## FMSP Lectures

Seminar information archive ～10/09｜Next seminar｜Future seminars 10/10～

**Seminar information archive**

### 2015/01/27

13:00-16:20 Room #126 (Graduate School of Math. Sci. Bldg.)

Limit order books II

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Abergel.pdf

**Frédéric Abergel**(École Centrale Paris)Limit order books II

[ Abstract ]

In this series of lectures, I will present results pertaining to the empirical properties, mathematical modeling and analysis of limit order books, an object that is now central in modern financial markets. Part of the lectures will be devoted to a survey of the quantitative finance and financial mathematics literature on the subject. I will also present some rather recent results related to the long time behaviour and stationarity of the limit order book.

[ Reference URL ]In this series of lectures, I will present results pertaining to the empirical properties, mathematical modeling and analysis of limit order books, an object that is now central in modern financial markets. Part of the lectures will be devoted to a survey of the quantitative finance and financial mathematics literature on the subject. I will also present some rather recent results related to the long time behaviour and stationarity of the limit order book.

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Abergel.pdf

### 2015/01/26

12:30-13:30 Room #128 (Graduate School of Math. Sci. Bldg.)

Unitary representations of reductive Lie groups III

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Trapa.pdf

**Peter Trapa**(University of Utah)Unitary representations of reductive Lie groups III

[ Abstract ]

Let G be a real reductive group. I will describe an algorithm to determine the unitary dual of G. More precisely, I will describe an algorithm to determine if an irreducible (g,K) module (specified in the Langlands classification) is unitary in the sense that it admits a positive definite invariant Hermitian form.

This is joint work with Jeffrey Adams, Marc van Leeuwen, and David Vogan.

[ Reference URL ]Let G be a real reductive group. I will describe an algorithm to determine the unitary dual of G. More precisely, I will describe an algorithm to determine if an irreducible (g,K) module (specified in the Langlands classification) is unitary in the sense that it admits a positive definite invariant Hermitian form.

This is joint work with Jeffrey Adams, Marc van Leeuwen, and David Vogan.

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Trapa.pdf

### 2015/01/26

9:30-10:30 Room #122 (Graduate School of Math. Sci. Bldg.)

The Geometry of Nontempered Characters

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Harris.pdf

**Benjamin Harris**(Oklahoma State University)The Geometry of Nontempered Characters

[ Abstract ]

In this talk, we will survey the results of Rossmann and Schmid-Vilonen on geometric formulas for nontempered characters of reductive groups, and we will mention an old result of Barbasch-Vogan on the special case A_q(lambda). We will discuss what nontempered character formulas would be necessary to generalize the main formula of the second talk, and we will make conjectures.

[ Reference URL ]In this talk, we will survey the results of Rossmann and Schmid-Vilonen on geometric formulas for nontempered characters of reductive groups, and we will mention an old result of Barbasch-Vogan on the special case A_q(lambda). We will discuss what nontempered character formulas would be necessary to generalize the main formula of the second talk, and we will make conjectures.

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Harris.pdf

### 2015/01/26

13:00-17:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Limit order books I

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Abergel.pdf

**Frédéric Abergel**(École Centrale Paris)Limit order books I

[ Abstract ]

In this series of lectures, I will present results pertaining to the empirical properties, mathematical modeling and analysis of limit order books, an object that is now central in modern financial markets. Part of the lectures will be devoted to a survey of the quantitative finance and financial mathematics literature on the subject. I will also present some rather recent results related to the long time behaviour and stationarity of the limit order book.

[ Reference URL ]In this series of lectures, I will present results pertaining to the empirical properties, mathematical modeling and analysis of limit order books, an object that is now central in modern financial markets. Part of the lectures will be devoted to a survey of the quantitative finance and financial mathematics literature on the subject. I will also present some rather recent results related to the long time behaviour and stationarity of the limit order book.

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Abergel.pdf

### 2015/01/26

11:00-12:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Local Theta lifting of generalized Whittaker models.

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Gomez.pdf

**Raul Gomez**(Cornell University)Local Theta lifting of generalized Whittaker models.

[ Abstract ]

In this talk, we describe the behavior of the space of generalized Whittaker models attached to a nilpotent orbit under the local theta correspondence. This description is a generalization of a result of Moeglin in the p-adic setting. This is joint work with Chengbo Zhu.

[ Reference URL ]In this talk, we describe the behavior of the space of generalized Whittaker models attached to a nilpotent orbit under the local theta correspondence. This description is a generalization of a result of Moeglin in the p-adic setting. This is joint work with Chengbo Zhu.

