## Seminar information archive

Seminar information archive ～02/18｜Today's seminar 02/19 | Future seminars 02/20～

#### Tuesday Seminar on Topology

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Homotopy theory of differential graded Lie algebras (ENGLISH)

**Aniceto Murillo**(Universidad de Malaga)Homotopy theory of differential graded Lie algebras (ENGLISH)

[ Abstract ]

Having as motivation the Deligne's principle by which every deformation functor is governed by a differential graded Lie algebra, we build a homotopy theory for these algebras which extend the classical Quillen approach and let us model any (non necessarily 1-connected nor path connected) complex. This is joint work with Urtzi Buijs, Yves Félix and Daniel Tanré.

Having as motivation the Deligne's principle by which every deformation functor is governed by a differential graded Lie algebra, we build a homotopy theory for these algebras which extend the classical Quillen approach and let us model any (non necessarily 1-connected nor path connected) complex. This is joint work with Urtzi Buijs, Yves Félix and Daniel Tanré.

#### Tuesday Seminar of Analysis

16:50-18:20 Room #126 (Graduate School of Math. Sci. Bldg.)

Scattering matrices and Dirichlet-to-Neumann maps (English)

**Jussi Behrndt**(Graz University of Technology)Scattering matrices and Dirichlet-to-Neumann maps (English)

[ Abstract ]

In this talk we discuss a recent result on the representation of the scattering matrix in terms of an abstract Titchmarsh-Weyl m-function. The general result can be applied to scattering problems for Schrödinger operators with $\delta$-type interactions on curves and hypersurfaces, and scattering problems involving Neumann and Robin realizations of Schrödinger operators on unbounded domains. In both applications we obtain formulas for the corresponding scattering matrices in terms of Dirichlet-to-Neumann maps. This talk is based on joint work with Mark Malamud and Hagen Neidhardt.

In this talk we discuss a recent result on the representation of the scattering matrix in terms of an abstract Titchmarsh-Weyl m-function. The general result can be applied to scattering problems for Schrödinger operators with $\delta$-type interactions on curves and hypersurfaces, and scattering problems involving Neumann and Robin realizations of Schrödinger operators on unbounded domains. In both applications we obtain formulas for the corresponding scattering matrices in terms of Dirichlet-to-Neumann maps. This talk is based on joint work with Mark Malamud and Hagen Neidhardt.

### 2016/04/11

#### Algebraic Geometry Seminar

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

Gysin maps, duality and Schubert classes (English)

https://www.impan.pl/~pragacz/main.htm

**Piotr Pragacz**(Institute of Mathematics, Polish Academy of Sciences )Gysin maps, duality and Schubert classes (English)

[ Abstract ]

We establish a Gysin formula for Schubert bundles

and a strong version of the duality theorem in Schubert calculus

on Grassmann bundles. We then combine them to compute the fundamental

classes of Schubert bundles in Grassmann bundles, which yields a new

proof of the Giambelli formula for vector bundles. This is a joint

work with Lionel Darondeau.

[ Reference URL ]We establish a Gysin formula for Schubert bundles

and a strong version of the duality theorem in Schubert calculus

on Grassmann bundles. We then combine them to compute the fundamental

classes of Schubert bundles in Grassmann bundles, which yields a new

proof of the Giambelli formula for vector bundles. This is a joint

work with Lionel Darondeau.

https://www.impan.pl/~pragacz/main.htm

#### Operator Algebra Seminars

16:４５-18:15 Room #118 (Graduate School of Math. Sci. Bldg.)

On analytic construction of the group three-cocycles (English)

**Ryszard Nest**(Univ. Copenhagen)On analytic construction of the group three-cocycles (English)

#### FMSP Lectures

15:30-17:00 Room #Lecture Hall, Kavli IPMU (Graduate School of Math. Sci. Bldg.)

Lecture 1: Special subspaces in symplectic vector spaces (ENGLISH)

[ Reference URL ]

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Weinstein.pdf

**Alan Weinstein**(University of California, Berkeley)Lecture 1: Special subspaces in symplectic vector spaces (ENGLISH)

[ Reference URL ]

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Weinstein.pdf

#### Seminar on Geometric Complex Analysis

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

Defining the Julia sets on CP^2 (JAPANESE)

**Taro Asuke**(The University of Tokyo)Defining the Julia sets on CP^2 (JAPANESE)

[ Abstract ]

The Julia sets play a central role in the study of complex dynamical systems as well as Kleinian groups where they appear as limit sets. They are also known to be meaningful for complex foliations without singularities, however still not defined for singular ones. In this talk, I will discuss some expected properties of the Julia sets for singular foliations and difficulties for defining them.

