## Seminar information archive

Seminar information archive ～09/10｜Today's seminar 09/11 | Future seminars 09/12～

### 2012/10/17

#### Geometry Colloquium

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

p-Kazhdan constants and non-expanders (JAPANESE)

**Masato Mimura**(Tohoku University)p-Kazhdan constants and non-expanders (JAPANESE)

[ Abstract ]

In study of graphs and finitely generated groups (as Cayley graphs) as metric spaces with the path metrics, one basic idea is to "linearize" them, more precisely, to embed them into certain Banach spaces in some nice way. Special attention has been paid to embeddings of graphs into Hilbert spaces or l^p spaces. It is a well-known result that a "family of expanders", namely, a family of finite graphs (of unifromly bounded degree) with uniform lower bound of spectral gaps (equivalently, of Cheeger constants), does not coarsely embed into Hilbert spaces, or l^p spaces.

In this talk, we investigate a "family of NON-expanders" coming from Cayley graphs of a family of finitely generated groups. In this setting we define l^p-version of the Kazhdan constant and of the property tau constant for groups, and study the decay rate of p-spectral gap of non-expanders in terms of them. This gives some metric geometrical information on the family. Our main example will be the family of (Cayley graphs of SL_n(Z/k_nZ)), indexed by n>2, for (k_n)_n a sequence of natural numbers>2 and with respect to standard 4-element generating sets. We will start from basic definitions, such as ones of Cayley graphs, expander families, and Kazhdan constants.

In study of graphs and finitely generated groups (as Cayley graphs) as metric spaces with the path metrics, one basic idea is to "linearize" them, more precisely, to embed them into certain Banach spaces in some nice way. Special attention has been paid to embeddings of graphs into Hilbert spaces or l^p spaces. It is a well-known result that a "family of expanders", namely, a family of finite graphs (of unifromly bounded degree) with uniform lower bound of spectral gaps (equivalently, of Cheeger constants), does not coarsely embed into Hilbert spaces, or l^p spaces.

In this talk, we investigate a "family of NON-expanders" coming from Cayley graphs of a family of finitely generated groups. In this setting we define l^p-version of the Kazhdan constant and of the property tau constant for groups, and study the decay rate of p-spectral gap of non-expanders in terms of them. This gives some metric geometrical information on the family. Our main example will be the family of (Cayley graphs of SL_n(Z/k_nZ)), indexed by n>2, for (k_n)_n a sequence of natural numbers>2 and with respect to standard 4-element generating sets. We will start from basic definitions, such as ones of Cayley graphs, expander families, and Kazhdan constants.

#### Lectures

15:00-16:00 Room #370 (Graduate School of Math. Sci. Bldg.)

Inverse problems for differential operators on spatial networks (ENGLISH)

**Vjacheslav Yurko**(Saratov University)Inverse problems for differential operators on spatial networks (ENGLISH)

### 2012/10/16

#### Tuesday Seminar on Topology

17:10-18:10 Room #056 (Graduate School of Math. Sci. Bldg.)

Analytic torsion of log-Enriques surfaces (JAPANESE)

**Ken-Ichi Yoshikawa**(Kyoto University)Analytic torsion of log-Enriques surfaces (JAPANESE)

[ Abstract ]

Log-Enriques surfaces are rational surfaces with nowhere vanishing

pluri-canonical forms. We report the recent progress on the computation

of analytic torsion of log-Enriques surfaces.

Log-Enriques surfaces are rational surfaces with nowhere vanishing

pluri-canonical forms. We report the recent progress on the computation

of analytic torsion of log-Enriques surfaces.

### 2012/10/15

#### Seminar on Geometric Complex Analysis

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

An L^2 estimate on domains and application to Levi-flat surfaces (JAPANESE)

**Takeo Ohsawa**(Nagoya University)An L^2 estimate on domains and application to Levi-flat surfaces (JAPANESE)

#### Algebraic Geometry Seminar

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

On the moduli b-divisors of lc-trivial fibrations (JAPANESE)

**Yoshinori Gongyo**(University of Tokyo)On the moduli b-divisors of lc-trivial fibrations (JAPANESE)

[ Abstract ]

Roughly speaking, by using the semi-stable minimal model program, we prove that the moduli part of an lc-trivial fibration coincides with that of a klt-trivial fibration induced by adjunction after taking a suitable generically finite cover. As an application, we obtain that the moduli part of an lc-trivial fibration is b-nef and abundant by Ambro's result on klt-trivial fibrations. Moreover I may explain some applications of canonical bundle formulas. These are joint works with Osamu Fujino.