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Gomez.pdf

### 2015/01/25

12:30-13:30 Room #126 (Graduate School of Math. Sci. Bldg.)

Unitary representations of reductive Lie groups II

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Trapa.pdf

**Peter Trapa**(University of Utah)Unitary representations of reductive Lie groups II

[ Abstract ]

Let G be a real reductive group. I will describe an algorithm to determine the unitary dual of G. More precisely, I will describe an algorithm to determine if an irreducible (g,K) module (specified in the Langlands classification) is unitary in the sense that it admits a positive definite invariant Hermitian form.

This is joint work with Jeffrey Adams, Marc van Leeuwen, and David Vogan.

[ Reference URL ]Let G be a real reductive group. I will describe an algorithm to determine the unitary dual of G. More precisely, I will describe an algorithm to determine if an irreducible (g,K) module (specified in the Langlands classification) is unitary in the sense that it admits a positive definite invariant Hermitian form.

This is joint work with Jeffrey Adams, Marc van Leeuwen, and David Vogan.

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Trapa.pdf

### 2015/01/25

11:00-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

The Geometry of Harmonic Analysis

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Harris.pdf

**Benjamin Harris**(Oklahoma State University)The Geometry of Harmonic Analysis

[ Abstract ]

In this talk, we will present recent joint work with Tobias Weich. When G is a real, reductive algebraic group and X is a homogeneous space for G with an invariant measure, we will completely describe the regular, semisimple asymptotics of the support of the Plancherel measure for L^2(X). We will give concrete examples of this theorem, describing what can and cannot be deduced from this result.

[ Reference URL ]In this talk, we will present recent joint work with Tobias Weich. When G is a real, reductive algebraic group and X is a homogeneous space for G with an invariant measure, we will completely describe the regular, semisimple asymptotics of the support of the Plancherel measure for L^2(X). We will give concrete examples of this theorem, describing what can and cannot be deduced from this result.

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Harris.pdf

### 2015/01/25

9:30-10:30 Room #126 (Graduate School of Math. Sci. Bldg.)

Generalized and degenerate Whittaker models associated to nilpotent orbits.

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Gomez.pdf

**Raul Gomez**(Cornell University)Generalized and degenerate Whittaker models associated to nilpotent orbits.

[ Abstract ]

In this talk, we examine the relation between the different spaces of Whittaker models that can be attached to a nilpotent orbit. We will also explore their relation to other nilpotent invariants (like the wave front set) and show some examples and applications.

This is joint work with Dmitry Gourevitch and Siddhartha Sahi.

[ Reference URL ]In this talk, we examine the relation between the different spaces of Whittaker models that can be attached to a nilpotent orbit. We will also explore their relation to other nilpotent invariants (like the wave front set) and show some examples and applications.

This is joint work with Dmitry Gourevitch and Siddhartha Sahi.

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Gomez.pdf

### 2015/01/24

13:00-14:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Unitary representations of reductive Lie groups I

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Trapa.pdf

**Peter Trapa**(University of Utah)Unitary representations of reductive Lie groups I

[ Abstract ]

Let G be a real reductive group. I will describe an algorithm to determine the unitary dual of G. More precisely, I will describe an algorithm to determine if an irreducible (g,K) module (specified in the Langlands classification) is unitary in the sense that it admits a positive definite invariant Hermitian form.

This is joint work with Jeffrey Adams, Marc van Leeuwen, and David Vogan.

[ Reference URL ]Let G be a real reductive group. I will describe an algorithm to determine the unitary dual of G. More precisely, I will describe an algorithm to determine if an irreducible (g,K) module (specified in the Langlands classification) is unitary in the sense that it admits a positive definite invariant Hermitian form.

This is joint work with Jeffrey Adams, Marc van Leeuwen, and David Vogan.

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Trapa.pdf

### 2015/01/24

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

The Geometry of Tempered Characters

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Harris.pdf

**Benjamin Harris**(Oklahoma State University)The Geometry of Tempered Characters

[ Abstract ]

In this introductory talk, we will briefly recall parts of Harish-Chandra's theory of characters for reductive groups and the geometric formula of Rossmann and Duflo for tempered characters of reductive groups. Examples will be given in the case G=SL(2,R).

[ Reference URL ]In this introductory talk, we will briefly recall parts of Harish-Chandra's theory of characters for reductive groups and the geometric formula of Rossmann and Duflo for tempered characters of reductive groups. Examples will be given in the case G=SL(2,R).

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Harris.pdf

### 2015/01/24

14:30-15:30 Room #126 (Graduate School of Math. Sci. Bldg.)

The Tor and Ext functors for smooth representations of real algebraic groups.

