## FMSPレクチャーズ

過去の記録 ～02/19｜次回の予定｜今後の予定 02/20～

担当者 | 河野俊丈 |
---|

**過去の記録**

### 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.

[ 講演参考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)

[ 講演参考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.

[ 講演参考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

[ 講演参考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.

[ 講演参考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.

[ 講演参考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.

[ 講演参考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.

[ 講演参考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.

[ 講演参考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.

[ 講演参考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.

[ 講演参考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.
[ 講演参考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

### 2018年05月07日(月)

15:00-17:00 数理科学研究科棟(駒場) 123号室

全5回講演の(1)

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.
[ 講演参考URL ]

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

全5回講演の(1)

**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

### 2018年03月26日(月)

10:00-12:00 数理科学研究科棟(駒場) 002号室

全2回講演の(2)

Geometric Recursion (ENGLISH)

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

全2回講演の(2)

**Jørgen Ellegaard Andersen 氏**(Aarhus University)Geometric Recursion (ENGLISH)

[ 講演概要 ]

Geometric Recursion is a very general machinery for constructing mapping class group invariants objects associated to two dimensional surfaces. After presenting the general abstract definition we shall see how a number of constructions in low dimensional geometry and topology fits into this setting. These will include the Mirzakhani-McShane identies, mapping class group invariant closed forms on Teichmüller space (including the Weil-Petterson symplectic form) and the Goldman symplectic form on moduli spaces of flat connections for general compact simple Lie groups. We shall also discuss the process which establishes that any application of Topological Recursion can be lifted to a Geometric Recursion setting involving continuous functions on Teichmüller space, where the connection back to Topological Recursion is obtained by integration over the moduli space of curve. The work

presented is joint with G. Borot and N. Orantin.

[ 講演参考URL ]Geometric Recursion is a very general machinery for constructing mapping class group invariants objects associated to two dimensional surfaces. After presenting the general abstract definition we shall see how a number of constructions in low dimensional geometry and topology fits into this setting. These will include the Mirzakhani-McShane identies, mapping class group invariant closed forms on Teichmüller space (including the Weil-Petterson symplectic form) and the Goldman symplectic form on moduli spaces of flat connections for general compact simple Lie groups. We shall also discuss the process which establishes that any application of Topological Recursion can be lifted to a Geometric Recursion setting involving continuous functions on Teichmüller space, where the connection back to Topological Recursion is obtained by integration over the moduli space of curve. The work

presented is joint with G. Borot and N. Orantin.

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

### 2018年03月23日(金)

10:00-12:00 数理科学研究科棟(駒場) 002号室

全2回講演の(1)

Geometric Recursion (ENGLISH)

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

全2回講演の(1)

**Jørgen Ellegaard Andersen 氏**(Aarhus University)Geometric Recursion (ENGLISH)

[ 講演概要 ]

Geometric Recursion is a very general machinery for constructing mapping class group invariants objects associated to two dimensional surfaces. After presenting the general abstract definition we shall see how a number of constructions in low dimensional geometry and topology fits into this setting. These will include the

Mirzakhani-McShane identies, mapping class group invariant closed forms on Teichmüller space (including the Weil-Petterson symplectic form) and the Goldman symplectic form on moduli spaces of flat connections for general compact simple Lie groups. We shall also discuss the process which establishes that any application of Topological Recursion can be lifted to a Geometric Recursion setting involving continuous functions on Teichmüller space, where the connection back to Topological Recursion is obtained by integration over the moduli space of curve. The work presented is joint with G. Borot and N. Orantin.

[ 講演参考URL ]Geometric Recursion is a very general machinery for constructing mapping class group invariants objects associated to two dimensional surfaces. After presenting the general abstract definition we shall see how a number of constructions in low dimensional geometry and topology fits into this setting. These will include the

Mirzakhani-McShane identies, mapping class group invariant closed forms on Teichmüller space (including the Weil-Petterson symplectic form) and the Goldman symplectic form on moduli spaces of flat connections for general compact simple Lie groups. We shall also discuss the process which establishes that any application of Topological Recursion can be lifted to a Geometric Recursion setting involving continuous functions on Teichmüller space, where the connection back to Topological Recursion is obtained by integration over the moduli space of curve. The work presented is joint with G. Borot and N. Orantin.

