## Seminar information archive

Seminar information archive ～02/01｜Today's seminar 02/02 | Future seminars 02/03～

### 2012/06/27

#### Classical Analysis

16:00-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)

Moduli space of meromorphic connections with ramified irregular singularities on principal bundles (JAPANESE)

**Daisuke Yamakawa**(Tokyo Institute of Technology)Moduli space of meromorphic connections with ramified irregular singularities on principal bundles (JAPANESE)

### 2012/06/26

#### Tuesday Seminar of Analysis

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

Absence of embedded eigenvalues for the Schr\\"odinger operator on manifold with ends (JAPANESE)

**Kenichi Ito**(Division of Mathematics, University of Tsukuba)Absence of embedded eigenvalues for the Schr\\"odinger operator on manifold with ends (JAPANESE)

[ Abstract ]

We consider a Riemannian manifold with, at least, one expanding end, and prove the absence of $L^2$-eigenvalues for the Schr\\"odinger operator above some critical value. The critical value is computed from the volume growth rate of the end and the potential behavior at infinity. The end structure is formulated abstractly in terms of some convex function, and the examples include asymptotically Euclidean and hyperbolic ends. The proof consists of a priori superexponential decay estimate for eigenfunctions and the absence of superexponentially decaying eigenfunctions, both of which employs the Mourre-type commutator argument. This talk is based on the recent joint work with E.Skibsted (Aarhus University).

We consider a Riemannian manifold with, at least, one expanding end, and prove the absence of $L^2$-eigenvalues for the Schr\\"odinger operator above some critical value. The critical value is computed from the volume growth rate of the end and the potential behavior at infinity. The end structure is formulated abstractly in terms of some convex function, and the examples include asymptotically Euclidean and hyperbolic ends. The proof consists of a priori superexponential decay estimate for eigenfunctions and the absence of superexponentially decaying eigenfunctions, both of which employs the Mourre-type commutator argument. This talk is based on the recent joint work with E.Skibsted (Aarhus University).

### 2012/06/25

#### Algebraic Geometry Seminar

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

Automorphism groups of Calabi-Yau manifolds of Picard number two (JAPANESE)

**Keiji Oguiso**(Osaka University)Automorphism groups of Calabi-Yau manifolds of Picard number two (JAPANESE)

[ Abstract ]

We prove that the automorphism group of an odd dimensional Calabi-Yau manifold of Picard number two is always a finite group. This makes a sharp contrast to the automorphism groups of K3 surfaces and hyperk\\"ahler manifolds and birational automorphism groups, as I shall explain. We also clarify the relation between finiteness of the automorphism group (resp. birational automorphism group) and the rationality of the nef cone (resp. movable cone) for a hyperk\\"ahler manifold of Picard number two. We will also discuss a similar conjectual relation for a Calabi-Yau threefold of Picard number two, together with exsistence of rational curve, expected by the cone conjecture.

We prove that the automorphism group of an odd dimensional Calabi-Yau manifold of Picard number two is always a finite group. This makes a sharp contrast to the automorphism groups of K3 surfaces and hyperk\\"ahler manifolds and birational automorphism groups, as I shall explain. We also clarify the relation between finiteness of the automorphism group (resp. birational automorphism group) and the rationality of the nef cone (resp. movable cone) for a hyperk\\"ahler manifold of Picard number two. We will also discuss a similar conjectual relation for a Calabi-Yau threefold of Picard number two, together with exsistence of rational curve, expected by the cone conjecture.

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

CR equivalence problem of CR manifolds with slice structure (JAPANESE)

**Atsushi Hayashimoto**(Nagano National College of Technology)CR equivalence problem of CR manifolds with slice structure (JAPANESE)

### 2012/06/20

#### Number Theory Seminar

16:40-17:40 Room #056 (Graduate School of Math. Sci. Bldg.)

