## Seminar information archive

Seminar information archive ～02/01｜Today's seminar 02/02 | Future seminars 02/03～

#### Number Theory Seminar

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

Equidistribution theorems in Arakelov geometry

**Antoine Chambert-Loir**(Universite de Rennes 1)Equidistribution theorems in Arakelov geometry

[ Abstract ]

The proof of Bogomolov's conjecture by Zhang made a crucial use

of an equidistribution property for the Galois orbits of points of small

heights in Abelian varieties defined over number fields.

Such an equidistribution property is proved using a method invented

by Szpiro, Ullmo and Zhang, and makes use of Arakelov theory.

This equidistribution theorem takes place in the complex torus

associated to the Abelian variety. I will show how a similar

equidistribution theorem can be proven for the p-adic topology ;

we have to use Berkovich space. Thanks to recent results of Yuan

about `big line bundles' in Arakelov geometry, the situation

is now very well understood.

The proof of Bogomolov's conjecture by Zhang made a crucial use

of an equidistribution property for the Galois orbits of points of small

heights in Abelian varieties defined over number fields.

Such an equidistribution property is proved using a method invented

by Szpiro, Ullmo and Zhang, and makes use of Arakelov theory.

This equidistribution theorem takes place in the complex torus

associated to the Abelian variety. I will show how a similar

equidistribution theorem can be proven for the p-adic topology ;

we have to use Berkovich space. Thanks to recent results of Yuan

about `big line bundles' in Arakelov geometry, the situation

is now very well understood.

#### Seminar on Probability and Statistics

14:50-16:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Statistical analysis of fragmentation chains

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2007/15.html

**Marc HOFFMANN**(Universite Paris-est Marne la vallee)Statistical analysis of fragmentation chains

[ Abstract ]

We address statistical inference in self-similar conservative fragmentation chains, when only observations on the size of the fragments below a given threshold are available. (Possibly, the measurement of the fragments themselves are subject to further systematic experimental noise.) This framework, introduced by Bertoin and Martinez is motivated by mineral crushing in mining industry. We compute upper and lower rates of estimation for several functionals of the dislocation measure, both in a semi-parametric and a non-parametric framework. The underlying estimated object is the step distribution of the random walk associated to a randomly tagged fragment that evolves along the genealogical tree representation of the fragmentation process. We establish a formal link with the statistical problem of estimating the overshoot of the distribution as the crossing level goes to infinity with the size of the dataset; in particular the difficulty of the estimation problem in the non-parametric case is comparable to ill-posed linear inverse problems of order 1 in signal denoising.

[ Reference URL ]We address statistical inference in self-similar conservative fragmentation chains, when only observations on the size of the fragments below a given threshold are available. (Possibly, the measurement of the fragments themselves are subject to further systematic experimental noise.) This framework, introduced by Bertoin and Martinez is motivated by mineral crushing in mining industry. We compute upper and lower rates of estimation for several functionals of the dislocation measure, both in a semi-parametric and a non-parametric framework. The underlying estimated object is the step distribution of the random walk associated to a randomly tagged fragment that evolves along the genealogical tree representation of the fragmentation process. We establish a formal link with the statistical problem of estimating the overshoot of the distribution as the crossing level goes to infinity with the size of the dataset; in particular the difficulty of the estimation problem in the non-parametric case is comparable to ill-posed linear inverse problems of order 1 in signal denoising.

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2007/15.html

### 2008/01/15

#### Tuesday Seminar on Topology

16:30-17:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Adiabatic limits of eta-invariants and the Meyer functions

**飯田 修一**(東京大学大学院数理科学研究科)Adiabatic limits of eta-invariants and the Meyer functions

[ Abstract ]

The Meyer function is the function defined on the hyperelliptic

mapping class group, which gives a signature formula for surface

bundles over surfaces.

In this talk, we introduce certain generalizations of the Meyer

function by using eta-invariants and we discuss the uniqueness of this

function and compute the values for Dehn twists.

The Meyer function is the function defined on the hyperelliptic

mapping class group, which gives a signature formula for surface

bundles over surfaces.

In this talk, we introduce certain generalizations of the Meyer

function by using eta-invariants and we discuss the uniqueness of this

function and compute the values for Dehn twists.

