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Seminar information archive
Seminar information archive ～09/10｜Today's seminar 09/11 | Future seminars 09/12～
2010/11/25
Operator Algebra Seminars
16:30-18:00 Room #122 (Graduate School of Math. Sci. Bldg.)
Reiji Tomatsu (Tokyo Univ. Science)
Classification of actions of Kac algebras (JAPANESE)
Reiji Tomatsu (Tokyo Univ. Science)
Classification of actions of Kac algebras (JAPANESE)
2010/11/18
Operator Algebra Seminars
16:30-18:00 Room #122 (Graduate School of Math. Sci. Bldg.)
Jean Roydor (Univ. Tokyo)
Perturbation of dual operator algebras and similarity (ENGLISH)
Jean Roydor (Univ. Tokyo)
Perturbation of dual operator algebras and similarity (ENGLISH)
2010/11/17
Number Theory Seminar
16:30-17:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Shin Harase (University of Tokyo)
Fast lattice reduction for F_2-linear pseudorandom number generators (JAPANESE)
Shin Harase (University of Tokyo)
Fast lattice reduction for F_2-linear pseudorandom number generators (JAPANESE)
2010/11/16
Numerical Analysis Seminar
16:30-18:00 Room #002 (Graduate School of Math. Sci. Bldg.)
Xuefeng Liu (Waseda University/CREST, JST)
On verified evaluation of eigenvalues for elliptic operator over arbitrary polygonal domain (JAPANESE)
[ Reference URL ]
http://www.infsup.jp/utnas/
Xuefeng Liu (Waseda University/CREST, JST)
On verified evaluation of eigenvalues for elliptic operator over arbitrary polygonal domain (JAPANESE)
[ Reference URL ]
http://www.infsup.jp/utnas/
Tuesday Seminar on Topology
16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)
Noboru Ito (Waseda University)
On a colored Khovanov bicomplex (JAPANESE)
Noboru Ito (Waseda University)
On a colored Khovanov bicomplex (JAPANESE)
[ Abstract ]
We discuss the existence of a bicomplex which is a Khovanov-type
complex associated with categorification of a colored Jones polynomial.
This is an answer to the question proposed by A. Beliakova and S. Wehrli.
Then the second term of the spectral sequence of the bicomplex corresponds
to the Khovanov-type homology group. In this talk, we explain how to define
the bicomplex. If time permits, we also define a colored Rasmussen invariant
by using another spectral sequence of the colored Jones polynomial.
We discuss the existence of a bicomplex which is a Khovanov-type
complex associated with categorification of a colored Jones polynomial.
This is an answer to the question proposed by A. Beliakova and S. Wehrli.
Then the second term of the spectral sequence of the bicomplex corresponds
to the Khovanov-type homology group. In this talk, we explain how to define
the bicomplex. If time permits, we also define a colored Rasmussen invariant
by using another spectral sequence of the colored Jones polynomial.
Algebraic Geometry Seminar
16:30-18:00 Room #122 (Graduate School of Math. Sci. Bldg.)
Viacheslav Nikulin (Univ Liverpool and Steklov Moscow)
Self-corresponences of K3 surfaces via moduli of sheaves (ENGLISH)
Viacheslav Nikulin (Univ Liverpool and Steklov Moscow)
Self-corresponences of K3 surfaces via moduli of sheaves (ENGLISH)
[ Abstract ]
In series of our papers with Carlo Madonna (2002--2008) we described self-correspondences via moduli of sheaves with primitive isotropic Mukai vectors for K3 surfaces with Picard number one or two. Here, we give a natural and functorial answer to the same problem for arbitrary Picard number of K3 surfaces. As an application, we characterize in terms of self-correspondences via moduli of sheaves K3 surfaces with reflective Picard lattices, that is when the automorphism group of the lattice is generated by reflections up to finite index. See some details in arXiv:0810.2945.
In series of our papers with Carlo Madonna (2002--2008) we described self-correspondences via moduli of sheaves with primitive isotropic Mukai vectors for K3 surfaces with Picard number one or two. Here, we give a natural and functorial answer to the same problem for arbitrary Picard number of K3 surfaces. As an application, we characterize in terms of self-correspondences via moduli of sheaves K3 surfaces with reflective Picard lattices, that is when the automorphism group of the lattice is generated by reflections up to finite index. See some details in arXiv:0810.2945.