The Julia sets play a central role in the study of complex dynamical systems as well as Kleinian groups where they appear as limit sets. They are also known to be meaningful for complex foliations without singularities, however still not defined for singular ones. In this talk, I will discuss some expected properties of the Julia sets for singular foliations and difficulties for defining them.

### 2016/04/08

#### Colloquium

15:30-16:30 Room #123 (Graduate School of Math. Sci. Bldg.)

Using mathematical objects (ENGLISH)

**François Apery**(l'IRMA à Strasbourg)Using mathematical objects (ENGLISH)

[ Abstract ]

Mathematical models are not only teaching tools or pieces of museum but can also have inspiring influence to discovering new truths in mathematics. Through some examples including the Boy surface we will show how models have played a major role in the emergence of new results.

Mathematical models are not only teaching tools or pieces of museum but can also have inspiring influence to discovering new truths in mathematics. Through some examples including the Boy surface we will show how models have played a major role in the emergence of new results.

### 2016/04/05

#### Tuesday Seminar on Topology

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Torsion invariants and representation varieties for non-positively curved cube complexes (JAPANESE)

**Takahiro Kitayama**(The University of Tokyo)Torsion invariants and representation varieties for non-positively curved cube complexes (JAPANESE)

[ Abstract ]

Applications of torsion invariants and representation varieties have been extensively studied for 3-manifolds. Twisted Alexander polynomials are known to detect the Thurston norm and fiberedness of a 3-manifold. Ideal points of character varieties are known to detect essential surfaces in a 3-manifold in a certain extension of Culler-Shalen theory. In view of cubulation of 3-manifolds one can expect that these results naturally extend to a wider framework and, in particular, the case of virtually special cube complexes. We formulate and discuss such analogous questions for non-positively curved cube complexes.

Applications of torsion invariants and representation varieties have been extensively studied for 3-manifolds. Twisted Alexander polynomials are known to detect the Thurston norm and fiberedness of a 3-manifold. Ideal points of character varieties are known to detect essential surfaces in a 3-manifold in a certain extension of Culler-Shalen theory. In view of cubulation of 3-manifolds one can expect that these results naturally extend to a wider framework and, in particular, the case of virtually special cube complexes. We formulate and discuss such analogous questions for non-positively curved cube complexes.

### 2016/04/04

#### Numerical Analysis Seminar

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

Staggered discontinuous Galerkin methods for the incompressible Navier-Stokes equations (English)

**Eric Chung**(Chinese University of Hong Kong)Staggered discontinuous Galerkin methods for the incompressible Navier-Stokes equations (English)

[ Abstract ]

In this talk, we present a staggered discontinuous Galerkin method for the approximation of the incompressible Navier-Stokes equations. Our new method combines the advantages of discontinuous Galerkin methods and staggered meshes, and results in many good properties, namely local and global conservations, optimal convergence and superconvergence through the use of a local postprocessing technique. Another key feature is that our method provides a skew-symmetric discretization of the convection term, with the aim of giving a better conservation property compared with existing discretizations. We also analyze the stability and convergence of the method. In addition, we will present some numerical results to show the performance of the proposed method.

In this talk, we present a staggered discontinuous Galerkin method for the approximation of the incompressible Navier-Stokes equations. Our new method combines the advantages of discontinuous Galerkin methods and staggered meshes, and results in many good properties, namely local and global conservations, optimal convergence and superconvergence through the use of a local postprocessing technique. Another key feature is that our method provides a skew-symmetric discretization of the convection term, with the aim of giving a better conservation property compared with existing discretizations. We also analyze the stability and convergence of the method. In addition, we will present some numerical results to show the performance of the proposed method.

### 2016/03/29

#### Number Theory Seminar

17:30-18:30 Room #002 (Graduate School of Math. Sci. Bldg.)

Motivic cohomology of formal schemes in characteristic p

(English)

**Matthew Morrow**(Universität Bonn)Motivic cohomology of formal schemes in characteristic p

(English)

[ Abstract ]

The logarithmic Hodge-Witt sheaves of Illusie, Milne, Kato, et al. of a smooth variety in characteristic p provide a concrete realisation of its p-adic motivic cohomology, thanks to results of Geisser-Levine and Bloch-Kato-Gabber which link them to algebraic K-theory. I will explain an analogous theory for formal schemes, as well as applications to algebraic cycles, such as a weak Lefschetz theorem for formal Chow groups.

