## Seminar information archive

Seminar information archive ～08/09｜Today's seminar 08/10 | Future seminars 08/11～

### 2011/11/02

#### Number Theory Seminar

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

Hypergeometric series and arithmetic-geometric mean over 2-adic fields (JAPANESE)

**Kensaku Kinjo**(University of Tokyo)Hypergeometric series and arithmetic-geometric mean over 2-adic fields (JAPANESE)

[ Abstract ]

Dwork proved that the Gaussian hypergeometric function on p-adic numbers

can be extended to a function which takes values of the unit roots of

ordinary elliptic curves over a finite field of characteristic p>2.

We present an analogous theory in the case p=2.

As an application, we give a relation between the canonical lift

and the unit root of an elliptic curve over a finite field of

characteristic 2

by using the 2-adic arithmetic-geometric mean.

Dwork proved that the Gaussian hypergeometric function on p-adic numbers

can be extended to a function which takes values of the unit roots of

ordinary elliptic curves over a finite field of characteristic p>2.

We present an analogous theory in the case p=2.

As an application, we give a relation between the canonical lift

and the unit root of an elliptic curve over a finite field of

characteristic 2

by using the 2-adic arithmetic-geometric mean.

### 2011/11/01

#### Tuesday Seminar on Topology

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

Motivic Milnor fibers and Jordan normal forms of monodromies (JAPANESE)

**Kiyoshi Takeuchi**(University of Tsukuba)Motivic Milnor fibers and Jordan normal forms of monodromies (JAPANESE)

[ Abstract ]

We introduce a method to calculate the equivariant

Hodge-Deligne numbers of toric hypersurfaces.

Then we apply it to motivic Milnor

fibers introduced by Denef-Loeser and study the Jordan

normal forms of the local and global monodromies

of polynomials maps in various situations.

Especially we focus our attention on monodromies

at infinity studied by many people. The results will be

explicitly described by the ``convexity" of

the Newton polyhedra of polynomials. This is a joint work

with Y. Matsui and A. Esterov.

We introduce a method to calculate the equivariant

Hodge-Deligne numbers of toric hypersurfaces.

Then we apply it to motivic Milnor

fibers introduced by Denef-Loeser and study the Jordan

normal forms of the local and global monodromies

of polynomials maps in various situations.

Especially we focus our attention on monodromies

at infinity studied by many people. The results will be

explicitly described by the ``convexity" of

the Newton polyhedra of polynomials. This is a joint work

with Y. Matsui and A. Esterov.

### 2011/10/31

#### Algebraic Geometry Seminar

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

Minimal model theory for log surfaces (JAPANESE)

**Osamu Fujino**(Kyoto University)Minimal model theory for log surfaces (JAPANESE)

[ Abstract ]

We discuss the log minimal model theory for log sur- faces. We show that the log minimal model program, the finite generation of log canonical rings, and the log abundance theorem for log surfaces hold true under assumptions weaker than the usual framework of the log minimal model theory.

We discuss the log minimal model theory for log sur- faces. We show that the log minimal model program, the finite generation of log canonical rings, and the log abundance theorem for log surfaces hold true under assumptions weaker than the usual framework of the log minimal model theory.

#### PDE Real Analysis Seminar

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

Stationary Weak Solutions of the Navier-Stokes Equations Past an Obstacle (ENGLISH)

**Horst Heck**

(Technische Universität Darmstadt)Stationary Weak Solutions of the Navier-Stokes Equations Past an Obstacle (ENGLISH)

[ Abstract ]

Consider the stationary Navier-Stokes equations in an exterior smooth domain $\\Omega$. If the flow condition $u_\\infty$ for $u$ at infinity is non-zero and the external force $f\\in \\dot H^{-1}_2(\\Omega)$ is given Leray constructed a weak solution $u\\in \\dot H^1_2(\\Omega)$, the homogeneous Sobolev space, with $u-u_\\infty\\in L^6(\\Omega)$.

We show that if in addition $f\\in \\dot H^{-1}_q(\\Omega)$ for some $q\\in (4/3,4)$ then the weak solution has the property $u-u_\\infty\\in L^{4q/(4-q)}(\\Omega)$.

This additional integrability implies that $u$ satisfies the energy identity. Further consequences are uniqueness results for small $u_\\infty$ and $f$ and continuous dependence of the solution with respect to $u_\\infty$.

The presented results are joint work with Hyunseok Kim and Hideo Kozono.

