## Seminar information archive

Seminar information archive ～05/21｜Today's seminar 05/22 | Future seminars 05/23～

#### Algebraic Geometry Seminar

15:00-16:00 Room #122 (Graduate School of Math. Sci. Bldg.)

p-adic Hodge theory in the non-commutative setting

**Dmitry Kaledin**(Steklov Institute)p-adic Hodge theory in the non-commutative setting

[ Abstract ]

We will explain what is the natural replacement of the notion of Hodge structure in the p-adic setting, and how to construct such a structure for non-commutative manifolds (something which at present cannot be done for the usual Hodge structures, but works perfectly well for the p-adic analog).

We will explain what is the natural replacement of the notion of Hodge structure in the p-adic setting, and how to construct such a structure for non-commutative manifolds (something which at present cannot be done for the usual Hodge structures, but works perfectly well for the p-adic analog).

### 2007/10/09

#### Tuesday Seminar on Topology

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

Classification of codimension-one locally free actions of the affine group of the real line.

**浅岡 正幸**(京都大学大学院理学研究科)Classification of codimension-one locally free actions of the affine group of the real line.

[ Abstract ]

By GA, we denote the group of affine and orientation-preserving transformations

of the real line. In this talk, I will report on classification of locally free action of

GA on closed three manifolds, which I obtained recently. In 1979, E.Ghys proved

that if such an action preserves a volume, then it is smoothly conjugate to a homogeneous action. However, it was unknown whether non-homogeneous action exists. As a consequence of the classification, we will see that the unit tangent bundle of a closed surface of higher genus admits a finite-parameter family of

non-homogeneous actions.

By GA, we denote the group of affine and orientation-preserving transformations

of the real line. In this talk, I will report on classification of locally free action of

GA on closed three manifolds, which I obtained recently. In 1979, E.Ghys proved

that if such an action preserves a volume, then it is smoothly conjugate to a homogeneous action. However, it was unknown whether non-homogeneous action exists. As a consequence of the classification, we will see that the unit tangent bundle of a closed surface of higher genus admits a finite-parameter family of

non-homogeneous actions.

#### Lie Groups and Representation Theory

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

Rankin-Cohen brackets and covariant quantization

http://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html

**Michael Pevzner**(Reims University and University of Tokyo)Rankin-Cohen brackets and covariant quantization

[ Abstract ]

The particular geometric structure of causal symmetric spaces permits the definition of a covariant quantization of these homogeneous manifolds.

Composition formulae (#-products) of quantizad operators give rise to a new interpretation of Rankin-Cohen brackets and allow to connect them with the branching laws of tensor products of holomorphic discrete series representations.

[ Reference URL ]The particular geometric structure of causal symmetric spaces permits the definition of a covariant quantization of these homogeneous manifolds.

Composition formulae (#-products) of quantizad operators give rise to a new interpretation of Rankin-Cohen brackets and allow to connect them with the branching laws of tensor products of holomorphic discrete series representations.

http://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html

### 2007/10/04

#### Operator Algebra Seminars

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

Local nets of von Neumann algebras and modular theory

**Gandalf Lechner**(Erwin Schroedinger Institute)Local nets of von Neumann algebras and modular theory

### 2007/10/02

#### Lie Groups and Representation Theory

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

Invariant integral operators on affine G-varieties and their kernels

http://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html

**Pablo Ramacher**(Gottingen University)Invariant integral operators on affine G-varieties and their kernels

[ Abstract ]

We consider certain invariant integral operators on a smooth affine variety M carrying the action of a reductive algebraic group G, and assume that G acts on M with an open orbit. Then M is isomorphic to a homogeneous vector bundle, and can locally be described via the theory of prehomogenous vector spaces. We then study the Schwartz kernels of the considered operators, and give a description of their singularities using the calculus of b-pseudodifferential operators developed by Melrose. In particular, the restrictions of the kernels to the diagonal can be described in terms of local zeta functions.