The logarithmic Hodge-Witt sheaves of Illusie, Milne, Kato, et al. of a smooth variety in characteristic p provide a concrete realisation of its p-adic motivic cohomology, thanks to results of Geisser-Levine and Bloch-Kato-Gabber which link them to algebraic K-theory. I will explain an analogous theory for formal schemes, as well as applications to algebraic cycles, such as a weak Lefschetz theorem for formal Chow groups.

### 2016/03/22

#### Colloquium

16:50-17:50 Room #大講義室 (Graduate School of Math. Sci. Bldg.)

Singularities and Jet schemes (JAPANESE)

[ Reference URL ]

http://faculty.ms.u-tokyo.ac.jp/~shihoko/

**Shihoko Ishii**(Graduate School of Mathematical Sciences, University of Tokyo)Singularities and Jet schemes (JAPANESE)

[ Reference URL ]

http://faculty.ms.u-tokyo.ac.jp/~shihoko/

#### FMSP Lectures

10:00-11:00 Room #118 (Graduate School of Math. Sci. Bldg.)

Control and stabilization of degenerate wave equations (ENGLISH)

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Alabau.pdf

**Fatiha Alabau**(Université de Lorraine)Control and stabilization of degenerate wave equations (ENGLISH)

[ Abstract ]

The control of degenerate PDE's arise in many applications such as cloaking, climatology, population genetics, and vision.

For such models, the diffusion operator degenerates on some subset of the spatial domain. We present some recent results on observability, control and stabilization of these equations.

This is a joint work with Piermarco Cannarsa (University di Roma Tor Vergata, Italy) and G\"unter Leugering (Friedrich-Alexander-Universit\"at Erlangen-N\"urnberg, Erlangen, Germany).

[ Reference URL ]The control of degenerate PDE's arise in many applications such as cloaking, climatology, population genetics, and vision.

For such models, the diffusion operator degenerates on some subset of the spatial domain. We present some recent results on observability, control and stabilization of these equations.

This is a joint work with Piermarco Cannarsa (University di Roma Tor Vergata, Italy) and G\"unter Leugering (Friedrich-Alexander-Universit\"at Erlangen-N\"urnberg, Erlangen, Germany).

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Alabau.pdf

#### FMSP Lectures

11:00-12:00 Room #118 (Graduate School of Math. Sci. Bldg.)

Stable determination of coefficients in the dynamical Schrödinger equation in a magnetic field (ENGLISH)

[ Reference URL ]

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Bellassoued.pdf

**Mourad Bellassoued**(Université de Tunis El Manar)Stable determination of coefficients in the dynamical Schrödinger equation in a magnetic field (ENGLISH)

[ Reference URL ]

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Bellassoued.pdf

### 2016/03/19

#### FMSP Lectures

11:00-12:00 Room #370 (Graduate School of Math. Sci. Bldg.)

About the Landis conjecture (ENGLISH)

[ Reference URL ]

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Emanouilov160319.pdf

**Oleg Emanouilov**(Colorado State Univ.)About the Landis conjecture (ENGLISH)

[ Reference URL ]

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Emanouilov160319.pdf

### 2016/03/18

#### FMSP Lectures

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

Control of degenerate parabolic equations: old and new (ENGLISH)

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Cannarsa.pdf

**Piermarco Cannarsa**(Università degli Studi di Roma Tor Vergata)Control of degenerate parabolic equations: old and new (ENGLISH)

[ Abstract ]

This talk will survey the main results obtained in the last fifteen years or so for the null controllability of degenerate parabolic operators.

We shall describe the state of the art for operators which degenerate at the boundary as well as for certain classes of interior degeneracy of hypoelliptic type.

[ Reference URL ]This talk will survey the main results obtained in the last fifteen years or so for the null controllability of degenerate parabolic operators.

We shall describe the state of the art for operators which degenerate at the boundary as well as for certain classes of interior degeneracy of hypoelliptic type.