On the reduction modulo p of representations of a quaternion

division algebra over a p-adic field (JAPANESE)

**Kazuki Tokimoto**(University of Tokyo)On the reduction modulo p of representations of a quaternion

division algebra over a p-adic field (JAPANESE)

[ Abstract ]

The p-adic Langlands correspondence and the mod p Langlands correspondence for GL_2(Q_p) are known to be compatible with the reduction modulo p in many cases.

In this talk, we examine whether a similar compatibility exists for the composition of the local Langlands correspondence and the local Jacquet-Langlands correspondence.

The simplest case has already been treated by Vign¥'eras. We deal with more cases.

The p-adic Langlands correspondence and the mod p Langlands correspondence for GL_2(Q_p) are known to be compatible with the reduction modulo p in many cases.

In this talk, we examine whether a similar compatibility exists for the composition of the local Langlands correspondence and the local Jacquet-Langlands correspondence.

The simplest case has already been treated by Vign¥'eras. We deal with more cases.

#### PDE Real Analysis Seminar

10:30-11:30 Room #056 (Graduate School of Math. Sci. Bldg.)

On the Navier-Stokes Cauchy problem with nondecaying data (ENGLISH)

**Paolo Maremonti**(Seconda Università degli Studi di Napoli)On the Navier-Stokes Cauchy problem with nondecaying data (ENGLISH)

[ Abstract ]

We prove the well posedeness of the Navier-Stokes Cauchy problem for nondecaying initial data u_0 \\in C (R^n) \\cap L^\\infty (R^n), n >= 3. This problem is studied by Giga, Inui and Matsui for n >= 3, and Giga, Matsui and Sawada in the two dimensional case. The aims of our paper are slight different since we also find pointwise estimates for the pressure field. Via a uniqueness theorem, we give a sort of structure theorem to GIM solutions.

We prove the well posedeness of the Navier-Stokes Cauchy problem for nondecaying initial data u_0 \\in C (R^n) \\cap L^\\infty (R^n), n >= 3. This problem is studied by Giga, Inui and Matsui for n >= 3, and Giga, Matsui and Sawada in the two dimensional case. The aims of our paper are slight different since we also find pointwise estimates for the pressure field. Via a uniqueness theorem, we give a sort of structure theorem to GIM solutions.

### 2012/06/19

#### Tuesday Seminar on Topology

17:10-18:10 Room #056 (Graduate School of Math. Sci. Bldg.)

On the universal degenerating family of Riemann surfaces

over the D-M compactification of moduli space (JAPANESE)

**Yukio Matsumoto**(Gakushuin University)On the universal degenerating family of Riemann surfaces

over the D-M compactification of moduli space (JAPANESE)

[ Abstract ]

It is usually understood that over the Deligne-

Mumford compactification of moduli space of Riemann surfaces of

genus > 1, there is a family of stable curves. However, if one tries to

construct this family precisely, he/she must first take a disjoint union

of various types of smooth families of stable curves, and then divide

them by their automorphisms to paste them together. In this talk we will

show that once the smooth families are divided, the resulting quotient

family contains not only stable curves but virtually all types of

degeneration of Riemann surfaces, becoming a kind of universal

degenerating family of Riemann surfaces.

It is usually understood that over the Deligne-

Mumford compactification of moduli space of Riemann surfaces of

genus > 1, there is a family of stable curves. However, if one tries to

construct this family precisely, he/she must first take a disjoint union

of various types of smooth families of stable curves, and then divide

them by their automorphisms to paste them together. In this talk we will

show that once the smooth families are divided, the resulting quotient

family contains not only stable curves but virtually all types of

degeneration of Riemann surfaces, becoming a kind of universal

degenerating family of Riemann surfaces.

