## Seminar information archive

Seminar information archive ～02/15｜Today's seminar 02/16 | Future seminars 02/17～

#### Algebraic Geometry Seminar

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

Perverse sheaves of categories and birational geometry (English)

**Will Donovan**(IPMU)Perverse sheaves of categories and birational geometry (English)

[ Abstract ]

Kapranov and Schechtman have initiated a program to study perverse sheaves of categories, or perverse schobers. It is expected that examples arise from birational geometry, in particular from webs of flops. I explain progress towards constructing these objects for Grothendieck resolutions (work of the above authors with Bondal), and for 3-folds (joint work of myself and Wemyss).

Kapranov and Schechtman have initiated a program to study perverse sheaves of categories, or perverse schobers. It is expected that examples arise from birational geometry, in particular from webs of flops. I explain progress towards constructing these objects for Grothendieck resolutions (work of the above authors with Bondal), and for 3-folds (joint work of myself and Wemyss).

#### Seminar on Geometric Complex Analysis

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

Kähler-Ricci soliton, K-stability and moduli space of Fano

manifolds (JAPANESE)

**Eiji Inoue**(The University of Tokyo)Kähler-Ricci soliton, K-stability and moduli space of Fano

manifolds (JAPANESE)

[ Abstract ]

Kähler-Ricci soliton is a kind of canonical metrics on Fano manifolds and is a natural generalization of Kähler-Einstein metric in

view of Kähler-Ricci flow.

In this talk, I will explain the following good geometric features of Fano manifolds admitting Kähler-Ricci solitons:

1. Volume minimization, reductivity and uniqueness results established by Tian&Zhu.

2. Relation to algebraic (modified) K-stability estabilished by Berman&Witt-Niström and Datar&Székelyhidi.

3. Moment map picture for Kähler-Ricci soliton (‘real side’)

4. Moduli stack (‘virtual side’) and moduli space of them

A result in 1 is indispensable for the formulation in 3 and 4, and explains why we should consider solitons, beyond Einstein metrics. I also show an essential idea in the construction of the moduli space of Fano manifolds admitting Kähler-Ricci solitons and give some remarks on technical key point.

Kähler-Ricci soliton is a kind of canonical metrics on Fano manifolds and is a natural generalization of Kähler-Einstein metric in

view of Kähler-Ricci flow.

In this talk, I will explain the following good geometric features of Fano manifolds admitting Kähler-Ricci solitons:

1. Volume minimization, reductivity and uniqueness results established by Tian&Zhu.

2. Relation to algebraic (modified) K-stability estabilished by Berman&Witt-Niström and Datar&Székelyhidi.

3. Moment map picture for Kähler-Ricci soliton (‘real side’)

4. Moduli stack (‘virtual side’) and moduli space of them

A result in 1 is indispensable for the formulation in 3 and 4, and explains why we should consider solitons, beyond Einstein metrics. I also show an essential idea in the construction of the moduli space of Fano manifolds admitting Kähler-Ricci solitons and give some remarks on technical key point.

### 2018/05/16

#### FMSP Lectures

14:45-15:45 Room #122 (Graduate School of Math. Sci. Bldg.)

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.**(Novosibirsk State University)Some strongly degenerate parabolic equations (joint with Prof. A. Tani) (ENGLISH)

[ Abstract ]

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.

[ Reference 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

### 2018/05/15

#### Tuesday Seminar on Topology

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

On the singularity theory of mixed hypersurfaces and some conjecture (JAPANESE)

**Mutsuo Oka**(Tokyo University of Science)On the singularity theory of mixed hypersurfaces and some conjecture (JAPANESE)

[ Abstract ]

Consider a real algebraic variety of real codimension 2 defined by $V:=\{g(\mathbf x,\mathbf y)=h(\mathbf x,\mathbf y)=0\}$ in $\mathbb C^n=\mathbb R^n\times \mathbb R^n$. Put $\mathbf z=\mathbf x+i\mathbf y$ and consider complex valued real analytic function $f=g+ih$. Replace the variables $x_1,y_1\dots, x_n,y_n$ using the equality $x_j=(z_j+\bar z_j)/2,\, y_j=(z_j-\bar z_j)/2i$. Then $f$ can be understood to be an analytic functions of $z_j,\bar z_j$. We call $f$ a mixed function. In this way, $V=\{f(\mathbf z,\bar{\mathbf z})=0\}$ and we can use the techniques of complex analytic functions and the singularity theory developed there. In this talk, we explain basic properties of the singularity of mixed hyper surface $V(f)$ and give several open questions.

