## Seminar information archive

Seminar information archive ～02/22｜Today's seminar 02/23 | Future seminars 02/24～

#### Classical Analysis

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

The third Painlev¥'e equation and quiver varieties (JAPANESE)

**Daisuke Yamakawa**(Kobe University)The third Painlev¥'e equation and quiver varieties (JAPANESE)

### 2010/12/01

#### Number Theory Seminar

16:30-18:45 Room #056 (Graduate School of Math. Sci. Bldg.)

On a problem of Matsumoto and Tamagawa concerning monodromic fullness of hyperbolic curves (JAPANESE)

Galois theory for schemes (ENGLISH)

**Yuichiro Hoshi**(RIMS, Kyoto University) 16:30-17:30On a problem of Matsumoto and Tamagawa concerning monodromic fullness of hyperbolic curves (JAPANESE)

[ Abstract ]

In this talk, we will discuss the following problem posed by Makoto Matsumoto and Akio Tamagawa concerning monodromic fullness of hyperbolic curves.

For a hyperbolic curve X over a number field, are the following three conditions equivalent?

(A) For any prime number l, X is quasi-l-monodromically full.

(B) There exists a prime number l such that X is l-monodromically full.

(C) X is l-monodromically full for all but finitely many prime numbers l.

The property of being (quasi-)monodromically full may be regarded as an analogue for hyperbolic curves of the property of not admitting complex multiplication for elliptic curves, and the above equivalence may be regarded as an analogue for hyperbolic curves of the following result concerning the Galois representation on the Tate module of an elliptic curve over a number field proven by Jean-Pierre Serre.

For an elliptic curve E over a number field, the following four conditions are equivalent:

(0) E does not admit complex multiplication.

(1) For any prime number l, the image of the l-adic Galois representation associated to E is open.

(2) There exists a prime number l such that the l-adic Galois representation associated to E is surjective.

(3) The l-adic Galois representation associated to E is surjective for all but finitely many prime numbers l.

In this talk, I will present some results concerning the above problem in the case where the given hyperbolic curve is of genus zero. In particular, I will give an example of a hyperbolic curve of type (0,4) over a number field which satisfies condition (C) but does not satisfy condition (A).

In this talk, we will discuss the following problem posed by Makoto Matsumoto and Akio Tamagawa concerning monodromic fullness of hyperbolic curves.

For a hyperbolic curve X over a number field, are the following three conditions equivalent?

(A) For any prime number l, X is quasi-l-monodromically full.

(B) There exists a prime number l such that X is l-monodromically full.

(C) X is l-monodromically full for all but finitely many prime numbers l.

The property of being (quasi-)monodromically full may be regarded as an analogue for hyperbolic curves of the property of not admitting complex multiplication for elliptic curves, and the above equivalence may be regarded as an analogue for hyperbolic curves of the following result concerning the Galois representation on the Tate module of an elliptic curve over a number field proven by Jean-Pierre Serre.

For an elliptic curve E over a number field, the following four conditions are equivalent:

(0) E does not admit complex multiplication.

(1) For any prime number l, the image of the l-adic Galois representation associated to E is open.

(2) There exists a prime number l such that the l-adic Galois representation associated to E is surjective.

(3) The l-adic Galois representation associated to E is surjective for all but finitely many prime numbers l.

In this talk, I will present some results concerning the above problem in the case where the given hyperbolic curve is of genus zero. In particular, I will give an example of a hyperbolic curve of type (0,4) over a number field which satisfies condition (C) but does not satisfy condition (A).

**Marco Garuti**(University of Padova) 17:45-18:45Galois theory for schemes (ENGLISH)

[ Abstract ]

We discuss some aspects of finite group scheme actions: the Galois correspondence and the notion of Galois closure.

We discuss some aspects of finite group scheme actions: the Galois correspondence and the notion of Galois closure.