#### Lie Groups and Representation Theory

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

Group contractions, invariant differential operators and the matrix Radon transform

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

**Fulton Gonzalez**(Tufts University)Group contractions, invariant differential operators and the matrix Radon transform

[ Abstract ]

Let $M_{n,k}$ denote the vector space of real $n\\times k$ matrices.

The matrix motion group is the semidirect product $(\\text O(n)\\times \\text O(k))\\ltimes M_{n,k}$, and is the Cartan motion group

associated with the real Grassmannian $G_{n,n+k}$.

The matrix Radon transform is an

integral transform associated with a double fibration involving

homogeneous spaces of this group. We provide a set of

algebraically independent generators of the subalgebra of its

universal enveloping algebra invariant under the Adjoint

representation. One of the elements of this set characterizes the range of the matrix Radon transform.

[ Reference URL ]Let $M_{n,k}$ denote the vector space of real $n\\times k$ matrices.

The matrix motion group is the semidirect product $(\\text O(n)\\times \\text O(k))\\ltimes M_{n,k}$, and is the Cartan motion group

associated with the real Grassmannian $G_{n,n+k}$.

The matrix Radon transform is an

integral transform associated with a double fibration involving

homogeneous spaces of this group. We provide a set of

algebraically independent generators of the subalgebra of its

universal enveloping algebra invariant under the Adjoint

representation. One of the elements of this set characterizes the range of the matrix Radon transform.

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

#### Algebraic Geometry Seminar

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

Homological methods in non-commutative geometry, part 9

[ Reference URL ]

http://imperium.lenin.ru/~kaledin/math/tokyo/

**Dmitry KALEDIN**(Steklov研究所, 東大数理)Homological methods in non-commutative geometry, part 9

[ Reference URL ]

http://imperium.lenin.ru/~kaledin/math/tokyo/

### 2008/01/09

#### Seminar on Probability and Statistics

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

Parameter estimated standardized U-statistics with degenerate kernel for weakly dependent random variables

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2007/14.html

**金川 秀也**(武蔵工業大学)Parameter estimated standardized U-statistics with degenerate kernel for weakly dependent random variables

[ Abstract ]

In this paper, extending the results of Gombay and Horv'{a}th (1998), we prove limit theorems for the maximum of standardized degenerate U-statistics defined by some absolutely regular sequences or functionals of them.

[ Reference URL ]In this paper, extending the results of Gombay and Horv'{a}th (1998), we prove limit theorems for the maximum of standardized degenerate U-statistics defined by some absolutely regular sequences or functionals of them.

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2007/14.html

### 2008/01/08

#### Tuesday Seminar of Analysis

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

On the restrictions of Laplace-Beltrami eigenfunctions to curves

**Nikolay Tzvetkov**(Lille大学)On the restrictions of Laplace-Beltrami eigenfunctions to curves

#### Algebraic Geometry Seminar

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

Homological methods in non-commutative geometry, part 8

**Dmitry KALEDIN**(Steklov研究所, 東大数理)Homological methods in non-commutative geometry, part 8

### 2008/01/07

#### Seminar on Mathematics for various disciplines

13:30-14:30 Room #056 (Graduate School of Math. Sci. Bldg.)

An Optimal Feedback Solution to Quantum Control Problems.

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

**伊藤一文**(North Carolina State University)An Optimal Feedback Solution to Quantum Control Problems.

[ Abstract ]

Control of quantum systems described by Schrodinger equation is considered. Feedback control laws are developed for the orbit tracking via a controled Hamiltonian. Asymptotic tracking properties of the feedback laws are analyzed. Numerical integrations via time-splitting are also analyzed and used to demonstrate the feasibility of the proposed feedback laws.

[ Reference URL ]Control of quantum systems described by Schrodinger equation is considered. Feedback control laws are developed for the orbit tracking via a controled Hamiltonian. Asymptotic tracking properties of the feedback laws are analyzed. Numerical integrations via time-splitting are also analyzed and used to demonstrate the feasibility of the proposed feedback laws.