Consider a real algebraic variety of real codimension 2 defined by $V:=\{g(\mathbf x,\mathbf y)=h(\mathbf x,\mathbf y)=0\}$ in $\mathbb C^n=\mathbb R^n\times \mathbb R^n$. Put $\mathbf z=\mathbf x+i\mathbf y$ and consider complex valued real analytic function $f=g+ih$. Replace the variables $x_1,y_1\dots, x_n,y_n$ using the equality $x_j=(z_j+\bar z_j)/2,\, y_j=(z_j-\bar z_j)/2i$. Then $f$ can be understood to be an analytic functions of $z_j,\bar z_j$. We call $f$ a mixed function. In this way, $V=\{f(\mathbf z,\bar{\mathbf z})=0\}$ and we can use the techniques of complex analytic functions and the singularity theory developed there. In this talk, we explain basic properties of the singularity of mixed hyper surface $V(f)$ and give several open questions.

### 2018/05/14

#### Seminar on Geometric Complex Analysis

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

Harmonic map and the Einstein equation in five dimension (JAPANESE)

**Sumio Yamada**(Gakushuin University)Harmonic map and the Einstein equation in five dimension (JAPANESE)

[ Abstract ]

We present a new method in constructing 5-dimensional stationary solutions to the vacuum Einstein equation. In 1917, H. Weyl expressed the Schwarzschild black hole solution using a cylindical coordinate system, and consequently realized that the metric is completely determined by a harmonic function. Since then, the relation between harmonic maps and the Einstein equation has been explored mostly by physicists, which they call the sigma model of the Einstein equation. In this talk, after explaining the historical background, we demonstrate that in 5D, the Einstein spacetimes can have a wide range of black hole horizons in their topological types. In particular we establish an existence theorem of harmonic maps, which subsequently leads to constructions of 5D spacetimes with black hole horizons of positive Yamabe types, namely $S^3$, $S^2 \times S^1$, and the lens space $L(p,q)$. This is a joint work with Marcus Khuri and Gilbert Weinstein.

We present a new method in constructing 5-dimensional stationary solutions to the vacuum Einstein equation. In 1917, H. Weyl expressed the Schwarzschild black hole solution using a cylindical coordinate system, and consequently realized that the metric is completely determined by a harmonic function. Since then, the relation between harmonic maps and the Einstein equation has been explored mostly by physicists, which they call the sigma model of the Einstein equation. In this talk, after explaining the historical background, we demonstrate that in 5D, the Einstein spacetimes can have a wide range of black hole horizons in their topological types. In particular we establish an existence theorem of harmonic maps, which subsequently leads to constructions of 5D spacetimes with black hole horizons of positive Yamabe types, namely $S^3$, $S^2 \times S^1$, and the lens space $L(p,q)$. This is a joint work with Marcus Khuri and Gilbert Weinstein.

#### Tokyo Probability Seminar

16:00-17:30 Room #126 (Graduate School of Math. Sci. Bldg.)

Chordal Komatu-Loewner equation for a family of continuously growing hulls (JAPANESE)

**Takuya MURAYAMA**(Graduate School of Science, Kyoto University)Chordal Komatu-Loewner equation for a family of continuously growing hulls (JAPANESE)

### 2018/05/11

#### FMSP Lectures

15:00-17:00 Room #123 (Graduate School of Math. Sci. Bldg.)

Introduction to the Langlands-Rapoport conjecture (ENGLISH)

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

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

[ Abstract ]

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.

[ Reference 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

#### Lectures

13:00-14:00 Room #123 (Graduate School of Math. Sci. Bldg.)

