## FMSPレクチャーズ

過去の記録 ～04/19｜次回の予定｜今後の予定 04/20～

担当者 | 河野俊丈 |
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**過去の記録**

### 2020年02月14日(金)

17:00-18:00 数理科学研究科棟(駒場) 128号室

Bilinear control for evolution equations of parabolic type (ENGLISH)

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Cannarsa200214.pdf

**Piermarco Cannarsa 氏**(University of Rome Tor Vergata)Bilinear control for evolution equations of parabolic type (ENGLISH)

[ 講演概要 ]

Recently, in a series of joint papers with F. Alabau-Boussouira and C. Urbani, I have studied the response of an evolution equation on a Hilbert space to the action of a bilinear control. As is well-known, a bilinear control is a scalar function of time multiplying one of the coefficient of the equation (usually, a lower order term). Therefore, this is a nonlinear control problem, even if the equation is linear in the state variable.

For such a problem, exact controllability is out of question, due to a well-known negative result by Ball, Marsden, and Slemrod back in the 80’s.

In this talk, equations of parabolic type will be considered, meaning that the infinitesimal generator - of the strongly continuous semigroup which drives the system - is assumed to be a self-adjoint accretive operator. It will be explained how, under some conditions relating the spectrum of the generator to the control coefficient, one can locally stabilise the system to the solution associated with the ground state at a doubly exponential speed, or even attain such a ground-state solution in finite time. Applications to concrete parabolic problems will also be provided.

[ 参考URL ]Recently, in a series of joint papers with F. Alabau-Boussouira and C. Urbani, I have studied the response of an evolution equation on a Hilbert space to the action of a bilinear control. As is well-known, a bilinear control is a scalar function of time multiplying one of the coefficient of the equation (usually, a lower order term). Therefore, this is a nonlinear control problem, even if the equation is linear in the state variable.

For such a problem, exact controllability is out of question, due to a well-known negative result by Ball, Marsden, and Slemrod back in the 80’s.

In this talk, equations of parabolic type will be considered, meaning that the infinitesimal generator - of the strongly continuous semigroup which drives the system - is assumed to be a self-adjoint accretive operator. It will be explained how, under some conditions relating the spectrum of the generator to the control coefficient, one can locally stabilise the system to the solution associated with the ground state at a doubly exponential speed, or even attain such a ground-state solution in finite time. Applications to concrete parabolic problems will also be provided.

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Cannarsa200214.pdf

### 2020年01月22日(水)

17:00-18:00 数理科学研究科棟(駒場) 128号室

Complex principal type operators in inverse conductivity problem (ENGLISH)

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

**Samuli Siltanen 氏**(University of Helsinki)Complex principal type operators in inverse conductivity problem (ENGLISH)

[ 講演概要 ]

Stroke is a leading cause of death all around the world. There are two main types of stroke: ischemic (blood clot preventing blood flow to a part of the brain) and hemorrhagic (bleeding in the brain). The symptoms are the same, but treatments very different. A portable "stroke classifier" would be a life-saving equipment to have in ambulances, but so far it does not exist. Electrical Impedance Tomography (EIT) is a promising and harmless imaging method for stroke classification. In EIT one attempts to recover the electric conductivity inside a domain from electric boundary measurements. This is a nonlinear and ill-posed inverse problem. The so-called Complex Geometric Optics (CGO) solutions have proven to be a useful computational tool for reconstruction tasks in EIT. A new property of CGO solutions is presented, showing that a one-dimensional Fourier transform in the spectral variable provides a connection to parallel-beam Xray tomography of the conductivity. One of the consequences of this “nonlinear Fourier slice theorem” is a novel capability to recover inclusions within inclusions in EIT. In practical imaging, measurement noise causes strong blurring in the recovered profile functions. However, machine learning algorithms can be combined with the nonlinear PDE techniques in a fruitful way. As an example, simulated strokes are classified into hemorrhagic and ischemic using EIT measurements.

[ 参考URL ]Stroke is a leading cause of death all around the world. There are two main types of stroke: ischemic (blood clot preventing blood flow to a part of the brain) and hemorrhagic (bleeding in the brain). The symptoms are the same, but treatments very different. A portable "stroke classifier" would be a life-saving equipment to have in ambulances, but so far it does not exist. Electrical Impedance Tomography (EIT) is a promising and harmless imaging method for stroke classification. In EIT one attempts to recover the electric conductivity inside a domain from electric boundary measurements. This is a nonlinear and ill-posed inverse problem. The so-called Complex Geometric Optics (CGO) solutions have proven to be a useful computational tool for reconstruction tasks in EIT. A new property of CGO solutions is presented, showing that a one-dimensional Fourier transform in the spectral variable provides a connection to parallel-beam Xray tomography of the conductivity. One of the consequences of this “nonlinear Fourier slice theorem” is a novel capability to recover inclusions within inclusions in EIT. In practical imaging, measurement noise causes strong blurring in the recovered profile functions. However, machine learning algorithms can be combined with the nonlinear PDE techniques in a fruitful way. As an example, simulated strokes are classified into hemorrhagic and ischemic using EIT measurements.

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

### 2019年12月10日(火)

17:00-18:00 数理科学研究科棟(駒場) 118号室

A priori and a posteriori error estimation for solutions of ill-posed problems (ENGLISH)

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

**Anatoly G. Yagola 氏**(Lomonosov Moscow State University)A priori and a posteriori error estimation for solutions of ill-posed problems (ENGLISH)

[ 講演概要 ]

In order to calculate a priori or a posteriori error estimates for solutions of an ill-posed operator equation with an injective operator we need to describe a set of approximate solutions that contains an exact solution. After that we have to calculate a diameter of this set or maximal distance from a fixed approximate solution to any element of this set. I will describe three approaches for constructing error estimates and also their practical applications.

