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過去の記録 ～04/25｜本日 04/26 | 今後の予定 04/27～
2011年11月24日(木)
講演会
16:30-18:00 数理科学研究科棟(駒場) 117号室
Spyridon Michalakis 氏 (Caltech)
Stability of topological phases of matter (ENGLISH)
Spyridon Michalakis 氏 (Caltech)
Stability of topological phases of matter (ENGLISH)
2011年11月22日(火)
作用素環セミナー
16:30-18:00 数理科学研究科棟(駒場) 126号室
Spyridon Michalakis 氏 ( Institute for Quantum Information and Matter (Caltech))
Stability of topological phases of matter (ENGLISH)
Spyridon Michalakis 氏 ( Institute for Quantum Information and Matter (Caltech))
Stability of topological phases of matter (ENGLISH)
[ 講演概要 ]
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.
トポロジー火曜セミナー
16:30-18:00 数理科学研究科棟(駒場) 056号室
Tea: 16:00 - 16:30 コモンルーム
河野俊丈 氏 (東京大学大学院数理科学研究科)
Quantum and homological representations of braid groups (JAPANESE)
Tea: 16:00 - 16:30 コモンルーム
河野俊丈 氏 (東京大学大学院数理科学研究科)
Quantum and homological representations of braid groups (JAPANESE)
[ 講演概要 ]
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群論・表現論セミナー
16:30-18:00 数理科学研究科棟(駒場) 002号室
奥田隆幸 氏 (Graduate School of Mathematical Sciences, the University of Tokyo)
Smallest complex nilpotent orbit with real points (JAPANESE)
奥田隆幸 氏 (Graduate School of Mathematical Sciences, the University of Tokyo)
Smallest complex nilpotent orbit with real points (JAPANESE)
[ 講演概要 ]
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年11月21日(月)
複素解析幾何セミナー
10:30-12:00 数理科学研究科棟(駒場) 128号室
糟谷久矢 氏 (東大数理)
Techniques of computations of Dolbeault cohomology of solvmanifolds (JAPANESE)
糟谷久矢 氏 (東大数理)
Techniques of computations of Dolbeault cohomology of solvmanifolds (JAPANESE)
諸分野のための数学研究会
13:30-14:30 数理科学研究科棟(駒場) 056号室
北海道大学のHPには、第1回(2005年6月22日)～第22回(2009年2月18日)までの情報が掲載されております。
Ernie Esser 氏 (University of California, Irvine)
A convex model for non-negative matrix factorization and dimensionality reduction on physical space (ENGLISH)
北海道大学のHPには、第1回(2005年6月22日)～第22回(2009年2月18日)までの情報が掲載されております。
Ernie Esser 氏 (University of California, Irvine)
A convex model for non-negative matrix factorization and dimensionality reduction on physical space (ENGLISH)
[ 講演概要 ]
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 数理科学研究科棟(駒場) 002号室
Siu-Cheong Lau 氏 (IPMU)
Enuemerative meaning of mirror maps for toric Calabi-Yau manifolds (ENGLISH)
Siu-Cheong Lau 氏 (IPMU)
Enuemerative meaning of mirror maps for toric Calabi-Yau manifolds (ENGLISH)
[ 講演概要 ]
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年11月19日(土)
保型形式の整数論月例セミナー
10:15-12:30 数理科学研究科棟(駒場) 117号室
部屋、開始時間がいつもと違います。午前中です
関口次郎 氏 (東京農工大学) 10:15-11:15
二面体群に関連する超楕円積分 (JAPANESE)
高い種数の代数関数体に対する、一般化された Bernoulli-Hurwitz 数と、関連する諸問題（概説） (JAPANESE)
部屋、開始時間がいつもと違います。午前中です
関口次郎 氏 (東京農工大学) 10:15-11:15
二面体群に関連する超楕円積分 (JAPANESE)
[ 講演概要 ]
正2面体群に関連した3次元空間の自由因子(ある条件をみたす超曲面)をあつかう。この超曲面にそって特異点をもち解空間が3次元である微分方程式を構成する。
この微分方程式の基本解が超楕円積分と超幾何関数を使って表示できることを示しさらに関連した話題を説明する。
大西良博 氏 (山梨大学) 11:30-12:30正2面体群に関連した3次元空間の自由因子(ある条件をみたす超曲面)をあつかう。この超曲面にそって特異点をもち解空間が3次元である微分方程式を構成する。
この微分方程式の基本解が超楕円積分と超幾何関数を使って表示できることを示しさらに関連した話題を説明する。
高い種数の代数関数体に対する、一般化された Bernoulli-Hurwitz 数と、関連する諸問題（概説） (JAPANESE)
[ 講演概要 ]
種数の高いある種の代数曲線に付随した代数函数 (楕円函数の自然な一般化) の Laurent 展開係数である generalized Bernoulli-Hurwitz 数が Bernoulli 数やBernoulli-Hurwitz 数と強く類似した性質 (Clausen von Staudt, von Staudt 第 2, Kummer 自身の得た合同式) を持つことについて survey をし, 最後に, その先にある問題について言及する.