The Langlands-Kottwitz method for deformation spaces of Hodge type

**Alex Youcis**(University of California, Berkeley)The Langlands-Kottwitz method for deformation spaces of Hodge type

[ Abstract ]

Cohomology of global Shimura varieties is an object of universal importance in the Langlands program. Given a Shimura datum (G,X) and a (sufficiently nice) representation ¥xi of G, one obtains an l-adic sheaf F_{¥xi,l} on Sh(G,X) with a G(A_f)-structure. Thus, in the standard way, the cohomology group H^*(Sh(G,X),F_¥xi) has an admissible action of Gal(¥overline{E}/E) ¥times G(A_f), where E=E(G,X) is the reflex field of (G,X). Extending work of Kottwitz, Scholze, and others we discuss a method for computing the traces of this action, more specifically of an element ¥tau ¥times g where ¥tau ¥in W_{E_¥mathfrak{p}} for some prime ¥mathfrak{p} of E dividing p and g ¥in G(A_f^p) ¥times G(Z_p), in terms of a weighted point count on the Shimura variety's special fiber, as well as the traces of various local Shimura varieties over E_¥mathfrak{p}, at least in the case when (G,X) is a abelian-type Shimura datum unramified at p.

Cohomology of global Shimura varieties is an object of universal importance in the Langlands program. Given a Shimura datum (G,X) and a (sufficiently nice) representation ¥xi of G, one obtains an l-adic sheaf F_{¥xi,l} on Sh(G,X) with a G(A_f)-structure. Thus, in the standard way, the cohomology group H^*(Sh(G,X),F_¥xi) has an admissible action of Gal(¥overline{E}/E) ¥times G(A_f), where E=E(G,X) is the reflex field of (G,X). Extending work of Kottwitz, Scholze, and others we discuss a method for computing the traces of this action, more specifically of an element ¥tau ¥times g where ¥tau ¥in W_{E_¥mathfrak{p}} for some prime ¥mathfrak{p} of E dividing p and g ¥in G(A_f^p) ¥times G(Z_p), in terms of a weighted point count on the Shimura variety's special fiber, as well as the traces of various local Shimura varieties over E_¥mathfrak{p}, at least in the case when (G,X) is a abelian-type Shimura datum unramified at p.

#### Colloquium

15:30-16:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Generic density theorems for periodic Reeb orbits and minimal hypersurfaces (日本語)

**Kei IRIE**(The University of Tokyo)Generic density theorems for periodic Reeb orbits and minimal hypersurfaces (日本語)

### 2018/05/10

#### FMSP Lectures

15:00-17:00 Room #123 (Graduate School of Math. Sci. Bldg.)

Introduction to the Langlands-Rapoport conjecture (ENGLISH)

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

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

[ Abstract ]

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.

[ Reference 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

#### Lectures

11:00-12:00 Room #123 (Graduate School of Math. Sci. Bldg.)

The Cohomology of Rapoport-Zink Spaces of EL-Type

**Alexander Bertoloni Meli**(University of California, Berkeley)The Cohomology of Rapoport-Zink Spaces of EL-Type

[ Abstract ]

I will discuss Rapoport-Zink spaces of EL-type and how to explicitly compute a certain variant of their cohomology in terms of the local Langlands correspondence for general linear groups. I will then show how this computation can be used to resolve certain cases of a conjecture of Harris.

I will discuss Rapoport-Zink spaces of EL-type and how to explicitly compute a certain variant of their cohomology in terms of the local Langlands correspondence for general linear groups. I will then show how this computation can be used to resolve certain cases of a conjecture of Harris.

### 2018/05/09

#### Number Theory Seminar

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

Endoscopy and cohomology of U(n-1,1) (ENGLISH)

**Sug Woo Shin**(University of California, Berkeley)Endoscopy and cohomology of U(n-1,1) (ENGLISH)

[ Abstract ]

We apply the endoscopic classification of automorphic representations for inner forms of unitary groups to bound the growth of cohomology in congruence towers of locally symmetric spaces associated with U(n-1,1). Our bound is sharper than the bound predicted by Sarnak-Xue for general locally symmetric spaces. This is joint work with Simon Marshall.