[ 参考URL ]In order to calculate a priori or a posteriori error estimates for solutions of an ill-posed operator equation with an injective operator we need to describe a set of approximate solutions that contains an exact solution. After that we have to calculate a diameter of this set or maximal distance from a fixed approximate solution to any element of this set. I will describe three approaches for constructing error estimates and also their practical applications.

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

### 2019年10月31日(木)

13:00-15:05 数理科学研究科棟(駒場) 002号室

全6回：9/26～10/31の毎週（木）13:00-15:05

Topic on minimal submanifolds (6/6) (ENGLISH)

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

全6回：9/26～10/31の毎週（木）13:00-15:05

**Chung-jun Tsai 氏**(National Taiwan University)Topic on minimal submanifolds (6/6) (ENGLISH)

[ 講演概要 ]

The main theme of these lectures will be theory about minimal submanifolds, which are higher dimensional generalizations of geodesics. A naive motivation is that one tries to understand the geometry from its special submanifolds (minimal, etc.).

For minimal submanifolds, the equations are no longer ODEs, but elliptic PDEs. This increases the difficulties. The study are very good examples for the application of methods from PDEs and calculus of variations. We will try to explain some important results in this theory, which stimulate many of the researches today.

Here are some specific materials we plan to cover: Simon’s work based on the second variational formula, Sacks - Uhlenback theorem on the existence of minimal 2-spheres, the theory of stable minimal hypersurfaces by Schoen-Simon-Yau.

[ 参考URL ]The main theme of these lectures will be theory about minimal submanifolds, which are higher dimensional generalizations of geodesics. A naive motivation is that one tries to understand the geometry from its special submanifolds (minimal, etc.).

For minimal submanifolds, the equations are no longer ODEs, but elliptic PDEs. This increases the difficulties. The study are very good examples for the application of methods from PDEs and calculus of variations. We will try to explain some important results in this theory, which stimulate many of the researches today.

Here are some specific materials we plan to cover: Simon’s work based on the second variational formula, Sacks - Uhlenback theorem on the existence of minimal 2-spheres, the theory of stable minimal hypersurfaces by Schoen-Simon-Yau.

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

### 2019年10月24日(木)

13:00-15:05 数理科学研究科棟(駒場) 002号室

全6回：9/26～10/31の毎週（木）13:00-15:05

Topic on minimal submanifolds (5/6) (ENGLISH)

全6回：9/26～10/31の毎週（木）13:00-15:05

**Chung-jun Tsai 氏**(National Taiwan University)Topic on minimal submanifolds (5/6) (ENGLISH)

[ 講演概要 ]

The main theme of these lectures will be theory about minimal submanifolds, which are higher dimensional generalizations of geodesics. A naive motivation is that one tries to understand the geometry from its special submanifolds (minimal, etc.).

For minimal submanifolds, the equations are no longer ODEs, but elliptic PDEs. This increases the difficulties. The study are very good examples for the application of methods from PDEs and calculus of variations. We will try to explain some important results in this theory, which stimulate many of the researches today.

Here are some specific materials we plan to cover: Simon’s work based on the second variational formula, Sacks - Uhlenback theorem on the existence of minimal 2-spheres, the theory of stable minimal hypersurfaces by Schoen-Simon-Yau.

The main theme of these lectures will be theory about minimal submanifolds, which are higher dimensional generalizations of geodesics. A naive motivation is that one tries to understand the geometry from its special submanifolds (minimal, etc.).

For minimal submanifolds, the equations are no longer ODEs, but elliptic PDEs. This increases the difficulties. The study are very good examples for the application of methods from PDEs and calculus of variations. We will try to explain some important results in this theory, which stimulate many of the researches today.

Here are some specific materials we plan to cover: Simon’s work based on the second variational formula, Sacks - Uhlenback theorem on the existence of minimal 2-spheres, the theory of stable minimal hypersurfaces by Schoen-Simon-Yau.

### 2019年10月17日(木)

13:00-15:05 数理科学研究科棟(駒場) 002号室

全6回：9/26～10/31の毎週（木）13:00-15:05

Topic on minimal submanifolds (4/6) (ENGLISH)

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

全6回：9/26～10/31の毎週（木）13:00-15:05

**Chung-jun Tsai 氏**(National Taiwan University)Topic on minimal submanifolds (4/6) (ENGLISH)

[ 講演概要 ]

The main theme of these lectures will be theory about minimal submanifolds, which are higher dimensional generalizations of geodesics. A naive motivation is that one tries to understand the geometry from its special submanifolds (minimal, etc.).

For minimal submanifolds, the equations are no longer ODEs, but elliptic PDEs. This increases the difficulties. The study are very good examples for the application of methods from PDEs and calculus of variations. We will try to explain some important results in this theory, which stimulate many of the researches today.

Here are some specific materials we plan to cover: Simon’s work based on the second variational formula, Sacks - Uhlenback theorem on the existence of minimal 2-spheres, the theory of stable minimal hypersurfaces by Schoen-Simon-Yau.

[ 参考URL ]The main theme of these lectures will be theory about minimal submanifolds, which are higher dimensional generalizations of geodesics. A naive motivation is that one tries to understand the geometry from its special submanifolds (minimal, etc.).

For minimal submanifolds, the equations are no longer ODEs, but elliptic PDEs. This increases the difficulties. The study are very good examples for the application of methods from PDEs and calculus of variations. We will try to explain some important results in this theory, which stimulate many of the researches today.

Here are some specific materials we plan to cover: Simon’s work based on the second variational formula, Sacks - Uhlenback theorem on the existence of minimal 2-spheres, the theory of stable minimal hypersurfaces by Schoen-Simon-Yau.

