## Seminar information archive

Seminar information archive ～08/21｜Today's seminar 08/22 | Future seminars 08/23～

### 2011/12/07

#### Lectures

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

"Inverse problems associated with linear and non-linear parabolic systems " (ENGLISH)

"Hautus test for the approximate controllability of linear systems" (ENGLISH)

**Michel Cristofol**(マルセイユ大学) 16:00-17:00"Inverse problems associated with linear and non-linear parabolic systems " (ENGLISH)

[ Abstract ]

In this talk, I present several inverse reconstruction results for linear and non linear parabolic systems with different coupling terms : for linear systems with reaction-convection terms and for cooperative systems like Lotka Volterra systems with strong coupling terms. I will show different approaches to prove uniqueness of the coefficients via Carleman inequalities or via regularities properties of the solutions.

In this talk, I present several inverse reconstruction results for linear and non linear parabolic systems with different coupling terms : for linear systems with reaction-convection terms and for cooperative systems like Lotka Volterra systems with strong coupling terms. I will show different approaches to prove uniqueness of the coefficients via Carleman inequalities or via regularities properties of the solutions.

**Guillaume Olive**(マルセイユ大学) 17:00-18:00"Hautus test for the approximate controllability of linear systems" (ENGLISH)

[ Abstract ]

We will introduce some generalization of the Hautus test to linear parabolic systems and give some applications to the distributed and boundary approximate controllability of such systems.

We will introduce some generalization of the Hautus test to linear parabolic systems and give some applications to the distributed and boundary approximate controllability of such systems.

### 2011/12/06

#### Numerical Analysis Seminar

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

Development of high order CFD solver for aerospace applications (JAPANESE)

[ Reference URL ]

http://www.infsup.jp/saito/

**Kanako Yasue**(JAXA)Development of high order CFD solver for aerospace applications (JAPANESE)

[ Reference URL ]

http://www.infsup.jp/saito/

### 2011/12/05

#### Algebraic Geometry Seminar

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

Obstructions to deforming curves on a uniruled 3-fold (JAPANESE)

**Hirokazu Nasu**(Tokai University)Obstructions to deforming curves on a uniruled 3-fold (JAPANESE)

[ Abstract ]

In this talk, I review some results from a joint work with Mukai:

1. a generalization of Mumford's example of a non-reduced component of the Hilbert scheme, and

2. a sufficient condition for a first order deformation of a curve on a uniruled 3-fold to be obstructed.

As a sequel of the study, we will discuss some obstructed deformations of degenerate curves on a higher dimensional scroll.

In this talk, I review some results from a joint work with Mukai:

1. a generalization of Mumford's example of a non-reduced component of the Hilbert scheme, and

2. a sufficient condition for a first order deformation of a curve on a uniruled 3-fold to be obstructed.

As a sequel of the study, we will discuss some obstructed deformations of degenerate curves on a higher dimensional scroll.

### 2011/12/01

#### Lectures

16:30-18:00 Room #117 (Graduate School of Math. Sci. Bldg.)

Stability of topological phases of matter (ENGLISH)

**Spyridon Michalakis**(Caltech)Stability of topological phases of matter (ENGLISH)

#### Algebraic Geometry Seminar

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

Cyclic K-theory (ENGLISH)

**Dmitry Kaledin**(Steklov Mathematics Institute/ KIAS)Cyclic K-theory (ENGLISH)

[ Abstract ]

Cyclic K-theory is a variant of algebraic K-theory introduced by Goodwillie 25 years ago and more-or-less forgotten by now. I will try to convince the audience that cyclic K-theory is actually quite useful -- in particular, it can be effectively computed for varieties over a finite field.

Cyclic K-theory is a variant of algebraic K-theory introduced by Goodwillie 25 years ago and more-or-less forgotten by now. I will try to convince the audience that cyclic K-theory is actually quite useful -- in particular, it can be effectively computed for varieties over a finite field.