### 2010/11/30

#### Numerical Analysis Seminar

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

Finite volume element method for singular solutions of elliptic PDEs

(JAPANESE)

[ Reference URL ]

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

**Yasunori Aoki**(University of Waterloo/NII)Finite volume element method for singular solutions of elliptic PDEs

(JAPANESE)

[ Reference URL ]

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

#### Tuesday Seminar on Topology

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

Pin^-(2)-monopole equations and intersection forms with local coefficients of 4-manifolds (JAPANESE)

**Nobuhiro Nakamura**(The University of Tokyo)Pin^-(2)-monopole equations and intersection forms with local coefficients of 4-manifolds (JAPANESE)

[ Abstract ]

We introduce a variant of the Seiberg-Witten equations, Pin^-(2)-monopole equations, and explain its applications to intersection forms with local coefficients of 4-manifolds.

The first application is an analogue of Froyshov's results on 4-manifolds which have definite forms with local coefficients.

The second one is a local coefficient version of Furuta's 10/8-inequality.

As a corollary, we construct nonsmoothable spin 4-manifolds satisfying Rohlin's theorem and the 10/8-inequality.

We introduce a variant of the Seiberg-Witten equations, Pin^-(2)-monopole equations, and explain its applications to intersection forms with local coefficients of 4-manifolds.

The first application is an analogue of Froyshov's results on 4-manifolds which have definite forms with local coefficients.

The second one is a local coefficient version of Furuta's 10/8-inequality.

As a corollary, we construct nonsmoothable spin 4-manifolds satisfying Rohlin's theorem and the 10/8-inequality.

#### Operator Algebra Seminars

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

Noncommutative geometry and Rankin-Cohen brackets (ENGLISH)

**Yi-Jun Yao**(Fudan Univ.)Noncommutative geometry and Rankin-Cohen brackets (ENGLISH)

### 2010/11/29

#### Kavli IPMU Komaba Seminar

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

Borcherds products in monstrous moonshine. (ENGLISH)

**Scott Carnahan**(IPMU)Borcherds products in monstrous moonshine. (ENGLISH)

[ Abstract ]

During the 1980s, Koike, Norton, and Zagier independently found an

infinite product expansion for the difference of two modular j-functions

on a product of half planes. Borcherds showed that this product identity

is the Weyl denominator formula for an infinite dimensional Lie algebra

that has an action of the monster simple group by automorphisms, and used

this action to prove the monstrous moonshine conjectures.

I will describe a more general construction that yields an infinite

product identity and an infinite dimensional Lie algebra for each element

of the monster group. The above objects then arise as the special cases

assigned to the identity element. Time permitting, I will attempt to

describe a connection to conformal field theory.

During the 1980s, Koike, Norton, and Zagier independently found an

infinite product expansion for the difference of two modular j-functions

on a product of half planes. Borcherds showed that this product identity

is the Weyl denominator formula for an infinite dimensional Lie algebra

that has an action of the monster simple group by automorphisms, and used

this action to prove the monstrous moonshine conjectures.

I will describe a more general construction that yields an infinite

product identity and an infinite dimensional Lie algebra for each element

of the monster group. The above objects then arise as the special cases

assigned to the identity element. Time permitting, I will attempt to

describe a connection to conformal field theory.

#### Algebraic Geometry Seminar

16:40-18:10 Room #126 (Graduate School of Math. Sci. Bldg.)

K3 surfaces and log del Pezzo surfaces of index three (JAPANESE)

**Hisanori Ohashi**(Nagoya Univ. )K3 surfaces and log del Pezzo surfaces of index three (JAPANESE)

[ Abstract ]

Alexeev and Nikulin have classified log del Pezzo surfaces of index 1 and 2 by using the classification of non-symplectic involutions on K3 surfaces. We want to discuss the generalization of this result to the index 3 cases. In this case we are also able to construct log del Pezzos $Z$ from K3 surfaces $X$, but the converse is not necessarily true. The condition on $Z$ is exactly the "multiple smooth divisor property", which we will define. Our theorem is the classification of log del Pezzo surfaces of index 3 with this property.