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

### 2019年10月10日(木)

13:00-15:05 数理科学研究科棟(駒場) 002号室

全6回：9/26～10/31の毎週（木）13:00-15:05

Topic on minimal submanifolds (3/6) (ENGLISH)

[ 講演概要 ]

The main theme of these lectures will be theory about minimal submanifolds, which are higher dimensional generalizations of geodesics. A naive motivation is that one tries to understand the geometry from its special submanifolds (minimal, etc.).

For minimal submanifolds, the equations are no longer ODEs, but elliptic PDEs. This increases the difficulties. The study are very good examples for the application of methods from PDEs and calculus of variations. We will try to explain some important results in this theory, which stimulate many of the researches today.

Here are some specific materials we plan to cover: Simon’s work based on the second variational formula, Sacks - Uhlenback theorem on the existence of minimal 2-spheres, the theory of stable minimal hypersurfaces by Schoen-Simon-Yau.
[ 参考URL ]

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

全6回：9/26～10/31の毎週（木）13:00-15:05

**Chung-jun Tsai 氏**(National Taiwan University)Topic on minimal submanifolds (3/6) (ENGLISH)

The main theme of these lectures will be theory about minimal submanifolds, which are higher dimensional generalizations of geodesics. A naive motivation is that one tries to understand the geometry from its special submanifolds (minimal, etc.).

For minimal submanifolds, the equations are no longer ODEs, but elliptic PDEs. This increases the difficulties. The study are very good examples for the application of methods from PDEs and calculus of variations. We will try to explain some important results in this theory, which stimulate many of the researches today.

Here are some specific materials we plan to cover: Simon’s work based on the second variational formula, Sacks - Uhlenback theorem on the existence of minimal 2-spheres, the theory of stable minimal hypersurfaces by Schoen-Simon-Yau.

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

### 2019年10月03日(木)

13:00-15:05 数理科学研究科棟(駒場) 002号室

全6回：9/26～10/31の毎週（木）13:00-15:05

Topic on minimal submanifolds (2/6) (ENGLISH)

[ 講演概要 ]

The main theme of these lectures will be theory about minimal submanifolds, which are higher dimensional generalizations of geodesics. A naive motivation is that one tries to understand the geometry from its special submanifolds (minimal, etc.).

For minimal submanifolds, the equations are no longer ODEs, but elliptic PDEs. This increases the difficulties. The study are very good examples for the application of methods from PDEs and calculus of variations. We will try to explain some important results in this theory, which stimulate many of the researches today.

Here are some specific materials we plan to cover: Simon’s work based on the second variational formula, Sacks - Uhlenback theorem on the existence of minimal 2-spheres, the theory of stable minimal hypersurfaces by Schoen-Simon-Yau.
[ 参考URL ]

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

全6回：9/26～10/31の毎週（木）13:00-15:05

**Chung-jun Tsai 氏**(National Taiwan University)Topic on minimal submanifolds (2/6) (ENGLISH)

The main theme of these lectures will be theory about minimal submanifolds, which are higher dimensional generalizations of geodesics. A naive motivation is that one tries to understand the geometry from its special submanifolds (minimal, etc.).

For minimal submanifolds, the equations are no longer ODEs, but elliptic PDEs. This increases the difficulties. The study are very good examples for the application of methods from PDEs and calculus of variations. We will try to explain some important results in this theory, which stimulate many of the researches today.

Here are some specific materials we plan to cover: Simon’s work based on the second variational formula, Sacks - Uhlenback theorem on the existence of minimal 2-spheres, the theory of stable minimal hypersurfaces by Schoen-Simon-Yau.

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

### 2019年09月26日(木)

13:00-15:05 数理科学研究科棟(駒場) 002号室

全6回：9/26～10/31の毎週（木）13:00-15:05

Topic on minimal submanifolds (1/6) (ENGLISH)

[ 講演概要 ]

The main theme of these lectures will be theory about minimal submanifolds, which are higher dimensional generalizations of geodesics. A naive motivation is that one tries to understand the geometry from its special submanifolds (minimal, etc.).

For minimal submanifolds, the equations are no longer ODEs, but elliptic PDEs. This increases the difficulties. The study are very good examples for the application of methods from PDEs and calculus of variations. We will try to explain some important results in this theory, which stimulate many of the researches today.

Here are some specific materials we plan to cover: Simon’s work based on the second variational formula, Sacks - Uhlenback theorem on the existence of minimal 2-spheres, the theory of stable minimal hypersurfaces by Schoen-Simon-Yau.
[ 参考URL ]

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

全6回：9/26～10/31の毎週（木）13:00-15:05

**Chung-jun Tsai 氏**(National Taiwan University)Topic on minimal submanifolds (1/6) (ENGLISH)

The main theme of these lectures will be theory about minimal submanifolds, which are higher dimensional generalizations of geodesics. A naive motivation is that one tries to understand the geometry from its special submanifolds (minimal, etc.).

For minimal submanifolds, the equations are no longer ODEs, but elliptic PDEs. This increases the difficulties. The study are very good examples for the application of methods from PDEs and calculus of variations. We will try to explain some important results in this theory, which stimulate many of the researches today.

Here are some specific materials we plan to cover: Simon’s work based on the second variational formula, Sacks - Uhlenback theorem on the existence of minimal 2-spheres, the theory of stable minimal hypersurfaces by Schoen-Simon-Yau.

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

### 2019年05月15日(水)

17:30-18:30 数理科学研究科棟(駒場) 122号室

*The date and room have changed.

'Oumuamua, the Gömböc and the Pebbles of Mars (ENGLISH)

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

*The date and room have changed.