種数の高いある種の代数曲線に付随した代数函数 (楕円函数の自然な一般化) の Laurent 展開係数である generalized Bernoulli-Hurwitz 数が Bernoulli 数やBernoulli-Hurwitz 数と強く類似した性質 (Clausen von Staudt, von Staudt 第 2, Kummer 自身の得た合同式) を持つことについて survey をし, 最後に, その先にある問題について言及する.
2011年11月18日(金)
講演会
15:00-16:00 数理科学研究科棟(駒場) 052号室
磯崎 洋 氏 (筑波大学)
Inverse problems for heat equations with discontinuous conductivities
(JAPANESE)
磯崎 洋 氏 (筑波大学)
Inverse problems for heat equations with discontinuous conductivities
(JAPANESE)
[ 講演概要 ]
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年11月17日(木)
GCOEレクチャーズ
17:00-18:00 数理科学研究科棟(駒場) 370号室
Oleg Emanouilov 氏 (Colorado State University)
Recovery of weakly coupled system from partial Cauchy data (ENGLISH)
Oleg Emanouilov 氏 (Colorado State University)
Recovery of weakly coupled system from partial Cauchy data (ENGLISH)
[ 講演概要 ]
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.
作用素環セミナー
16:30-18:00 数理科学研究科棟(駒場) 117号室
山ノ内毅彦 氏 (東京学芸大学)
エルゴード的測度付き同値関係におけるヘッケペアーについて (JAPANESE)
山ノ内毅彦 氏 (東京学芸大学)
エルゴード的測度付き同値関係におけるヘッケペアーについて (JAPANESE)
2011年11月16日(水)
複素解析幾何セミナー
16:30-18:00 数理科学研究科棟(駒場) 128号室
Franc Forstneric 氏 (University of Ljubljana)
Disc functionals and Siciak-Zaharyuta extremal functions on singular varieties (ENGLISH)
Franc Forstneric 氏 (University of Ljubljana)
Disc functionals and Siciak-Zaharyuta extremal functions on singular varieties (ENGLISH)
[ 講演概要 ]
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セミナー
10:00-11:00 数理科学研究科棟(駒場) 270号室
Alfred Ramani 氏 (Ecole Polytechnique)
All you never really wanted to know about QRT, but were foolhardy enough to ask (ENGLISH)
Alfred Ramani 氏 (Ecole Polytechnique)
All you never really wanted to know about QRT, but were foolhardy enough to ask (ENGLISH)
[ 講演概要 ]
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年11月15日(火)
トポロジー火曜セミナー
16:30-18:00 数理科学研究科棟(駒場) 056号室
Tea: 16:00 - 16:30 コモンルーム
Francois Laudenbach 氏 (Univ. de Nantes)
Singular codimension-one foliations
and twisted open books in dimension 3.
(joint work with G. Meigniez)
(ENGLISH)
Tea: 16:00 - 16:30 コモンルーム
Francois Laudenbach 氏 (Univ. de Nantes)
Singular codimension-one foliations
and twisted open books in dimension 3.
(joint work with G. Meigniez)
(ENGLISH)
[ 講演概要 ]
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.