Consider the stationary Navier-Stokes equations in an exterior smooth domain $\\Omega$. If the flow condition $u_\\infty$ for $u$ at infinity is non-zero and the external force $f\\in \\dot H^{-1}_2(\\Omega)$ is given Leray constructed a weak solution $u\\in \\dot H^1_2(\\Omega)$, the homogeneous Sobolev space, with $u-u_\\infty\\in L^6(\\Omega)$.

We show that if in addition $f\\in \\dot H^{-1}_q(\\Omega)$ for some $q\\in (4/3,4)$ then the weak solution has the property $u-u_\\infty\\in L^{4q/(4-q)}(\\Omega)$.

This additional integrability implies that $u$ satisfies the energy identity. Further consequences are uniqueness results for small $u_\\infty$ and $f$ and continuous dependence of the solution with respect to $u_\\infty$.

The presented results are joint work with Hyunseok Kim and Hideo Kozono.

#### Seminar on Geometric Complex Analysis

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

Classification of Moishezon twistor spaces on 4CP^2 (JAPANESE)

**Nobuhiro Honda**(Tohoku Univeristy)Classification of Moishezon twistor spaces on 4CP^2 (JAPANESE)

### 2011/10/29

#### Infinite Analysis Seminar Tokyo

11:00-14:30 Room #117 (Graduate School of Math. Sci. Bldg.)

CKP Hierarchy, Bosonic Tau Function, Bosonization Formulae and Orthogonal Polynomials both in Odd and Even Variables

(based on a joint work with Johan van de Leur and Takahiro Shiota) (ENGLISH)

Kernel function identities associated with van Diejen's $q$-difference operators

and transformation formulas for multiple $q$-hypergeometric series (JAPANESE)

**Alexander Orlov**(Nonlinear Wave Processes Laboratory, Oceanology Institute (Moscow)) 11:00-12:00CKP Hierarchy, Bosonic Tau Function, Bosonization Formulae and Orthogonal Polynomials both in Odd and Even Variables

(based on a joint work with Johan van de Leur and Takahiro Shiota) (ENGLISH)

[ Abstract ]

We develop the theory of CKP hierarchy introduced in the papers of Kyoto school where the CKP tau function is written as a vacuum expectation value in terms of free bosons. We show that a sort of odd currents naturaly appear. We consider bosonization formulae which relate bosonic Fock vectors to polynomials in even and odd Grassmannian variables, where both sets play a role of higher times.

We develop the theory of CKP hierarchy introduced in the papers of Kyoto school where the CKP tau function is written as a vacuum expectation value in terms of free bosons. We show that a sort of odd currents naturaly appear. We consider bosonization formulae which relate bosonic Fock vectors to polynomials in even and odd Grassmannian variables, where both sets play a role of higher times.

**Yasuho Masuda**(Kobe Univ. ) 13:30-14:30Kernel function identities associated with van Diejen's $q$-difference operators

and transformation formulas for multiple $q$-hypergeometric series (JAPANESE)

### 2011/10/26

#### Seminar on Probability and Statistics

15:00-16:10 Room #000 (Graduate School of Math. Sci. Bldg.)

Statistical models constructed by optimal stationary coupling (JAPANESE)

[ Reference URL ]

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2011/04.html

**SEI, Tomonari**(Department of Mathematics, Keio University)Statistical models constructed by optimal stationary coupling (JAPANESE)

[ Reference URL ]

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2011/04.html

### 2011/10/25

#### Tuesday Seminar on Topology

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

Circle-valued Morse theory for complex hyperplane arrangements (ENGLISH)

**Andrei Pajitnov**(Univ. de Nantes, Univ. of Tokyo)Circle-valued Morse theory for complex hyperplane arrangements (ENGLISH)

[ Abstract ]

Let A be a complex hyperplane arrangement

in an n-dimensional complex vector space V.

Denote by H the union of the hyperplanes

and by M the complement to H in V.

We develop the real-valued and circle-valued Morse

theory on M. We prove that if A is essential then

M has the homotopy type of a space

obtained from a finite n-dimensional

CW complex fibered over a circle,

by attaching several cells of dimension n.

We compute the Novikov homology of M and show

that its structure is similar to the

homology with generic local coefficients:

it vanishes for all dimensions except n.

This is a joint work with Toshitake Kohno.

Let A be a complex hyperplane arrangement

in an n-dimensional complex vector space V.

Denote by H the union of the hyperplanes

and by M the complement to H in V.

We develop the real-valued and circle-valued Morse

theory on M. We prove that if A is essential then

M has the homotopy type of a space

obtained from a finite n-dimensional

CW complex fibered over a circle,

by attaching several cells of dimension n.