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

### 2008/01/06

#### Tuesday Seminar of Analysis

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

野海・山田方程式系のWKB解に付随する幾何的構造

**青木 貴史**(近畿大理工)野海・山田方程式系のWKB解に付随する幾何的構造

[ Abstract ]

本多尚文氏、梅田陽子氏との共同研究

本多尚文氏、梅田陽子氏との共同研究

### 2007/12/26

#### Operator Algebra Seminars

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

Pairs of intermediate subfactors

**Pinhas Grossman**(Vanderbilt University)Pairs of intermediate subfactors

### 2007/12/25

#### Tuesday Seminar of Analysis

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

Inverse boundary value problems for the Schrodinger equation with time-dependent electromagnetic potentials and the Aharonov-Bohm effect

**Gregory Eskin**(UCLA)Inverse boundary value problems for the Schrodinger equation with time-dependent electromagnetic potentials and the Aharonov-Bohm effect

[ Abstract ]

We consider the determination of the time-dependent magnetic and electric potentials (modulo gauge transforamtions) by the boundary measurements in domains with obstacles. We use the geometric optics and the tomography of broken rays. The presence of the obstacles leads to the Aharonov-Bohm effect caused by the magnetic and electric fluxes.

We consider the determination of the time-dependent magnetic and electric potentials (modulo gauge transforamtions) by the boundary measurements in domains with obstacles. We use the geometric optics and the tomography of broken rays. The presence of the obstacles leads to the Aharonov-Bohm effect caused by the magnetic and electric fluxes.

### 2007/12/22

#### Infinite Analysis Seminar Tokyo

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

Double Schubert polynomials for the classical Lie groups

Nichols-Woronowicz model of the K-ring of flag vaieties G/B

**池田岳**(岡山理大理) 13:00-14:30Double Schubert polynomials for the classical Lie groups

[ Abstract ]

For each infinite series of the classical Lie groups of type $B$,

$C$ or $D$, we introduce a family of polynomials parametrized by the

elements of the corresponding Weyl group of infinite rank. These

polynomials

represent the Schubert classes in the equivariant cohomology of the

corresponding

flag variety. When indexed by maximal Grassmannian elements of the Weyl

group,

these polynomials are equal to the factorial analogues of Schur $Q$- or

$P$-functions defined earlier by Ivanov. This talk is based on joint work

with L. Mihalcea and H. Naruse.

For each infinite series of the classical Lie groups of type $B$,

$C$ or $D$, we introduce a family of polynomials parametrized by the

elements of the corresponding Weyl group of infinite rank. These

polynomials

represent the Schubert classes in the equivariant cohomology of the

corresponding

flag variety. When indexed by maximal Grassmannian elements of the Weyl

group,

these polynomials are equal to the factorial analogues of Schur $Q$- or

$P$-functions defined earlier by Ivanov. This talk is based on joint work

with L. Mihalcea and H. Naruse.

**前野 俊昭**(京大工) 15:00-16:30Nichols-Woronowicz model of the K-ring of flag vaieties G/B

[ Abstract ]

We give a model of the equivariant $K$-ring $K_T(G/B)$ for

generalized flag varieties $G/B$ in the braided Hopf algebra

called Nichols-Woronowicz algebra. Our model is based on

the Chevalley-type formula for $K_T(G/B)$ due to Lenart

and Postnikov, which is described in terms of alcove paths.

We also discuss a conjecture on the model of the quantum

$K$-ring $QK(G/B)$.

We give a model of the equivariant $K$-ring $K_T(G/B)$ for

generalized flag varieties $G/B$ in the braided Hopf algebra

called Nichols-Woronowicz algebra. Our model is based on

the Chevalley-type formula for $K_T(G/B)$ due to Lenart

and Postnikov, which is described in terms of alcove paths.

We also discuss a conjecture on the model of the quantum

$K$-ring $QK(G/B)$.

### 2007/12/21

#### Colloquium

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

Plato's Cave: what we still don't know about generic projections

**D. Eisenbud**(Univ. of California, Berkeley)Plato's Cave: what we still don't know about generic projections

[ Abstract ]

Riemann Surfaces were first studied algebraically by first projecting them into the complex projective plan; later the same idea was applied to surfaces and higher dimensional varieties, projecting them to hypersurfaces. How much damage is done in this process? For example, what can the fibers of a generic linear projection look like? This is pretty easy for smooth curves and surfaces (though there are still open questions), not so easy in the higher-dimensional case. I'll explain some of what's known, including recent work of mine with Roya Beheshti, Joe Harris, and Craig Huneke.