**Gábor Domokos 氏**(Hungarian Academy of Sciences/Budapest University of Technology and Economics)'Oumuamua, the Gömböc and the Pebbles of Mars (ENGLISH)

[ 講演概要 ]

In this talk I will concentrate on two examples from planetary science, which made the headlines in recent years to highlight the power and significance of nonlinear geometric partial differential equations (PDEs) explaining puzzles presented by Nature. One key link between PDE theory of shape evolution and natural phenomena is the Gömböc, the first mono-monostatic object whose existence was first conjectured by V.I. Arnold in 1995. I will explain the connection and illustrate the process how mathematical models of Nature may be identified.

[ 参考URL ]In this talk I will concentrate on two examples from planetary science, which made the headlines in recent years to highlight the power and significance of nonlinear geometric partial differential equations (PDEs) explaining puzzles presented by Nature. One key link between PDE theory of shape evolution and natural phenomena is the Gömböc, the first mono-monostatic object whose existence was first conjectured by V.I. Arnold in 1995. I will explain the connection and illustrate the process how mathematical models of Nature may be identified.

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

### 2019年05月15日(水)

15:00-17:20 数理科学研究科棟(駒場) 122号室

Part 1 : Categorical analogues of surface singularities

Part 2 : Prismatic Homology (ENGLISH)

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

**J. Scott Carter 氏**(University of South Alabama / Osaka City University)Part 1 : Categorical analogues of surface singularities

Part 2 : Prismatic Homology (ENGLISH)

[ 講演概要 ]

Part 1 :

Isotopy classes of surfaces that are embedded in 3-space can be described as a free 4-category that has one object and one weakly invertible arrow. That description coincides with a fundamental higher homotopy group. The surface singularities that correspond to cusps and optimal points on folds can be used to develop categorical analogues of swallow-tails and horizontal cusps. In this talk, the 4-category will be constructed from the ground up, and the general structure will be described.

Part 2 :

A qualgebra is a set that has two binary operations whose relationships to each other are similar to the relations between group multiplication and conjugation. The axioms themselves are described in terms of isotopies of knotted trivalent graphs and the handle-body knots that are represented. The moves naturally live in prisms. By using a generalization of the tensor product of chain complexes, a homology theory is presented that encapsulates these axioms and the higher order relations between them. We show how to use this homology theory to give a solution a system of tensor equations related to the Yang-Baxter relation.

[ 参考URL ]Part 1 :

Isotopy classes of surfaces that are embedded in 3-space can be described as a free 4-category that has one object and one weakly invertible arrow. That description coincides with a fundamental higher homotopy group. The surface singularities that correspond to cusps and optimal points on folds can be used to develop categorical analogues of swallow-tails and horizontal cusps. In this talk, the 4-category will be constructed from the ground up, and the general structure will be described.

Part 2 :

A qualgebra is a set that has two binary operations whose relationships to each other are similar to the relations between group multiplication and conjugation. The axioms themselves are described in terms of isotopies of knotted trivalent graphs and the handle-body knots that are represented. The moves naturally live in prisms. By using a generalization of the tensor product of chain complexes, a homology theory is presented that encapsulates these axioms and the higher order relations between them. We show how to use this homology theory to give a solution a system of tensor equations related to the Yang-Baxter relation.

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

### 2018年10月31日(水)

15:00-16:30 数理科学研究科棟(駒場) 122号室

K-THEORY AND THE DIRAC OPERATOR (4/4)

Lecture 4. BEYOND ELLIPTICITY or K-HOMOLOGY AND INDEX THEORY ON CONTACT MANIFOLDS (ENGLISH)

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

**Paul Baum 氏**(The Pennsylvania State University)K-THEORY AND THE DIRAC OPERATOR (4/4)

Lecture 4. BEYOND ELLIPTICITY or K-HOMOLOGY AND INDEX THEORY ON CONTACT MANIFOLDS (ENGLISH)

[ 講演概要 ]

K-homology is the dual theory to K-theory. The BD (Baum-Douglas) isomorphism of Atiyah-Kasparov K-homology and K-cycle K-homology provides a framework within which the Atiyah-Singer index theorem can be extended to certain differential operators which are hypoelliptic but not elliptic. This talk will consider such a class of differential operators on compact contact manifolds. These operators have been studied by a number of mathematicians (e.g. C.Epstein and R.Melrose).

Operators with similar analytical properties have also been studied (e.g. by Alain Connes and Henri Moscovici --- also Michel Hilsum and Georges Skandalis). Working within the BD framework, the index problem will be solved for these differential operators on compact contact manifolds.

This is joint work with Erik van Erp.

[ 参考URL ]K-homology is the dual theory to K-theory. The BD (Baum-Douglas) isomorphism of Atiyah-Kasparov K-homology and K-cycle K-homology provides a framework within which the Atiyah-Singer index theorem can be extended to certain differential operators which are hypoelliptic but not elliptic. This talk will consider such a class of differential operators on compact contact manifolds. These operators have been studied by a number of mathematicians (e.g. C.Epstein and R.Melrose).

Operators with similar analytical properties have also been studied (e.g. by Alain Connes and Henri Moscovici --- also Michel Hilsum and Georges Skandalis). Working within the BD framework, the index problem will be solved for these differential operators on compact contact manifolds.

This is joint work with Erik van Erp.