Roughly speaking, by using the semi-stable minimal model program, we prove that the moduli part of an lc-trivial fibration coincides with that of a klt-trivial fibration induced by adjunction after taking a suitable generically finite cover. As an application, we obtain that the moduli part of an lc-trivial fibration is b-nef and abundant by Ambro's result on klt-trivial fibrations. Moreover I may explain some applications of canonical bundle formulas. These are joint works with Osamu Fujino.

### 2012/10/12

#### Colloquium

16:30-17:30 Room #123 (Graduate School of Math. Sci. Bldg.)

Homogenization on arbitrary manifolds (ENGLISH)

**Antonio Siconolfi**(La Sapienza - University of Rome)Homogenization on arbitrary manifolds (ENGLISH)

[ Abstract ]

We show that results on periodic homogenization for Hamilton-Jacobi equations can be generalized replacing the torus by an arbitrary compact manifold. This allows to reach a deeper understanding of the matter and unveils phenomena somehow hidden in the periodic case, for instance the fact that the ambient spaces of oscillating equations and that of the limit problem are different, and possess even different dimensions. Repetition structure for the base manifold, changes of scale in it and asymptotic analysis, which are the basic ingredients of homogenization, need substantial modification to work in the new frame, and this task is partially accomplished using tools from algebraic topology. An adapted notion of convergence allowing approximating entities and limit to lie in different spaces is also provided.

We show that results on periodic homogenization for Hamilton-Jacobi equations can be generalized replacing the torus by an arbitrary compact manifold. This allows to reach a deeper understanding of the matter and unveils phenomena somehow hidden in the periodic case, for instance the fact that the ambient spaces of oscillating equations and that of the limit problem are different, and possess even different dimensions. Repetition structure for the base manifold, changes of scale in it and asymptotic analysis, which are the basic ingredients of homogenization, need substantial modification to work in the new frame, and this task is partially accomplished using tools from algebraic topology. An adapted notion of convergence allowing approximating entities and limit to lie in different spaces is also provided.

### 2012/10/09

#### Tuesday Seminar on Topology

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

The growth series of pure Artin groups of dihedral type (JAPANESE)

**Michihiko Fujii**(Kyoto University)The growth series of pure Artin groups of dihedral type (JAPANESE)

[ Abstract ]

In this talk, I consider the kernel of the natural projection from

the Artin group of dihedral type to the corresponding Coxeter group,

that we call a pure Artin group of dihedral type,

and present rational function expressions for both the spherical and

geodesic growth series

of the pure Artin group of dihedral type with respect to a natural

generating set.

Also, I show that their growth rates are Pisot numbers.

This talk is partially based on a joint work with Takao Satoh.

In this talk, I consider the kernel of the natural projection from

the Artin group of dihedral type to the corresponding Coxeter group,

that we call a pure Artin group of dihedral type,

and present rational function expressions for both the spherical and

geodesic growth series

of the pure Artin group of dihedral type with respect to a natural

generating set.

Also, I show that their growth rates are Pisot numbers.

This talk is partially based on a joint work with Takao Satoh.

#### Numerical Analysis Seminar

16:30-18:00 Room #002 (Graduate School of Math. Sci. Bldg.)

On the numerical verification method for parabolic problems (JAPANESE)

[ Reference URL ]

http://www.infsup.jp/utnas/

**Takuma Kimura**(Waseda University)On the numerical verification method for parabolic problems (JAPANESE)

[ Reference URL ]

http://www.infsup.jp/utnas/

### 2012/10/06

#### Harmonic Analysis Komaba Seminar

13:30-17:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Method of rotations with weight for nonisotropic dilations (JAPANESE)

TBA (JAPANESE)

**Shuichi Sato**(Kanazawa university) 13:30-15:00Method of rotations with weight for nonisotropic dilations (JAPANESE)

**( ) 15:30-17:00**TBA (JAPANESE)

### 2012/10/05

#### Seminar on Probability and Statistics

14:50-16:00 Room #006 (Graduate School of Math. Sci. Bldg.)