Riemann Surfaces were first studied algebraically by first projecting them into the complex projective plan; later the same idea was applied to surfaces and higher dimensional varieties, projecting them to hypersurfaces. How much damage is done in this process? For example, what can the fibers of a generic linear projection look like? This is pretty easy for smooth curves and surfaces (though there are still open questions), not so easy in the higher-dimensional case. I'll explain some of what's known, including recent work of mine with Roya Beheshti, Joe Harris, and Craig Huneke.

### 2007/12/20

#### Operator Algebra Seminars

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

Gauge-invariant ideal structure of ultragraph $C^*$-algebras

**崎山理史**(東大数理)Gauge-invariant ideal structure of ultragraph $C^*$-algebras

#### Lectures

10:40-12:10 Room #128 (Graduate School of Math. Sci. Bldg.)

Topics in ergodic theory, von Neumann algebras, and rigidity

[ Reference URL ]

https://www.ms.u-tokyo.ac.jp/~yasuyuki/pichot.htm

**Mikael Pichot**(東大数理)Topics in ergodic theory, von Neumann algebras, and rigidity

[ Reference URL ]

https://www.ms.u-tokyo.ac.jp/~yasuyuki/pichot.htm

### 2007/12/19

#### Seminar on Probability and Statistics

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

Sequential Tests for Criticality of Branching Processes.

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2007/13.html

**永井 圭二**(横浜国立大学)Sequential Tests for Criticality of Branching Processes.

[ Abstract ]

We consider sequential testing procedures for detection of

criticality of Galton-Watson branching process with or without

immigration. We develop a t-test from fixed accuracy estimation

theory and a sequential probability ratio test (SPRT). We provide

local asymptotic normality (LAN) of the t-test and some asymptotic

optimality of the SPRT. We consider a general framework of

diffusion approximations from discrete-time processes and develop

sequential tests for one-dimensional diffusion processes to

investigate the operating characteristics of sequential tests

of the discrete-time processes. Especially the Bessel process with

constant drift plays a important role for the sequential test

of criticality of branching process with immigration.

(Joint work with K. Hitomi (Kyoto Institute of Technology)

and Y. Nishiyama (Kyoto Univ.))

[ Reference URL ]We consider sequential testing procedures for detection of

criticality of Galton-Watson branching process with or without

immigration. We develop a t-test from fixed accuracy estimation

theory and a sequential probability ratio test (SPRT). We provide

local asymptotic normality (LAN) of the t-test and some asymptotic

optimality of the SPRT. We consider a general framework of

diffusion approximations from discrete-time processes and develop

sequential tests for one-dimensional diffusion processes to

investigate the operating characteristics of sequential tests

of the discrete-time processes. Especially the Bessel process with

constant drift plays a important role for the sequential test

of criticality of branching process with immigration.

(Joint work with K. Hitomi (Kyoto Institute of Technology)

and Y. Nishiyama (Kyoto Univ.))

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2007/13.html

### 2007/12/18

#### Tuesday Seminar on Topology

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

Groupoid lifts of representations of mapping classes

**R.C. Penner**(USC and Aarhus University)Groupoid lifts of representations of mapping classes

[ Abstract ]

The "Ptolemy groupoid" is the fundamental path groupoid of the dual to the ideal cell decomposition of the decorated Teichmueller space of a punctured or bordered surface, and the "Torelli groupoid" is thesimilar discretization of the fundamental path groupoid of the quotient

by the Torelli subgroup of mapping classes that acts identically on the first integral homology of the surface. Mapping classes can be represented as appropriate elements of the Ptolemy groupoid and likewise for elements of the Torelli group in the Torelli groupoid.