Algebraic Geometry Seminar
16:30-18:00 Room #122 (Graduate School of Math. Sci. Bldg.)
Viacheslav Nikulin (Univ Liverpool and Steklov Moscow)
Self-corresponences of K3 surfaces via moduli of sheaves (ENGLISH)
Viacheslav Nikulin (Univ Liverpool and Steklov Moscow)
Self-corresponences of K3 surfaces via moduli of sheaves (ENGLISH)
[ Abstract ]
In series of our papers with Carlo Madonna (2002--2008) we described self-correspondences via moduli of sheaves with primitive isotropic Mukai vectors for K3 surfaces with Picard number one or two. Here, we give a natural and functorial answer to the same problem for arbitrary Picard number of K3 surfaces. As an application, we characterize in terms of self-correspondences via moduli of sheaves K3 surfaces with reflective Picard lattices, that is when the automorphism group of the lattice is generated by reflections up to finite index. See some details in arXiv:0810.2945.
In series of our papers with Carlo Madonna (2002--2008) we described self-correspondences via moduli of sheaves with primitive isotropic Mukai vectors for K3 surfaces with Picard number one or two. Here, we give a natural and functorial answer to the same problem for arbitrary Picard number of K3 surfaces. As an application, we characterize in terms of self-correspondences via moduli of sheaves K3 surfaces with reflective Picard lattices, that is when the automorphism group of the lattice is generated by reflections up to finite index. See some details in arXiv:0810.2945.
Tuesday Seminar of Analysis
16:00-18:30 Room #123 (Graduate School of Math. Sci. Bldg.)
Keisuke Uchikoshi (National Defense Academy of Japan) 16:00-16:45
Hyperfunctions and vortex sheets (ENGLISH)
L. Boutet de Monvel (University of Paris 6) 17:00-18:30
Residual trace and equivariant asymptotic trace of Toeplitz operators (ENGLISH)
Keisuke Uchikoshi (National Defense Academy of Japan) 16:00-16:45
Hyperfunctions and vortex sheets (ENGLISH)
L. Boutet de Monvel (University of Paris 6) 17:00-18:30
Residual trace and equivariant asymptotic trace of Toeplitz operators (ENGLISH)
2010/11/15
Seminar on Geometric Complex Analysis
10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)
Tatsuo Suwa (Hokkaido Univ*)
Excess intersections and residues in improper dimension (JAPANESE)
Tatsuo Suwa (Hokkaido Univ*)
Excess intersections and residues in improper dimension (JAPANESE)
[ Abstract ]
This talk concerns localization of characteristic classes and associated residues, in the context of intersection theory and residue theory of singular holomorphic foliations. The localization comes from the vanishing of certain characteristic forms, usually caused by the existence of some geometric object, away from the "singular set" of the object. This gives rise to residues in the homology of the singular set and residue theorems relating local and global invariants. In the generic situation, i.e., if the dimension of the singular set is "proper", we have a reasonable understanding of the residues. We indicate how to cope with the problem when the dimension is "excessive" (partly a joint work with F. Bracci).
This talk concerns localization of characteristic classes and associated residues, in the context of intersection theory and residue theory of singular holomorphic foliations. The localization comes from the vanishing of certain characteristic forms, usually caused by the existence of some geometric object, away from the "singular set" of the object. This gives rise to residues in the homology of the singular set and residue theorems relating local and global invariants. In the generic situation, i.e., if the dimension of the singular set is "proper", we have a reasonable understanding of the residues. We indicate how to cope with the problem when the dimension is "excessive" (partly a joint work with F. Bracci).
Algebraic Geometry Seminar
16:40-18:10 Room #126 (Graduate School of Math. Sci. Bldg.)