Quasi-likelihood analysis for stochastic regression models from nonsynchronous observations (JAPANESE)

http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2012/07.html

**OGIHARA, Teppei**(Center for the Study of Finance and Insurance, Osaka University)Quasi-likelihood analysis for stochastic regression models from nonsynchronous observations (JAPANESE)

[ Abstract ]

高頻度金融時系列データの解析時に, 二資産価格データの共変動を解析する上での問題として

"観測の非同期性"がある. データの線形補完や直前データを用いた補完などによるシンプルな

"同期化"を行ったデータに対する共分散推定量は深刻なバイアスが存在することが知られている.

Hayashi and Yoshida (2005)では, 非同期観測下での共分散のノンパラメトリックな不偏推定量を提案し,

推定量の一致性, 漸近(混合)正規性などを示している.

本発表ではパラメータ付2次元拡散過程の非同期観測の問題に対する, 尤度解析を用いたアプローチを紹介し,

最尤型推定量, ベイズ型推定量の構築とその一致性, 漸近混合正規性に関する結果を紹介する.

[ Reference URL ]高頻度金融時系列データの解析時に, 二資産価格データの共変動を解析する上での問題として

"観測の非同期性"がある. データの線形補完や直前データを用いた補完などによるシンプルな

"同期化"を行ったデータに対する共分散推定量は深刻なバイアスが存在することが知られている.

Hayashi and Yoshida (2005)では, 非同期観測下での共分散のノンパラメトリックな不偏推定量を提案し,

推定量の一致性, 漸近(混合)正規性などを示している.

本発表ではパラメータ付2次元拡散過程の非同期観測の問題に対する, 尤度解析を用いたアプローチを紹介し,

最尤型推定量, ベイズ型推定量の構築とその一致性, 漸近混合正規性に関する結果を紹介する.

http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2012/07.html

### 2012/10/02

#### Tuesday Seminar on Topology

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

Geometric flows and their self-similar solutions

(JAPANESE)

**Akito Futaki**(The University of Tokyo)Geometric flows and their self-similar solutions

(JAPANESE)

[ Abstract ]

In the first half of this expository talk we consider the Ricci flow and its self-similar solutions,

namely the Ricci solitons. We then specialize in the K\\"ahler case and discuss on the K\\"ahler-Einstein

problem. In the second half of this talk we consider the mean curvature flow and its self-similar

solutions, and see common aspects of the two geometric flows.

In the first half of this expository talk we consider the Ricci flow and its self-similar solutions,

namely the Ricci solitons. We then specialize in the K\\"ahler case and discuss on the K\\"ahler-Einstein

problem. In the second half of this talk we consider the mean curvature flow and its self-similar

solutions, and see common aspects of the two geometric flows.

### 2012/10/01

#### Algebraic Geometry Seminar

13:30-15:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Weak Lefschetz for divisors (ENGLISH)

**Robert Laterveer**(CNRS, IRMA, Université de Strasbourg)Weak Lefschetz for divisors (ENGLISH)

[ Abstract ]

Let $X$ be a complex projective variety (possibly singular), and $Y\\subset X$ a generic hyperplane section. We prove several weak Lefschetz results concerning the restriction $A^1(X)_{\\qq}\\to A^1(Y)_{\\qq}$, where $A^1$ denotes Fulton--MacPherson's operational Chow cohomology group. In addition, we reprove (and slightly extend) a weak Lefschetz result concerning the Chow group of Weil divisors first proven by Ravindra and Srinivas. As an application of these weak Lefschetz results, we can say something about when the natural map from the Picard group to $A^1$ is an isomorphism.

Let $X$ be a complex projective variety (possibly singular), and $Y\\subset X$ a generic hyperplane section. We prove several weak Lefschetz results concerning the restriction $A^1(X)_{\\qq}\\to A^1(Y)_{\\qq}$, where $A^1$ denotes Fulton--MacPherson's operational Chow cohomology group. In addition, we reprove (and slightly extend) a weak Lefschetz result concerning the Chow group of Weil divisors first proven by Ravindra and Srinivas. As an application of these weak Lefschetz results, we can say something about when the natural map from the Picard group to $A^1$ is an isomorphism.

#### Algebraic Geometry Seminar

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Frobenius morphisms and derived categories on two dimensional toric Deligne--Mumford stacks (JAPANESE)

**Ryo Ohkawa**(RIMS, Kyoto University)Frobenius morphisms and derived categories on two dimensional toric Deligne--Mumford stacks (JAPANESE)

[ Abstract ]

For a toric Deligne-Mumford (DM) stack over the complex number field, we can consider a certain generalization of the Frobenius endomorphism. For such an endomorphism of a two-dimensional toric DM stack, we show that the push-forward of the structure sheaf generates the bounded derived category of coherent sheaves on the stack. This is joint work with Hokuto Uehara.