#### Numerical Analysis Seminar

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

Advances in the charge simulation method (JAPANESE)

[ Reference URL ]

http://www.infsup.jp/utnas/

**Hidenori Ogata**(The University of Electro-Communications)Advances in the charge simulation method (JAPANESE)

[ Reference URL ]

http://www.infsup.jp/utnas/

### 2012/06/18

#### Algebraic Geometry Seminar

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

アルバネーゼ次元最大の複素射影多様体の特殊集合について (JAPANESE)

**Katsutoshi Yamanoi**(Tokyo Institute of Technology)アルバネーゼ次元最大の複素射影多様体の特殊集合について (JAPANESE)

[ Abstract ]

アルバネーゼ次元が最大の複素射影多様体の中に含まれる代数的あるいは超越的な複

素曲線について、

高次元ネヴァンリンナ理論の立場からお話します。

アルバネーゼ次元が最大の複素射影多様体の中に含まれる代数的あるいは超越的な複

素曲線について、

高次元ネヴァンリンナ理論の立場からお話します。

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

CR Q-curvature flow and CR Paneitz operator on 3-dimensional CR manifolds (JAPANESE)

**Takanari Saotome**(Academia Sinica)CR Q-curvature flow and CR Paneitz operator on 3-dimensional CR manifolds (JAPANESE)

#### Lectures

09:45-12:30 Room #002 (Graduate School of Math. Sci. Bldg.)

Fluctuation of solutions to PDEs with random coefficients (Part 2) (ENGLISH)

https://www.ms.u-tokyo.ac.jp/~matano/SDE2012/

Weak coupling limits for particles and PDEs (Part 2) (ENGLISH)

https://www.ms.u-tokyo.ac.jp/~matano/SDE2012/

**James Nolen**(Duke University) 09:45-10:45Fluctuation of solutions to PDEs with random coefficients (Part 2) (ENGLISH)

[ Abstract ]

This is continuation of the previous day's lecture.

[ Reference URL ]This is continuation of the previous day's lecture.

https://www.ms.u-tokyo.ac.jp/~matano/SDE2012/

**Leonid Ryzhik**(Stanford University) 11:00-12:30Weak coupling limits for particles and PDEs (Part 2) (ENGLISH)

[ Abstract ]

This is continuation of the previous day's lecture.

[ Reference URL ]This is continuation of the previous day's lecture.

https://www.ms.u-tokyo.ac.jp/~matano/SDE2012/

### 2012/06/17

#### Lectures

09:45-17:30 Room #002 (Graduate School of Math. Sci. Bldg.)

Fluctuation of solutions to PDEs with random coefficients (Part 1) (JAPANESE)

https://www.ms.u-tokyo.ac.jp/~matano/SDE2012/

Weak coupling limits for particles and PDEs (Part 1) (JAPANESE)

https://www.ms.u-tokyo.ac.jp/~matano/SDE2012/

Asymptotic spreading for heterogeneous Fisher-KPP reaction-diffusion equations (JAPANESE)

https://www.ms.u-tokyo.ac.jp/~matano/SDE2012/

**James Nolen**(Duke University) 09:45-17:30Fluctuation of solutions to PDEs with random coefficients (Part 1) (JAPANESE)

[ Abstract ]

For PDEs with random coefficients, it is interesting to understand whether the solutions exhibit some universal statistical behavior that is independent of the details of the coefficients. In particular, how do solutions fluctuate around the mean behavior? We will discuss this issue in the context of three examples:

(1) Traveling fronts in random media in one dimension.

(2) Elliptic homogenization problems.

(3) Random Hamilton-Jacobi equations.

The relation between PDE tools and probabilistic ideas will be

explained.

[ Reference URL ]For PDEs with random coefficients, it is interesting to understand whether the solutions exhibit some universal statistical behavior that is independent of the details of the coefficients. In particular, how do solutions fluctuate around the mean behavior? We will discuss this issue in the context of three examples:

(1) Traveling fronts in random media in one dimension.

(2) Elliptic homogenization problems.

(3) Random Hamilton-Jacobi equations.