We apply the endoscopic classification of automorphic representations for inner forms of unitary groups to bound the growth of cohomology in congruence towers of locally symmetric spaces associated with U(n-1,1). Our bound is sharper than the bound predicted by Sarnak-Xue for general locally symmetric spaces. This is joint work with Simon Marshall.

#### FMSP Lectures

15:00-17:00 Room #123 (Graduate School of Math. Sci. Bldg.)

Introduction to the Langlands-Rapoport conjecture (ENGLISH)

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

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

[ Abstract ]

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.

[ Reference 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

#### Operator Algebra Seminars

16:45-18:15 Room #126 (Graduate School of Math. Sci. Bldg.)

Factoriality, Connes' invariants and fullness of amalgamated free products (English)

**Yusuke Isono**(RIMS, Kyoto Univ.)Factoriality, Connes' invariants and fullness of amalgamated free products (English)

### 2018/05/08

#### FMSP Lectures

**Sug Woo Shin**(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

#### Tuesday Seminar on Topology

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

Beyond the Weinstein conjecture (ENGLISH)

**Dan Cristofaro-Gardiner**(University of California, Santa Cruz)Beyond the Weinstein conjecture (ENGLISH)

[ Abstract ]

The Weinstein conjecture states that any Reeb vector field on a closed manifold has at least one closed orbit. The three-dimensional case of this conjecture was proved by Taubes in 2007, and Hutchings and I later showed that in this case there are always at least 2 orbits. While examples exist with exactly two orbits, one expects that this lower bound can be significantly improved with additional assumptions. For example, a theorem of Hofer, Wysocki, and Zehnder states that a generic nondegenerate Reeb vector field associated to the "standard" contact structure on $S^3$ has either 2, or infinitely many, closed orbits. We prove that any nondegenerate Reeb vector field has 2 or infinitely many closed orbits as long as the associated contact structure has torsion first Chern class. This is joint work with Mike Hutchings and Dan Pomerleano.

The Weinstein conjecture states that any Reeb vector field on a closed manifold has at least one closed orbit. The three-dimensional case of this conjecture was proved by Taubes in 2007, and Hutchings and I later showed that in this case there are always at least 2 orbits. While examples exist with exactly two orbits, one expects that this lower bound can be significantly improved with additional assumptions. For example, a theorem of Hofer, Wysocki, and Zehnder states that a generic nondegenerate Reeb vector field associated to the "standard" contact structure on $S^3$ has either 2, or infinitely many, closed orbits. We prove that any nondegenerate Reeb vector field has 2 or infinitely many closed orbits as long as the associated contact structure has torsion first Chern class. This is joint work with Mike Hutchings and Dan Pomerleano.

#### Algebraic Geometry Seminar

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

Higher order families of lines and Fano manifolds covered by linear

spaces

(Japanese (writing in English))

**Taku Suzuki**(Utsunomiya)Higher order families of lines and Fano manifolds covered by linear

spaces

(Japanese (writing in English))

[ Abstract ]

In this talk, for an embedded Fano manifold $X$, we introduce higher

order families of lines and a new invariant $S_X$. They are line

versions of higher order minimal families of rational curves and the

invariant $N_X$ which were introduced in my previous talk on 4th

November 2016. In addition, $S_X$ is related to the dimension of

covering linear spaces. Our goal is to classify Fano manifolds $X$ which

have large $S_X$.

In this talk, for an embedded Fano manifold $X$, we introduce higher

order families of lines and a new invariant $S_X$. They are line

versions of higher order minimal families of rational curves and the

invariant $N_X$ which were introduced in my previous talk on 4th

November 2016. In addition, $S_X$ is related to the dimension of

covering linear spaces. Our goal is to classify Fano manifolds $X$ which

have large $S_X$.

#### Lectures

13:00-14:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Langlands-Rapoport for the Modular Curve

**Sander Mack-Crane**(University of California, Berkeley)Langlands-Rapoport for the Modular Curve

[ Abstract ]

We discuss a concrete version of the Langlands-Rapoport conjecture in the case of the modular curve, and use this case to illuminate some of the more abstract features of the Langlands-Rapoport conjecture for general (abelian type) Shimura varieties.

We discuss a concrete version of the Langlands-Rapoport conjecture in the case of the modular curve, and use this case to illuminate some of the more abstract features of the Langlands-Rapoport conjecture for general (abelian type) Shimura varieties.