### 2011/11/30

#### Seminar on Probability and Statistics

15:00-16:10 Room #002 (Graduate School of Math. Sci. Bldg.)

Information criteria for parametric and semi-parametric models (JAPANESE)

http://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2011/05.html

**HIROSE, Yuichi**(Victoria University of Wellington)Information criteria for parametric and semi-parametric models (JAPANESE)

[ Abstract ]

Since Akaike proposed an Information Criteria, this approach to

model selection has been important part of Statistical data analysis.

Since then many Information Criteria have been proposed and it is still

an active field of research. Despite there are many contributors in this

topic, we have not have proper Information Criteria for semiparametric

models. In this talk, we give ideas to develop an Information Criteria

for semiparametric models.

[ Reference URL ]Since Akaike proposed an Information Criteria, this approach to

model selection has been important part of Statistical data analysis.

Since then many Information Criteria have been proposed and it is still

an active field of research. Despite there are many contributors in this

topic, we have not have proper Information Criteria for semiparametric

models. In this talk, we give ideas to develop an Information Criteria

for semiparametric models.

http://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2011/05.html

### 2011/11/29

#### Lectures

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

Stability of topological phases of matter (ENGLISH)

**Spyridon Michalakis**(Caltech)Stability of topological phases of matter (ENGLISH)

#### Tuesday Seminar on Topology

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

Mapping class group actions (ENGLISH)

**Athanase Papadopoulos**(IRMA, Univ. de Strasbourg)Mapping class group actions (ENGLISH)

[ Abstract ]

I will describe and present some rigidity results on mapping

class group actions on spaces of foliations on surfaces, equipped with various topologies.

I will describe and present some rigidity results on mapping

class group actions on spaces of foliations on surfaces, equipped with various topologies.

#### Lie Groups and Representation Theory

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

Symmetries, (their) deformations, and physics: some perspectives and open problems from half a century of personal experience (ENGLISH)

**Daniel Sternheimer**(Rikkyo Univertiry and Université de Bourgogne)Symmetries, (their) deformations, and physics: some perspectives and open problems from half a century of personal experience (ENGLISH)

[ Abstract ]

This is a flexible general framework, based on quite a number of papers, some of which are reviewed in:

MR2285047 (2008c:53079) Sternheimer, Daniel. The geometry of space-time and its deformations from a physical perspective. From geometry to quantum mechanics, 287–301, Progr. Math., 252, Birkhäuser Boston, Boston, MA, 2007

http://monge.u-bourgogne.fr/d.sternh/papers/PiMOmori-DS.pdf

This is a flexible general framework, based on quite a number of papers, some of which are reviewed in:

MR2285047 (2008c:53079) Sternheimer, Daniel. The geometry of space-time and its deformations from a physical perspective. From geometry to quantum mechanics, 287–301, Progr. Math., 252, Birkhäuser Boston, Boston, MA, 2007

http://monge.u-bourgogne.fr/d.sternh/papers/PiMOmori-DS.pdf

### 2011/11/28

#### Algebraic Geometry Seminar

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

Comparison with Gieseker stability and slope stability via Bridgeland's stability (JAPANESE)

**Kotaro Kawatani**(Kyoto University)Comparison with Gieseker stability and slope stability via Bridgeland's stability (JAPANESE)

[ Abstract ]

In this talk we compare two classical notions of stability (Gieseker stability and slope stability) for sheaves on K3 surfaces by using stability conditions which was introduced by Bridgeland. As a consequence of this work, we give a classification of 2 dimensional moduli spaces of sheaves on K3 surface depending on the rank of the sheaves.

In this talk we compare two classical notions of stability (Gieseker stability and slope stability) for sheaves on K3 surfaces by using stability conditions which was introduced by Bridgeland. As a consequence of this work, we give a classification of 2 dimensional moduli spaces of sheaves on K3 surface depending on the rank of the sheaves.