Lie群論・表現論セミナー
16:30-18:00 数理科学研究科棟(駒場) 126号室
Laurant Demonet 氏 (Nagoya University)
Categorification of cluster algebras arising from unipotent subgroups of non-simply laced Lie groups (ENGLISH)
Laurant Demonet 氏 (Nagoya University)
Categorification of cluster algebras arising from unipotent subgroups of non-simply laced Lie groups (ENGLISH)
[ 講演概要 ]
We introduce an abstract framework to categorify some antisymetrizable cluster algebras by using actions of finite groups on stably 2-Calabi-Yau exact categories. We introduce the notion of the equivariant category and, with similar technics as in [K], [CK], [GLS1], [GLS2], [DK], [FK], [P], we construct some examples of such categorifications. For example, if we let Z/2Z act on the category of representations of the preprojective algebra of type A2n-1 via the only non trivial action on the diagram, we obtain the cluster structure on the coordinate ring of the maximal unipotent subgroup of the semi-simple Lie group of type Bn [D]. Hence, we can get relations between the cluster algebras categorified by some exact subcategories of these two categories. We also prove by the same methods as in [FK] a conjecture of Fomin and Zelevinsky stating that the cluster monomials are linearly independent.
References
[CK] P. Caldero, B. Keller, From triangulated categories to cluster algebras, Invent. Math. 172 (2008), no. 1, 169--211.
[DK] R. Dehy, B. Keller, On the combinatorics of rigid objects in 2-Calabi-Yau categories, arXiv: 0709.0882.
[D] L. Demonet, Cluster algebras and preprojective algebras: the non simply-laced case, C. R. Acad. Sci. Paris, Ser. I 346 (2008), 379--384.
[FK] C. Fu, B. Keller, On cluster algebras with coefficients and 2-Calabi-Yau categories, arXiv: 0710.3152.
[GLS1] C. Geiss, B. Leclerc, J. Schröer, Rigid modules over preprojective algebras, Invent. Math. 165 (2006), no. 3, 589--632.
[GLS2] C. Geiss, B. Leclerc, J. Schröer, Cluster algebra structures and semicanoncial bases for unipotent groups, arXiv: math/0703039.
[K] B. Keller, Categorification of acyclic cluster algebras: an introduction, arXiv: 0801.3103.
[P] Y. Palu, Cluster characters for triangulated 2-Calabi--Yau categories, arXiv: math/0703540.
We introduce an abstract framework to categorify some antisymetrizable cluster algebras by using actions of finite groups on stably 2-Calabi-Yau exact categories. We introduce the notion of the equivariant category and, with similar technics as in [K], [CK], [GLS1], [GLS2], [DK], [FK], [P], we construct some examples of such categorifications. For example, if we let Z/2Z act on the category of representations of the preprojective algebra of type A2n-1 via the only non trivial action on the diagram, we obtain the cluster structure on the coordinate ring of the maximal unipotent subgroup of the semi-simple Lie group of type Bn [D]. Hence, we can get relations between the cluster algebras categorified by some exact subcategories of these two categories. We also prove by the same methods as in [FK] a conjecture of Fomin and Zelevinsky stating that the cluster monomials are linearly independent.
References
[CK] P. Caldero, B. Keller, From triangulated categories to cluster algebras, Invent. Math. 172 (2008), no. 1, 169--211.
[DK] R. Dehy, B. Keller, On the combinatorics of rigid objects in 2-Calabi-Yau categories, arXiv: 0709.0882.
[D] L. Demonet, Cluster algebras and preprojective algebras: the non simply-laced case, C. R. Acad. Sci. Paris, Ser. I 346 (2008), 379--384.
[FK] C. Fu, B. Keller, On cluster algebras with coefficients and 2-Calabi-Yau categories, arXiv: 0710.3152.
[GLS1] C. Geiss, B. Leclerc, J. Schröer, Rigid modules over preprojective algebras, Invent. Math. 165 (2006), no. 3, 589--632.
[GLS2] C. Geiss, B. Leclerc, J. Schröer, Cluster algebra structures and semicanoncial bases for unipotent groups, arXiv: math/0703039.
[K] B. Keller, Categorification of acyclic cluster algebras: an introduction, arXiv: 0801.3103.
[P] Y. Palu, Cluster characters for triangulated 2-Calabi--Yau categories, arXiv: math/0703540.