#### Numerical Analysis Seminar

16:50-18:20 Room #002 (Graduate School of Math. Sci. Bldg.)

Various aspects of numerical analysis (Japanese)

**Norikazu Saito**(University of Tokyo)Various aspects of numerical analysis (Japanese)

#### Seminar on Probability and Statistics

15:00-16:10 Room #052 (Graduate School of Math. Sci. Bldg.)

### 2018/05/07

#### Seminar on Geometric Complex Analysis

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

Proper holomorphic mappings and generalized pseudoellipsoids (JAPANESE)

**Atsushi Hayashimoto**(National Institute of Technology, Nagano College)Proper holomorphic mappings and generalized pseudoellipsoids (JAPANESE)

[ Abstract ]

We study the classification of proper holomorphic mappings between generalized pseudoellipsoids of different dimensions.

Huang proved some classification theorems of proper holomorphic mappings between balls of different dimensions, which are called gap theorems. Our present theorems are their weakly pseudoconvex versions.

In the theorem, classified mapping is so-called a variables splitting mapping and each component is derived from a homogeneous proper polynomial mapping between balls.

The essential methods are the ''good'' decompositions of CR vector bundle and reduction the mapping under consideration to the mapping of balls. By this reduction, we can apply Huang's gap theorem.

We study the classification of proper holomorphic mappings between generalized pseudoellipsoids of different dimensions.

Huang proved some classification theorems of proper holomorphic mappings between balls of different dimensions, which are called gap theorems. Our present theorems are their weakly pseudoconvex versions.

In the theorem, classified mapping is so-called a variables splitting mapping and each component is derived from a homogeneous proper polynomial mapping between balls.

The essential methods are the ''good'' decompositions of CR vector bundle and reduction the mapping under consideration to the mapping of balls. By this reduction, we can apply Huang's gap theorem.

#### Tokyo Probability Seminar

16:00-17:30 Room #126 (Graduate School of Math. Sci. Bldg.)

(JAPANESE)

[ Reference URL ]

http://www.taksagawa.com

**Takahiro SAGAWA**(Faculty of Engineering, The University of Tokyo)(JAPANESE)

[ Reference URL ]

http://www.taksagawa.com

#### FMSP Lectures

**Sug Woo Shin**(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

### 2018/05/02

#### Operator Algebra Seminars

16:45-18:15 Room #126 (Graduate School of Math. Sci. Bldg.)

Classification of Rokhlin flows (English)

**Gabor Szabo**(Copenhagen Univ.)Classification of Rokhlin flows (English)

### 2018/04/24

#### Algebraic Geometry Seminar

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

BIRATIONAL BOUNDEDNESS OF RATIONALLY CONNECTED CALABI–YAU 3-FOLDS

(English)

**Wei-Chung Chen**(Tokyo)BIRATIONAL BOUNDEDNESS OF RATIONALLY CONNECTED CALABI–YAU 3-FOLDS

(English)

[ Abstract ]

Firstly, we show that rationally connected Calabi–Yau 3- folds with kawamata log terminal (klt) singularities form a birationally bounded family, or more generally, rationally connected 3-folds of ε-CY type form a birationally bounded family for ε > 0. Then we focus on ε-lc log Calabi–Yau pairs (X, B) such that coefficients of B are bounded from below away from zero. We show that such pairs are log bounded modulo flops. As a consequence, we show that rationally connected klt Calabi–Yau 3-folds with mld bounding away from 1 are bounded modulo flops.

Firstly, we show that rationally connected Calabi–Yau 3- folds with kawamata log terminal (klt) singularities form a birationally bounded family, or more generally, rationally connected 3-folds of ε-CY type form a birationally bounded family for ε > 0. Then we focus on ε-lc log Calabi–Yau pairs (X, B) such that coefficients of B are bounded from below away from zero. We show that such pairs are log bounded modulo flops. As a consequence, we show that rationally connected klt Calabi–Yau 3-folds with mld bounding away from 1 are bounded modulo flops.

< Previous 123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130131132133134135136137138139140141142143 Next >