We compute the Novikov homology of M and show

that its structure is similar to the

homology with generic local coefficients:

it vanishes for all dimensions except n.

This is a joint work with Toshitake Kohno.

#### Lie Groups and Representation Theory

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

Localization of Cohomological Induction (ENGLISH)

**Yoshiki Oshima**(Graduate School of Mathematical Sciences, the University of Tokyo)Localization of Cohomological Induction (ENGLISH)

[ Abstract ]

Cohomological induction is defined for (g,K)-modules in an algebraic way and construct important representations such as (Harish-Chandra modules of) discrete series representations,

principal series representations and Zuckerman's modules of

semisimple Lie groups.

Hecht, Milicic, Schmid, and Wolf proved that modules induced from

one-dimensional representations of Borel subalgebra can be realized as D-modules on the flag variety.

In this talk, we show a similar result for modules induced from

more general representations.

Cohomological induction is defined for (g,K)-modules in an algebraic way and construct important representations such as (Harish-Chandra modules of) discrete series representations,

principal series representations and Zuckerman's modules of

semisimple Lie groups.

Hecht, Milicic, Schmid, and Wolf proved that modules induced from

one-dimensional representations of Borel subalgebra can be realized as D-modules on the flag variety.

In this talk, we show a similar result for modules induced from

more general representations.

### 2011/10/22

#### Infinite Analysis Seminar Tokyo

13:30-16:00 Room #117 (Graduate School of Math. Sci. Bldg.)

Department of Mathematics) 13:30-14:30

Quantization of Quasimaps' Spaces (joint work with M. Finkelberg) (ENGLISH)

Instituteof Biochemical Physics) 15:00-16:00

Quantum integrable models with elliptic R-matrices

and elliptic hypergeometric series (ENGLISH)

**Leonid Rybnikov**(IITP, and State University Higher School of Economics,Department of Mathematics) 13:30-14:30

Quantization of Quasimaps' Spaces (joint work with M. Finkelberg) (ENGLISH)

[ Abstract ]

Quasimaps' space Z_d (also known as Drinfeld's Zastava space) is a

remarkable compactification of the space of based degree d maps from

the projective line to the flag variety of type A. The space Z_d has a

natural Poisson structure,

which goes back to Atiyah and Hitchin. We describe

the Quasimaps' space as some quiver variety, and define the

Atiyah-Hitchin Poisson structure in quiver terms.

This gives a natural way to quantize this Poisson structure.

The quantization of the coordinate ring of the Quasimaps' space turns

to be some natural subquotient of the Yangian of type A.

I will also discuss some generalization of this result to the BCD types.

Quasimaps' space Z_d (also known as Drinfeld's Zastava space) is a

remarkable compactification of the space of based degree d maps from

the projective line to the flag variety of type A. The space Z_d has a

natural Poisson structure,

which goes back to Atiyah and Hitchin. We describe

the Quasimaps' space as some quiver variety, and define the

Atiyah-Hitchin Poisson structure in quiver terms.

This gives a natural way to quantize this Poisson structure.

The quantization of the coordinate ring of the Quasimaps' space turns

to be some natural subquotient of the Yangian of type A.

I will also discuss some generalization of this result to the BCD types.

**Anton Zabrodin**(Instituteof Biochemical Physics) 15:00-16:00

Quantum integrable models with elliptic R-matrices

and elliptic hypergeometric series (ENGLISH)

[ Abstract ]

Intertwining operators for infinite-dimensional representations of the

Sklyanin algebra with spins l and -l-1 are constructed using the technique of

intertwining vectors for elliptic L-operator. They are expressed in

terms of

elliptic hypergeometric series with operator argument. The intertwining

operators obtained (W-operators) serve as building blocks for the

elliptic R-matrix

which intertwines tensor product of two L-operators taken in

infinite-dimensional

representations of the Sklyanin algebra with arbitrary spin. The

Yang-Baxter equation

for this R-matrix follows from simpler equations of the star-triangle

type for the

W-operators. A natural graphic representation of the objects and

equations involved

in the construction is used.

Intertwining operators for infinite-dimensional representations of the

Sklyanin algebra with spins l and -l-1 are constructed using the technique of

intertwining vectors for elliptic L-operator. They are expressed in

terms of

elliptic hypergeometric series with operator argument. The intertwining

operators obtained (W-operators) serve as building blocks for the

elliptic R-matrix

which intertwines tensor product of two L-operators taken in

infinite-dimensional

representations of the Sklyanin algebra with arbitrary spin. The

Yang-Baxter equation

for this R-matrix follows from simpler equations of the star-triangle

type for the

W-operators. A natural graphic representation of the objects and

equations involved

in the construction is used.