[ Reference URL ]We consider certain invariant integral operators on a smooth affine variety M carrying the action of a reductive algebraic group G, and assume that G acts on M with an open orbit. Then M is isomorphic to a homogeneous vector bundle, and can locally be described via the theory of prehomogenous vector spaces. We then study the Schwartz kernels of the considered operators, and give a description of their singularities using the calculus of b-pseudodifferential operators developed by Melrose. In particular, the restrictions of the kernels to the diagonal can be described in terms of local zeta functions.

http://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html

### 2007/09/28

#### Colloquium

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

Irreducible unitary representations and automorphic forms

**Marko Tadic'**(University of Zagreb)Irreducible unitary representations and automorphic forms

[ Abstract ]

Unitary representations of adelic groups in the spaces of automorphic forms are big source of important irreducible unitary representations of classical groups over local fields.

We shall present classifications of some classes of irreducible unitary representations (older, as well as quite new), describe

isolated unitary representations among them, and discuss which of them can be obtained from spaces of automorphic forms.

Unitary representations of adelic groups in the spaces of automorphic forms are big source of important irreducible unitary representations of classical groups over local fields.

We shall present classifications of some classes of irreducible unitary representations (older, as well as quite new), describe

isolated unitary representations among them, and discuss which of them can be obtained from spaces of automorphic forms.

### 2007/09/26

#### Algebraic Geometry Seminar

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

Floor diagrams and enumeration of tropical curves

**Grigory Mikhalkin**(Toronto大学)Floor diagrams and enumeration of tropical curves

[ Abstract ]

The enumerative problems considered in this talk are finding the number of curves in projective spaces (over complex, real and tropical numbers) of given genus and degree constrained by certain incidence conditions (e.g. passing via points or lines). Floor diagrams are a combinatorial tool that reduces an enumerative problem in dimension n to the corresponding problem n dimension n-1. Floor diagrams give a constructive (and rather efficient) way to find all tropical curves for a given enumerative problem. And once we have a tropical solution of the problem we can use it to solve the corresponding problems over the complex and real numbers.

The enumerative problems considered in this talk are finding the number of curves in projective spaces (over complex, real and tropical numbers) of given genus and degree constrained by certain incidence conditions (e.g. passing via points or lines). Floor diagrams are a combinatorial tool that reduces an enumerative problem in dimension n to the corresponding problem n dimension n-1. Floor diagrams give a constructive (and rather efficient) way to find all tropical curves for a given enumerative problem. And once we have a tropical solution of the problem we can use it to solve the corresponding problems over the complex and real numbers.

### 2007/09/19

#### Number Theory Seminar

16:30-17:30 Room #117 (Graduate School of Math. Sci. Bldg.)

Etale cobordism

**Gereon Quick**(Universitaet Muenster)Etale cobordism

[ Abstract ]

We define and study a new candidate of etale topological cohomology theories for schemes over a field of abritrary characteristic: etale cobordism. As etale K-theory is related to algebraic K-theory, etale cobordism is related to algebraic cobordism of Voevodsky and Levine/Morel. It shares some nice properties of topological theories, e.g. it is equipped with an Atiyah-Hirzebruch spectral sequence from etale cohomology. We discuss in particular a comparison theorem between etale and algebraic cobordism after inverting a Bott element and, finally, we give an outlook to further possible applications of this theory.

We define and study a new candidate of etale topological cohomology theories for schemes over a field of abritrary characteristic: etale cobordism. As etale K-theory is related to algebraic K-theory, etale cobordism is related to algebraic cobordism of Voevodsky and Levine/Morel. It shares some nice properties of topological theories, e.g. it is equipped with an Atiyah-Hirzebruch spectral sequence from etale cohomology. We discuss in particular a comparison theorem between etale and algebraic cobordism after inverting a Bott element and, finally, we give an outlook to further possible applications of this theory.

### 2007/09/15

#### Monthly Seminar on Arithmetic of Automorphic Forms

13:30-16:00 Room #117 (Graduate School of Math. Sci. Bldg.)