A natural series of questions is to wonder which representations of mapping class groups, Torelli groups, and their subgroups can be lifted to the groupoid level. In a series of joint works with J. Andersen, A. Bene, N. Kawazumi, and S. Morita, we have given explicit lifts of a number of classical representations: The Johnson representations of the classical and higher Torelli groups

and the symplectic representation of the mapping class group all lift to the Torelli groupoid. Furthermore, the Nielsen representation of the mapping class group as automorphisms of a

free group lifts to the Ptolemy groupoid, and hence so too does any representation

of the mapping class group that factors through its action on the fundamental group of

the surface such as the Magnus representation. We shall survey these various groupoid lifts and discuss current and potential future applications.

The "Ptolemy groupoid" is the fundamental path groupoid of the dual to the ideal cell decomposition of the decorated Teichmueller space of a punctured or bordered surface, and the "Torelli groupoid" is thesimilar discretization of the fundamental path groupoid of the quotient

by the Torelli subgroup of mapping classes that acts identically on the first integral homology of the surface. Mapping classes can be represented as appropriate elements of the Ptolemy groupoid and likewise for elements of the Torelli group in the Torelli groupoid.

A natural series of questions is to wonder which representations of mapping class groups, Torelli groups, and their subgroups can be lifted to the groupoid level. In a series of joint works with J. Andersen, A. Bene, N. Kawazumi, and S. Morita, we have given explicit lifts of a number of classical representations: The Johnson representations of the classical and higher Torelli groups

and the symplectic representation of the mapping class group all lift to the Torelli groupoid. Furthermore, the Nielsen representation of the mapping class group as automorphisms of a

free group lifts to the Ptolemy groupoid, and hence so too does any representation

of the mapping class group that factors through its action on the fundamental group of

the surface such as the Magnus representation. We shall survey these various groupoid lifts and discuss current and potential future applications.

#### Lie Groups and Representation Theory

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

On the existence of homomorphisms between principal series of complex

semisimple Lie groups

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

**阿部 紀行**(東京大学)On the existence of homomorphisms between principal series of complex

semisimple Lie groups

[ Abstract ]

The principal series representations of a semisimple Lie group play an important role in the representation theory. We study the principal series representation of a complex semisimple Lie group and determine when there exists a nonzero homomorphism between the representations.

[ Reference URL ]The principal series representations of a semisimple Lie group play an important role in the representation theory. We study the principal series representation of a complex semisimple Lie group and determine when there exists a nonzero homomorphism between the representations.

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

### 2007/12/17

#### Seminar on Geometric Complex Analysis

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

種数3の曲線とあるCalabi-Yau threefoldの代数的対応(松本圭司氏との共同研究)

**寺杣友秀**(東京大学)種数3の曲線とあるCalabi-Yau threefoldの代数的対応(松本圭司氏との共同研究)

#### Kavli IPMU Komaba Seminar

17:00-18:30 Room #002 (Graduate School of Math. Sci. Bldg.)

Analytic torsion for Calabi-Yau threefolds

**Ken-Ichi Yoshikawa**(The University of Tokyo)Analytic torsion for Calabi-Yau threefolds

[ Abstract ]

In 1994, Bershadky-Cecotti-Ooguri-Vafa conjectured that analytic torsion

gives rise to a function on the moduli space of Calabi-Yau threefolds and

that it coincides with the quantity $F_{1}$ in string theory.

Since the holomorphic part of $F_{1}$ is conjecturally the generating function

of the counting problem of elliptic curves in the mirror Calabi-Yau threefold,

this implies the conjectural equivalence of analytic torsion and the counting

problem of elliptic curves for Calabi-Yau threefolds through mirror symmetry.

After Bershadsky-Cecotti-Ooguri-Vafa, we introduced an invariant of

Calabi-Yau threefolds, which we obtained using analytic torsion and

a Bott-Chern secondary class. In this talk, we will talk about the construction

and some explicit formulae of this analytic torsion invariant.

Some part of this talk is based on the joint work with H. Fang and Z. Lu.

In 1994, Bershadky-Cecotti-Ooguri-Vafa conjectured that analytic torsion

gives rise to a function on the moduli space of Calabi-Yau threefolds and

that it coincides with the quantity $F_{1}$ in string theory.

Since the holomorphic part of $F_{1}$ is conjecturally the generating function

of the counting problem of elliptic curves in the mirror Calabi-Yau threefold,

this implies the conjectural equivalence of analytic torsion and the counting

problem of elliptic curves for Calabi-Yau threefolds through mirror symmetry.