The relation between PDE tools and probabilistic ideas will be

explained.

https://www.ms.u-tokyo.ac.jp/~matano/SDE2012/

**Leonid Ryzhik**(Stanford Univeristy) 13:00-14:45Weak coupling limits for particles and PDEs (Part 1) (JAPANESE)

[ Abstract ]

Weak random fluctuations in medium parameters may lead to a non-trivial effect after large times and propagation over long distances. We will consider several examples when the large time limit can be treated:

(1) a particle in a weakly random velocity field.

(2) weak random fluctuations of Hamilton equations, and

(3) the linear Scrhoedinger equation with a weak random potential.

The role of long range correlation of the random fluctuations will also be discussed.

[ Reference URL ]Weak random fluctuations in medium parameters may lead to a non-trivial effect after large times and propagation over long distances. We will consider several examples when the large time limit can be treated:

(1) a particle in a weakly random velocity field.

(2) weak random fluctuations of Hamilton equations, and

(3) the linear Scrhoedinger equation with a weak random potential.

The role of long range correlation of the random fluctuations will also be discussed.

https://www.ms.u-tokyo.ac.jp/~matano/SDE2012/

**Gregoire Nadin**(CNRS / Paris 6) 15:15-17:15Asymptotic spreading for heterogeneous Fisher-KPP reaction-diffusion equations (JAPANESE)

[ Abstract ]

The solutions of the heterogeneous Fisher-KPP equation associated with compactly supported initial data are known to take off from the unstable steady state 0 and to converge to the steady state 1 for large times. The aim of this lecture is to estimate the speed at which the interface between 0 and 1 spreads.

Using the new notion of generalized principal eigenvalues for non-compact elliptic operators, we will derive such estimates which will be proved to be optimal for several classes of heterogeneity such as periodic, almost periodic or random stationary ergodic ones.

[ Reference URL ]The solutions of the heterogeneous Fisher-KPP equation associated with compactly supported initial data are known to take off from the unstable steady state 0 and to converge to the steady state 1 for large times. The aim of this lecture is to estimate the speed at which the interface between 0 and 1 spreads.

Using the new notion of generalized principal eigenvalues for non-compact elliptic operators, we will derive such estimates which will be proved to be optimal for several classes of heterogeneity such as periodic, almost periodic or random stationary ergodic ones.

https://www.ms.u-tokyo.ac.jp/~matano/SDE2012/

### 2012/06/16

#### Lectures

13:15-17:00 Room #002 (Graduate School of Math. Sci. Bldg.)

A viewpoint of the stochastic analysis in differential equations (JAPANESE)

[ Reference URL ]

https://www.ms.u-tokyo.ac.jp/~matano/SDE2012/

Stochastic (partial) differential equations from a functional analytic point of view (JAPANESE)

[ Reference URL ]

https://www.ms.u-tokyo.ac.jp/~matano/SDE2012/

**Tadahisa Funaki**(University of Tokyo) 13:15-14:45A viewpoint of the stochastic analysis in differential equations (JAPANESE)

[ Reference URL ]

https://www.ms.u-tokyo.ac.jp/~matano/SDE2012/

**Yoshiki Otobe**(Shinshu University) 15:00-17:00Stochastic (partial) differential equations from a functional analytic point of view (JAPANESE)

[ Reference URL ]

https://www.ms.u-tokyo.ac.jp/~matano/SDE2012/

### 2012/06/14

#### Algebraic Geometry Seminar

13:30-15:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Vanishing theorems for perverse sheaves on abelian varieties (ENGLISH)

**Christian Schnell**(IPMU)Vanishing theorems for perverse sheaves on abelian varieties (ENGLISH)

[ Abstract ]

I will describe a few results, due to Kraemer-Weissauer and myself, about perverse sheaves on complex abelian varieties; they are natural generalizations of the generic vanishing theorem of Green-Lazarsfeld.