### 2011/10/20

#### GCOE Seminars

16:30-18:00 Room #270 (Graduate School of Math. Sci. Bldg.)

On reconstruction of Lame coefficients from partial Cauchy data (ENGLISH)

**O. Emanouilov**(Colorado State University)On reconstruction of Lame coefficients from partial Cauchy data (ENGLISH)

[ Abstract ]

For the isotropic Lame system, we prove that if the Lame coefficient ¥mu is a positive constant, then both Lame coefficients can be recovered from the partial Cauchy data.

For the isotropic Lame system, we prove that if the Lame coefficient ¥mu is a positive constant, then both Lame coefficients can be recovered from the partial Cauchy data.

### 2011/10/19

#### Number Theory Seminar

17:30-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

K_2 of the biquaternion algebra (ENGLISH)

[ Reference URL ]

http://www.ihes.fr/~abbes/SGA/suslin.pdf

**Andrei Suslin**(Northwestern University)K_2 of the biquaternion algebra (ENGLISH)

[ Reference URL ]

http://www.ihes.fr/~abbes/SGA/suslin.pdf

### 2011/10/17

#### Seminar on Geometric Complex Analysis

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

Compact locally homogeneous Kähler manifolds $\Gamma\backslash G/K$ (JAPANESE)

**Yoshinobu Kamishima**(Tokyo Metropolitan University)Compact locally homogeneous Kähler manifolds $\Gamma\backslash G/K$ (JAPANESE)

### 2011/10/12

#### Seminar on Probability and Statistics

15:00-16:10 Room #000 (Graduate School of Math. Sci. Bldg.)

On Convergence Rate of Multiple Kernel Learning with Various Regularization Types (JAPANESE)

[ Reference URL ]

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2011/03.html

**SUZUKI, Taiji**(University of Tokyo)On Convergence Rate of Multiple Kernel Learning with Various Regularization Types (JAPANESE)

[ Reference URL ]

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2011/03.html

### 2011/10/11

#### Tuesday Seminar on Topology

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

Making foliations of codimension one,

thirty years after Thurston's works

(ENGLISH)

**Gael Meigniez**(Univ. de Bretagne-Sud, Chuo Univ.)Making foliations of codimension one,

thirty years after Thurston's works

(ENGLISH)

[ Abstract ]

In 1976 Thurston proved that every closed manifold M whose

Euler characteristic is null carries a smooth foliation F of codimension

one. He actually established a h-principle allowing the regularization of

Haefliger structures through homotopy. I shall give some accounts of a new,

simpler proof of Thurston's result, not using Mather's homology equivalence; and also show that this proof allows to make F have dense leaves if dim M is at least 4. The emphasis will be put on the high dimensions.

In 1976 Thurston proved that every closed manifold M whose

Euler characteristic is null carries a smooth foliation F of codimension

one. He actually established a h-principle allowing the regularization of

Haefliger structures through homotopy. I shall give some accounts of a new,

simpler proof of Thurston's result, not using Mather's homology equivalence; and also show that this proof allows to make F have dense leaves if dim M is at least 4. The emphasis will be put on the high dimensions.

#### Tuesday Seminar of Analysis

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

On the best constant of the weighted Trudinger-Moser

type inequality (JAPANESE)

**Hidemitsu Wadade**(Waseda University (JSPS-PD))On the best constant of the weighted Trudinger-Moser

type inequality (JAPANESE)

### 2011/10/07

#### Operator Algebra Seminars

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

Towards the classification of non-simple $C^*$-algebras of real rank zero (ENGLISH)

**Takeshi Katsura**(Keio University)Towards the classification of non-simple $C^*$-algebras of real rank zero (ENGLISH)

### 2011/10/05

#### Seminar on Mathematics for various disciplines

10:00-11:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Energetic Variational Approaches for Ionic Fluids (ENGLISH)

**Chun Liu**(University of Tokyo / Pennsylvania State University)Energetic Variational Approaches for Ionic Fluids (ENGLISH)

[ Abstract ]

In this talk, I will present our recent study on the ionic transport through ion channels in cell membranes. Motivated by our earlier work on energetic variational approaches, developed for various complex fluids, especially electrorheological (ER) fluids (Phys. Rev. Lett. 101, 194503 (2008)), we derived/proposed a coupled system for ionic solutions, which takes into account of the solvent water, the diffusion and electro-static interaction of different ions. In particular, I will emphasize on the selectivity effects of the ion channels, under the simplest geometric and molecular structures.