#### Seminar on Geometric Complex Analysis

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

An ampleness criterion with the extendability of singular positive metrics (JAPANESE)

**Shin-ichi Matsumura**(University of Tokyo)An ampleness criterion with the extendability of singular positive metrics (JAPANESE)

[ Abstract ]

Coman, Guedj and Zeriahi proved that, for an ample line bundle $L$ on a projective manifold $X$, any singular positive metric on the line bundle $L|_{V}$ along a subvariety $V \subset X$ can be extended to a global singular positive metric of $L$. In this talk, we prove that the extendability of singular positive metrics on a line bundle along a subvariety implies the ampleness of the line bundle.

Coman, Guedj and Zeriahi proved that, for an ample line bundle $L$ on a projective manifold $X$, any singular positive metric on the line bundle $L|_{V}$ along a subvariety $V \subset X$ can be extended to a global singular positive metric of $L$. In this talk, we prove that the extendability of singular positive metrics on a line bundle along a subvariety implies the ampleness of the line bundle.

### 2011/11/25

#### Colloquium

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

Nonlinear dispersive evolution equations (JAPANESE)

**SHIMOMURA Akihiro**(Graduate School of mathematical Sciences, University of Tokyo)Nonlinear dispersive evolution equations (JAPANESE)

[ Abstract ]

I will talk about the time evolution of solutions to nonlinear dispersive equations.

I will talk about the time evolution of solutions to nonlinear dispersive equations.

### 2011/11/24

#### Lectures

16:30-18:00 Room #117 (Graduate School of Math. Sci. Bldg.)

Stability of topological phases of matter (ENGLISH)

**Spyridon Michalakis**(Caltech)Stability of topological phases of matter (ENGLISH)

### 2011/11/22

#### Operator Algebra Seminars

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

Stability of topological phases of matter (ENGLISH)

**( Institute for Quantum Information and Matter (Caltech))**

Spyridon MichalakisSpyridon Michalakis

Stability of topological phases of matter (ENGLISH)

[ Abstract ]

The first lecture will be an introduction to quantum mechanics and a proof of Lieb-Robinson bounds for constant range interaction Hamiltonians. The second lecture will build on the first to prove a powerful lemma on the transformation of the interactions of generic gapped Hamiltonians to a new set of rapidly-decaying interactions that commute with the groundstate subspace. I call this "The Energy Filtering Lemma". Then, the third lecture will be on the construction of the Spectral Flow unitary (Quasi-adiabatic evolution) and its properties; in particular, the perfect simulation of the evolution of the groundstate subspace within a gapped path. I will end with a presentation of the recent result on the stability of the spectral gap for frustration-free Hamiltonians, highlighting how the previous three lectures fit into the proof.

The first lecture will be an introduction to quantum mechanics and a proof of Lieb-Robinson bounds for constant range interaction Hamiltonians. The second lecture will build on the first to prove a powerful lemma on the transformation of the interactions of generic gapped Hamiltonians to a new set of rapidly-decaying interactions that commute with the groundstate subspace. I call this "The Energy Filtering Lemma". Then, the third lecture will be on the construction of the Spectral Flow unitary (Quasi-adiabatic evolution) and its properties; in particular, the perfect simulation of the evolution of the groundstate subspace within a gapped path. I will end with a presentation of the recent result on the stability of the spectral gap for frustration-free Hamiltonians, highlighting how the previous three lectures fit into the proof.

#### Tuesday Seminar on Topology

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

Quantum and homological representations of braid groups (JAPANESE)

**Toshitake Kohno**(The University of Tokyo)Quantum and homological representations of braid groups (JAPANESE)

[ Abstract ]

Homological representations of braid groups are defined as

the action of homeomorphisms of a punctured disk on

the homology of an abelian covering of its configuration space.