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

### 2018年10月29日(月)

15:00-16:30 数理科学研究科棟(駒場) 117号室

K-THEORY AND THE DIRAC OPERATOR (3/4)

Lecture 3. THE RIEMANN-ROCH THEOREM (ENGLISH)

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

**Paul Baum 氏**(The Pennsylvania State University)K-THEORY AND THE DIRAC OPERATOR (3/4)

Lecture 3. THE RIEMANN-ROCH THEOREM (ENGLISH)

[ 講演概要 ]

Topics in this talk :

1. Classical Riemann-Roch

2. Hirzebruch-Riemann-Roch (HRR)

3. Grothendieck-Riemann-Roch (GRR)

4. RR for possibly singular varieties (Baum-Fulton-MacPherson)

[ 参考URL ]Topics in this talk :

1. Classical Riemann-Roch

2. Hirzebruch-Riemann-Roch (HRR)

3. Grothendieck-Riemann-Roch (GRR)

4. RR for possibly singular varieties (Baum-Fulton-MacPherson)

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

### 2018年10月24日(水)

15:00-16:30 数理科学研究科棟(駒場) 123号室

K-THEORY AND THE DIRAC OPERATOR (2/4)

Lecture 2. THE DIRAC OPERATOR (ENGLISH)

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

**Paul Baum 氏**(The Pennsylvania State University)K-THEORY AND THE DIRAC OPERATOR (2/4)

Lecture 2. THE DIRAC OPERATOR (ENGLISH)

[ 講演概要 ]

The Dirac operator of R^n will be defined. This is a first order elliptic differential operator with constant coefficients.

Next, the class of differentiable manifolds which come equipped with an order one differential operator which (at the symbol level)is locally isomorphic to the Dirac operator of R^n will be considered. These are the Spin-c manifolds.

Spin-c is slightly stronger than oriented, so Spin-c can be viewed as "oriented plus epsilon". Most of the oriented manifolds that occur in practice are Spin-c. The Dirac operator of a closed Spin-c manifold is the basic example for the Hirzebruch-Riemann-Roch theorem and the Atiyah-Singer index theorem.

[ 参考URL ]The Dirac operator of R^n will be defined. This is a first order elliptic differential operator with constant coefficients.

Next, the class of differentiable manifolds which come equipped with an order one differential operator which (at the symbol level)is locally isomorphic to the Dirac operator of R^n will be considered. These are the Spin-c manifolds.

Spin-c is slightly stronger than oriented, so Spin-c can be viewed as "oriented plus epsilon". Most of the oriented manifolds that occur in practice are Spin-c. The Dirac operator of a closed Spin-c manifold is the basic example for the Hirzebruch-Riemann-Roch theorem and the Atiyah-Singer index theorem.

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

### 2018年10月22日(月)

15:00-16:30 数理科学研究科棟(駒場) 123号室

K-THEORY AND THE DIRAC OPERATOR (1/4)

Lecture 1. WHAT IS K-THEORY AND WHAT IS IT GOOD FOR? (ENGLISH)

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

**Paul Baum 氏**(The Pennsylvania State University)K-THEORY AND THE DIRAC OPERATOR (1/4)

Lecture 1. WHAT IS K-THEORY AND WHAT IS IT GOOD FOR? (ENGLISH)

[ 講演概要 ]

This talk will consist of four points.

1. The basic definition of K-theory

2. A brief history of K-theory

3. Algebraic versus topological K-theory

4. The unity of K-theory

[ 参考URL ]This talk will consist of four points.

1. The basic definition of K-theory

2. A brief history of K-theory

3. Algebraic versus topological K-theory

4. The unity of K-theory

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

### 2018年07月25日(水)

10:15-12:15 数理科学研究科棟(駒場) 118号室

集中講義901-118「数物先端科学X」として行われます（7/18,20,23,24,25の全5回）。

Singular hermitian metrics and morphisms to abelian varieties (ENGLISH)

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

集中講義901-118「数物先端科学X」として行われます（7/18,20,23,24,25の全5回）。

**Christian Schnell 氏**(Stony Book University)Singular hermitian metrics and morphisms to abelian varieties (ENGLISH)

[ 講演概要 ]

Consider a morphism from a smooth projective variety to an abelian variety (over the field of complex numbers). After reviewing what is known about the pushforward of the canonical bundle under such a morphism, we will try to extend these results to the case of pluricanonical bundles (= the tensor powers of the canonical bundle). Along the way, we will learn about three important tools: generic vanishing theory; Viehweg's cyclic covering trick; and some new results from complex analysis about metrics with singularities. As an application, we will discuss the proof of Iitaka's conjecture (about the subadditivity of the Kodaira dimension in algebraic fiber spaces) over abelian varieties, following Cao and Paun.

[ 参考URL ]Consider a morphism from a smooth projective variety to an abelian variety (over the field of complex numbers). After reviewing what is known about the pushforward of the canonical bundle under such a morphism, we will try to extend these results to the case of pluricanonical bundles (= the tensor powers of the canonical bundle). Along the way, we will learn about three important tools: generic vanishing theory; Viehweg's cyclic covering trick; and some new results from complex analysis about metrics with singularities. As an application, we will discuss the proof of Iitaka's conjecture (about the subadditivity of the Kodaira dimension in algebraic fiber spaces) over abelian varieties, following Cao and Paun.

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

### 2018年07月24日(火)

10:15-12:15 数理科学研究科棟(駒場) 118号室

集中講義901-118「数物先端科学X」として行われます（7/18,20,23,24,25の全5回）。

Singular hermitian metrics and morphisms to abelian varieties (ENGLISH)

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

集中講義901-118「数物先端科学X」として行われます（7/18,20,23,24,25の全5回）。

**Christian Schnell 氏**(Stony Book University)Singular hermitian metrics and morphisms to abelian varieties (ENGLISH)

[ 講演概要 ]

Consider a morphism from a smooth projective variety to an abelian variety (over the field of complex numbers). After reviewing what is known about the pushforward of the canonical bundle under such a morphism, we will try to extend these results to the case of pluricanonical bundles (= the tensor powers of the canonical bundle). Along the way, we will learn about three important tools: generic vanishing theory; Viehweg's cyclic covering trick; and some new results from complex analysis about metrics with singularities. As an application, we will discuss the proof of Iitaka's conjecture (about the subadditivity of the Kodaira dimension in algebraic fiber spaces) over abelian varieties, following Cao and Paun.