Shuhei Yoshitomi (Univ. of Tokyo)
Generators of tropical modules (JAPANESE)
Shuhei Yoshitomi (Univ. of Tokyo)
Generators of tropical modules (JAPANESE)
2010/11/09
Tuesday Seminar on Topology
17:00-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)
Ken'ichi Ohshika (Osaka University)
Characterising bumping points on deformation spaces of Kleinian groups (JAPANESE)
Ken'ichi Ohshika (Osaka University)
Characterising bumping points on deformation spaces of Kleinian groups (JAPANESE)
[ Abstract ]
It is known that components of the interior of a deformation space of a Kleinian group can bump, and a component of the interior can bump itself, on the boundary of the deformation space.
Anderson-Canary-McCullogh gave a necessary and sufficient condition for two components to bump.
In this talk, I shall give a criterion for points on the boundary to be bumping points.
It is known that components of the interior of a deformation space of a Kleinian group can bump, and a component of the interior can bump itself, on the boundary of the deformation space.
Anderson-Canary-McCullogh gave a necessary and sufficient condition for two components to bump.
In this talk, I shall give a criterion for points on the boundary to be bumping points.
2010/11/08
Seminar on Geometric Complex Analysis
10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)
Hajime Tsuji (Sophia Univ)
Variation of canonical measures under Kaehler deformations (JAPANESE)
Hajime Tsuji (Sophia Univ)
Variation of canonical measures under Kaehler deformations (JAPANESE)
GCOE lecture series
16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)
Michael Eastwood (Australian National University)
Invariant differential operators on the sphere (ENGLISH)
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar2010.html#20101102eastwood
Michael Eastwood (Australian National University)
Invariant differential operators on the sphere (ENGLISH)
[ Abstract ]
The circle is acted upon by the rotation group SO(2) and there are plenty of differential operators invariant under this action. But the circle is also acted upon by SL(2,R) and this larger symmetry group cuts down the list of invariant differential operators to something smaller but more interesting! I shall explain what happens and how this phenomenon generalises to spheres. These constructions are part of a general theory but have numerous unexpected applications, for example in suggesting a new stable finite-element scheme in linearised elasticity (due to Arnold, Falk, and Winther).
[ Reference URL ]The circle is acted upon by the rotation group SO(2) and there are plenty of differential operators invariant under this action. But the circle is also acted upon by SL(2,R) and this larger symmetry group cuts down the list of invariant differential operators to something smaller but more interesting! I shall explain what happens and how this phenomenon generalises to spheres. These constructions are part of a general theory but have numerous unexpected applications, for example in suggesting a new stable finite-element scheme in linearised elasticity (due to Arnold, Falk, and Winther).
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar2010.html#20101102eastwood
2010/11/05
GCOE lecture series
16:30-18:00 Room #123 (Graduate School of Math. Sci. Bldg.)
Michael Eastwood (Australian National University)
How to recognise the geodesics of a metric connection (ENGLISH)
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar2010.html#20101102eastwood
Michael Eastwood (Australian National University)
How to recognise the geodesics of a metric connection (ENGLISH)
[ Abstract ]
The geodesics on a Riemannian manifold form a distinguished family of curves, one in every direction through every point. Sometimes two metrics can provide the same family of curves: the Euclidean metric and the round metric on the hemisphere have this property. It is also possible that a family of curves does not arise from a metric at all. Following a classical procedure due to Roger Liouville, I shall explain how to tell these cases apart on a surface. This is joint work with Robert Bryant and Maciej Dunajski.
[ Reference URL ]The geodesics on a Riemannian manifold form a distinguished family of curves, one in every direction through every point. Sometimes two metrics can provide the same family of curves: the Euclidean metric and the round metric on the hemisphere have this property. It is also possible that a family of curves does not arise from a metric at all. Following a classical procedure due to Roger Liouville, I shall explain how to tell these cases apart on a surface. This is joint work with Robert Bryant and Maciej Dunajski.
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar2010.html#20101102eastwood
2010/11/04
Operator Algebra Seminars
16:30-18:00 Room #122 (Graduate School of Math. Sci. Bldg.)
Yoshiko Ogata (Univ.Tokyo)
Nonequilibrium Statistical Mechanics (JAPANESE)
Yoshiko Ogata (Univ.Tokyo)
Nonequilibrium Statistical Mechanics (JAPANESE)
Lectures
10:40-12:10 Room #123 (Graduate School of Math. Sci. Bldg.)