These representations were extensively studied by Lawrence,

Krammer and Bigelow. In this talk we show that specializations

of the homological representations of braid groups

are equivalent to the monodromy of the KZ equation with

values in the space of null vectors in the tensor product

of Verma modules when the parameters are generic.

To prove this we use representations of the solutions of the

KZ equation by hypergeometric integrals due to Schechtman,

Varchenko and others.

In the case of special parameters these representations

are extended to quantum representations of mapping

class groups. We describe the images of such representations

and show that the images of any Johnson subgroups

contain non-abelian free groups if the genus and the

level are sufficiently large. The last part is a joint

work with Louis Funar.

Homological representations of braid groups are defined as

the action of homeomorphisms of a punctured disk on

the homology of an abelian covering of its configuration space.

These representations were extensively studied by Lawrence,

Krammer and Bigelow. In this talk we show that specializations

of the homological representations of braid groups

are equivalent to the monodromy of the KZ equation with

values in the space of null vectors in the tensor product

of Verma modules when the parameters are generic.

To prove this we use representations of the solutions of the

KZ equation by hypergeometric integrals due to Schechtman,

Varchenko and others.

In the case of special parameters these representations

are extended to quantum representations of mapping

class groups. We describe the images of such representations

and show that the images of any Johnson subgroups

contain non-abelian free groups if the genus and the

level are sufficiently large. The last part is a joint

work with Louis Funar.

#### Lie Groups and Representation Theory

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

Smallest complex nilpotent orbit with real points (JAPANESE)

**Takayuki Okuda**(東京大学大学院 数理科学研究科)Smallest complex nilpotent orbit with real points (JAPANESE)

[ Abstract ]

Let $\\mathfrak{g}$ be a non-compact simple Lie algebra with no complex

structures.

In this talk, we show that there exists a complex nilpotent orbit

$\\mathcal{O}^{G_\\mathbb{C}}_{\\text{min},\\mathfrak{g}}$ in

$\\mathfrak{g}_\\mathbb{C}$ ($:=\\mathfrak{g} \\otimes \\mathbb{C}$)

containing all of real nilpotent orbits in $\\mathfrak{g}$ of minimal

positive dimension.

For many $\\mathfrak{g}$, the orbit

$\\mathcal{O}^{G_\\mathbb{C}}_{\\text{min},\\mathfrak{g}}$ is just the

complex minimal nilpotent orbit in $\\mathfrak{g}_\\mathbb{C}$.

However, for the cases where $\\mathfrak{g}$ is isomorphic to

$\\mathfrak{su}^*(2k)$, $\\mathfrak{so}(n-1,1)$, $\\mathfrak{sp}(p,q)$,

$\\mathfrak{e}_{6(-26)}$ or $\\mathfrak{f}_{4(-20)}$,

the orbit $\\mathcal{O}^{G_\\mathbb{C}}_{\\text{min},\\mathfrak{g}}$ is not

the complex minimal nilpotent orbit in $\\mathfrak{g}_\\mathbb{C}$.

We also determine $\\mathcal{O}^{G_\\mathbb{C}}_{\\text{min},\\mathfrak{g}}$

by describing the weighted Dynkin diagrams of these for such cases.

Let $\\mathfrak{g}$ be a non-compact simple Lie algebra with no complex

structures.

In this talk, we show that there exists a complex nilpotent orbit

$\\mathcal{O}^{G_\\mathbb{C}}_{\\text{min},\\mathfrak{g}}$ in

$\\mathfrak{g}_\\mathbb{C}$ ($:=\\mathfrak{g} \\otimes \\mathbb{C}$)

containing all of real nilpotent orbits in $\\mathfrak{g}$ of minimal

positive dimension.

For many $\\mathfrak{g}$, the orbit

$\\mathcal{O}^{G_\\mathbb{C}}_{\\text{min},\\mathfrak{g}}$ is just the

complex minimal nilpotent orbit in $\\mathfrak{g}_\\mathbb{C}$.