After Bershadsky-Cecotti-Ooguri-Vafa, we introduced an invariant of

Calabi-Yau threefolds, which we obtained using analytic torsion and

a Bott-Chern secondary class. In this talk, we will talk about the construction

and some explicit formulae of this analytic torsion invariant.

Some part of this talk is based on the joint work with H. Fang and Z. Lu.

### 2007/12/13

#### Lectures

10:40-12:10 Room #128 (Graduate School of Math. Sci. Bldg.)

Topics in ergodic theory, von Neumann algebras, and rigidity

[ Reference URL ]

https://www.ms.u-tokyo.ac.jp/~yasuyuki/pichot.htm

**Mikael Pichot**(東大数理)Topics in ergodic theory, von Neumann algebras, and rigidity

[ Reference URL ]

https://www.ms.u-tokyo.ac.jp/~yasuyuki/pichot.htm

#### Applied Analysis

16:00-17:30 Room #126 (Graduate School of Math. Sci. Bldg.)

Singular limit of a competition-diffusion system

**Danielle Hilhorst**(CNRS / パリ第11大学)Singular limit of a competition-diffusion system

[ Abstract ]

We revisit a competition-diffusion system for the densities of biological populations, and (i) prove the strong convergence in L^2 of the densities of the biological species (joint work with Iida, Mimura and Ninomiya); (ii) derive the singular limit of some reaction terms as the reaction coefficient tends to infinity (joint work with Martin and Mimura).

We revisit a competition-diffusion system for the densities of biological populations, and (i) prove the strong convergence in L^2 of the densities of the biological species (joint work with Iida, Mimura and Ninomiya); (ii) derive the singular limit of some reaction terms as the reaction coefficient tends to infinity (joint work with Martin and Mimura).

### 2007/12/12

#### Seminar on Probability and Statistics

15:20-16:30 Room #122 (Graduate School of Math. Sci. Bldg.)

Inference problems for the telegraph process observed at discrete times

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2007/12.html

**Stefano IACUS**(Department of Economics, Business and Statistics, University of Milan)Inference problems for the telegraph process observed at discrete times

[ Abstract ]

The telegraph process {X(t), t>0}, has been introduced (see

Goldstein, 1951) as an alternative model to the Brownian motion B(t).

This process describes a motion of a particle on the real line which

alternates its velocity, at Poissonian times, from +v to -v. The

density of the distribution of the position of the particle at time t

solves the hyperbolic differential equation called telegraph equation

and hence the name of the process.

Contrary to B(t) the process X(t) has finite variation and

continuous and differentiable paths. At the same time it is

mathematically challenging to handle. Several variation of this

process have been recently introduced in the context of Finance.

In this talk we will discuss pseudo-likelihood and moment type

estimators of the intensity of the Poisson process, from discrete

time observations of standard telegraph process X(t). We also

discuss the problem of change point estimation for the intensity of

the underlying Poisson process and show the performance of this

estimator on real data.

[ Reference URL ]The telegraph process {X(t), t>0}, has been introduced (see

Goldstein, 1951) as an alternative model to the Brownian motion B(t).

This process describes a motion of a particle on the real line which

alternates its velocity, at Poissonian times, from +v to -v. The

density of the distribution of the position of the particle at time t

solves the hyperbolic differential equation called telegraph equation

and hence the name of the process.

Contrary to B(t) the process X(t) has finite variation and

continuous and differentiable paths. At the same time it is

mathematically challenging to handle. Several variation of this

process have been recently introduced in the context of Finance.

In this talk we will discuss pseudo-likelihood and moment type

estimators of the intensity of the Poisson process, from discrete

time observations of standard telegraph process X(t). We also

discuss the problem of change point estimation for the intensity of

the underlying Poisson process and show the performance of this

estimator on real data.

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2007/12.html

### 2007/12/11

#### Tuesday Seminar on Topology

16:30-18:40 Room #056 (Graduate School of Math. Sci. Bldg.)

