian groups on subfactors with index less than 4 (ENGLISH)

### 2014/10/14

#### Seminar on Mathematics for various disciplines

10:30-11:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Fluid flow and electromagnetic fields, from viewpoint of theoretical physics -- Is the Navier-Stokes Equation sufficient to describe turbulence at very high Reynolds numbers? -- (JAPANESE)

**Tsutomu Kambe**(University of Tokyo)Fluid flow and electromagnetic fields, from viewpoint of theoretical physics -- Is the Navier-Stokes Equation sufficient to describe turbulence at very high Reynolds numbers? -- (JAPANESE)

[ Abstract ]

There exists analogy between the fluid flow and electromagnetic fields with respect to their mathematical representations. This is reasonable because both are continuous physical fields having energy and momentum in space-time. In particular, fluid’s vorticity is analogous to magnetic field.

On the other hand, for simulation of atmospheric global motion on the giant computer Earth Simulator, many empirical physical parameters must be introduced in order to obtain realistic results for weather prediction, etc. This implies that the present system of equations of fluid flows may not be sufficient to describe fluid motions of large scales at very high Reynolds numbers. We consider whether the above-mentioned analogy is useful for improvement of the theory of turbulence at very high Reynolds numbers.

There exists analogy between the fluid flow and electromagnetic fields with respect to their mathematical representations. This is reasonable because both are continuous physical fields having energy and momentum in space-time. In particular, fluid’s vorticity is analogous to magnetic field.

On the other hand, for simulation of atmospheric global motion on the giant computer Earth Simulator, many empirical physical parameters must be introduced in order to obtain realistic results for weather prediction, etc. This implies that the present system of equations of fluid flows may not be sufficient to describe fluid motions of large scales at very high Reynolds numbers. We consider whether the above-mentioned analogy is useful for improvement of the theory of turbulence at very high Reynolds numbers.

#### Number Theory Seminar

17:30-18:30 Room #002 (Graduate School of Math. Sci. Bldg.)

A p-adic criterion for good reduction of curves (ENGLISH)

**Fabrizio Andreatta**(Università Statale di Milano)A p-adic criterion for good reduction of curves (ENGLISH)

[ Abstract ]

Given a curve over a dvr of mixed characteristic 0-p with smooth generic fiber and with semistable reduction, I will present a criterion for good reduction in terms of the (unipotent) p-adic étale fundamental group of its generic fiber.

Given a curve over a dvr of mixed characteristic 0-p with smooth generic fiber and with semistable reduction, I will present a criterion for good reduction in terms of the (unipotent) p-adic étale fundamental group of its generic fiber.

### 2014/10/11

#### Monthly Seminar on Arithmetic of Automorphic Forms

13:30-16:00 Room #123 (Graduate School of Math. Sci. Bldg.)

The "secondary spherical function" of the discrete series representations of $SU(3,1)$ (following the memo of Tadashi Miyazaki) (JAPANESE)

Toward Fourier expansion of automorphic forms on $Sp(2, R)$ (JAPANESE)

**Takayuki Oda**(Grad. School of Math.-Sci., Univ. of Tokyo) 13:30-14:30The "secondary spherical function" of the discrete series representations of $SU(3,1)$ (following the memo of Tadashi Miyazaki) (JAPANESE)

**Hideshi Takayanagi**(Sakushin-Gakuin University) 15:00-16:00Toward Fourier expansion of automorphic forms on $Sp(2, R)$ (JAPANESE)

### 2014/10/10

#### Colloquium

16:30-17:30 Room #002 (Graduate School of Math. Sci. Bldg.)

Etale cohomology of local Shimura varieties and the local Langlands correspondence (JAPANESE)

**Yoichi Mieda**(Graduate School of Mathematical Sciences, University of Tokyo)Etale cohomology of local Shimura varieties and the local Langlands correspondence (JAPANESE)

### 2014/10/08

#### Operator Algebra Seminars

16:30-18:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Realization of hyperbolic group $C^*$-algebras as decreasing intersection of Cuntz algebras $O_2$ (ENGLISH)

**Yuhei Suzuki**(Univ. Tokyo/RIMS, Kyoto Univ.)Realization of hyperbolic group $C^*$-algebras as decreasing intersection of Cuntz algebras $O_2$ (ENGLISH)

#### Mathematical Biology Seminar

14:50-16:20 Room #128 (Graduate School of Math. Sci. Bldg.)