However, for the cases where $\\mathfrak{g}$ is isomorphic to

$\\mathfrak{su}^*(2k)$, $\\mathfrak{so}(n-1,1)$, $\\mathfrak{sp}(p,q)$,

$\\mathfrak{e}_{6(-26)}$ or $\\mathfrak{f}_{4(-20)}$,

the orbit $\\mathcal{O}^{G_\\mathbb{C}}_{\\text{min},\\mathfrak{g}}$ is not

the complex minimal nilpotent orbit in $\\mathfrak{g}_\\mathbb{C}$.

We also determine $\\mathcal{O}^{G_\\mathbb{C}}_{\\text{min},\\mathfrak{g}}$

by describing the weighted Dynkin diagrams of these for such cases.

### 2011/11/21

#### Seminar on Geometric Complex Analysis

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

Techniques of computations of Dolbeault cohomology of solvmanifolds (JAPANESE)

**Hisashi Kasuya**(University of Tokyo)Techniques of computations of Dolbeault cohomology of solvmanifolds (JAPANESE)

#### Seminar on Mathematics for various disciplines

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

A convex model for non-negative matrix factorization and dimensionality reduction on physical space (ENGLISH)

**Ernie Esser**(University of California, Irvine)A convex model for non-negative matrix factorization and dimensionality reduction on physical space (ENGLISH)

[ Abstract ]

A collaborative convex framework for factoring a data matrix X into a non-negative product AS, with a sparse coefficient matrix S, is proposed. We restrict the columns of the dictionary matrix A to coincide with certain columns of the data matrix X, thereby guaranteeing a physically meaningful dictionary and dimensionality reduction. We focus on applications of the proposed framework to hyperspectral endmember and abundances identification and also show an application to blind source separation of NMR data.

This talk is based on joint work with Michael Moeller, Stanley Osher, Guillermo Sapiro and Jack Xin.

A collaborative convex framework for factoring a data matrix X into a non-negative product AS, with a sparse coefficient matrix S, is proposed. We restrict the columns of the dictionary matrix A to coincide with certain columns of the data matrix X, thereby guaranteeing a physically meaningful dictionary and dimensionality reduction. We focus on applications of the proposed framework to hyperspectral endmember and abundances identification and also show an application to blind source separation of NMR data.

This talk is based on joint work with Michael Moeller, Stanley Osher, Guillermo Sapiro and Jack Xin.

#### Kavli IPMU Komaba Seminar

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

Enuemerative meaning of mirror maps for toric Calabi-Yau manifolds (ENGLISH)

**Siu-Cheong Lau**(IPMU)Enuemerative meaning of mirror maps for toric Calabi-Yau manifolds (ENGLISH)

[ Abstract ]

For a mirror pair of smooth manifolds X and Y, mirror symmetry associates a complex structure on Y to each Kaehler structure on X, and this association is called the mirror map. Traditionally mirror maps are defined by solving Picard-Fuchs equations and its geometric meaning was unclear. In this talk I explain a recent joint work with K.W. Chan, N.C. Leung and H.H. Tseng which proves that mirror maps can be obtained by taking torus duality (the SYZ approach) and disk-counting for a class of toric Calabi-Yau manifolds in any dimensions. As a consequence we can compute disk-counting invariants by solving Picard-Fuchs equations.

For a mirror pair of smooth manifolds X and Y, mirror symmetry associates a complex structure on Y to each Kaehler structure on X, and this association is called the mirror map. Traditionally mirror maps are defined by solving Picard-Fuchs equations and its geometric meaning was unclear. In this talk I explain a recent joint work with K.W. Chan, N.C. Leung and H.H. Tseng which proves that mirror maps can be obtained by taking torus duality (the SYZ approach) and disk-counting for a class of toric Calabi-Yau manifolds in any dimensions. As a consequence we can compute disk-counting invariants by solving Picard-Fuchs equations.