[ 参考URL ]Consider a morphism from a smooth projective variety to an abelian variety (over the field of complex numbers). After reviewing what is known about the pushforward of the canonical bundle under such a morphism, we will try to extend these results to the case of pluricanonical bundles (= the tensor powers of the canonical bundle). Along the way, we will learn about three important tools: generic vanishing theory; Viehweg's cyclic covering trick; and some new results from complex analysis about metrics with singularities. As an application, we will discuss the proof of Iitaka's conjecture (about the subadditivity of the Kodaira dimension in algebraic fiber spaces) over abelian varieties, following Cao and Paun.

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

### 2018年07月23日(月)

10:15-12:15 数理科学研究科棟(駒場) 118号室

集中講義901-118「数物先端科学X」として行われます（7/18,20,23,24,25の全5回）。

Singular hermitian metrics and morphisms to abelian varieties (ENGLISH)

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

集中講義901-118「数物先端科学X」として行われます（7/18,20,23,24,25の全5回）。

**Christian Schnell 氏**(Stony Book University)Singular hermitian metrics and morphisms to abelian varieties (ENGLISH)

[ 講演概要 ]

Consider a morphism from a smooth projective variety to an abelian variety (over the field of complex numbers). After reviewing what is known about the pushforward of the canonical bundle under such a morphism, we will try to extend these results to the case of pluricanonical bundles (= the tensor powers of the canonical bundle). Along the way, we will learn about three important tools: generic vanishing theory; Viehweg's cyclic covering trick; and some new results from complex analysis about metrics with singularities. As an application, we will discuss the proof of Iitaka's conjecture (about the subadditivity of the Kodaira dimension in algebraic fiber spaces) over abelian varieties, following Cao and Paun.

[ 参考URL ]Consider a morphism from a smooth projective variety to an abelian variety (over the field of complex numbers). After reviewing what is known about the pushforward of the canonical bundle under such a morphism, we will try to extend these results to the case of pluricanonical bundles (= the tensor powers of the canonical bundle). Along the way, we will learn about three important tools: generic vanishing theory; Viehweg's cyclic covering trick; and some new results from complex analysis about metrics with singularities. As an application, we will discuss the proof of Iitaka's conjecture (about the subadditivity of the Kodaira dimension in algebraic fiber spaces) over abelian varieties, following Cao and Paun.

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

### 2018年07月20日(金)

集中講義901-118「数物先端科学X」として行われます（7/18,20,23,24,25の全5回）。

**Christian Schnell 氏**(Stony Book University)

Singular hermitian metrics and morphisms to abelian varieties (ENGLISH)

Consider a morphism from a smooth projective variety to an abelian variety (over the field of complex numbers). After reviewing what is known about the pushforward of the canonical bundle under such a morphism, we will try to extend these results to the case of pluricanonical bundles (= the tensor powers of the canonical bundle). Along the way, we will learn about three important tools: generic vanishing theory; Viehweg's cyclic covering trick; and some new results from complex analysis about metrics with singularities. As an application, we will discuss the proof of Iitaka's conjecture (about the subadditivity of the Kodaira dimension in algebraic fiber spaces) over abelian varieties, following Cao and Paun.

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

### 2018年07月18日(水)

集中講義901-118「数物先端科学X」として行われます（7/18,20,23,24,25の全5回）。

**Christian Schnell 氏**(Stony Book University)

Singular hermitian metrics and morphisms to abelian varieties (ENGLISH)

Consider a morphism from a smooth projective variety to an abelian variety (over the field of complex numbers). After reviewing what is known about the pushforward of the canonical bundle under such a morphism, we will try to extend these results to the case of pluricanonical bundles (= the tensor powers of the canonical bundle). Along the way, we will learn about three important tools: generic vanishing theory; Viehweg's cyclic covering trick; and some new results from complex analysis about metrics with singularities. As an application, we will discuss the proof of Iitaka's conjecture (about the subadditivity of the Kodaira dimension in algebraic fiber spaces) over abelian varieties, following Cao and Paun.

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

### 2018年05月16日(水)

14:45-15:45 数理科学研究科棟(駒場) 122号室

Some strongly degenerate parabolic equations (joint with Prof. A. Tani) (ENGLISH)

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

**M.M. Lavrentʼev, Jr. 氏**(Novosibirsk State University)Some strongly degenerate parabolic equations (joint with Prof. A. Tani) (ENGLISH)

[ 講演概要 ]

We consider some nonlinear 1D parabolic equations with the positive leading coefficient which is not away from zero. "Hyperbolic phenomena" (gradient blowing up phenomena) were reported in literature for such models. We describe special cases of regular solvability for degenerate equations under study.

[ 参考URL ]We consider some nonlinear 1D parabolic equations with the positive leading coefficient which is not away from zero. "Hyperbolic phenomena" (gradient blowing up phenomena) were reported in literature for such models. We describe special cases of regular solvability for degenerate equations under study.

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

### 2018年05月11日(金)

15:00-17:00 数理科学研究科棟(駒場) 123号室

全5回講演の(5)

Introduction to the Langlands-Rapoport conjecture (ENGLISH)

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

全5回講演の(5)

**Sug Woo Shin 氏**(University of California, Berkeley)Introduction to the Langlands-Rapoport conjecture (ENGLISH)

[ 講演概要 ]

In 1970s Langlands envisioned a program to compute the Hasse-Weil zeta functions of Shimura varieties as an alternating product of automorphic L-functions, which in particular implies the meromorphic continuation and functional equation for the zeta functions. In 1987, Langlands and Rapoport formulated a precise and far-reaching conjecture describing the set of points of Shimura varieties modulo p as an essential step towards the goal. The program has been largely carried out by Langlands, Kottwitz, and others for PEL-type Shimura varieties with striking applications to the local and global Langlands correspondences (which in turn led to further applications). We have started to understand the more general Hodge-type and abelian-type cases only recently, thanks to Kisin's work on the Langlands-Rapoport conjecture in the good reduction case. The lecture aims to give a gentle introduction to his seminal paper. After a brief introduction, the lecture is divided into four parts.