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Cannarsa.pdf

### 2016/03/16

#### PDE Real Analysis Seminar

16:00-17:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Fluids, vortex membranes, and skew-mean-curvature flows (English)

**Boris Khesin**(University of Toronto)Fluids, vortex membranes, and skew-mean-curvature flows (English)

[ Abstract ]

We show that an approximation of the hydrodynamical Euler equation describes the skew-mean-curvature flow on vortex membranes in any dimension. This generalizes the classical binormal, or vortex filament, equation in 3D. We present a Hamiltonian framework for dynamics of higher-dimensional vortex filaments and vortex sheets as singular 2-forms (Green currents) with support of codimensions 2 and 1, respectively.

We show that an approximation of the hydrodynamical Euler equation describes the skew-mean-curvature flow on vortex membranes in any dimension. This generalizes the classical binormal, or vortex filament, equation in 3D. We present a Hamiltonian framework for dynamics of higher-dimensional vortex filaments and vortex sheets as singular 2-forms (Green currents) with support of codimensions 2 and 1, respectively.

### 2016/03/11

#### Operator Algebra Seminars

16:45-18:15 Room #118 (Graduate School of Math. Sci. Bldg.)

Convergence to the boundary for random walks on discrete quantum groups

(English)

**Bas Jordans**(Univ. Oslo)Convergence to the boundary for random walks on discrete quantum groups

(English)

### 2016/03/08

#### Operator Algebra Seminars

16:45-18:15 Room #118 (Graduate School of Math. Sci. Bldg.)

Equilibrium and non-equilibrium states in Conformal Field Theory (英語)

**Roberto Longo**(Univ. Rome "Tor Vergata")Equilibrium and non-equilibrium states in Conformal Field Theory (英語)

### 2016/02/22

#### FMSP Lectures

17:00-18:00 Room #118 (Graduate School of Math. Sci. Bldg.)

Inverse problems for parabolic operators : comparison of three different approaches (ENGLISH)

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Cristofol.pdf

**Michel Cristofol**(Aix-Marseille Univ.)Inverse problems for parabolic operators : comparison of three different approaches (ENGLISH)

[ Abstract ]

Basing my talk around a toy problem : reconstruction of the potential for a linear parabolic problem, I will recall the two main classical following methods : Dirichlet to Neumann map and Carleman estimates. Then I will present a diﬀerent and recent approach based on pointwise observations and I will underline some points to be improved for each method. Then, I will finish my talk by a review of some results related to this technic of pointwise observations.

[ Reference URL ]Basing my talk around a toy problem : reconstruction of the potential for a linear parabolic problem, I will recall the two main classical following methods : Dirichlet to Neumann map and Carleman estimates. Then I will present a diﬀerent and recent approach based on pointwise observations and I will underline some points to be improved for each method. Then, I will finish my talk by a review of some results related to this technic of pointwise observations.

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Cristofol.pdf

### 2016/02/19

#### FMSP Lectures

10:30-11:30 Room #118 (Graduate School of Math. Sci. Bldg.)

Recovering time-dependent inclusion in heat conductive bodies by a dynamical probe method (ENGLISH)

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Poisson.pdf

**Olivier Poisson**(Aix-Marseille University)Recovering time-dependent inclusion in heat conductive bodies by a dynamical probe method (ENGLISH)

[ Abstract ]

Many articles solve a version of the Calder'on inverse problem for the heat equation. The biggest part of them assume that the unknown conductivity do not depend on time t. But they are very few results concerning the time de- pendent situation, and they are based on the computation of an ansatz for the parabolic equation:

- A reconstruction method of an unknown moving inclusion by a dynamical probe method was performed by Daido-Kang-Nakamura in 2007, but it works for the one dimensional spatial space only,

- An energy estimate for x-multidimensional convex inclusions.

In the talk I will present a dynamical probe method based on special fundamental solutions of the heat equation and basic inequalities :

this approach is very close to the probe method for the elliptic Calderon inverse problem, and does not require regularity of the inclusion.

[ Reference URL ]Many articles solve a version of the Calder'on inverse problem for the heat equation. The biggest part of them assume that the unknown conductivity do not depend on time t. But they are very few results concerning the time de- pendent situation, and they are based on the computation of an ansatz for the parabolic equation:

- A reconstruction method of an unknown moving inclusion by a dynamical probe method was performed by Daido-Kang-Nakamura in 2007, but it works for the one dimensional spatial space only,

- An energy estimate for x-multidimensional convex inclusions.

In the talk I will present a dynamical probe method based on special fundamental solutions of the heat equation and basic inequalities :

this approach is very close to the probe method for the elliptic Calderon inverse problem, and does not require regularity of the inclusion.