Siegl principal series Whittaker functions on $Sp(2,\\mathbf{R})$

(部屋は056室)

A higher rank version of Abel-Jacobi's theorem (Room 056)

**長谷川泰子**(東京大学数理科学) 13:30-14:30Siegl principal series Whittaker functions on $Sp(2,\\mathbf{R})$

(部屋は056室)

[ Abstract ]

2次シンプレクティック群のSiegel極大放物型部分群から誘導された一般型主系列表現に対するWhittaker関数の級数表示と積分表示を与えることを目的とし,Whittaker関数の満たす微分評定式を与え,その解の構成に向けて現在進めている研究の方針を述べる.

(部屋は,冷房効く056室に変更です)

2次シンプレクティック群のSiegel極大放物型部分群から誘導された一般型主系列表現に対するWhittaker関数の級数表示と積分表示を与えることを目的とし,Whittaker関数の満たす微分評定式を与え,その解の構成に向けて現在進めている研究の方針を述べる.

(部屋は,冷房効く056室に変更です)

**市川尚志**(佐賀大学理工学部) 15:00-16:00A higher rank version of Abel-Jacobi's theorem (Room 056)

[ Abstract ]

極大退化曲線に近いリーマン面上のベクトル束とそのモジュライについて話す.次数0の安定ベクトル束が,リーマン面を一意化するショットキー群の線形表現から得られることを述べ,ショットキー群の線形表現の空間とベクトル束のモジュライ空間の関係を,アーベル・ヤコビの定理,フェアリンデ公式を用いて考察する.

(部屋は117号室です)

極大退化曲線に近いリーマン面上のベクトル束とそのモジュライについて話す.次数0の安定ベクトル束が,リーマン面を一意化するショットキー群の線形表現から得られることを述べ,ショットキー群の線形表現の空間とベクトル束のモジュライ空間の関係を,アーベル・ヤコビの定理,フェアリンデ公式を用いて考察する.

(部屋は117号室です)

### 2007/09/12

#### Algebraic Geometry Seminar

15:00-18:00 Room #117 (Graduate School of Math. Sci. Bldg.)

Classification of p-divisible groups by displays and duality

Applications of the theory of displays

Presentation of mapping class groups from algebraic geometry

**E. Lau**(Univ. of Bielefeld) 15:00-15:45Classification of p-divisible groups by displays and duality

**T. Zink**(Univ. of Bielefeld) 16:00-16:45Applications of the theory of displays

**E. Looijenga**(Univ. of Utrecht) 17:00-18:00Presentation of mapping class groups from algebraic geometry

[ Abstract ]

A presentation of the mapping class group of a genus g surface with one hole is due to Wajnryb with later improvements due to M. Matsumoto. The generators are Dehn twists defined by 2g+1 closed curves on the surface. The relations involving only two Dehn twists are the familiar Artin relations, we show that those involving more than two can be derived from algebro-geometry considerations.

A presentation of the mapping class group of a genus g surface with one hole is due to Wajnryb with later improvements due to M. Matsumoto. The generators are Dehn twists defined by 2g+1 closed curves on the surface. The relations involving only two Dehn twists are the familiar Artin relations, we show that those involving more than two can be derived from algebro-geometry considerations.

#### Number Theory Seminar

15:00-18:00 Room #117 (Graduate School of Math. Sci. Bldg.)

Classification of p-divisible groups by displays and duality

Applications of the theory of displays

Presentation of mapping class groups from algebraic geometry

**E. Lau**(Univ. of Bielefeld) 15:00-15:45Classification of p-divisible groups by displays and duality

**T. Zink**(Univ. of Bielefeld) 16:00-16:45Applications of the theory of displays

**E. Looijenga**(Univ. of Utrecht) 17:00-18:00Presentation of mapping class groups from algebraic geometry

[ Abstract ]

A presentation of the mapping class group of a genus g surface with one hole is due to Wajnryb with later improvements due to M. Matsumoto. The generators are Dehn twists defined by 2g+1 closed curves on the surface. The relations involving only two Dehn twists are the familiar Artin relations, we show that those involving more than two can be derived from algebro-geometry considerations.

A presentation of the mapping class group of a genus g surface with one hole is due to Wajnryb with later improvements due to M. Matsumoto. The generators are Dehn twists defined by 2g+1 closed curves on the surface. The relations involving only two Dehn twists are the familiar Artin relations, we show that those involving more than two can be derived from algebro-geometry considerations.