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Gomez.pdf

**Raul Gomez**(Cornell University)The Tor and Ext functors for smooth representations of real algebraic groups.

[ Abstract ]

Inspired by the recent work of Dipendra Prasad in the $p$-adic setting, we define the Tor and Ext functors for an appropriate category of smooth representations of a real algebraic group $G$, and give some applications. This is joint work with Birgit Speh.

[ Reference URL ]Inspired by the recent work of Dipendra Prasad in the $p$-adic setting, we define the Tor and Ext functors for an appropriate category of smooth representations of a real algebraic group $G$, and give some applications. This is joint work with Birgit Speh.

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Gomez.pdf

### 2014/12/10

16:00-17:30 Room #118 (Graduate School of Math. Sci. Bldg.)

Singular limit analysis of a reaction-diffusion system with precipitation and dissolution in a porous medium

**Danielle Hilhorst**(CNRS / Univ. Paris-Sud)Singular limit analysis of a reaction-diffusion system with precipitation and dissolution in a porous medium

[ Abstract ]

This talk is concerned with a mathematical model for the storage of radioactive waste. The model which we study deals with the diffusion of chemical species transported by water, with possible dissolution or precipitation and for a rather general kinetics law. In this talk, we consider a three-component reaction-diffusion system with a fast precipitation and dissolution reaction term. We investigate its singular limit as the reaction rate tends to infinity. The limit problem is described by the combination of a Stefan problem and a linear heat equation. The rate of convergence with respect to the reaction rate is established in a specific case. This is joint work with Hideki Murakawa.

This talk is concerned with a mathematical model for the storage of radioactive waste. The model which we study deals with the diffusion of chemical species transported by water, with possible dissolution or precipitation and for a rather general kinetics law. In this talk, we consider a three-component reaction-diffusion system with a fast precipitation and dissolution reaction term. We investigate its singular limit as the reaction rate tends to infinity. The limit problem is described by the combination of a Stefan problem and a linear heat equation. The rate of convergence with respect to the reaction rate is established in a specific case. This is joint work with Hideki Murakawa.

### 2014/11/10

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

Discrete Painlevé equations with periodic coefficients (ENGLISH)

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Ramani.pdf

**Alfred Ramani**(Ecole Polytechnique)Discrete Painlevé equations with periodic coefficients (ENGLISH)

[ Abstract ]

We present a series of results on discrete Painlevé equations with coefficients which are periodic functions of the independent variable. We show by explicit construction that for each affine Weyl group there exists an equation the coefficients of which have maximal periodicity. New results on equations associated to the affine Weyl group E_8 are also presented.

[ Reference URL ]We present a series of results on discrete Painlevé equations with coefficients which are periodic functions of the independent variable. We show by explicit construction that for each affine Weyl group there exists an equation the coefficients of which have maximal periodicity. New results on equations associated to the affine Weyl group E_8 are also presented.

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Ramani.pdf

### 2014/09/19

14:30-16:00 Room #128 (Graduate School of Math. Sci. Bldg.)

A bridge between knotted graphs and axiomatizations of groups (ENGLISH)

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Lebed.pdf

**Victoria Lebed**(Osaka City University, JSPS)A bridge between knotted graphs and axiomatizations of groups (ENGLISH)

[ Abstract ]

This talk will be devoted to a new algebraic structure called qualgebra. From the topological viewpoint, our construction is motivated by a study of knotted 3-valent graphs via combinatorially defined coloring invariants. From the algebraic viewpoint, it gives a part of an alternative axiomatization of groups, describing the properties of the conjugation operation and its interactions with the group multiplication. Explicit examples of qualgebras and associated graph invariants will be given.

[ Reference URL ]This talk will be devoted to a new algebraic structure called qualgebra. From the topological viewpoint, our construction is motivated by a study of knotted 3-valent graphs via combinatorially defined coloring invariants. From the algebraic viewpoint, it gives a part of an alternative axiomatization of groups, describing the properties of the conjugation operation and its interactions with the group multiplication. Explicit examples of qualgebras and associated graph invariants will be given.