2011年11月14日(月)
GCOEレクチャーズ
17:00-18:00 数理科学研究科棟(駒場) 470号室
Oleg Emanouilov 氏 (Colorado State University)
Recovery of weakly coupled system from partial Cauchy data (ENGLISH)
Oleg Emanouilov 氏 (Colorado State University)
Recovery of weakly coupled system from partial Cauchy data (ENGLISH)
[ 講演概要 ]
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.
代数幾何学セミナー
15:30-17:00 数理科学研究科棟(駒場) 122号室
渡辺 究 氏 (東京大学数理科学研究科)
On projective manifolds swept out by cubic varieties (JAPANESE)
渡辺 究 氏 (東京大学数理科学研究科)
On projective manifolds swept out by cubic varieties (JAPANESE)
[ 講演概要 ]
The structures of embedded complex projective manifolds swept out by varieties with preassigned properties have been studied by several authors. In this talk, we study structures of embedded projective manifolds swept out by cubic varieties.
The structures of embedded complex projective manifolds swept out by varieties with preassigned properties have been studied by several authors. In this talk, we study structures of embedded projective manifolds swept out by cubic varieties.
2011年11月10日(木)
GCOEレクチャーズ
17:00-18:00 数理科学研究科棟(駒場) 370号室
Oleg Emanouilov 氏 (Colorado State University)
Inverse boundary value problem for Schroedinger equation in two dimensions (ENGLISH)
Oleg Emanouilov 氏 (Colorado State University)
Inverse boundary value problem for Schroedinger equation in two dimensions (ENGLISH)
[ 講演概要 ]
We relax the regularity condition on potentials of Schroedinger equations in uniqueness results on the inverse boundary value problem recently proved in A.Bukhgeim (2008) and O. Imanuvilov, G.Uhlmann and M. Yamamoto (2010).
We relax the regularity condition on potentials of Schroedinger equations in uniqueness results on the inverse boundary value problem recently proved in A.Bukhgeim (2008) and O. Imanuvilov, G.Uhlmann and M. Yamamoto (2010).
応用解析セミナー
15:00-16:00 数理科学研究科棟(駒場) 128号室
今回はダブルヘッダーのため,開始時間が通常と異なりますのでご注意ください.
森洋一朗 氏 (ミネソタ大学)
電解質および浸透圧調節の細胞生理学とその数理モデル (JAPANESE)
今回はダブルヘッダーのため,開始時間が通常と異なりますのでご注意ください.
森洋一朗 氏 (ミネソタ大学)
電解質および浸透圧調節の細胞生理学とその数理モデル (JAPANESE)
[ 講演概要 ]
細胞体積とイオン濃度の調節は細胞生理学、特に上皮細胞の生理学
において中心的な課題である。細胞体積調節を記述する標準的な数理
モデルはpump-leak model と呼ばれ、数学的には常微分方程式に
代数的な拘束条件の加わった系である。60年代から現在に至るまで
多くの研究者が用いてきたにもかかわらず、その数理的な性質は全く
知られていなかった。本講演では、pump-leak model には熱力学的な
構造があることを解説し、これを利用することで最近得られた解析的な
結果を紹介する。さらにpump-leak model を拡張して得られる偏微分
方程式系とその熱力学的構造について解説する。
細胞体積とイオン濃度の調節は細胞生理学、特に上皮細胞の生理学
において中心的な課題である。細胞体積調節を記述する標準的な数理
モデルはpump-leak model と呼ばれ、数学的には常微分方程式に
代数的な拘束条件の加わった系である。60年代から現在に至るまで
多くの研究者が用いてきたにもかかわらず、その数理的な性質は全く
知られていなかった。本講演では、pump-leak model には熱力学的な
構造があることを解説し、これを利用することで最近得られた解析的な
結果を紹介する。さらにpump-leak model を拡張して得られる偏微分
方程式系とその熱力学的構造について解説する。
応用解析セミナー
16:30-17:30 数理科学研究科棟(駒場) 128号室
この日はダブルヘッダーです。
Bernold Fiedler 氏 (Free University of Berlin)
Schoenflies spheres in Sturm attractors (ENGLISH)
この日はダブルヘッダーです。
Bernold Fiedler 氏 (Free University of Berlin)
Schoenflies spheres in Sturm attractors (ENGLISH)
[ 講演概要 ]
In gradient systems on compact manifolds the boundary of the unstable manifold of an equilibrium need not be homeomorphic to a sphere, or to any compact manifold.