### 2011/11/19

#### Monthly Seminar on Arithmetic of Automorphic Forms

10:15-12:30 Room #117 (Graduate School of Math. Sci. Bldg.)

Hyperelliptic integrals related with dihedral groups (JAPANESE)

Survey on the generalized Bernoulli-Hurwitz numbers for a higher genus algebraic function, and some problems (JAPANESE)

**Jiro Sekiguchi**(Tokyo Univ. of Agriculture) 10:15-11:15Hyperelliptic integrals related with dihedral groups (JAPANESE)

**Yoshihiro Ohnishi**(Yamanashi University) 11:30-12:30Survey on the generalized Bernoulli-Hurwitz numbers for a higher genus algebraic function, and some problems (JAPANESE)

### 2011/11/18

#### Lectures

15:00-16:00 Room #052 (Graduate School of Math. Sci. Bldg.)

Inverse problems for heat equations with discontinuous conductivities

(JAPANESE)

**Hiroshi Isozaki**(University of Tsukuba)Inverse problems for heat equations with discontinuous conductivities

(JAPANESE)

[ Abstract ]

In a bounded domain $\\Omega \\subset {\\bf R}^n$, consider the heat

equation $\\partial_tu = \\nabla(\\gamma(t,x)\\nabla u)$. The heat

conductivity is assumed to be piecewise constant : $\\gamma = k^2$ on

$\\Omaga_1(t) \\subset\\subset \\Omega$, $\\gamma(t,x) = 1$ on

$\\Omega\\setminus\\Omega_1(t)$. In this talk, we present recent results

for the inverse problems of reconstructing $\\gamma(t,x)$ from the

Dirichlet-to-Neumann map :

$u(t)|_{\\partial\\Omega} \\to $\\partial_{\\nu}u|_{\\partial\\Omega}$ for a time

interval $(0,T)$. These are the joint works with P.Gaitan, O.Poisson,

S.Siltanen, J.Tamminen.

In a bounded domain $\\Omega \\subset {\\bf R}^n$, consider the heat

equation $\\partial_tu = \\nabla(\\gamma(t,x)\\nabla u)$. The heat

conductivity is assumed to be piecewise constant : $\\gamma = k^2$ on

$\\Omaga_1(t) \\subset\\subset \\Omega$, $\\gamma(t,x) = 1$ on

$\\Omega\\setminus\\Omega_1(t)$. In this talk, we present recent results

for the inverse problems of reconstructing $\\gamma(t,x)$ from the

Dirichlet-to-Neumann map :

$u(t)|_{\\partial\\Omega} \\to $\\partial_{\\nu}u|_{\\partial\\Omega}$ for a time

interval $(0,T)$. These are the joint works with P.Gaitan, O.Poisson,

S.Siltanen, J.Tamminen.

### 2011/11/17

#### GCOE lecture series

17:00-18:00 Room #370 (Graduate School of Math. Sci. Bldg.)

Recovery of weakly coupled system from partial Cauchy data (ENGLISH)

**Oleg Emanouilov**(Colorado State University)Recovery of weakly coupled system from partial Cauchy data (ENGLISH)

[ Abstract ]

We consider the inverse problem for recovery of coefficients of weakly coupled system of elliptic equations in a bounded 2D domain.

We consider the inverse problem for recovery of coefficients of weakly coupled system of elliptic equations in a bounded 2D domain.

#### Operator Algebra Seminars

16:30-18:00 Room #117 (Graduate School of Math. Sci. Bldg.)