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

### 2018年02月23日(金)

13:30-15:00 数理科学研究科棟(駒場) 002号室

全3回講演の(3)

The topology of singular points of real analytic curves (ENGLISH)

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

全3回講演の(3)

**Etienne Ghys 氏**(ENS de Lyon)The topology of singular points of real analytic curves (ENGLISH)

[ 講演概要 ]

In the neighborhood of a singular point, a germ of real analytic curve in the plane consists of a finite number of branches. Each of these branches intersects a small circle around the singular point in two points. Therefore, the local topology is described by a chord diagram : an even number of points on a circle paired two by two. Not all chord diagrams come from a singular point. The main purpose of this mini course is to give an complete description of those ‘’analytic ? chord diagrams. On our way, we shall meet some interesting concepts from computer science, graph theory and operads.

[ 講演参考URL ]In the neighborhood of a singular point, a germ of real analytic curve in the plane consists of a finite number of branches. Each of these branches intersects a small circle around the singular point in two points. Therefore, the local topology is described by a chord diagram : an even number of points on a circle paired two by two. Not all chord diagrams come from a singular point. The main purpose of this mini course is to give an complete description of those ‘’analytic ? chord diagrams. On our way, we shall meet some interesting concepts from computer science, graph theory and operads.

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

### 2018年02月22日(木)

15:00-16:30 数理科学研究科棟(駒場) 117号室

全3回講演の(2)

The topology of singular points of real analytic curves (ENGLISH)

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

全3回講演の(2)

**Etienne Ghys 氏**(ENS de Lyon)The topology of singular points of real analytic curves (ENGLISH)

[ 講演概要 ]

In the neighborhood of a singular point, a germ of real analytic curve in the plane consists of a finite number of branches. Each of these branches intersects a small circle around the singular point in two points. Therefore, the local topology is described by a chord diagram : an even number of points on a circle paired two by two. Not all chord diagrams come from a singular point. The main purpose of this mini course is to give an complete description of those ‘’analytic ? chord diagrams. On our way, we shall meet some interesting concepts from computer science, graph theory and operads.

[ 講演参考URL ]In the neighborhood of a singular point, a germ of real analytic curve in the plane consists of a finite number of branches. Each of these branches intersects a small circle around the singular point in two points. Therefore, the local topology is described by a chord diagram : an even number of points on a circle paired two by two. Not all chord diagrams come from a singular point. The main purpose of this mini course is to give an complete description of those ‘’analytic ? chord diagrams. On our way, we shall meet some interesting concepts from computer science, graph theory and operads.

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

### 2018年02月21日(水)

15:00-16:30 数理科学研究科棟(駒場) 117号室

全3回講演の(1)

The topology of singular points of real analytic curves (ENGLISH)

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

全3回講演の(1)

**Etienne Ghys 氏**(ENS de Lyon)The topology of singular points of real analytic curves (ENGLISH)

[ 講演概要 ]

In the neighborhood of a singular point, a germ of real analytic curve in the plane consists of a finite number of branches. Each of these branches intersects a small circle around the singular point in two points. Therefore, the local topology is described by a chord diagram : an even number of points on a circle paired two by two. Not all chord diagrams come from a singular point. The main purpose of this mini course is to give an complete description of those ‘’analytic ? chord diagrams. On our way, we shall meet some interesting concepts from computer science, graph theory and operads.

[ 講演参考URL ]In the neighborhood of a singular point, a germ of real analytic curve in the plane consists of a finite number of branches. Each of these branches intersects a small circle around the singular point in two points. Therefore, the local topology is described by a chord diagram : an even number of points on a circle paired two by two. Not all chord diagrams come from a singular point. The main purpose of this mini course is to give an complete description of those ‘’analytic ? chord diagrams. On our way, we shall meet some interesting concepts from computer science, graph theory and operads.