For a toric Deligne-Mumford (DM) stack over the complex number field, we can consider a certain generalization of the Frobenius endomorphism. For such an endomorphism of a two-dimensional toric DM stack, we show that the push-forward of the structure sheaf generates the bounded derived category of coherent sheaves on the stack. This is joint work with Hokuto Uehara.

### 2012/09/20

#### Applied Analysis

16:00-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)

Fusco-Rocha meanders: from Temperley-Lieb algebras to black holes

(ENGLISH)

**Bernold Fiedler**(Free University of Berlin)Fusco-Rocha meanders: from Temperley-Lieb algebras to black holes

(ENGLISH)

[ Abstract ]

Fusco and Rocha studied Neumann boundary value problems for ODEs of second order via a shooting approach. They introduced the notion of what we now call Sturm permutation. These permutation relate, on the one hand, to a special class of meandering curves as introduced by Arnol'd in a singularity context. On the other hand, their special class became central in the study of global attractors of parabolic PDEs of Sturm type.

We discuss relations of Fusco-Rocha meanders with further areas: the multiplicative and trace structure in Temperley-Lieb algebras, discrete versions of Cartesian billiards, and the problem of constructing initial conditions for black hole dynamics which satisfy the Einstein constraints. We also risk a brief glimpse at the long and meandric history of meander patterns themselves.

This is joint work with Juliette Hell, Brian Smith, Carlos Rocha, Pablo Castaneda, and Matthias Wolfrum.

Fusco and Rocha studied Neumann boundary value problems for ODEs of second order via a shooting approach. They introduced the notion of what we now call Sturm permutation. These permutation relate, on the one hand, to a special class of meandering curves as introduced by Arnol'd in a singularity context. On the other hand, their special class became central in the study of global attractors of parabolic PDEs of Sturm type.

We discuss relations of Fusco-Rocha meanders with further areas: the multiplicative and trace structure in Temperley-Lieb algebras, discrete versions of Cartesian billiards, and the problem of constructing initial conditions for black hole dynamics which satisfy the Einstein constraints. We also risk a brief glimpse at the long and meandric history of meander patterns themselves.

This is joint work with Juliette Hell, Brian Smith, Carlos Rocha, Pablo Castaneda, and Matthias Wolfrum.

### 2012/09/15

#### Monthly Seminar on Arithmetic of Automorphic Forms

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

Matrix coefficients of the large discrete series of SU(2,1) and SU(3,1) (JAPANESE)

Strict positivity of the central values of some Rankin L-functions of GSp(1,1) and special values of hypergeometric functions (JAPANESE)

**Tadashi Miyazaki**(Kitazato University) 13:30-14:30Matrix coefficients of the large discrete series of SU(2,1) and SU(3,1) (JAPANESE)

**Hiro-aki Narita**(Kumamoto University) 15:00-16:00Strict positivity of the central values of some Rankin L-functions of GSp(1,1) and special values of hypergeometric functions (JAPANESE)

[ Abstract ]

We discuss the strict positivity of the central values of certain convolution type L-functions for several theta lifts to GSp(1,1). Such strict positivity is closely related to special values of some hypergeometric functions.

We discuss the strict positivity of the central values of certain convolution type L-functions for several theta lifts to GSp(1,1). Such strict positivity is closely related to special values of some hypergeometric functions.

### 2012/09/04

#### Tuesday Seminar on Topology

17:00-18:00 Room #002 (Graduate School of Math. Sci. Bldg.)

Poincare inequalities, rigid groups and applications (ENGLISH)

**Piotr Nowak**(the Institute of Mathematics, Polish Academy of Sciences)Poincare inequalities, rigid groups and applications (ENGLISH)

[ Abstract ]

Kazhdan’s property (T) for a group G can be expressed as a

fixed point property for affine isometric actions of G on a Hilbert

space. This definition generalizes naturally to other normed spaces. In

this talk we will focus on the spectral (aka geometric) method for

proving property (T), based on the work of Garland and studied earlier

by Pansu, Zuk, Ballmann-Swiatkowski, Dymara-Januszkiewicz

(“lambda_1>1/2” conditions) and we generalize it to to the setting of

all reflexive Banach spaces.

As applications we will show estimates of the conformal dimension of the

boundary of random hyperbolic groups in the Gromov density model and

present progress on Shalom’s conjecture on vanishing of 1-cohomology

with coefficients in uniformly bounded representations on Hilbert spaces.