### 2007/09/05

#### PDE Real Analysis Seminar

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

Reguarity of Weak Solutions to the Navier-Stokes System beyond Serrin's Criterion

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html

**Reinhard Farwig**(Darmstadt University of Technology)Reguarity of Weak Solutions to the Navier-Stokes System beyond Serrin's Criterion

[ Abstract ]

Consider a weak instationary solution $u(x,t)$ of the Navier-Stokes equations in a domain $\\Omega \\subset \\mathbb{R}^3$ in the sense of Leray-Hopf. As is well-known, $u$ is is unique and regular if $u\\in L^s(0,T;L^q(\\Omega))$ satisfies the {\\it strong energy inequality} and $s,q$ satisfy Serrin's condition $\\frac{2}{s} + \\frac{3}{q}=1$, $s>2,\\, q>3$. Now consider $u$ such that $$u\\in L^r(0,T;L^q(\\Omega))\\quad \\mbox{ where }\\quad \\frac{2}{r} + \\frac{3}{q}>1$$ and has a sufficiently small norm in $L^r(0,T;L^q(\\Omega))$. Then we will prove that $u$ is regular. Similar results of local rather than global type in space will be proved provided that $u$ satisfies the {\\it localized energy inequality}. Finally H\\"older continuity of the kinetic energy in time will imply regularity.

The proofs use local in time regularity results which are based on the {\\it theory of very weak solutions} and on uniqueness arguments for weak solutions.

[ Reference URL ]Consider a weak instationary solution $u(x,t)$ of the Navier-Stokes equations in a domain $\\Omega \\subset \\mathbb{R}^3$ in the sense of Leray-Hopf. As is well-known, $u$ is is unique and regular if $u\\in L^s(0,T;L^q(\\Omega))$ satisfies the {\\it strong energy inequality} and $s,q$ satisfy Serrin's condition $\\frac{2}{s} + \\frac{3}{q}=1$, $s>2,\\, q>3$. Now consider $u$ such that $$u\\in L^r(0,T;L^q(\\Omega))\\quad \\mbox{ where }\\quad \\frac{2}{r} + \\frac{3}{q}>1$$ and has a sufficiently small norm in $L^r(0,T;L^q(\\Omega))$. Then we will prove that $u$ is regular. Similar results of local rather than global type in space will be proved provided that $u$ satisfies the {\\it localized energy inequality}. Finally H\\"older continuity of the kinetic energy in time will imply regularity.

The proofs use local in time regularity results which are based on the {\\it theory of very weak solutions} and on uniqueness arguments for weak solutions.

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html

### 2007/08/29

#### Algebraic Geometry Seminar

17:00-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Computations on the moduli spaces of weighted log pairs

**Valery Alexeev**(Georgia大学)Computations on the moduli spaces of weighted log pairs

### 2007/08/27

#### Number Theory Seminar

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

The reductive Borel-Serre motive

**Steven Zucker**(Johns Hopkins大学)The reductive Borel-Serre motive

### 2007/08/02

#### Algebraic Geometry Seminar

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

Dynamics of automorphisms on algebraic varieties

**De-Qi Zhang**(Singapore大学)Dynamics of automorphisms on algebraic varieties

[ Abstract ]

The building blocks of automorphisms / endomorphisms of compact varieties are determined --- an algebro geometric approach towards dynamics.

The building blocks of automorphisms / endomorphisms of compact varieties are determined --- an algebro geometric approach towards dynamics.

### 2007/07/25

#### Seminar on Probability and Statistics

16:20-17:30 Room #122 (Graduate School of Math. Sci. Bldg.)