Qualitative analysis of disease transmission dynamics for renewal equations (JAPANESE)

**Yoichi Enatsu**(Graduate School of Math. Sci. Bldg.)Qualitative analysis of disease transmission dynamics for renewal equations (JAPANESE)

### 2014/10/07

#### Tuesday Seminar on Topology

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Transversality problems in string topology and de Rham chains (JAPANESE)

**Kei Irie**(RIMS, Kyoto University)Transversality problems in string topology and de Rham chains (JAPANESE)

[ Abstract ]

The starting point of string topology is the work of Chas-Sullivan, which uncovered the Batalin-Vilkovisky(BV) structure on homology of the free loop space of a manifold.

It is important to define chain level structures beneath the BV structure on homology, however this problem is yet to be settled.

One of difficulties is that, to define intersection products on chain level, we have to address the transversality issue.

In this talk, we introduce a notion of "de Rham chain" to bypass this trouble, and partially realize expected chain level structures.

The starting point of string topology is the work of Chas-Sullivan, which uncovered the Batalin-Vilkovisky(BV) structure on homology of the free loop space of a manifold.

It is important to define chain level structures beneath the BV structure on homology, however this problem is yet to be settled.

One of difficulties is that, to define intersection products on chain level, we have to address the transversality issue.

In this talk, we introduce a notion of "de Rham chain" to bypass this trouble, and partially realize expected chain level structures.

### 2014/10/03

#### Infinite Analysis Seminar Tokyo

13:30-17:00 Room #002 (Graduate School of Math. Sci. Bldg.)

KPZ equation and Macdonald process (JAPANESE)

Entanglement spectra in topological phases and coupled Tomonaga-Luttinger liquids (JAPANESE)

**Tomohiro SASAMOTO**(Department of Physics, Tokyo Institute of Technology) 13:30-15:00KPZ equation and Macdonald process (JAPANESE)

**Shunsuke FURUKAWA**(Department of Physics, the Tokyo University) 15:30-17:00Entanglement spectra in topological phases and coupled Tomonaga-Luttinger liquids (JAPANESE)

[ Abstract ]

The entanglement spectrum (ES) has been found to provide useful probes of topological phases of matter and other exotic strongly correlated states. For the system's ground state, the ES is defined as the full eigenvalue spectrum of the reduced density matrix obtained by tracing out the degrees of freedom in part of the system. A key result observed in various topological phases and other gapped systems has been the remarkable correspondence between the ES and the edge-state spectrum. While this correspondence has been analytically proven for some topological phases, it is interesting to ask what systems show this correspondence more generally and how the ES changes when the bulk energy gap closes.

We here study the ES in two coupled Tomonaga-Luttinger liquids (TLLs) on parallel periodic chains. In addition to having direct applications to ladder systems, this problem is closely related to the entanglement properties of two-dimensional topological phases. Based on the calculation for coupled chiral TLLs, we provide a simple physical proof for the correspondence between edge states and the ES in quantum Hall systems consistent with previous numerical and analytical studies. We also discuss violations of this correspondence in gapped and gapless phases of coupled non-chiral TLLs.

Reference: R. Lundgren, Y. Fuji, SF, and M. Oshikawa, Phys. Rev. B 88, 245137 (2013).

The entanglement spectrum (ES) has been found to provide useful probes of topological phases of matter and other exotic strongly correlated states. For the system's ground state, the ES is defined as the full eigenvalue spectrum of the reduced density matrix obtained by tracing out the degrees of freedom in part of the system. A key result observed in various topological phases and other gapped systems has been the remarkable correspondence between the ES and the edge-state spectrum. While this correspondence has been analytically proven for some topological phases, it is interesting to ask what systems show this correspondence more generally and how the ES changes when the bulk energy gap closes.

We here study the ES in two coupled Tomonaga-Luttinger liquids (TLLs) on parallel periodic chains. In addition to having direct applications to ladder systems, this problem is closely related to the entanglement properties of two-dimensional topological phases. Based on the calculation for coupled chiral TLLs, we provide a simple physical proof for the correspondence between edge states and the ES in quantum Hall systems consistent with previous numerical and analytical studies. We also discuss violations of this correspondence in gapped and gapless phases of coupled non-chiral TLLs.

Reference: R. Lundgren, Y. Fuji, SF, and M. Oshikawa, Phys. Rev. B 88, 245137 (2013).

### 2014/10/02

#### Operator Algebra Seminars

16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Entropy and $L^2$-torsion (ENGLISH)

**Hanfeng Li**(SUNY Buffalo)Entropy and $L^2$-torsion (ENGLISH)

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