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Seminar information archive
Seminar information archive ～10/22｜Today's seminar 10/23 | Future seminars 10/24～
2011/12/06
Numerical Analysis Seminar
16:30-18:00 Room #002 (Graduate School of Math. Sci. Bldg.)
Kanako Yasue (JAXA)
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.)
Hirokazu Nasu (Tokai University)
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.)
Spyridon Michalakis (Caltech)
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.)
Dmitry Kaledin (Steklov Mathematics Institute/ KIAS)
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.)
HIROSE, Yuichi (Victoria University of Wellington)
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.)
Spyridon Michalakis (Caltech)
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.)
Athanase Papadopoulos (IRMA, Univ. de Strasbourg)
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.)
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)
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.)
Kotaro Kawatani (Kyoto University)
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.)
Shin-ichi Matsumura (University of Tokyo)
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.)
SHIMOMURA Akihiro (Graduate School of mathematical Sciences, University of Tokyo)
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.)
Spyridon Michalakis (Caltech)
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.)
Spyridon Michalakis ( Institute for Quantum Information and Matter (Caltech))
Stability of topological phases of matter (ENGLISH)
Spyridon Michalakis ( Institute for Quantum Information and Matter (Caltech))
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.)
Toshitake Kohno (The University of Tokyo)
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.)
Takayuki Okuda (東京大学大学院 数理科学研究科)
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.)
Hisashi Kasuya (University of Tokyo)
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.)
Ernie Esser (University of California, Irvine)
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.)
Siu-Cheong Lau (IPMU)
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.)
Jiro Sekiguchi (Tokyo Univ. of Agriculture) 10:15-11:15
Hyperelliptic integrals related with dihedral groups (JAPANESE)
Yoshihiro Ohnishi (Yamanashi University) 11:30-12:30
Survey on the generalized Bernoulli-Hurwitz numbers for a higher genus algebraic function, and some problems (JAPANESE)
Jiro Sekiguchi (Tokyo Univ. of Agriculture) 10:15-11:15
Hyperelliptic integrals related with dihedral groups (JAPANESE)
Yoshihiro Ohnishi (Yamanashi University) 11:30-12:30
Survey 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.)
Hiroshi Isozaki (University of Tsukuba)
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.)
Oleg Emanouilov (Colorado State University)
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.)
Takehiko Yamanouchi (Tokyo Gakugei University)
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.)
Franc Forstneric (University of Ljubljana)
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.)
Alfred Ramani (Ecole Polytechnique)
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.
2011/11/15
Tuesday Seminar on Topology
16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)
Francois Laudenbach (Univ. de Nantes)
Singular codimension-one foliations
and twisted open books in dimension 3.
(joint work with G. Meigniez)
(ENGLISH)
Francois Laudenbach (Univ. de Nantes)
Singular codimension-one foliations
and twisted open books in dimension 3.
(joint work with G. Meigniez)
(ENGLISH)
[ Abstract ]
The allowed singularities are those of functions.
According to A. Haefliger (1958),
such structures on manifolds, called $\\Gamma_1$-structures,
are objects of a cohomological
theory with a classifying space $B\\Gamma_1$.
The problem of cancelling the singularities
(or regularization problem)
arise naturally.
For a closed manifold, it was solved by W.Thurston in a famous paper
(1976), with a proof relying on Mather's isomorphism (1971):
Diff$^\\infty(\\mathbb R)$ as a discrete group has the same homology
as the based loop space
$\\Omega B\\Gamma_1^+$.
For further extension to contact geometry, it is necessary
to solve the regularization problem
without using Mather's isomorphism.
That is what we have done in dimension 3. Our result is the following.
{\\it Every $\\Gamma_1$-structure $\\xi$ on a 3-manifold $M$ whose
normal bundle
embeds into the tangent bundle to $M$ is $\\Gamma_1$-homotopic
to a regular foliation
carried by a (possibily twisted) open book.}
The proof is elementary and relies on the dynamics of a (twisted)
pseudo-gradient of $\\xi$.
All the objects will be defined in the talk, in particular the notion
of twisted open book which is a central object in the reported paper.
The allowed singularities are those of functions.
According to A. Haefliger (1958),
such structures on manifolds, called $\\Gamma_1$-structures,
are objects of a cohomological
theory with a classifying space $B\\Gamma_1$.
The problem of cancelling the singularities
(or regularization problem)
arise naturally.
For a closed manifold, it was solved by W.Thurston in a famous paper
(1976), with a proof relying on Mather's isomorphism (1971):
Diff$^\\infty(\\mathbb R)$ as a discrete group has the same homology
as the based loop space
$\\Omega B\\Gamma_1^+$.
For further extension to contact geometry, it is necessary
to solve the regularization problem
without using Mather's isomorphism.
That is what we have done in dimension 3. Our result is the following.
{\\it Every $\\Gamma_1$-structure $\\xi$ on a 3-manifold $M$ whose
normal bundle
embeds into the tangent bundle to $M$ is $\\Gamma_1$-homotopic
to a regular foliation
carried by a (possibily twisted) open book.}
The proof is elementary and relies on the dynamics of a (twisted)
pseudo-gradient of $\\xi$.
All the objects will be defined in the talk, in particular the notion
of twisted open book which is a central object in the reported paper.
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