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

### 2018年01月25日(木)

15:00-16:30 数理科学研究科棟(駒場) 002号室

NUMERICAL ANALYSIS, COBORDISM OF MANIFOLDS AND MONODROMY. (ENGLISH)

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

**Norbert A'Campo 氏**(University of Basel)NUMERICAL ANALYSIS, COBORDISM OF MANIFOLDS AND MONODROMY. (ENGLISH)

[ 講演概要 ]

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

[ 講演参考URL ]http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_ACampo_abst.pdf

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

### 2018年01月16日(火)

10:30-11:30 数理科学研究科棟(駒場) 056号室

FMSP Tokyo-Princeton joint student seminar

Large data global solutions for the shallow water system in one space dimension

[ 講演参考URL ]

http://fmsp.ms.u-tokyo.ac.jp/FMSP_180116.pdf

Introduction to the maximal Lp-regularity and its applications to the quasi-linear parabolic equations

[ 講演参考URL ]

http://fmsp.ms.u-tokyo.ac.jp/FMSP_180116.pdf

FMSP Tokyo-Princeton joint student seminar

**Federico Pasqualotto 氏**(Princeton) -Large data global solutions for the shallow water system in one space dimension

[ 講演参考URL ]

http://fmsp.ms.u-tokyo.ac.jp/FMSP_180116.pdf

**Naoto Kaziwara 氏**(U. Tokyo) -Introduction to the maximal Lp-regularity and its applications to the quasi-linear parabolic equations

[ 講演参考URL ]

http://fmsp.ms.u-tokyo.ac.jp/FMSP_180116.pdf

### 2017年12月13日(水)

17:00-17:45 数理科学研究科棟(駒場) 470号室

An approach to numerical solution to inverse source problems with nonlocal conditions (ENGLISH)

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

**Anar Rahimov 氏**(The Institute of Control Systems of ANAS and Baku State University)An approach to numerical solution to inverse source problems with nonlocal conditions (ENGLISH)

[ 講演概要 ]

We consider two inverse source problems for a parabolic equation under nonlocal, final, and boundary conditions. A numerical method is proposed to solve the inverse source problems, which is based on the use of the method of lines. The initial problems are reduced to a system of ordinary differential equations with unknown parameters. To solve this system, we propose an approach based on the sweep method type. We present the results of numerical experiments on test problems. This is joint work with Prof. K. Aida-zade.

[ 講演参考URL ]We consider two inverse source problems for a parabolic equation under nonlocal, final, and boundary conditions. A numerical method is proposed to solve the inverse source problems, which is based on the use of the method of lines. The initial problems are reduced to a system of ordinary differential equations with unknown parameters. To solve this system, we propose an approach based on the sweep method type. We present the results of numerical experiments on test problems. This is joint work with Prof. K. Aida-zade.

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

### 2017年11月06日(月)

17:00-18:00 数理科学研究科棟(駒場) 118号室

Phaseless inverse problems for Maxwell equations (ENGLISH)

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Romanov2.pdf

**V. G. Romanov 氏**(Sobolev Institute of Mathematics)Phaseless inverse problems for Maxwell equations (ENGLISH)

[ 講演概要 ]

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Romanov2.pdf

[ 講演参考URL ]http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Romanov2.pdf

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Romanov2.pdf

### 2017年10月31日(火)

16:00-17:00 数理科学研究科棟(駒場) 118号室

Some Geometric Aspects in Inverse Problems (ENGLISH)

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

**V. G. Romanov 氏**(Sobolev Institute of Mathematics)Some Geometric Aspects in Inverse Problems (ENGLISH)

[ 講演概要 ]

We consider inverse problems related to recovering coefficients in partial differential equations of the second order. It is supposed that some measurements of solutions to direct problems are produced on convenient sets. A study of some inverse problems for hyperbolic equations leads to geometric problems: recovering a function from its integrals along geodesic lines of the Riemannian metric or recovering the Riemannian metric inside a domain from given distances between arbitrary points of the domain boundary. Our main goal here is to demonstrate how such geometric problems arise for equations of parabolic and elliptic types.

[ 講演参考URL ]We consider inverse problems related to recovering coefficients in partial differential equations of the second order. It is supposed that some measurements of solutions to direct problems are produced on convenient sets. A study of some inverse problems for hyperbolic equations leads to geometric problems: recovering a function from its integrals along geodesic lines of the Riemannian metric or recovering the Riemannian metric inside a domain from given distances between arbitrary points of the domain boundary. Our main goal here is to demonstrate how such geometric problems arise for equations of parabolic and elliptic types.

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