The idea of the proof is similar to that of Alexeev and Nikulin, but the methods are different because of the existence of singularities: although the singularity is mild, the description of nef cone by reflection groups cannot be used. Instead

we construct and analyze good elliptic fibrations on K3 surfaces $X$ and use it to obtain the classification. It includes a partial but geometric generalization of the classification of non-symplectic automorphisms of order three, recently done by Artebani, Sarti and Taki.

Alexeev and Nikulin have classified log del Pezzo surfaces of index 1 and 2 by using the classification of non-symplectic involutions on K3 surfaces. We want to discuss the generalization of this result to the index 3 cases. In this case we are also able to construct log del Pezzos $Z$ from K3 surfaces $X$, but the converse is not necessarily true. The condition on $Z$ is exactly the "multiple smooth divisor property", which we will define. Our theorem is the classification of log del Pezzo surfaces of index 3 with this property.

The idea of the proof is similar to that of Alexeev and Nikulin, but the methods are different because of the existence of singularities: although the singularity is mild, the description of nef cone by reflection groups cannot be used. Instead

we construct and analyze good elliptic fibrations on K3 surfaces $X$ and use it to obtain the classification. It includes a partial but geometric generalization of the classification of non-symplectic automorphisms of order three, recently done by Artebani, Sarti and Taki.

### 2010/11/26

#### Kavli IPMU Komaba Seminar

14:40-16:10 Room #002 (Graduate School of Math. Sci. Bldg.)

Hamiltonian torus actions on orbifolds and orbifold-GKM theorem (joint

work with T. Holm) (JAPANESE)

**Tomoo Matsumura**(Cornell University)Hamiltonian torus actions on orbifolds and orbifold-GKM theorem (joint

work with T. Holm) (JAPANESE)

[ Abstract ]

When a symplectic manifold M carries a Hamiltonian torus R action, the

injectivity theorem states that the R-equivariant cohomology of M is a

subring of the one of the fixed points and the GKM theorem allows us

to compute this subring by only using the data of 1-dimensional

orbits. The results in the first part of this talk are a

generalization of this technique to Hamiltonian R actions on orbifolds

and an application to the computation of the equivariant cohomology of

toric orbifolds. In the second part, we will introduce the equivariant

Chen-Ruan cohomology ring which is a symplectic invariant of the

action on the orbifold and explain the injectivity/GKM theorem for this ring.

When a symplectic manifold M carries a Hamiltonian torus R action, the

injectivity theorem states that the R-equivariant cohomology of M is a

subring of the one of the fixed points and the GKM theorem allows us

to compute this subring by only using the data of 1-dimensional

orbits. The results in the first part of this talk are a

generalization of this technique to Hamiltonian R actions on orbifolds

and an application to the computation of the equivariant cohomology of

toric orbifolds. In the second part, we will introduce the equivariant

Chen-Ruan cohomology ring which is a symplectic invariant of the

action on the orbifold and explain the injectivity/GKM theorem for this ring.

### 2010/11/25

#### Operator Algebra Seminars

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

Classification of actions of Kac algebras (JAPANESE)

**Reiji Tomatsu**(Tokyo Univ. Science)Classification of actions of Kac algebras (JAPANESE)

### 2010/11/18

#### Operator Algebra Seminars

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

Perturbation of dual operator algebras and similarity (ENGLISH)

**Jean Roydor**(Univ. Tokyo)Perturbation of dual operator algebras and similarity (ENGLISH)

### 2010/11/17

#### Number Theory Seminar

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

Fast lattice reduction for F_2-linear pseudorandom number generators (JAPANESE)

**Shin Harase**(University of Tokyo)Fast lattice reduction for F_2-linear pseudorandom number generators (JAPANESE)

### 2010/11/16

#### Numerical Analysis Seminar

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

On verified evaluation of eigenvalues for elliptic operator over arbitrary polygonal domain (JAPANESE)

[ Reference URL ]

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

**Xuefeng Liu**(Waseda University/CREST, JST)On verified evaluation of eigenvalues for elliptic operator over arbitrary polygonal domain (JAPANESE)

[ Reference URL ]

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

#### Tuesday Seminar on Topology

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

On a colored Khovanov bicomplex (JAPANESE)

**Noboru Ito**(Waseda University)On a colored Khovanov bicomplex (JAPANESE)

[ Abstract ]

We discuss the existence of a bicomplex which is a Khovanov-type

complex associated with categorification of a colored Jones polynomial.