Jean Meyer, Yasuko HISAMATSU (Risk Capital Market Tokyo, BNP Paribas)
Market, Liquidity and Counterparty Risk (ENGLISH)
Jean Meyer, Yasuko HISAMATSU (Risk Capital Market Tokyo, BNP Paribas)
Market, Liquidity and Counterparty Risk (ENGLISH)
[ Abstract ]
1. Introduction to the market risk
- Introduction to the Risk Management
in the Financial institutions
- Overview of the main market risks
2. Market & Liquidity Risks –Basics
-Presentation of the main Greeks
-Focus on volatility risk
-Focus on correlation risk
-Conclusion (common features of the market risks)
3. Risk measure
- Stress test
- Value at risk
- Risks measure for counterparty risk
1. Introduction to the market risk
- Introduction to the Risk Management
in the Financial institutions
- Overview of the main market risks
2. Market & Liquidity Risks –Basics
-Presentation of the main Greeks
-Focus on volatility risk
-Focus on correlation risk
-Conclusion (common features of the market risks)
3. Risk measure
- Stress test
- Value at risk
- Risks measure for counterparty risk
2010/11/02
Lectures
13:00-16:10 Room #122 (Graduate School of Math. Sci. Bldg.)
Vladimir Bogachev (Moscow)
The Malliavin calculus on configuration spaces and applications (ENGLISH)
Vladimir Bogachev (Moscow)
The Malliavin calculus on configuration spaces and applications (ENGLISH)
[ Abstract ]
It is planned to discuss first a general scheme of the Malliavin
calculus on an abstract measurable
manifold with minimal assumptions about the manifold.
Then a practical realization of this scheme will be discussed in
several concrete examples with emphasis
on configuration spaces, i.e., spaces of locally finite configurations
in a given manifold (for example, just
a finite-dimensional Euclidean space), which can be alternatively
described as the spaces of integer-valued
discrete measures equipped with suitable differential structures.
No acquaintance with the Malliavin calculus and differential geometry
is assumed.
It is planned to discuss first a general scheme of the Malliavin
calculus on an abstract measurable
manifold with minimal assumptions about the manifold.
Then a practical realization of this scheme will be discussed in
several concrete examples with emphasis
on configuration spaces, i.e., spaces of locally finite configurations
in a given manifold (for example, just
a finite-dimensional Euclidean space), which can be alternatively
described as the spaces of integer-valued
discrete measures equipped with suitable differential structures.
No acquaintance with the Malliavin calculus and differential geometry
is assumed.
Tuesday Seminar on Topology
16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)
Daniel Ruberman (Brandeis University)
Periodic-end manifolds and SW theory (ENGLISH)
Daniel Ruberman (Brandeis University)
Periodic-end manifolds and SW theory (ENGLISH)
[ Abstract ]
We study an extension of Seiberg-Witten invariants to
4-manifolds with the homology of S^1 \\times S^3. This extension has
many potential applications in low-dimensional topology, including the
study of the homology cobordism group. Because b_2^+ =0, the usual
strategy for defining invariants does not work--one cannot disregard
reducible solutions. In fact, the count of solutions can jump in a
family of metrics or perturbations. To remedy this, we define an
index-theoretic counter-term that jumps by the same amount. The
counterterm is the index of the Dirac operator on a manifold with a
periodic end modeled at infinity by the infinite cyclic cover of the
manifold. This is joint work with Tomasz Mrowka and Nikolai Saveliev.
We study an extension of Seiberg-Witten invariants to
4-manifolds with the homology of S^1 \\times S^3. This extension has
many potential applications in low-dimensional topology, including the
study of the homology cobordism group. Because b_2^+ =0, the usual
strategy for defining invariants does not work--one cannot disregard
reducible solutions. In fact, the count of solutions can jump in a
family of metrics or perturbations. To remedy this, we define an
index-theoretic counter-term that jumps by the same amount. The
counterterm is the index of the Dirac operator on a manifold with a
periodic end modeled at infinity by the infinite cyclic cover of the
manifold. This is joint work with Tomasz Mrowka and Nikolai Saveliev.