In this talk, I will present our recent study on the ionic transport through ion channels in cell membranes. Motivated by our earlier work on energetic variational approaches, developed for various complex fluids, especially electrorheological (ER) fluids (Phys. Rev. Lett. 101, 194503 (2008)), we derived/proposed a coupled system for ionic solutions, which takes into account of the solvent water, the diffusion and electro-static interaction of different ions. In particular, I will emphasize on the selectivity effects of the ion channels, under the simplest geometric and molecular structures.

### 2011/10/04

#### Tuesday Seminar on Topology

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

Relatively quasiconvex subgroups of relatively hyperbolic groups (JAPANESE)

**Yoshifumi Matsuda**(The University of Tokyo)Relatively quasiconvex subgroups of relatively hyperbolic groups (JAPANESE)

[ Abstract ]

Relative hyperbolicity of groups was introduced by Gromov as a

generalization of word hyperbolicity. Motivating examples of relatively

hyperbolic groups are fundamental groups of noncompact complete

hyperbolic manifolds of finite volume. The class of relatively

quasiconvex subgroups of a realtively hyperbolic group is defined as a

genaralization of that of quasicovex subgroups of a word hyperbolic

group. The notion of hyperbolically embedded subgroups of a relatively

hyperbolic group was introduced by Osin and such groups are

characterized as relatively quasiconvex subgroups with additional

algebraic properties. In this talk I will present an introduction to

relatively quasiconvex subgroups and discuss recent joint work with Shin

-ichi Oguni and Saeko Yamagata on hyperbolically embedded subgroups.

Relative hyperbolicity of groups was introduced by Gromov as a

generalization of word hyperbolicity. Motivating examples of relatively

hyperbolic groups are fundamental groups of noncompact complete

hyperbolic manifolds of finite volume. The class of relatively

quasiconvex subgroups of a realtively hyperbolic group is defined as a

genaralization of that of quasicovex subgroups of a word hyperbolic

group. The notion of hyperbolically embedded subgroups of a relatively

hyperbolic group was introduced by Osin and such groups are

characterized as relatively quasiconvex subgroups with additional

algebraic properties. In this talk I will present an introduction to

relatively quasiconvex subgroups and discuss recent joint work with Shin

-ichi Oguni and Saeko Yamagata on hyperbolically embedded subgroups.

### 2011/09/20

#### Tuesday Seminar on Topology

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

Functorial semi-norms on singular homology (ENGLISH)

**Clara Loeh**(Univ. Regensburg)Functorial semi-norms on singular homology (ENGLISH)

[ Abstract ]

Functorial semi-norms on singular homology add metric information to

homology classes that is compatible with continuous maps. In particular,

functorial semi-norms give rise to degree theorems for certain classes

of manifolds; an invariant fitting into this context is Gromov's

simplicial volume. On the other hand, knowledge about mapping degrees

allows to construct functorial semi-norms with interesting properties;

for example, so-called inflexible simply connected manifolds give rise

to functorial semi-norms that are non-trivial on certain simply connected

spaces.

Functorial semi-norms on singular homology add metric information to

homology classes that is compatible with continuous maps. In particular,

functorial semi-norms give rise to degree theorems for certain classes

of manifolds; an invariant fitting into this context is Gromov's

simplicial volume. On the other hand, knowledge about mapping degrees

allows to construct functorial semi-norms with interesting properties;

for example, so-called inflexible simply connected manifolds give rise

to functorial semi-norms that are non-trivial on certain simply connected

spaces.

### 2011/08/12

#### GCOE Seminars

16:00-17:00 Room #370 (Graduate School of Math. Sci. Bldg.)

Kernel-based Approximation Methods for Cauchy Problems of Fractional Order Partial Differential Equations (ENGLISH)

**Benny Hon**(Department of Mathematics City University of Hong Kong)Kernel-based Approximation Methods for Cauchy Problems of Fractional Order Partial Differential Equations (ENGLISH)

[ Abstract ]

In this talk we present the recent development of meshless computational methods based on the use of kernel-based functions for solving various inverse and ill-posed problems. Properties of some special kernels such as harmonic kernels; kernels from the construction of fundamental and particular solutions; Green’s functions; and radial basis functions will be discussed. As an illustration, the recent work in using the method of fundamental solutions combined with the Laplace transform and the Tikhonov regularization techniques to solve Cauchy problems of Fractional Order Partial Differential Equations (FOPDEs) will be demonstrated. The main idea is to approximate the unknown solution by a linear combination of fundamental solutions whose singularities are located outside the solution domain. The Laplace transform technique is used to obtain a more accurate numerical approximation of the fundamental solutions and the L-curve method is adopted for searching an optimal regularization parameter in obtaining stable solution from measured data with noises.