This is an answer to the question proposed by A. Beliakova and S. Wehrli.

Then the second term of the spectral sequence of the bicomplex corresponds

to the Khovanov-type homology group. In this talk, we explain how to define

the bicomplex. If time permits, we also define a colored Rasmussen invariant

by using another spectral sequence of the colored Jones polynomial.

We discuss the existence of a bicomplex which is a Khovanov-type

complex associated with categorification of a colored Jones polynomial.

This is an answer to the question proposed by A. Beliakova and S. Wehrli.

Then the second term of the spectral sequence of the bicomplex corresponds

to the Khovanov-type homology group. In this talk, we explain how to define

the bicomplex. If time permits, we also define a colored Rasmussen invariant

by using another spectral sequence of the colored Jones polynomial.

#### Algebraic Geometry Seminar

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

Self-corresponences of K3 surfaces via moduli of sheaves (ENGLISH)

**Viacheslav Nikulin**(Univ Liverpool and Steklov Moscow)Self-corresponences of K3 surfaces via moduli of sheaves (ENGLISH)

[ Abstract ]

In series of our papers with Carlo Madonna (2002--2008) we described self-correspondences via moduli of sheaves with primitive isotropic Mukai vectors for K3 surfaces with Picard number one or two. Here, we give a natural and functorial answer to the same problem for arbitrary Picard number of K3 surfaces. As an application, we characterize in terms of self-correspondences via moduli of sheaves K3 surfaces with reflective Picard lattices, that is when the automorphism group of the lattice is generated by reflections up to finite index. See some details in arXiv:0810.2945.

In series of our papers with Carlo Madonna (2002--2008) we described self-correspondences via moduli of sheaves with primitive isotropic Mukai vectors for K3 surfaces with Picard number one or two. Here, we give a natural and functorial answer to the same problem for arbitrary Picard number of K3 surfaces. As an application, we characterize in terms of self-correspondences via moduli of sheaves K3 surfaces with reflective Picard lattices, that is when the automorphism group of the lattice is generated by reflections up to finite index. See some details in arXiv:0810.2945.

#### Algebraic Geometry Seminar

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

Self-corresponences of K3 surfaces via moduli of sheaves (ENGLISH)

**Viacheslav Nikulin**(Univ Liverpool and Steklov Moscow)Self-corresponences of K3 surfaces via moduli of sheaves (ENGLISH)

[ Abstract ]

In series of our papers with Carlo Madonna (2002--2008) we described self-correspondences via moduli of sheaves with primitive isotropic Mukai vectors for K3 surfaces with Picard number one or two. Here, we give a natural and functorial answer to the same problem for arbitrary Picard number of K3 surfaces. As an application, we characterize in terms of self-correspondences via moduli of sheaves K3 surfaces with reflective Picard lattices, that is when the automorphism group of the lattice is generated by reflections up to finite index. See some details in arXiv:0810.2945.

In series of our papers with Carlo Madonna (2002--2008) we described self-correspondences via moduli of sheaves with primitive isotropic Mukai vectors for K3 surfaces with Picard number one or two. Here, we give a natural and functorial answer to the same problem for arbitrary Picard number of K3 surfaces. As an application, we characterize in terms of self-correspondences via moduli of sheaves K3 surfaces with reflective Picard lattices, that is when the automorphism group of the lattice is generated by reflections up to finite index. See some details in arXiv:0810.2945.

#### Tuesday Seminar of Analysis

16:00-18:30 Room #123 (Graduate School of Math. Sci. Bldg.)