Lie Groups and Representation Theory
16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Michael Eastwood (University of Adelaide)
Twistor theory and the harmonic hull (ENGLISH)
Michael Eastwood (University of Adelaide)
Twistor theory and the harmonic hull (ENGLISH)
[ Abstract ]
Harmonic functions are real-analytic and so automatically extend from being functions of real variables to being functions of complex variables. But how far do they extend? This question may be answered by twistor theory, the Penrose transform, and associated geometry. I shall base the constructions on a formula of Bateman from 1904. This is joint work with Feng Xu.
Harmonic functions are real-analytic and so automatically extend from being functions of real variables to being functions of complex variables. But how far do they extend? This question may be answered by twistor theory, the Penrose transform, and associated geometry. I shall base the constructions on a formula of Bateman from 1904. This is joint work with Feng Xu.
2010/11/01
Algebraic Geometry Seminar
16:40-18:10 Room #126 (Graduate School of Math. Sci. Bldg.)
Atsushi Ito (Univ. of Tokyo)
How to estimate Seshadri constants (JAPANESE)
Atsushi Ito (Univ. of Tokyo)
How to estimate Seshadri constants (JAPANESE)
[ Abstract ]
Seshadri constant is an invariant which measures the positivities of ample line bundles. This relates with adjoint bundles, Nagata conjecture, slope stabilities, Gromov width (an invariant of symplectic manifolds) and so on. But it is very diffiult to compute or estimate Seshadri constants in general, especially in higher dimension.
In this talk, we first study Seshadri constants of toric varieties, and next consider about non-toric cases using toric degenerations. For example, good estimations are obtained for complete intersections in projective spaces.
Seshadri constant is an invariant which measures the positivities of ample line bundles. This relates with adjoint bundles, Nagata conjecture, slope stabilities, Gromov width (an invariant of symplectic manifolds) and so on. But it is very diffiult to compute or estimate Seshadri constants in general, especially in higher dimension.
In this talk, we first study Seshadri constants of toric varieties, and next consider about non-toric cases using toric degenerations. For example, good estimations are obtained for complete intersections in projective spaces.
Lectures
16:00-18:15 Room #270 (Graduate School of Math. Sci. Bldg.)
Michel Cristofol (マルセイユ大学) 16:00-17:00
Inverse problems in non linear parabolic equations : Two differents approaches (ENGLISH)
[ Reference URL ]
https://www.ms.u-tokyo.ac.jp/~kengok/abstractTokyo.pdf
Patricia Gaitan (マルセイユ大学) 17:15-18:15
Inverse Problems for parabolic System
(ENGLISH)
Michel Cristofol (マルセイユ大学) 16:00-17:00
Inverse problems in non linear parabolic equations : Two differents approaches (ENGLISH)
[ Reference URL ]
https://www.ms.u-tokyo.ac.jp/~kengok/abstractTokyo.pdf
Patricia Gaitan (マルセイユ大学) 17:15-18:15
Inverse Problems for parabolic System
(ENGLISH)
[ Abstract ]
I will present a review of stability and controllability results for linear parabolic coupled systems with coupling of first and zeroth-order terms by data of reduced number of components. The key ingredients are global Carleman estimates.
I will present a review of stability and controllability results for linear parabolic coupled systems with coupling of first and zeroth-order terms by data of reduced number of components. The key ingredients are global Carleman estimates.
2010/10/29
Colloquium
16:30-17:30 Room #002 (Graduate School of Math. Sci. Bldg.)
Robin Graham (University of Washington)
Ambient metrics and exceptional holonomy (ENGLISH)
Robin Graham (University of Washington)
Ambient metrics and exceptional holonomy (ENGLISH)
[ Abstract ]
The holonomy of a pseudo-Riemannian metric is a subgroup of the orthogonal group which measures the structure preserved by parallel translation. Construction of pseudo-Riemannian metrics whose holonomy is an exceptional Lie group has been of great interest in recent years. This talk will outline a construction of metrics in dimension 7 whose holonomy is contained in the split real form of the exceptional group $G_2$. The datum for the construction is a generic real-analytic 2-plane field on a manifold of dimension 5; the metric in dimension 7 arises as the ambient metric of a conformal structure on the 5-manifold defined by Nurowski in terms of the 2-plane field.