Kazhdan’s property (T) for a group G can be expressed as a

fixed point property for affine isometric actions of G on a Hilbert

space. This definition generalizes naturally to other normed spaces. In

this talk we will focus on the spectral (aka geometric) method for

proving property (T), based on the work of Garland and studied earlier

by Pansu, Zuk, Ballmann-Swiatkowski, Dymara-Januszkiewicz

(“lambda_1>1/2” conditions) and we generalize it to to the setting of

all reflexive Banach spaces.

As applications we will show estimates of the conformal dimension of the

boundary of random hyperbolic groups in the Gromov density model and

present progress on Shalom’s conjecture on vanishing of 1-cohomology

with coefficients in uniformly bounded representations on Hilbert spaces.

### 2012/08/29

#### thesis presentations

10:30-11:45 Room #123 (Graduate School of Math. Sci. Bldg.)

Studies on the asymptotic invariants of cohomology groups and the positivity in complex geometry (JAPANESE)

**Shinichi MATSUMURA**(Graduate School of Mathematical Sciences the University of Tokyo)Studies on the asymptotic invariants of cohomology groups and the positivity in complex geometry (JAPANESE)

### 2012/08/20

#### thesis presentations

11:00-12:15 Room #128 (Graduate School of Math. Sci. Bldg.)

The variational formulation of the fully parabolic Keller-Segel system with degenerate diffusion (JAPANESE)

**Yoshifumi MIMURA**(Graduate School of Mathematical Sciences the University of Tokyo)The variational formulation of the fully parabolic Keller-Segel system with degenerate diffusion (JAPANESE)

### 2012/07/30

#### Algebraic Geometry Seminar

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Log Bend-and-Break on Deligne-Mumford stacks (ENGLISH)

**Gianluca Pacienza**(Université de Strasbourg)Log Bend-and-Break on Deligne-Mumford stacks (ENGLISH)

[ Abstract ]

We prove a logarithmic Bend-and-Break lemma on a LCI Deligne-Mumford stacks with projective moduli space and integral boundary divisor. As a by-product we obtain a logarithmic version of the Miyaoka-Mori numerical criterion of uniruledness for DM stacks (under additional conditions on the boundary and on the non-schematic locus) and a Cone Theorem for Deligne-Mumford stacks with boundary. These results hold on an algebraically closed field of any characteristic. This is joint work with Michael McQuillan.

We prove a logarithmic Bend-and-Break lemma on a LCI Deligne-Mumford stacks with projective moduli space and integral boundary divisor. As a by-product we obtain a logarithmic version of the Miyaoka-Mori numerical criterion of uniruledness for DM stacks (under additional conditions on the boundary and on the non-schematic locus) and a Cone Theorem for Deligne-Mumford stacks with boundary. These results hold on an algebraically closed field of any characteristic. This is joint work with Michael McQuillan.

### 2012/07/27

#### Seminar on Probability and Statistics

14:00-17:00 Room #006 (Graduate School of Math. Sci. Bldg.)

General approach to reinforcement learning based on statistical inference (JAPANESE)

[ Reference URL ]

http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2012/06.html

**UENO, Tsuyoshi**(Minato Discrete Structure Manipulation System Project, Japan Science and Technology Agency)General approach to reinforcement learning based on statistical inference (JAPANESE)

[ Reference URL ]

http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2012/06.html

### 2012/07/25

#### GCOE lecture series

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

A survey of recent results on the classification of C*-algebras (ENGLISH)

**George Elliott**(University of Toronto)A survey of recent results on the classification of C*-algebras (ENGLISH)

### 2012/07/24

#### Tuesday Seminar on Topology

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

Orthospectra and identities (ENGLISH)

**Greg McShane**(Institut Fourier, Grenoble)Orthospectra and identities (ENGLISH)

[ Abstract ]

The orthospectra of a hyperbolic manifold with geodesic

boundary consists of the lengths of all geodesics perpendicular to the

boundary.

We discuss the properties of the orthospectra, asymptotics, multiplicity

and identities due to Basmajian, Bridgeman and Calegari. We will give

a proof that the identities of Bridgeman and Calegari are the same.

The orthospectra of a hyperbolic manifold with geodesic

boundary consists of the lengths of all geodesics perpendicular to the

boundary.