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Poisson.pdf

### 2016/02/18

#### FMSP Lectures

10:00-11:00 Room #002 (Graduate School of Math. Sci. Bldg.)

L^2 Extension and its applications: A survey (3) (ENGLISH)

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Varolin.pdf

**Dror Varolin**(Stony Brook)L^2 Extension and its applications: A survey (3) (ENGLISH)

[ Abstract ]

We discuss certain aspects of the theory of L^2 extension, going back to the famous and fundamental work of Ohsawa and Takegoshi, and even further back to work of Donnelly and Fefferman.

The first of three lectures will recall a number of L^2 extension theorems, to some extent following an incomplete history (as I see it), and ending with a sketch of a proof of one of these theorems.

The second lecture will discuss a number of my favorite applications of L^2 extension, from Bergman kernels to deformation invariance of plurigenera.

The third lecture will discuss a new proof of the extension theorem, due to Berndtsson and Lempert. The key tool is a theorem of Berndtsson on the positivity of direct images, which we will review. Berndtsson's theorem permits the introduction of degeneration techniques, whose surprising application to L^2 extension represents one of the most beautiful and fundamental recent breakthroughs in the subject.

[ Reference URL ]We discuss certain aspects of the theory of L^2 extension, going back to the famous and fundamental work of Ohsawa and Takegoshi, and even further back to work of Donnelly and Fefferman.

The first of three lectures will recall a number of L^2 extension theorems, to some extent following an incomplete history (as I see it), and ending with a sketch of a proof of one of these theorems.

The second lecture will discuss a number of my favorite applications of L^2 extension, from Bergman kernels to deformation invariance of plurigenera.

The third lecture will discuss a new proof of the extension theorem, due to Berndtsson and Lempert. The key tool is a theorem of Berndtsson on the positivity of direct images, which we will review. Berndtsson's theorem permits the introduction of degeneration techniques, whose surprising application to L^2 extension represents one of the most beautiful and fundamental recent breakthroughs in the subject.

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Varolin.pdf

#### FMSP Lectures

10:30-11:30 Room #118 (Graduate School of Math. Sci. Bldg.)

Inverse boundary value problem for a hyperbolic equation in an infinite guide (ENGLISH)

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Cristofol.pdf

**Michel Cristofol**(Aix-Marseille Univ.)Inverse boundary value problem for a hyperbolic equation in an infinite guide (ENGLISH)

[ Abstract ]

We consider the multidimensional inverse problem of determining the conductivity coefficient of a hyperbolic equation in an infinite cylindrical domain, from a single boundary observation of the solution. We prove Holder stability with the aid of a Carleman estimate specially designed for hyperbolic waveguides. I will provide numerical simulations in multiple backgrounds.

[ Reference URL ]We consider the multidimensional inverse problem of determining the conductivity coefficient of a hyperbolic equation in an infinite cylindrical domain, from a single boundary observation of the solution. We prove Holder stability with the aid of a Carleman estimate specially designed for hyperbolic waveguides. I will provide numerical simulations in multiple backgrounds.

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Cristofol.pdf

### 2016/02/17

#### FMSP Lectures

10:00-11:00 Room #002 (Graduate School of Math. Sci. Bldg.)

L^2 Extension and its applications: A survey (2) (ENGLISH)

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Varolin.pdf

**Dror Varolin**(Stony Brook)L^2 Extension and its applications: A survey (2) (ENGLISH)

[ Abstract ]

We discuss certain aspects of the theory of L^2 extension, going back to the famous and fundamental work of Ohsawa and Takegoshi, and even further back to work of Donnelly and Fefferman.

The first of three lectures will recall a number of L^2 extension theorems, to some extent following an incomplete history (as I see it), and ending with a sketch of a proof of one of these theorems.

The second lecture will discuss a number of my favorite applications of L^2 extension, from Bergman kernels to deformation invariance of plurigenera.

The third lecture will discuss a new proof of the extension theorem, due to Berndtsson and Lempert. The key tool is a theorem of Berndtsson on the positivity of direct images, which we will review. Berndtsson's theorem permits the introduction of degeneration techniques, whose surprising application to L^2 extension represents one of the most beautiful and fundamental recent breakthroughs in the subject.

[ Reference URL ]We discuss certain aspects of the theory of L^2 extension, going back to the famous and fundamental work of Ohsawa and Takegoshi, and even further back to work of Donnelly and Fefferman.