In this talk we present the recent development of meshless computational methods based on the use of kernel-based functions for solving various inverse and ill-posed problems. Properties of some special kernels such as harmonic kernels; kernels from the construction of fundamental and particular solutions; Green’s functions; and radial basis functions will be discussed. As an illustration, the recent work in using the method of fundamental solutions combined with the Laplace transform and the Tikhonov regularization techniques to solve Cauchy problems of Fractional Order Partial Differential Equations (FOPDEs) will be demonstrated. The main idea is to approximate the unknown solution by a linear combination of fundamental solutions whose singularities are located outside the solution domain. The Laplace transform technique is used to obtain a more accurate numerical approximation of the fundamental solutions and the L-curve method is adopted for searching an optimal regularization parameter in obtaining stable solution from measured data with noises.

### 2011/08/03

#### thesis presentations

10:00-11:15 Room #123 (Graduate School of Math. Sci. Bldg.)

Abundance conjecture and canonical bundle formula (JAPANESE)

**Yoshinori GONGYO**(Graduate School of Mathematical Sciences the University of Tokyo)Abundance conjecture and canonical bundle formula (JAPANESE)

### 2011/07/29

#### Colloquium

16:30-17:30 Room #123 (Graduate School of Math. Sci. Bldg.)

Arc spaces and algebraic geometry (JAPANESE)

**Shihoko Ishii**(Graduate School of Mathematical Sciences, University of Tokyo)Arc spaces and algebraic geometry (JAPANESE)

### 2011/07/27

#### Number Theory Seminar

16:00-18:15 Room #123 (Graduate School of Math. Sci. Bldg.)

Discriminants and determinant of a hypersurface of even dimension (ENGLISH)

Multiplicities of discriminants (ENGLISH)

**Takeshi Saito**(University of Tokyo) 16:00-17:00Discriminants and determinant of a hypersurface of even dimension (ENGLISH)

[ Abstract ]

The determinant of the cohomology of a smooth hypersurface

of even dimension as a quadratic character of the absolute

Galois group is computed by the discriminant of the de Rham

cohomology. They are also computed by the discriminant of a

defining polynomial. We determine the sign involved by testing

the formula for the Fermat hypersurfaces.

This is a joint work with J-P. Serre.

The determinant of the cohomology of a smooth hypersurface

of even dimension as a quadratic character of the absolute

Galois group is computed by the discriminant of the de Rham

cohomology. They are also computed by the discriminant of a

defining polynomial. We determine the sign involved by testing

the formula for the Fermat hypersurfaces.

This is a joint work with J-P. Serre.

**Dennis Eriksson**(University of Gothenburg) 17:15-18:15Multiplicities of discriminants (ENGLISH)

[ Abstract ]

The discriminant of a homogenous polynomial is another homogenous

polynomial in the coefficients of the polynomial, which is zero

if and only if the corresponding hypersurface is singular. In

case the coefficients are in a discrete valuation ring, the

order of the discriminant (if non-zero) measures the bad

reduction. We give some new results on this order, and in

particular tie it to Bloch's conjecture/the Kato-T.Saito formula

on equality of localized Chern classes and Artin conductors. We

can precisely compute all the numbers in the case of ternary

forms, giving a partial generalization of Ogg's formula for

elliptic curves.

The discriminant of a homogenous polynomial is another homogenous

polynomial in the coefficients of the polynomial, which is zero

if and only if the corresponding hypersurface is singular. In

case the coefficients are in a discrete valuation ring, the

order of the discriminant (if non-zero) measures the bad

reduction. We give some new results on this order, and in

particular tie it to Bloch's conjecture/the Kato-T.Saito formula

on equality of localized Chern classes and Artin conductors. We

can precisely compute all the numbers in the case of ternary

forms, giving a partial generalization of Ogg's formula for

elliptic curves.

### 2011/07/21

#### Operator Algebra Seminars

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

Almost completely isometric maps and applications (ENGLISH)

**Jean Roydor**(Univ. Tokyo)Almost completely isometric maps and applications (ENGLISH)

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