For scalar parabolic equations in one space dimension, however, we can exlude complications like Reidemeister torsion and the Alexander horned sphere. Instead the boundary is a Schoenflies embedded sphere. This is due to Sturm nodal properties related to the Matano lap number.
In gradient systems on compact manifolds the boundary of the unstable manifold of an equilibrium need not be homeomorphic to a sphere, or to any compact manifold.
For scalar parabolic equations in one space dimension, however, we can exlude complications like Reidemeister torsion and the Alexander horned sphere. Instead the boundary is a Schoenflies embedded sphere. This is due to Sturm nodal properties related to the Matano lap number.
2011年11月09日(水)
代数学コロキウム
18:00-19:00 数理科学研究科棟(駒場) 056号室
いつもと時間が異なりますのでご注意下さい.
志甫 淳 氏 (東京大学数理科学研究科)
On extension and restriction of overconvergent isocrystals (ENGLISH)
いつもと時間が異なりますのでご注意下さい.
志甫 淳 氏 (東京大学数理科学研究科)
On extension and restriction of overconvergent isocrystals (ENGLISH)
[ 講演概要 ]
First we explain two theorems concerning (log) extension of overconvergent isocrystals. One is a p-adic analogue of the theorem of logarithmic extension of regular integrable connections, and the other is a p-adic analogue of Zariski-Nagata purity. Next we explain a theorem which says that we can check certain property of overconvergent isocrystals by restricting them to curves.
(本講演は「東京パリ数論幾何セミナー」として、インターネットによる東大数理とIHESとの双方向同時中継で行います。)
First we explain two theorems concerning (log) extension of overconvergent isocrystals. One is a p-adic analogue of the theorem of logarithmic extension of regular integrable connections, and the other is a p-adic analogue of Zariski-Nagata purity. Next we explain a theorem which says that we can check certain property of overconvergent isocrystals by restricting them to curves.
(本講演は「東京パリ数論幾何セミナー」として、インターネットによる東大数理とIHESとの双方向同時中継で行います。)
2011年11月08日(火)
トポロジー火曜セミナー
16:30-18:00 数理科学研究科棟(駒場) 056号室
Tea: 16:00 - 16:30 コモンルーム
與倉 昭治 氏 (鹿児島大学)
Fiberwise bordism groups and related topics (JAPANESE)
Tea: 16:00 - 16:30 コモンルーム
與倉 昭治 氏 (鹿児島大学)
Fiberwise bordism groups and related topics (JAPANESE)
[ 講演概要 ]
We have recently introduced the notion of fiberwise bordism. In this talk, after a quick review of some of the classical (co)bordism theories, we will explain motivations of considering fiberwise bordism and some results and connections with other known works, such as M. Kreck's bordism groups of orientation preserving diffeomorphisms and Emerson-Meyer's bivariant K-theory etc. An essential motivation is our recent work towards constructing a bivariant-theoretic analogue (in the sense of Fulton-MacPherson) of Levine-Morel's or Levine-Pandharipande's algebraic cobordism.
We have recently introduced the notion of fiberwise bordism. In this talk, after a quick review of some of the classical (co)bordism theories, we will explain motivations of considering fiberwise bordism and some results and connections with other known works, such as M. Kreck's bordism groups of orientation preserving diffeomorphisms and Emerson-Meyer's bivariant K-theory etc. An essential motivation is our recent work towards constructing a bivariant-theoretic analogue (in the sense of Fulton-MacPherson) of Levine-Morel's or Levine-Pandharipande's algebraic cobordism.