A Singular Version of The Poincar\\'e-Hopf Theorem

Chen Ruan cohomology of cotangent orbifolds and Chas-Sullivan string topology

**Xavier G\'omez-Mont**(CIMAT, Mexico) 16:30-17:30A Singular Version of The Poincar\\'e-Hopf Theorem

[ Abstract ]

The Poincar\\'e-Hopf Theorem asserts that the Euler Characteristic of a compact manifold is the sum of the indices of any vector field on it with isolated singularities.

A hypersurface in real or complex number space may be considered as the limit of the smooth hypersurfaces obtained from nearby regular values. The singularity contains “hidden” topology, which is unfolded by a smooth regeneration. At the singularity one has an algebraic invariant, the Jacobi Algebra, which is obtained by considering analytic functions modulo the partial derivatives. It contains topological information of the singularity.

One may consider vector fields tangent to a hypersurface with isolated singularities, and define topologically an index, which coincides with the sum of the Poincar\\'e-Hopf indices of a regeneration of it tangent to a nearby smooth hypersurface.

I will explain how to compute the index of a vector field X tangent to an isolated hypersurface singularity V using Homological Algebra, as the Euler Characteristic of the homology of the complex obtained by contracting differential forms on V with the vector field X. The formula contains several terms, but the higher order terms may be translated from the invariants of the singular point to invariants in the Jacobi Algebra, making this translation a local version of the Poincar\\'e-Hopf Theorem.

I will also explain how some of these ideas can be extended to complete intersections.

The Poincar\\'e-Hopf Theorem asserts that the Euler Characteristic of a compact manifold is the sum of the indices of any vector field on it with isolated singularities.

A hypersurface in real or complex number space may be considered as the limit of the smooth hypersurfaces obtained from nearby regular values. The singularity contains “hidden” topology, which is unfolded by a smooth regeneration. At the singularity one has an algebraic invariant, the Jacobi Algebra, which is obtained by considering analytic functions modulo the partial derivatives. It contains topological information of the singularity.

One may consider vector fields tangent to a hypersurface with isolated singularities, and define topologically an index, which coincides with the sum of the Poincar\\'e-Hopf indices of a regeneration of it tangent to a nearby smooth hypersurface.

I will explain how to compute the index of a vector field X tangent to an isolated hypersurface singularity V using Homological Algebra, as the Euler Characteristic of the homology of the complex obtained by contracting differential forms on V with the vector field X. The formula contains several terms, but the higher order terms may be translated from the invariants of the singular point to invariants in the Jacobi Algebra, making this translation a local version of the Poincar\\'e-Hopf Theorem.

I will also explain how some of these ideas can be extended to complete intersections.

**Miguel A. Xicotencatl**(CINVESTAV, Mexico) 17:40-18:40Chen Ruan cohomology of cotangent orbifolds and Chas-Sullivan string topology

[ Abstract ]

(Joint with: A. Gonzalez, E. Lupercio, C. Segovia, and B. Uribe)

At the end of 90's, two theories of topology were invented roughly at the same time and attracted considerable interest in the mathematical community. One is the Chas-Sullivan's loop product on the homology of loop space and the second one is Chen-Ruan's stringy cohomology of orbifold. It was an observation of Chen that inertia orbifold (which carries Chen-Ruan cohomology) is the space of constant loops of an orbifold. Therefore, two theories should interact. In this work we show that for an interesting family of orbifolds, the virtual orbifold cohomology, turns out to be a subalgebra of the homology of the loop orbifold, and is isomorphic, as algebras, to the Chen-Ruan orbifold cohomology of its cotangent orbifold.

(Joint with: A. Gonzalez, E. Lupercio, C. Segovia, and B. Uribe)

At the end of 90's, two theories of topology were invented roughly at the same time and attracted considerable interest in the mathematical community. One is the Chas-Sullivan's loop product on the homology of loop space and the second one is Chen-Ruan's stringy cohomology of orbifold. It was an observation of Chen that inertia orbifold (which carries Chen-Ruan cohomology) is the space of constant loops of an orbifold. Therefore, two theories should interact. In this work we show that for an interesting family of orbifolds, the virtual orbifold cohomology, turns out to be a subalgebra of the homology of the loop orbifold, and is isomorphic, as algebras, to the Chen-Ruan orbifold cohomology of its cotangent orbifold.

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