I will describe a few results, due to Kraemer-Weissauer and myself, about perverse sheaves on complex abelian varieties; they are natural generalizations of the generic vanishing theorem of Green-Lazarsfeld.

### 2012/06/13

#### Number Theory Seminar

16:40-17:40 Room #056 (Graduate School of Math. Sci. Bldg.)

Singular homologies of non-Archimedean analytic spaces and integrals along cycles (JAPANESE)

**Tomoki Mihara**(University of Tokyo)Singular homologies of non-Archimedean analytic spaces and integrals along cycles (JAPANESE)

#### Lectures

17:00-18:00 Room #122 (Graduate School of Math. Sci. Bldg.)

A KPZ equation for zero-range interactions (ENGLISH)

**Sunder Sethuraman**(University of Arizona)A KPZ equation for zero-range interactions (ENGLISH)

[ Abstract ]

We derive a type of KPZ equation, in terms of a martingale problem, as a scaling limit of fluctuation fields in weakly asymmetric zero-range processes. Joint work (in progress) with Milton Jara and Patricia Goncalves.

We derive a type of KPZ equation, in terms of a martingale problem, as a scaling limit of fluctuation fields in weakly asymmetric zero-range processes. Joint work (in progress) with Milton Jara and Patricia Goncalves.

#### Lectures

11:00-15:30 Room #128 (Graduate School of Math. Sci. Bldg.)

Workshop for Quasi-Monte Carlo and Pseudo Random Number Generation (ENGLISH)

[ Reference URL ]

http://sites.google.com/a/craft.titech.ac.jp/workshop-on-qmc-and-prng-2012-utms/

**S. Harase, et. al.**(Tokyo Institute of Technology/JSPS)Workshop for Quasi-Monte Carlo and Pseudo Random Number Generation (ENGLISH)

[ Reference URL ]

http://sites.google.com/a/craft.titech.ac.jp/workshop-on-qmc-and-prng-2012-utms/

### 2012/06/12

#### Tuesday Seminar on Topology

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

Topological interpretation of the quandle cocycle invariants of links (JAPANESE)

**Takefumi Nosaka**(RIMS, Kyoto University, JSPS)Topological interpretation of the quandle cocycle invariants of links (JAPANESE)

[ Abstract ]

Carter et al. introduced many quandle cocycle invariants

combinatorially constructed from link-diagrams. For connected quandles of

finite order, we give a topological meaning of the invariants, without

some torsion parts. Precisely, this invariant equals a sum of "knot

colouring polynomial" and of a Z-equivariant part of the Dijkgraaf-Witten

invariant. Moreover, our approach involves applications to compute "good"

torsion subgroups of the 3-rd quandle homologies and the 2-nd homotopy

groups of rack spaces.

Carter et al. introduced many quandle cocycle invariants

combinatorially constructed from link-diagrams. For connected quandles of

finite order, we give a topological meaning of the invariants, without

some torsion parts. Precisely, this invariant equals a sum of "knot

colouring polynomial" and of a Z-equivariant part of the Dijkgraaf-Witten

invariant. Moreover, our approach involves applications to compute "good"

torsion subgroups of the 3-rd quandle homologies and the 2-nd homotopy

groups of rack spaces.

#### Lie Groups and Representation Theory

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

Conformally invariant systems of differential operators of non-Heisenberg parabolic type (ENGLISH)

**Toshihisa Kubo**(the University of Tokyo)Conformally invariant systems of differential operators of non-Heisenberg parabolic type (ENGLISH)

[ Abstract ]

The wave operator in Minkowski space is a classical example of a conformally invariant differential operator.

Recently, the notion of conformality of one operator has been

generalized by Barchini-Kable-Zierau to systems of differential operators.

Such systems yield homomrophisms between generalized Verma modules. In this talk we build such systems of second-order differential operators in the maximal non-Heisenberg parabolic setting.