Hyperfunctions and vortex sheets (ENGLISH)

Residual trace and equivariant asymptotic trace of Toeplitz operators (ENGLISH)

**Keisuke Uchikoshi**(National Defense Academy of Japan) 16:00-16:45Hyperfunctions and vortex sheets (ENGLISH)

**L. Boutet de Monvel**(University of Paris 6) 17:00-18:30Residual trace and equivariant asymptotic trace of Toeplitz operators (ENGLISH)

### 2010/11/15

#### Seminar on Geometric Complex Analysis

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

Excess intersections and residues in improper dimension (JAPANESE)

**Tatsuo Suwa**(Hokkaido Univ*)Excess intersections and residues in improper dimension (JAPANESE)

[ Abstract ]

This talk concerns localization of characteristic classes and associated residues, in the context of intersection theory and residue theory of singular holomorphic foliations. The localization comes from the vanishing of certain characteristic forms, usually caused by the existence of some geometric object, away from the "singular set" of the object. This gives rise to residues in the homology of the singular set and residue theorems relating local and global invariants. In the generic situation, i.e., if the dimension of the singular set is "proper", we have a reasonable understanding of the residues. We indicate how to cope with the problem when the dimension is "excessive" (partly a joint work with F. Bracci).

This talk concerns localization of characteristic classes and associated residues, in the context of intersection theory and residue theory of singular holomorphic foliations. The localization comes from the vanishing of certain characteristic forms, usually caused by the existence of some geometric object, away from the "singular set" of the object. This gives rise to residues in the homology of the singular set and residue theorems relating local and global invariants. In the generic situation, i.e., if the dimension of the singular set is "proper", we have a reasonable understanding of the residues. We indicate how to cope with the problem when the dimension is "excessive" (partly a joint work with F. Bracci).

#### Algebraic Geometry Seminar

16:40-18:10 Room #126 (Graduate School of Math. Sci. Bldg.)

Generators of tropical modules (JAPANESE)

**Shuhei Yoshitomi**(Univ. of Tokyo)Generators of tropical modules (JAPANESE)

### 2010/11/09

#### Tuesday Seminar on Topology

17:00-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Characterising bumping points on deformation spaces of Kleinian groups (JAPANESE)

**Ken'ichi Ohshika**(Osaka University)Characterising bumping points on deformation spaces of Kleinian groups (JAPANESE)

[ Abstract ]

It is known that components of the interior of a deformation space of a Kleinian group can bump, and a component of the interior can bump itself, on the boundary of the deformation space.

Anderson-Canary-McCullogh gave a necessary and sufficient condition for two components to bump.

In this talk, I shall give a criterion for points on the boundary to be bumping points.

It is known that components of the interior of a deformation space of a Kleinian group can bump, and a component of the interior can bump itself, on the boundary of the deformation space.

Anderson-Canary-McCullogh gave a necessary and sufficient condition for two components to bump.

In this talk, I shall give a criterion for points on the boundary to be bumping points.

### 2010/11/08

#### Seminar on Geometric Complex Analysis

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

Variation of canonical measures under Kaehler deformations (JAPANESE)

**Hajime Tsuji**(Sophia Univ)Variation of canonical measures under Kaehler deformations (JAPANESE)

#### GCOE lecture series

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

Invariant differential operators on the sphere (ENGLISH)

http://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar2010.html#20101102eastwood

**Michael Eastwood**(Australian National University)Invariant differential operators on the sphere (ENGLISH)

[ Abstract ]

The circle is acted upon by the rotation group SO(2) and there are plenty of differential operators invariant under this action. But the circle is also acted upon by SL(2,R) and this larger symmetry group cuts down the list of invariant differential operators to something smaller but more interesting! I shall explain what happens and how this phenomenon generalises to spheres. These constructions are part of a general theory but have numerous unexpected applications, for example in suggesting a new stable finite-element scheme in linearised elasticity (due to Arnold, Falk, and Winther).

[ Reference URL ]The circle is acted upon by the rotation group SO(2) and there are plenty of differential operators invariant under this action. But the circle is also acted upon by SL(2,R) and this larger symmetry group cuts down the list of invariant differential operators to something smaller but more interesting! I shall explain what happens and how this phenomenon generalises to spheres. These constructions are part of a general theory but have numerous unexpected applications, for example in suggesting a new stable finite-element scheme in linearised elasticity (due to Arnold, Falk, and Winther).

http://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar2010.html#20101102eastwood

### 2010/11/05

#### GCOE lecture series

16:30-18:00 Room #123 (Graduate School of Math. Sci. Bldg.)