大規模ランダム行列のスペクトル理論とデータ解析への応用(Review)

http://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2007/05.html

**小林 景**(統計数理研究所, 学振特別研究員)大規模ランダム行列のスペクトル理論とデータ解析への応用(Review)

[ Abstract ]

The empirical spectral distribution of random matrices have been studied since Wigner's pioneering work on the semicircular law in the 1950's. The result says that the empirical spectral distribution of a symmetric matrix with i.i.d. random elements converges to the semicircular law as the size of the matrix becomes large. Though this result is beautiful in theory, its application has been limited to a few problems in nuclear physics and coding theory. The next breakthrough was the Marcenko-Pastur (M-P) law for the asymptotic spectral distribution of sample covariance matrices. The M-P law has found more applications, in particular high dimensional statistical data analysis, than the semicircular law.

In this talk I will first review these two significant results. Each of them has three completely different proofs. Then I will explain several other theoretical results that have mostly been studied this decade. Finally, I will present some of the applications of these results. This review is partly based on lectures on random matrices given by P. Bickel, N. El-Karoui and A. Guionnet, and also some seminars at UC Berkeley.

(# This talk is almost the same as the talk I gave at ISM on June 1.)

[ Reference URL ]The empirical spectral distribution of random matrices have been studied since Wigner's pioneering work on the semicircular law in the 1950's. The result says that the empirical spectral distribution of a symmetric matrix with i.i.d. random elements converges to the semicircular law as the size of the matrix becomes large. Though this result is beautiful in theory, its application has been limited to a few problems in nuclear physics and coding theory. The next breakthrough was the Marcenko-Pastur (M-P) law for the asymptotic spectral distribution of sample covariance matrices. The M-P law has found more applications, in particular high dimensional statistical data analysis, than the semicircular law.

In this talk I will first review these two significant results. Each of them has three completely different proofs. Then I will explain several other theoretical results that have mostly been studied this decade. Finally, I will present some of the applications of these results. This review is partly based on lectures on random matrices given by P. Bickel, N. El-Karoui and A. Guionnet, and also some seminars at UC Berkeley.

(# This talk is almost the same as the talk I gave at ISM on June 1.)

http://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2007/05.html

### 2007/07/20

#### Colloquium

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

ソリトン物理はおもしろい---スピノル型ボーズ・アインシュタイン凝縮体におけるソリトン

**和達三樹**(東京理科大学理学部物理学科)ソリトン物理はおもしろい---スピノル型ボーズ・アインシュタイン凝縮体におけるソリトン

### 2007/07/19

#### Seminar for Mathematical Past of Asia

17:00-18:30 Room #123 (Graduate School of Math. Sci. Bldg.)

幕末・明治初期の日本における西洋数学の導入と漢訳西洋数学書籍

[ Reference URL ]

http://www.ms.u-tokyo.ac.jp/~kawazumi/asia.html

**李 佳女華**(東京大学大学院総合文化研究科)幕末・明治初期の日本における西洋数学の導入と漢訳西洋数学書籍

[ Reference URL ]

http://www.ms.u-tokyo.ac.jp/~kawazumi/asia.html

#### Operator Algebra Seminars

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

On a class of II$_1$ factors with at most one Cartan subalgebra

**小沢登高**(東大数理)On a class of II$_1$ factors with at most one Cartan subalgebra

### 2007/07/18

#### Number Theory Seminar

16:30-17:30 Room #117 (Graduate School of Math. Sci. Bldg.)

Tropical toric varieties

**梶原 健**(横浜国立大学)Tropical toric varieties

#### Seminar on Probability and Statistics

16:20-17:30 Room #122 (Graduate School of Math. Sci. Bldg.)

Easy full-joint estimators of stable processes

http://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2007/04.html

**増田 弘毅**(九州大学大学院数理学研究院)Easy full-joint estimators of stable processes

[ Abstract ]

安定レヴィ過程からの高頻度観測に基づいた同時推定に関しては,尤度比確率場の漸近挙動は安易に二次項の観測情報量まで見ただけでは解明されず,最尤推定量の典型的な“良い漸近挙動”が保証されないことが分かっている.本発表では,標本メディアンおよび標本メディアンのプラグイン型統計量の漸近挙動に基づき,モデルに入る全パラメータの同時推定を可能とする計算容易な推定量の構成法を紹介し,正則な漸近共分散行列を有する漸近正規性を導出する.推定量の有限標本での挙動を数値実験で検証する.時間があれば,非対称安定レヴィ過程の場合に関して分かっている事柄についても触れる.