Hecke pairs in ergodic measured equivalence relations (JAPANESE)

**Takehiko Yamanouchi**(Tokyo Gakugei University)Hecke pairs in ergodic measured equivalence relations (JAPANESE)

### 2011/11/16

#### Seminar on Geometric Complex Analysis

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

Disc functionals and Siciak-Zaharyuta extremal functions on singular varieties (ENGLISH)

**Franc Forstneric**(University of Ljubljana)Disc functionals and Siciak-Zaharyuta extremal functions on singular varieties (ENGLISH)

[ Abstract ]

A disc functional on a complex space, $X$, is a function P that assign a real number $P(f)$ (possibly minus infinity) to every analytic disc $f$ in $X$. An examples is the Poisson functional $P_u$ of an upper semicontinuous function $u$ on $X$: in that case $P_u(f)$ is the average of u over the boundary curve of the disc $f$. Other natural examples include the Lelong and the Riesz functionals. The envelope of a disc functional $P$ is a function on $X$ associating to every point $x$ of $X$ the infimum of the values $P(f)$ over all analytic discs $f$ in $X$ satisfying $f(0)=x$. The main point of interest is that the envelopes of many natural disc functionals are plurisubharmonic functions solving certain extremal problems. In the classical case when $X=\mathbf{C}^n$ this was first discovered by E. Poletsky in the early 1990's. In this talk I will discuss recent results on plurisubharmonicity of envelopes of all the classical disc functional mentioned above on locally irreducible complex spaces. In the second part of the talk I will give formulas expressing the classical Siciak-Zaharyuta maximal function of an open set in an affine algebraic variety as the envelope of certain disc functionals. We establish plurisubharmonicity of envelopes of certain classical disc functionals on locally irreducible complex spaces, thereby generalizing the corresponding results for complex manifolds. We also find new formulae expressing the Siciak-Zaharyuta extremal function of an open set in a locally irreducible affine algebraic variety as the envelope of certain disc functionals, similarly to what has been done for open sets in $\mathbf{C}^n$ by Lempert and by Larusson and Sigurdsson.

A disc functional on a complex space, $X$, is a function P that assign a real number $P(f)$ (possibly minus infinity) to every analytic disc $f$ in $X$. An examples is the Poisson functional $P_u$ of an upper semicontinuous function $u$ on $X$: in that case $P_u(f)$ is the average of u over the boundary curve of the disc $f$. Other natural examples include the Lelong and the Riesz functionals. The envelope of a disc functional $P$ is a function on $X$ associating to every point $x$ of $X$ the infimum of the values $P(f)$ over all analytic discs $f$ in $X$ satisfying $f(0)=x$. The main point of interest is that the envelopes of many natural disc functionals are plurisubharmonic functions solving certain extremal problems. In the classical case when $X=\mathbf{C}^n$ this was first discovered by E. Poletsky in the early 1990's. In this talk I will discuss recent results on plurisubharmonicity of envelopes of all the classical disc functional mentioned above on locally irreducible complex spaces. In the second part of the talk I will give formulas expressing the classical Siciak-Zaharyuta maximal function of an open set in an affine algebraic variety as the envelope of certain disc functionals. We establish plurisubharmonicity of envelopes of certain classical disc functionals on locally irreducible complex spaces, thereby generalizing the corresponding results for complex manifolds. We also find new formulae expressing the Siciak-Zaharyuta extremal function of an open set in a locally irreducible affine algebraic variety as the envelope of certain disc functionals, similarly to what has been done for open sets in $\mathbf{C}^n$ by Lempert and by Larusson and Sigurdsson.

#### GCOE Seminars

10:00-11:00 Room #270 (Graduate School of Math. Sci. Bldg.)

All you never really wanted to know about QRT, but were foolhardy enough to ask (ENGLISH)

**Alfred Ramani**(Ecole Polytechnique)All you never really wanted to know about QRT, but were foolhardy enough to ask (ENGLISH)

[ Abstract ]

We discuss various extensions of the famous QRT second order, first degree, integrable mapping. We show how these extensions can be combined. A discussion of integrable correspondences related to these extended QRT mappings is also presented.

We discuss various extensions of the famous QRT second order, first degree, integrable mapping. We show how these extensions can be combined. A discussion of integrable correspondences related to these extended QRT mappings is also presented.

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