(i) Shimura varieties: We introduce Shimura varieties of Hodge type and abelian type and their integral models.

(ii) Statement of the conjecture: After setting up the language of

Galois gerbs, we state the Langlands-Rapoport conjecture.

(iii) Sketch of Kisin's proof: We sketch Kisin's proof of the conjecture for Shimura varieties of Hodge type.

(iv) Counting fixed points: Following forthcoming work of Kisin, Y. Zhu, and the speaker, we explain how to apply the Langlands-Rapoport conjecture to count fixed-points of

Hecke-Frobenius correspondences.

[ 参考URL ]In 1970s Langlands envisioned a program to compute the Hasse-Weil zeta functions of Shimura varieties as an alternating product of automorphic L-functions, which in particular implies the meromorphic continuation and functional equation for the zeta functions. In 1987, Langlands and Rapoport formulated a precise and far-reaching conjecture describing the set of points of Shimura varieties modulo p as an essential step towards the goal. The program has been largely carried out by Langlands, Kottwitz, and others for PEL-type Shimura varieties with striking applications to the local and global Langlands correspondences (which in turn led to further applications). We have started to understand the more general Hodge-type and abelian-type cases only recently, thanks to Kisin's work on the Langlands-Rapoport conjecture in the good reduction case. The lecture aims to give a gentle introduction to his seminal paper. After a brief introduction, the lecture is divided into four parts.

(i) Shimura varieties: We introduce Shimura varieties of Hodge type and abelian type and their integral models.

(ii) Statement of the conjecture: After setting up the language of

Galois gerbs, we state the Langlands-Rapoport conjecture.

(iii) Sketch of Kisin's proof: We sketch Kisin's proof of the conjecture for Shimura varieties of Hodge type.

(iv) Counting fixed points: Following forthcoming work of Kisin, Y. Zhu, and the speaker, we explain how to apply the Langlands-Rapoport conjecture to count fixed-points of

Hecke-Frobenius correspondences.

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

### 2018年05月10日(木)

15:00-17:00 数理科学研究科棟(駒場) 123号室

全5回講演の(4)

Introduction to the Langlands-Rapoport conjecture (ENGLISH)

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

全5回講演の(4)

**Sug Woo Shin 氏**(University of California, Berkeley)Introduction to the Langlands-Rapoport conjecture (ENGLISH)

[ 講演概要 ]

In 1970s Langlands envisioned a program to compute the Hasse-Weil zeta functions of Shimura varieties as an alternating product of automorphic L-functions, which in particular implies the meromorphic continuation and functional equation for the zeta functions. In 1987, Langlands and Rapoport formulated a precise and far-reaching conjecture describing the set of points of Shimura varieties modulo p as an essential step towards the goal. The program has been largely carried out by Langlands, Kottwitz, and others for PEL-type Shimura varieties with striking applications to the local and global Langlands correspondences (which in turn led to further applications). We have started to understand the more general Hodge-type and abelian-type cases only recently, thanks to Kisin's work on the Langlands-Rapoport conjecture in the good reduction case. The lecture aims to give a gentle introduction to his seminal paper. After a brief introduction, the lecture is divided into four parts.

(i) Shimura varieties: We introduce Shimura varieties of Hodge type and abelian type and their integral models.

(ii) Statement of the conjecture: After setting up the language of

Galois gerbs, we state the Langlands-Rapoport conjecture.

(iii) Sketch of Kisin's proof: We sketch Kisin's proof of the conjecture for Shimura varieties of Hodge type.

(iv) Counting fixed points: Following forthcoming work of Kisin, Y. Zhu, and the speaker, we explain how to apply the Langlands-Rapoport conjecture to count fixed-points of

Hecke-Frobenius correspondences.

[ 参考URL ]In 1970s Langlands envisioned a program to compute the Hasse-Weil zeta functions of Shimura varieties as an alternating product of automorphic L-functions, which in particular implies the meromorphic continuation and functional equation for the zeta functions. In 1987, Langlands and Rapoport formulated a precise and far-reaching conjecture describing the set of points of Shimura varieties modulo p as an essential step towards the goal. The program has been largely carried out by Langlands, Kottwitz, and others for PEL-type Shimura varieties with striking applications to the local and global Langlands correspondences (which in turn led to further applications). We have started to understand the more general Hodge-type and abelian-type cases only recently, thanks to Kisin's work on the Langlands-Rapoport conjecture in the good reduction case. The lecture aims to give a gentle introduction to his seminal paper. After a brief introduction, the lecture is divided into four parts.

(i) Shimura varieties: We introduce Shimura varieties of Hodge type and abelian type and their integral models.

(ii) Statement of the conjecture: After setting up the language of

Galois gerbs, we state the Langlands-Rapoport conjecture.

(iii) Sketch of Kisin's proof: We sketch Kisin's proof of the conjecture for Shimura varieties of Hodge type.

(iv) Counting fixed points: Following forthcoming work of Kisin, Y. Zhu, and the speaker, we explain how to apply the Langlands-Rapoport conjecture to count fixed-points of

Hecke-Frobenius correspondences.