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Lebed.pdf

### 2014/09/12

10:30-15:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Stochastic homogenization for first order Hamilton-Jacobi equations(III) (ENGLISH)

[ Reference URL ]

http://fmsp.ms.u-tokyo.ac.jp/Tran2014_0908-0912.pdf

**Hung V. Tran**(The University of Chicago)Stochastic homogenization for first order Hamilton-Jacobi equations(III) (ENGLISH)

[ Reference URL ]

http://fmsp.ms.u-tokyo.ac.jp/Tran2014_0908-0912.pdf

### 2014/09/10

10:30-15:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Stochastic homogenization for first order Hamilton-Jacobi equations(II) (ENGLISH)

[ Reference URL ]

http://fmsp.ms.u-tokyo.ac.jp/Tran2014_0908-0912.pdf

**Hung V. Tran**(The University of Chicago)Stochastic homogenization for first order Hamilton-Jacobi equations(II) (ENGLISH)

[ Reference URL ]

http://fmsp.ms.u-tokyo.ac.jp/Tran2014_0908-0912.pdf

### 2014/09/09

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

Energy methods and blow-up rate for semilinear wave equations in the superconformal case (ENGLISH)

**Hatem Zaag**(CNRS/University of Paris Nord)Energy methods and blow-up rate for semilinear wave equations in the superconformal case (ENGLISH)

[ Abstract ]

In a series of papers with Mohamed Ali Hamza (University of Tunis-el Manar), we consider the semilinear wave equations with power nonlinearity.

In the subconformal and the conformal case, we consider perturbations with lower order terms and modify the Lyapunov functional Antonini and Merle designed for the unperturbed case. We also find a blow-up criterion for the equation. As a consequence, we bound the Lyapunov functional. Thanks to interpolations in Sobolev spaces and a Gagliardo-Nirenberg inequality, we bound the solution in the self-similar variable, which gives a sharp bound on the blow-up rate.

Surprisingly, our approach works in the superconformal case (still Sobolev subcritical), leading to a new bound on the blow-up rate, which improves the bound of Killip, Stoval and Visan.

In a series of papers with Mohamed Ali Hamza (University of Tunis-el Manar), we consider the semilinear wave equations with power nonlinearity.

In the subconformal and the conformal case, we consider perturbations with lower order terms and modify the Lyapunov functional Antonini and Merle designed for the unperturbed case. We also find a blow-up criterion for the equation. As a consequence, we bound the Lyapunov functional. Thanks to interpolations in Sobolev spaces and a Gagliardo-Nirenberg inequality, we bound the solution in the self-similar variable, which gives a sharp bound on the blow-up rate.

Surprisingly, our approach works in the superconformal case (still Sobolev subcritical), leading to a new bound on the blow-up rate, which improves the bound of Killip, Stoval and Visan.

### 2014/09/08

10:30-15:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Stochastic homogenization for first order Hamilton-Jacobi equations(I) (ENGLISH)

[ Reference URL ]

http://fmsp.ms.u-tokyo.ac.jp/Tran2014_0908-0912.pdf

**Hung V. Tran**(The University of Chicago)Stochastic homogenization for first order Hamilton-Jacobi equations(I) (ENGLISH)

[ Reference URL ]

http://fmsp.ms.u-tokyo.ac.jp/Tran2014_0908-0912.pdf

### 2014/07/11

15:00-16:00 Room #270 (Graduate School of Math. Sci. Bldg.)

Conditional stability estimate for the Calderon's problem in two dimensional case (ENGLISH)

[ Reference URL ]

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Emanouilov140711.pdf

**Oleg Emanouilov**(Colorado State Univ.)Conditional stability estimate for the Calderon's problem in two dimensional case (ENGLISH)

[ Reference URL ]

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Emanouilov140711.pdf

### 2014/07/03

16:00-18:00 Room #470 (Graduate School of Math. Sci. Bldg.)

Structure of rational orbits in prehomogeneous spaces. (ENGLISH)

[ Reference URL ]

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Savin.pdf

**Gordan Savin**(Univ. of Utah)Structure of rational orbits in prehomogeneous spaces. (ENGLISH)

[ Reference URL ]

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Savin.pdf

### 2014/05/15

16:30-17:30 Room #050 (Graduate School of Math. Sci. Bldg.)

Synthetic theory of Ricci curvature ― When Monge, Riemann and Boltzmann meet ― (ENGLISH)

[ Reference URL ]

http://faculty.ms.u-tokyo.ac.jp/Villani.html

**Cédric Villani**(Université de Lyon, Institut Henri Poincaré)Synthetic theory of Ricci curvature ― When Monge, Riemann and Boltzmann meet ― (ENGLISH)

[ Reference URL ]

http://faculty.ms.u-tokyo.ac.jp/Villani.html

### 2014/01/15

14:50-16:20 Room #056 (Graduate School of Math. Sci. Bldg.)

Lectures on quantum Teichmüller theory IV (ENGLISH)

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Kashaev.pdf

**Rinat Kashaev**(University of Geneva)Lectures on quantum Teichmüller theory IV (ENGLISH)

[ Abstract ]

Quantum Teichmüller theory leads to a family of unitary infinite dimensional projective representations of the mapping class groups of punctured surfaces. One of the recent applications of this theory is the construction of state integral three-manifold invariants related with hyperbolic geometry.