We discuss the properties of the orthospectra, asymptotics, multiplicity

and identities due to Basmajian, Bridgeman and Calegari. We will give

a proof that the identities of Bridgeman and Calegari are the same.

#### Lie Groups and Representation Theory

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

The Dynkin index and conformally invariant systems of Heisenberg parabolic type (ENGLISH)

**Toshihisa Kubo**(the University of Tokyo)The Dynkin index and conformally invariant systems of Heisenberg parabolic type (ENGLISH)

[ Abstract ]

Recently, Barchini-Kable-Zierau systematically constructed conformally invariant systems of differential operators using Heisenberg parabolic subalgebras. When they built such systems, two constants, which are defined as the constant of proportionality between two expressions,played an important role. In this talk we give concrete and uniform expressions for these constants. To do so the Dynkin index of a finite dimensional representation of a complex simple Lie algebra plays a key role.

Recently, Barchini-Kable-Zierau systematically constructed conformally invariant systems of differential operators using Heisenberg parabolic subalgebras. When they built such systems, two constants, which are defined as the constant of proportionality between two expressions,played an important role. In this talk we give concrete and uniform expressions for these constants. To do so the Dynkin index of a finite dimensional representation of a complex simple Lie algebra plays a key role.

### 2012/07/23

#### Algebraic Geometry Seminar

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Derived category of smooth proper Deligne-Mumford stack with p_g>0 (JAPANESE)

**Shinnosuke Okawa**(University of Tokyo)Derived category of smooth proper Deligne-Mumford stack with p_g>0 (JAPANESE)

[ Abstract ]

Semiorthogonal decomposition (SOD) of the derived category of coherent sheaves reflects interesting geometry of varieties (more generally stacks), such as minimal model program. We show that the global sections of the canonical line bundle (if exists) give restrictions on the possible form of SODs. As a special case, we see that the global generation of the canonical line bundle implies the non-existence of SODs. (joint work with Kotaro Kawatani)

Semiorthogonal decomposition (SOD) of the derived category of coherent sheaves reflects interesting geometry of varieties (more generally stacks), such as minimal model program. We show that the global sections of the canonical line bundle (if exists) give restrictions on the possible form of SODs. As a special case, we see that the global generation of the canonical line bundle implies the non-existence of SODs. (joint work with Kotaro Kawatani)

#### Lectures

16:30-17:30 Room #126 (Graduate School of Math. Sci. Bldg.)

Combinatorial Ergodicity (ENGLISH)

**Thomas W. Roby**(University of Connecticut)Combinatorial Ergodicity (ENGLISH)

[ Abstract ]

Many cyclic actions $\\tau$ on a finite set $S$ of

combinatorial objects, along with many natural

statistics $\\phi$ on $S$, exhibit``combinatorial ergodicity'':

the average of $\\phi$ over each $\\tau$-orbit in $S$ is

the same as the average of $\\phi$ over the whole set $S$.

One example is the case where $S$ is the set of

length $n$ binary strings $a_{1}\\dots a_{n}$

with exactly $k$ 1's,

$\\tau$ is the map that cyclically rotates them,

and $\\phi$ is the number of \\textit{inversions}

(i.e, pairs $(a_{i},a_{j})=(1,0)$ for $iJ$ less than $j$).

This phenomenon was first noticed by Panyushev

in 2007 in the context of antichains in root posets;

Armstrong, Stump, and Thomas proved his

conjecture in 2011.

We describe a theoretical framework for results of this kind,

and discuss old and new results for products of two chains.

This is joint work with Jim Propp.

Many cyclic actions $\\tau$ on a finite set $S$ of

combinatorial objects, along with many natural

statistics $\\phi$ on $S$, exhibit``combinatorial ergodicity'':

the average of $\\phi$ over each $\\tau$-orbit in $S$ is

the same as the average of $\\phi$ over the whole set $S$.

One example is the case where $S$ is the set of

length $n$ binary strings $a_{1}\\dots a_{n}$

with exactly $k$ 1's,

$\\tau$ is the map that cyclically rotates them,

and $\\phi$ is the number of \\textit{inversions}

(i.e, pairs $(a_{i},a_{j})=(1,0)$ for $iJ$ less than $j$).

This phenomenon was first noticed by Panyushev

in 2007 in the context of antichains in root posets;

Armstrong, Stump, and Thomas proved his

conjecture in 2011.

We describe a theoretical framework for results of this kind,

and discuss old and new results for products of two chains.

This is joint work with Jim Propp.

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