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

### 2018年05月09日(水)

15:00-17:00 数理科学研究科棟(駒場) 123号室

全5回講演の(3)

Introduction to the Langlands-Rapoport conjecture (ENGLISH)

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

全5回講演の(3)

**Sug Woo Shin 氏**(University of California, Berkeley)Introduction to the Langlands-Rapoport conjecture (ENGLISH)

[ 講演概要 ]

In 1970s Langlands envisioned a program to compute the Hasse-Weil zeta functions of Shimura varieties as an alternating product of automorphic L-functions, which in particular implies the meromorphic continuation and functional equation for the zeta functions. In 1987, Langlands and Rapoport formulated a precise and far-reaching conjecture describing the set of points of Shimura varieties modulo p as an essential step towards the goal. The program has been largely carried out by Langlands, Kottwitz, and others for PEL-type Shimura varieties with striking applications to the local and global Langlands correspondences (which in turn led to further applications). We have started to understand the more general Hodge-type and abelian-type cases only recently, thanks to Kisin's work on the Langlands-Rapoport conjecture in the good reduction case. The lecture aims to give a gentle introduction to his seminal paper. After a brief introduction, the lecture is divided into four parts.

(i) Shimura varieties: We introduce Shimura varieties of Hodge type and abelian type and their integral models.

(ii) Statement of the conjecture: After setting up the language of

Galois gerbs, we state the Langlands-Rapoport conjecture.

(iii) Sketch of Kisin's proof: We sketch Kisin's proof of the conjecture for Shimura varieties of Hodge type.

(iv) Counting fixed points: Following forthcoming work of Kisin, Y. Zhu, and the speaker, we explain how to apply the Langlands-Rapoport conjecture to count fixed-points of

Hecke-Frobenius correspondences.

[ 参考URL ]In 1970s Langlands envisioned a program to compute the Hasse-Weil zeta functions of Shimura varieties as an alternating product of automorphic L-functions, which in particular implies the meromorphic continuation and functional equation for the zeta functions. In 1987, Langlands and Rapoport formulated a precise and far-reaching conjecture describing the set of points of Shimura varieties modulo p as an essential step towards the goal. The program has been largely carried out by Langlands, Kottwitz, and others for PEL-type Shimura varieties with striking applications to the local and global Langlands correspondences (which in turn led to further applications). We have started to understand the more general Hodge-type and abelian-type cases only recently, thanks to Kisin's work on the Langlands-Rapoport conjecture in the good reduction case. The lecture aims to give a gentle introduction to his seminal paper. After a brief introduction, the lecture is divided into four parts.

(i) Shimura varieties: We introduce Shimura varieties of Hodge type and abelian type and their integral models.

(ii) Statement of the conjecture: After setting up the language of

Galois gerbs, we state the Langlands-Rapoport conjecture.

(iii) Sketch of Kisin's proof: We sketch Kisin's proof of the conjecture for Shimura varieties of Hodge type.

(iv) Counting fixed points: Following forthcoming work of Kisin, Y. Zhu, and the speaker, we explain how to apply the Langlands-Rapoport conjecture to count fixed-points of

Hecke-Frobenius correspondences.

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

### 2018年05月08日(火)

15:00-17:00 数理科学研究科棟(駒場) 123号室

全5回講演の(2)

Introduction to the Langlands-Rapoport conjecture (ENGLISH)

[ 講演概要 ]

In 1970s Langlands envisioned a program to compute the Hasse-Weil zeta functions of Shimura varieties as an alternating product of automorphic L-functions, which in particular implies the meromorphic continuation and functional equation for the zeta functions. In 1987, Langlands and Rapoport formulated a precise and far-reaching conjecture describing the set of points of Shimura varieties modulo p as an essential step towards the goal. The program has been largely carried out by Langlands, Kottwitz, and others for PEL-type Shimura varieties with striking applications to the local and global Langlands correspondences (which in turn led to further applications). We have started to understand the more general Hodge-type and abelian-type cases only recently, thanks to Kisin's work on the Langlands-Rapoport conjecture in the good reduction case. The lecture aims to give a gentle introduction to his seminal paper. After a brief introduction, the lecture is divided into four parts.

(i) Shimura varieties: We introduce Shimura varieties of Hodge type and abelian type and their integral models.

(ii) Statement of the conjecture: After setting up the language of

Galois gerbs, we state the Langlands-Rapoport conjecture.

(iii) Sketch of Kisin's proof: We sketch Kisin's proof of the conjecture for Shimura varieties of Hodge type.

(iv) Counting fixed points: Following forthcoming work of Kisin, Y. Zhu, and the speaker, we explain how to apply the Langlands-Rapoport conjecture to count fixed-points of

Hecke-Frobenius correspondences.
[ 参考URL ]

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

全5回講演の(2)

**Sug Woo Shin 氏**(University of California, Berkeley)Introduction to the Langlands-Rapoport conjecture (ENGLISH)

In 1970s Langlands envisioned a program to compute the Hasse-Weil zeta functions of Shimura varieties as an alternating product of automorphic L-functions, which in particular implies the meromorphic continuation and functional equation for the zeta functions. In 1987, Langlands and Rapoport formulated a precise and far-reaching conjecture describing the set of points of Shimura varieties modulo p as an essential step towards the goal. The program has been largely carried out by Langlands, Kottwitz, and others for PEL-type Shimura varieties with striking applications to the local and global Langlands correspondences (which in turn led to further applications). We have started to understand the more general Hodge-type and abelian-type cases only recently, thanks to Kisin's work on the Langlands-Rapoport conjecture in the good reduction case. The lecture aims to give a gentle introduction to his seminal paper. After a brief introduction, the lecture is divided into four parts.

(i) Shimura varieties: We introduce Shimura varieties of Hodge type and abelian type and their integral models.

(ii) Statement of the conjecture: After setting up the language of

Galois gerbs, we state the Langlands-Rapoport conjecture.

(iii) Sketch of Kisin's proof: We sketch Kisin's proof of the conjecture for Shimura varieties of Hodge type.

(iv) Counting fixed points: Following forthcoming work of Kisin, Y. Zhu, and the speaker, we explain how to apply the Langlands-Rapoport conjecture to count fixed-points of

Hecke-Frobenius correspondences.

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