The holonomy of a pseudo-Riemannian metric is a subgroup of the orthogonal group which measures the structure preserved by parallel translation. Construction of pseudo-Riemannian metrics whose holonomy is an exceptional Lie group has been of great interest in recent years. This talk will outline a construction of metrics in dimension 7 whose holonomy is contained in the split real form of the exceptional group $G_2$. The datum for the construction is a generic real-analytic 2-plane field on a manifold of dimension 5; the metric in dimension 7 arises as the ambient metric of a conformal structure on the 5-manifold defined by Nurowski in terms of the 2-plane field.
2010/10/28
Operator Algebra Seminars
16:30-18:00 Room #122 (Graduate School of Math. Sci. Bldg.)
Makoto Yamashita (Univ. Tokyo)
Type III representations of the infinite symmetric group (ENGLISH)
Makoto Yamashita (Univ. Tokyo)
Type III representations of the infinite symmetric group (ENGLISH)
[ Abstract ]
Based on earlier results about the structure of the II$_1$ representations of the infinite symmetric group, we investigate its type III representations and the related inclusion of von Neumann algebras of type III.
Based on earlier results about the structure of the II$_1$ representations of the infinite symmetric group, we investigate its type III representations and the related inclusion of von Neumann algebras of type III.
Lectures
10:40-12:10 Room #123 (Graduate School of Math. Sci. Bldg.)
Jean Meyer, Yasuko HISAMATSU (Risk Capital Market Tokyo, BNP Paribas)
Market, Liquidity and Counterparty Risk (ENGLISH)
Jean Meyer, Yasuko HISAMATSU (Risk Capital Market Tokyo, BNP Paribas)
Market, Liquidity and Counterparty Risk (ENGLISH)
[ Abstract ]
1. Introduction to the market risk
- Introduction to the Risk Management
in the Financial institutions
- Overview of the main market risks
2. Market & Liquidity Risks –Basics
-Presentation of the main Greeks
-Focus on volatility risk
-Focus on correlation risk
-Conclusion (common features of the market risks)
3. Risk measure
- Stress test
- Value at risk
- Risks measure for counterparty risk
1. Introduction to the market risk
- Introduction to the Risk Management
in the Financial institutions
- Overview of the main market risks
2. Market & Liquidity Risks –Basics
-Presentation of the main Greeks
-Focus on volatility risk
-Focus on correlation risk
-Conclusion (common features of the market risks)
3. Risk measure
- Stress test
- Value at risk
- Risks measure for counterparty risk
2010/10/26
Seminar on Geometric Complex Analysis
13:00-14:30 Room #123 (Graduate School of Math. Sci. Bldg.)
Dan Popovici (Toulouse)
Limits of Moishezon Manifolds under Holomorphic Deformations (ENGLISH)
Dan Popovici (Toulouse)
Limits of Moishezon Manifolds under Holomorphic Deformations (ENGLISH)
[ Abstract ]
We prove that if all the fibres, except one, of a holomorphic family of compact complex manifolds are supposed to be Moishezon (i.e. bimeromorphic to projective manifolds), then the remaining (limit) fibre is again Moishezon. The two ingredients of the proof are the relative Barlet space of divisors contained in the fibres for which we show properness over the base of the family and the "strongly Gauduchon" (sG) metrics that we have introduced for the purpose of controlling volumes of cycles. These new metrics enjoy stability properties under both deformations and modifications and play a crucial role in obtaining a uniform control on volumes of relative divisors that prove the above-mentioned properness.
We prove that if all the fibres, except one, of a holomorphic family of compact complex manifolds are supposed to be Moishezon (i.e. bimeromorphic to projective manifolds), then the remaining (limit) fibre is again Moishezon. The two ingredients of the proof are the relative Barlet space of divisors contained in the fibres for which we show properness over the base of the family and the "strongly Gauduchon" (sG) metrics that we have introduced for the purpose of controlling volumes of cycles. These new metrics enjoy stability properties under both deformations and modifications and play a crucial role in obtaining a uniform control on volumes of relative divisors that prove the above-mentioned properness.
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