The first of three lectures will recall a number of L^2 extension theorems, to some extent following an incomplete history (as I see it), and ending with a sketch of a proof of one of these theorems.

The second lecture will discuss a number of my favorite applications of L^2 extension, from Bergman kernels to deformation invariance of plurigenera.

The third lecture will discuss a new proof of the extension theorem, due to Berndtsson and Lempert. The key tool is a theorem of Berndtsson on the positivity of direct images, which we will review. Berndtsson's theorem permits the introduction of degeneration techniques, whose surprising application to L^2 extension represents one of the most beautiful and fundamental recent breakthroughs in the subject.

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Varolin.pdf

### 2016/02/16

#### Tuesday Seminar on Topology

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

String Topology, Euler Class and TNCZ free loop fibrations (ENGLISH)

**Luc Menichi**(University of Angers)String Topology, Euler Class and TNCZ free loop fibrations (ENGLISH)

[ Abstract ]

Let $M$ be a connected, closed oriented manifold.

Chas and Sullivan have defined a loop product and a loop coproduct on

$H_*(LM;¥mathbb{F})$, the homology of the

free loops on $M$ with coefficients in the field $¥mathbb{F}$.

By studying this loop coproduct, I will show that if the free loop

fibration

$¥Omega M¥buildrel{i}¥over¥hookrightarrow

LM¥buildrel{ev}¥over¥twoheadrightarrow M$

is homologically trivial, i.e. $i^*:H^*(LM;¥mathbb{F})¥twoheadrightarrow

H^*(¥Omega M;¥mathbb{F})$ is onto,

then the Euler characteristic of $M$ is divisible by the characteristic

of the field $¥mathbb{F}$

(or $M$ is a point).

Let $M$ be a connected, closed oriented manifold.

Chas and Sullivan have defined a loop product and a loop coproduct on

$H_*(LM;¥mathbb{F})$, the homology of the

free loops on $M$ with coefficients in the field $¥mathbb{F}$.

By studying this loop coproduct, I will show that if the free loop

fibration

$¥Omega M¥buildrel{i}¥over¥hookrightarrow

LM¥buildrel{ev}¥over¥twoheadrightarrow M$

is homologically trivial, i.e. $i^*:H^*(LM;¥mathbb{F})¥twoheadrightarrow

H^*(¥Omega M;¥mathbb{F})$ is onto,

then the Euler characteristic of $M$ is divisible by the characteristic

of the field $¥mathbb{F}$

(or $M$ is a point).

#### FMSP Lectures

10:00-11:00 Room #002 (Graduate School of Math. Sci. Bldg.)

L^2 Extension and its applications: A survey (1) (ENGLISH)

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Varolin.pdf

**Dror Varolin**(Stony Brook)L^2 Extension and its applications: A survey (1) (ENGLISH)

[ Abstract ]

We discuss certain aspects of the theory of L^2 extension, going back to the famous and fundamental work of Ohsawa and Takegoshi, and even further back to work of Donnelly and Fefferman.

The first of three lectures will recall a number of L^2 extension theorems, to some extent following an incomplete history (as I see it), and ending with a sketch of a proof of one of these theorems.

The second lecture will discuss a number of my favorite applications of L^2 extension, from Bergman kernels to deformation invariance of plurigenera.

The third lecture will discuss a new proof of the extension theorem, due to Berndtsson and Lempert. The key tool is a theorem of Berndtsson on the positivity of direct images, which we will review. Berndtsson's theorem permits the introduction of degeneration techniques, whose surprising application to L^2 extension represents one of the most beautiful and fundamental recent breakthroughs in the subject.

[ Reference URL ]We discuss certain aspects of the theory of L^2 extension, going back to the famous and fundamental work of Ohsawa and Takegoshi, and even further back to work of Donnelly and Fefferman.

The first of three lectures will recall a number of L^2 extension theorems, to some extent following an incomplete history (as I see it), and ending with a sketch of a proof of one of these theorems.

The second lecture will discuss a number of my favorite applications of L^2 extension, from Bergman kernels to deformation invariance of plurigenera.

The third lecture will discuss a new proof of the extension theorem, due to Berndtsson and Lempert. The key tool is a theorem of Berndtsson on the positivity of direct images, which we will review. Berndtsson's theorem permits the introduction of degeneration techniques, whose surprising application to L^2 extension represents one of the most beautiful and fundamental recent breakthroughs in the subject.

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Varolin.pdf

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