If time permits then we will discuss the corresponding homomorphisms between generalized Verma modules.

The wave operator in Minkowski space is a classical example of a conformally invariant differential operator.

Recently, the notion of conformality of one operator has been

generalized by Barchini-Kable-Zierau to systems of differential operators.

Such systems yield homomrophisms between generalized Verma modules. In this talk we build such systems of second-order differential operators in the maximal non-Heisenberg parabolic setting.

If time permits then we will discuss the corresponding homomorphisms between generalized Verma modules.

#### Lectures

09:50-17:10 Room #118 (Graduate School of Math. Sci. Bldg.)

Workshop for Quasi-Monte Carlo and Pseudo Random Number Generation (ENGLISH)

[ Reference URL ]

http://sites.google.com/a/craft.titech.ac.jp/workshop-on-qmc-and-prng-2012-utms/

**Josef Dick, et. al.**(Univ. New South Wales)Workshop for Quasi-Monte Carlo and Pseudo Random Number Generation (ENGLISH)

[ Reference URL ]

http://sites.google.com/a/craft.titech.ac.jp/workshop-on-qmc-and-prng-2012-utms/

### 2012/06/11

#### Kavli IPMU Komaba Seminar

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

Quantum cohomology of flag varieties (ENGLISH)

**Changzheng Li**(Kavli IPMU)Quantum cohomology of flag varieties (ENGLISH)

[ Abstract ]

In this talk, I will give a brief introduction to the quantum cohomology of flag varieties first. Then I will introduce a Z^2-filtration on the quantum cohomology of complete flag varieties. In the end, we will study the quantum Pieri rules for complex/symplectic Grassmannians, as applications of the Z^2-filtration.

In this talk, I will give a brief introduction to the quantum cohomology of flag varieties first. Then I will introduce a Z^2-filtration on the quantum cohomology of complete flag varieties. In the end, we will study the quantum Pieri rules for complex/symplectic Grassmannians, as applications of the Z^2-filtration.

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Differential forms on complete intersections (ENGLISH)

**Damian BROTBEK**(University of Tokyo)Differential forms on complete intersections (ENGLISH)

[ Abstract ]

Brückmann and Rackwitz proved a vanishing result for particular types of differential forms on complete intersection varieties. We will be interested in the cases not covered by their result. In some cases, we will show how the space $H^0(X,S^{m_1}\Omega_X\otimes \cdots \otimes S^{m_k}\Omega_X)$ depends on the equations defining $X$, and in particular we will prove that the theorem of Brückmann and Rackwitz is optimal. The proofs are based on simple, combinatorial, cohomology computations.

Brückmann and Rackwitz proved a vanishing result for particular types of differential forms on complete intersection varieties. We will be interested in the cases not covered by their result. In some cases, we will show how the space $H^0(X,S^{m_1}\Omega_X\otimes \cdots \otimes S^{m_k}\Omega_X)$ depends on the equations defining $X$, and in particular we will prove that the theorem of Brückmann and Rackwitz is optimal. The proofs are based on simple, combinatorial, cohomology computations.

### 2012/06/09

#### Harmonic Analysis Komaba Seminar

13:30-17:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Resent topics on the Cauchy integrals (the works of Muscalu and others) (JAPANESE)

Ill-posedness for the nonlinear Schr\\"odinger equations in one

space dimension

(JAPANESE)

**Yasuo Furuya**(Tokai University) 13:30-15:00Resent topics on the Cauchy integrals (the works of Muscalu and others) (JAPANESE)

**Tsukasa Iwabuchi**(Chuo University) 15:30-17:00Ill-posedness for the nonlinear Schr\\"odinger equations in one

space dimension

(JAPANESE)

[ Abstract ]

In this talk, we consider the Cauchy problems for the nonlinear Schr\\"odinger equations. In particular, we study the ill-posedness by showing that the continuous dependence on initial data does not hold. In the known results, Bejenaru-Tao (2006) considered the problem in the Sobolev spaces $H^s (\\mathbb R)$ and showed the ill-posedness when $s < -1 $. In this talk, we study the ill-posedness in the Besov space for one space dimension and in the Sobolev spaces for two space dimensions.