In these lectures it is planned to address the following subjects:

1) Penner’s coordinates in the decorated Teichmüller space.

2) Ratio coordinates.

3) Quantization.

4) The length spectrum of simple closed curves.

[ Reference URL ]Quantum Teichmüller theory leads to a family of unitary infinite dimensional projective representations of the mapping class groups of punctured surfaces. One of the recent applications of this theory is the construction of state integral three-manifold invariants related with hyperbolic geometry.

In these lectures it is planned to address the following subjects:

1) Penner’s coordinates in the decorated Teichmüller space.

2) Ratio coordinates.

3) Quantization.

4) The length spectrum of simple closed curves.

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Kashaev.pdf

### 2014/01/14

14:50-16:20 Room #056 (Graduate School of Math. Sci. Bldg.)

Lectures on quantum Teichmüller theory III (ENGLISH)

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Kashaev.pdf

**Rinat Kashaev**(University of Geneva)Lectures on quantum Teichmüller theory III (ENGLISH)

[ Abstract ]

Quantum Teichmüller theory leads to a family of unitary infinite dimensional projective representations of the mapping class groups of punctured surfaces. One of the recent applications of this theory is the construction of state integral three-manifold invariants related with hyperbolic geometry.

In these lectures it is planned to address the following subjects:

1) Penner’s coordinates in the decorated Teichmüller space.

2) Ratio coordinates.

3) Quantization.

4) The length spectrum of simple closed curves.

[ Reference URL ]Quantum Teichmüller theory leads to a family of unitary infinite dimensional projective representations of the mapping class groups of punctured surfaces. One of the recent applications of this theory is the construction of state integral three-manifold invariants related with hyperbolic geometry.

In these lectures it is planned to address the following subjects:

1) Penner’s coordinates in the decorated Teichmüller space.

2) Ratio coordinates.

3) Quantization.

4) The length spectrum of simple closed curves.

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Kashaev.pdf

### 2014/01/10

14:50-16:20 Room #056 (Graduate School of Math. Sci. Bldg.)

Lectures on quantum Teichmüller theory II (ENGLISH)

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Kashaev.pdf

**Rinat Kashaev**(University of Geneva)Lectures on quantum Teichmüller theory II (ENGLISH)

[ Abstract ]

Quantum Teichmüller theory leads to a family of unitary infinite dimensional projective representations of the mapping class groups of punctured surfaces. One of the recent applications of this theory is the construction of state integral three-manifold invariants related with hyperbolic geometry.

In these lectures it is planned to address the following subjects:

1) Penner’s coordinates in the decorated Teichmüller space.

2) Ratio coordinates.

3) Quantization.

4) The length spectrum of simple closed curves.

[ Reference URL ]Quantum Teichmüller theory leads to a family of unitary infinite dimensional projective representations of the mapping class groups of punctured surfaces. One of the recent applications of this theory is the construction of state integral three-manifold invariants related with hyperbolic geometry.

In these lectures it is planned to address the following subjects:

1) Penner’s coordinates in the decorated Teichmüller space.

2) Ratio coordinates.

3) Quantization.

4) The length spectrum of simple closed curves.

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Kashaev.pdf

### 2014/01/09

14:50-16:20 Room #056 (Graduate School of Math. Sci. Bldg.)

Lectures on quantum Teichmüller theory I (ENGLISH)

[ Abstract ]

Quantum Teichmüller theory leads to a family of unitary infinite dimensional projective representations of the mapping class groups of punctured surfaces. One of the recent applications of this theory is the construction of state integral three-manifold invariants related with hyperbolic geometry.

In these lectures it is planned to address the following subjects:

1) Penner’s coordinates in the decorated Teichmüller space.

2) Ratio coordinates.

3) Quantization.

4) The length spectrum of simple closed curves.
[ Reference URL ]

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Kashaev.pdf

**Rinat Kashaev**(University of Geneva)Lectures on quantum Teichmüller theory I (ENGLISH)

Quantum Teichmüller theory leads to a family of unitary infinite dimensional projective representations of the mapping class groups of punctured surfaces. One of the recent applications of this theory is the construction of state integral three-manifold invariants related with hyperbolic geometry.

In these lectures it is planned to address the following subjects:

1) Penner’s coordinates in the decorated Teichmüller space.

2) Ratio coordinates.

3) Quantization.

4) The length spectrum of simple closed curves.

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Kashaev.pdf