[ Reference URL ]安定レヴィ過程からの高頻度観測に基づいた同時推定に関しては,尤度比確率場の漸近挙動は安易に二次項の観測情報量まで見ただけでは解明されず,最尤推定量の典型的な“良い漸近挙動”が保証されないことが分かっている.本発表では,標本メディアンおよび標本メディアンのプラグイン型統計量の漸近挙動に基づき,モデルに入る全パラメータの同時推定を可能とする計算容易な推定量の構成法を紹介し,正則な漸近共分散行列を有する漸近正規性を導出する.推定量の有限標本での挙動を数値実験で検証する.時間があれば,非対称安定レヴィ過程の場合に関して分かっている事柄についても触れる.

http://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2007/04.html

### 2007/07/17

#### Tuesday Seminar on Topology

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

Orbifold Cohomology of Wreath Product Orbifolds and

Cohomological HyperKahler Resolution Conjecture

**松村 朝雄**(東京大学大学院数理科学研究科)Orbifold Cohomology of Wreath Product Orbifolds and

Cohomological HyperKahler Resolution Conjecture

[ Abstract ]

Chen-Ruan orbifold cohomology ring was introduced in 2000 as

the degree zero genus zero orbifold Gromov-Witten invariants with

three marked points. We will review its construction in the case of

global quotient orbifolds, following Fantechi-Gottsche and

Jarvis-Kaufmann-Kimura. We will describe the orbifold cohomology of

wreath product orbifolds and explain its application to Ruan's

cohomological hyperKahler resolution conjecture.

Chen-Ruan orbifold cohomology ring was introduced in 2000 as

the degree zero genus zero orbifold Gromov-Witten invariants with

three marked points. We will review its construction in the case of

global quotient orbifolds, following Fantechi-Gottsche and

Jarvis-Kaufmann-Kimura. We will describe the orbifold cohomology of

wreath product orbifolds and explain its application to Ruan's

cohomological hyperKahler resolution conjecture.

### 2007/07/12

#### Operator Algebra Seminars

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Normalizers of MASAs and irreducible subfactors

**酒匂宏樹**(東大数理)Normalizers of MASAs and irreducible subfactors

### 2007/07/11

#### Number Theory Seminar

16:30-17:30 Room #117 (Graduate School of Math. Sci. Bldg.)

Algebraic cycles on products of elliptic curves over p-adic fields

**Andreas Rosenschon**(University of Alberta)Algebraic cycles on products of elliptic curves over p-adic fields

### 2007/07/10

#### Tuesday Seminar on Topology

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

Combable functions, quasimorphisms, and the central limit theorem

(joint with Koji Fujiwara)

**Danny C. Calegari**(California Institute of Technology)Combable functions, quasimorphisms, and the central limit theorem

(joint with Koji Fujiwara)

[ Abstract ]

Quasimorphisms on groups are dual to stable commutator length,

and detect extremal phenomena in topology and dynamics. In typical groups

(even in a free group) stable commutator length is very difficult to

calculate, because the space of quasimorphisms is too large to study

directly without adding more structure.

In this talk, we show that a large class of quasimorphisms - the so-called

"counting quasimorphisms" on word-hyperbolic groups - can be effectively

described using simple machines called finite state automata. From this,

and from the ergodic theory of finite directed graphs, one can deduce a

number of properties about the statistical distribution of the values of a

counting quasimorphism on elements of the group.

Quasimorphisms on groups are dual to stable commutator length,

and detect extremal phenomena in topology and dynamics. In typical groups

(even in a free group) stable commutator length is very difficult to

calculate, because the space of quasimorphisms is too large to study

directly without adding more structure.

In this talk, we show that a large class of quasimorphisms - the so-called

"counting quasimorphisms" on word-hyperbolic groups - can be effectively

described using simple machines called finite state automata. From this,

and from the ergodic theory of finite directed graphs, one can deduce a

number of properties about the statistical distribution of the values of a

counting quasimorphism on elements of the group.

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