How to recognise the geodesics of a metric connection (ENGLISH)

http://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar2010.html#20101102eastwood

**Michael Eastwood**(Australian National University)How to recognise the geodesics of a metric connection (ENGLISH)

[ Abstract ]

The geodesics on a Riemannian manifold form a distinguished family of curves, one in every direction through every point. Sometimes two metrics can provide the same family of curves: the Euclidean metric and the round metric on the hemisphere have this property. It is also possible that a family of curves does not arise from a metric at all. Following a classical procedure due to Roger Liouville, I shall explain how to tell these cases apart on a surface. This is joint work with Robert Bryant and Maciej Dunajski.

[ Reference URL ]The geodesics on a Riemannian manifold form a distinguished family of curves, one in every direction through every point. Sometimes two metrics can provide the same family of curves: the Euclidean metric and the round metric on the hemisphere have this property. It is also possible that a family of curves does not arise from a metric at all. Following a classical procedure due to Roger Liouville, I shall explain how to tell these cases apart on a surface. This is joint work with Robert Bryant and Maciej Dunajski.

http://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar2010.html#20101102eastwood

### 2010/11/04

#### Operator Algebra Seminars

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

Nonequilibrium Statistical Mechanics (JAPANESE)

**Yoshiko Ogata**(Univ.Tokyo)Nonequilibrium Statistical Mechanics (JAPANESE)

#### Lectures

10:40-12:10 Room #123 (Graduate School of Math. Sci. Bldg.)

Market, Liquidity and Counterparty Risk (ENGLISH)

**Jean Meyer, Yasuko HISAMATSU**(Risk Capital Market Tokyo, BNP Paribas)Market, Liquidity and Counterparty Risk (ENGLISH)

[ Abstract ]

1. Introduction to the market risk

- Introduction to the Risk Management

in the Financial institutions

- Overview of the main market risks

2. Market & Liquidity Risks –Basics

-Presentation of the main Greeks

-Focus on volatility risk

-Focus on correlation risk

-Conclusion (common features of the market risks)

3. Risk measure

- Stress test

- Value at risk

- Risks measure for counterparty risk

1. Introduction to the market risk

- Introduction to the Risk Management

in the Financial institutions

- Overview of the main market risks

2. Market & Liquidity Risks –Basics

-Presentation of the main Greeks

-Focus on volatility risk

-Focus on correlation risk

-Conclusion (common features of the market risks)

3. Risk measure

- Stress test

- Value at risk

- Risks measure for counterparty risk

### 2010/11/02

#### Lectures

13:00-16:10 Room #122 (Graduate School of Math. Sci. Bldg.)

The Malliavin calculus on configuration spaces and applications (ENGLISH)

**Vladimir Bogachev**(Moscow)The Malliavin calculus on configuration spaces and applications (ENGLISH)

[ Abstract ]

It is planned to discuss first a general scheme of the Malliavin

calculus on an abstract measurable

manifold with minimal assumptions about the manifold.

Then a practical realization of this scheme will be discussed in

several concrete examples with emphasis

on configuration spaces, i.e., spaces of locally finite configurations

in a given manifold (for example, just

a finite-dimensional Euclidean space), which can be alternatively

described as the spaces of integer-valued

discrete measures equipped with suitable differential structures.

No acquaintance with the Malliavin calculus and differential geometry

is assumed.

It is planned to discuss first a general scheme of the Malliavin

calculus on an abstract measurable

manifold with minimal assumptions about the manifold.

Then a practical realization of this scheme will be discussed in

several concrete examples with emphasis

on configuration spaces, i.e., spaces of locally finite configurations

in a given manifold (for example, just

a finite-dimensional Euclidean space), which can be alternatively

described as the spaces of integer-valued

discrete measures equipped with suitable differential structures.

No acquaintance with the Malliavin calculus and differential geometry

is assumed.

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