解析学火曜セミナー
16:30-18:00 数理科学研究科棟(駒場) 128号室
寺澤 祐高 氏 (東京大数理（日本学術振興会特別研究員PD）)
確率的摂動項を持つ冪乗法則流体方程式の弱解の存在と 一意性について (JAPANESE)
寺澤 祐高 氏 (東京大数理（日本学術振興会特別研究員PD）)
確率的摂動項を持つ冪乗法則流体方程式の弱解の存在と 一意性について (JAPANESE)
[ 講演概要 ]
本講演では、非圧縮性非ニュートン流体の運動を記述する
偏微分方程式に加法的確率的摂動項を加えた確率偏微分方程式 の弱解
の存在と一意性について考察する。 非ニュートン流体としては、粘性
が変形速度テンソルの大きさの冪乗 の形で依存する冪乗法則流体を考
察し、確率的摂動項としては 有色雑音を考察する。 Necas-Malek-
Ruzicka('93)において、確率的外力項を伴わない、 決定方程式に関し
て示された弱解の存在と一意性の主張を、 確率的摂動項を持つ方程式
に対して示す。 解の存在の証明は、ガレルキン近似によって得られた
解の列に対して、 伊藤の公式、Birkholder-Davis-Gundyの不等式など
により、 解の列のコンパクト性を示すこと及び、解の部分列が収束し
、 その収束先が方程式を弱い意味で満たすことを示すことでなされる。
なお、本講演は、吉田伸生氏(京都大学)との共同研究に基づく。
本講演では、非圧縮性非ニュートン流体の運動を記述する
偏微分方程式に加法的確率的摂動項を加えた確率偏微分方程式 の弱解
の存在と一意性について考察する。 非ニュートン流体としては、粘性
が変形速度テンソルの大きさの冪乗 の形で依存する冪乗法則流体を考
察し、確率的摂動項としては 有色雑音を考察する。 Necas-Malek-
Ruzicka('93)において、確率的外力項を伴わない、 決定方程式に関し
て示された弱解の存在と一意性の主張を、 確率的摂動項を持つ方程式
に対して示す。 解の存在の証明は、ガレルキン近似によって得られた
解の列に対して、 伊藤の公式、Birkholder-Davis-Gundyの不等式など
により、 解の列のコンパクト性を示すこと及び、解の部分列が収束し
、 その収束先が方程式を弱い意味で満たすことを示すことでなされる。
なお、本講演は、吉田伸生氏(京都大学)との共同研究に基づく。
諸分野のための数学研究会
16:30-17:30 数理科学研究科棟(駒場) 052号室
北海道大学のHPには、第1回(2005年6月22日)～第22回(2009年2月18日)までの情報が掲載されております。
Ralph Bruckschen 氏 (ベルリン工科大学、MATHEON)
Interactive Data Visualization challenges, approaches and examples (ENGLISH)
北海道大学のHPには、第1回(2005年6月22日)～第22回(2009年2月18日)までの情報が掲載されております。
Ralph Bruckschen 氏 (ベルリン工科大学、MATHEON)
Interactive Data Visualization challenges, approaches and examples (ENGLISH)
[ 講演概要 ]
Data visualization is probably the most important method to analyze scientific datasets. In the time of petaflop supercomputers and high resolution sensors, the visualization of such datasets became a challenge because of the sheer magnitude. Using the latest technology I will describe some of the challenges and approaches to visualize large and massive datasets. The main bottle necks will be explained, as some algorithms and data structures to widen them. Finally I will show some examples of data visualization using a CAVE environment and virtual prototyping from the 3D Labor at the Technical University of Berlin.
Data visualization is probably the most important method to analyze scientific datasets. In the time of petaflop supercomputers and high resolution sensors, the visualization of such datasets became a challenge because of the sheer magnitude. Using the latest technology I will describe some of the challenges and approaches to visualize large and massive datasets. The main bottle necks will be explained, as some algorithms and data structures to widen them. Finally I will show some examples of data visualization using a CAVE environment and virtual prototyping from the 3D Labor at the Technical University of Berlin.
2011年11月07日(月)
GCOEレクチャーズ
17:00-18:00 数理科学研究科棟(駒場) 470号室
Oleg Emanouilov 氏 (Colorado State University)
Inverse boundary value problem for Schroedinger equation in two dimensions (ENGLISH)
Oleg Emanouilov 氏 (Colorado State University)
Inverse boundary value problem for Schroedinger equation in two dimensions (ENGLISH)
[ 講演概要 ]
We relax the regularity condition on potentials of Schroedinger equations in uniqueness results on the inverse boundary value problem recently proved in A.Bukhgeim (2008) and O. Imanuvilov, G.Uhlmann and M. Yamamoto (2010).
We relax the regularity condition on potentials of Schroedinger equations in uniqueness results on the inverse boundary value problem recently proved in A.Bukhgeim (2008) and O. Imanuvilov, G.Uhlmann and M. Yamamoto (2010).
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