In this talk, we consider the Cauchy problems for the nonlinear Schr\\"odinger equations. In particular, we study the ill-posedness by showing that the continuous dependence on initial data does not hold. In the known results, Bejenaru-Tao (2006) considered the problem in the Sobolev spaces $H^s (\\mathbb R)$ and showed the ill-posedness when $s < -1 $. In this talk, we study the ill-posedness in the Besov space for one space dimension and in the Sobolev spaces for two space dimensions.

### 2012/06/08

#### GCOE lecture series

14:00-15:30 Room #118 (Graduate School of Math. Sci. Bldg.)

Derived categories and cohomological invariants II (ENGLISH)

**Mihnea Popa**(University of Illinois at Chicago)Derived categories and cohomological invariants II (ENGLISH)

[ Abstract ]

(Abstract for both Parts I and II)

I will discuss results on the derived invariance of various cohomological quantities, like the Hodge numbers, a twisted version of Hochschild cohomology, the Picard variety, and cohomological support loci. I will include a small discussion of current work on orbifolds if time permits.

(Abstract for both Parts I and II)

I will discuss results on the derived invariance of various cohomological quantities, like the Hodge numbers, a twisted version of Hochschild cohomology, the Picard variety, and cohomological support loci. I will include a small discussion of current work on orbifolds if time permits.

#### Kavli IPMU Komaba Seminar

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

Period Integrals and Tautological Systems (ENGLISH)

**Bong Lian**(Brandeis University)Period Integrals and Tautological Systems (ENGLISH)

[ Abstract ]

We develop a global Poincar\\'e residue formula to study

period integrals of families of complex manifolds. For any compact

complex manifold $X$ equipped with a linear system $V^*$ of

generically smooth CY hypersurfaces, the formula expresses period

integrals in terms of a canonical global meromorphic top form on $X$.

Two important ingredients of this construction are the notion of a CY

principal bundle, and a classification of such rank one bundles.

We also generalize the construction to CY and general type complete

intersections. When $X$ is an algebraic manifold having a sufficiently

large automorphism group $G$ and $V^*$ is a linear representation of

$G$, we construct a holonomic D-module that governs the period

integrals. The construction is based in part on the theory of

tautological systems we have developed earlier. The approach allows us

to explicitly describe a Picard-Fuchs type system for complete

intersection varieties of general types, as well as CY, in any Fano

variety, and in a homogeneous space in particular. In addition, the

approach provides a new perspective of old examples such as CY

complete intersections in a toric variety or partial flag variety. The

talk is based on recent joint work with R. Song and S.T. Yau.

We develop a global Poincar\\'e residue formula to study

period integrals of families of complex manifolds. For any compact

complex manifold $X$ equipped with a linear system $V^*$ of

generically smooth CY hypersurfaces, the formula expresses period

integrals in terms of a canonical global meromorphic top form on $X$.

Two important ingredients of this construction are the notion of a CY

principal bundle, and a classification of such rank one bundles.

We also generalize the construction to CY and general type complete

intersections. When $X$ is an algebraic manifold having a sufficiently

large automorphism group $G$ and $V^*$ is a linear representation of

$G$, we construct a holonomic D-module that governs the period

integrals. The construction is based in part on the theory of

tautological systems we have developed earlier. The approach allows us

to explicitly describe a Picard-Fuchs type system for complete

intersection varieties of general types, as well as CY, in any Fano

variety, and in a homogeneous space in particular. In addition, the

approach provides a new perspective of old examples such as CY

complete intersections in a toric variety or partial flag variety. The

talk is based on recent joint work with R. Song and S.T. Yau.

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