## Future seminars

Seminar information archive ～03/22｜Today's seminar 03/23 | Future seminars 03/24～

### 2017/03/30

#### Number Theory Seminar

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

Logarithmic ramifications via pull-back to curves (English)

**Haoyu Hu**(University of Tokyo)Logarithmic ramifications via pull-back to curves (English)

[ Abstract ]

Let X be a smooth variety over a perfect field of characteristic p>0, D a strict normal crossing divisor of X, U the complement of D in X, j:U—>X the canonical injection, and F a locally constant and constructible sheaf of F_l-modules on U (l is a prime number different from p). Using Abbes and Saito’s logarithmic ramification theory, we define a Swan divisor SW(j_!F), which supported on D. Let i:C-->X be a quasi-finite morphism from a smooth curve C to X. Following T. Saito’s idea, we compare the pull-back of SW(j_!F) to C with the Swan divisor of the pull-back of j_!F to C. It answers an expectation of Esnault and Kerz and generalizes the same result of Barrientos for rank 1 sheaves. As an application, we obtain a lower semi-continuity property for Swan divisors of an l-adic sheaf on a smooth fibration, which gives a generalization of Deligne and Laumon’s lower semi-continuity property of Swan conductors of l-adic sheaves on relative curves to higher relative dimensions. This application is a supplement of the semi-continuity of total dimension of vanishing cycles due to T. Saito and the lower semi-continuity of total dimension divisors due to myself and E. Yang.

Let X be a smooth variety over a perfect field of characteristic p>0, D a strict normal crossing divisor of X, U the complement of D in X, j:U—>X the canonical injection, and F a locally constant and constructible sheaf of F_l-modules on U (l is a prime number different from p). Using Abbes and Saito’s logarithmic ramification theory, we define a Swan divisor SW(j_!F), which supported on D. Let i:C-->X be a quasi-finite morphism from a smooth curve C to X. Following T. Saito’s idea, we compare the pull-back of SW(j_!F) to C with the Swan divisor of the pull-back of j_!F to C. It answers an expectation of Esnault and Kerz and generalizes the same result of Barrientos for rank 1 sheaves. As an application, we obtain a lower semi-continuity property for Swan divisors of an l-adic sheaf on a smooth fibration, which gives a generalization of Deligne and Laumon’s lower semi-continuity property of Swan conductors of l-adic sheaves on relative curves to higher relative dimensions. This application is a supplement of the semi-continuity of total dimension of vanishing cycles due to T. Saito and the lower semi-continuity of total dimension divisors due to myself and E. Yang.

### 2017/04/10

#### Seminar on Geometric Complex Analysis

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

TBA

**Kengo Hirachi**(The University of Tokyo)TBA

[ Abstract ]

TBA

TBA

### 2017/04/11

#### Number Theory Seminar

17:30-18:30 Room #117 (Graduate School of Math. Sci. Bldg.)

The geometric Satake equivalence in mixed characteristic (ENGLISH)

**Peter Scholze**(University of Bonn)The geometric Satake equivalence in mixed characteristic (ENGLISH)

[ Abstract ]

In order to apply V. Lafforgue's ideas to the study of representations of p-adic groups, one needs a version of the geometric Satake equivalence in that setting. For the affine Grassmannian defined using the Witt vectors, this has been proven by Zhu. However, one actually needs a version for the affine Grassmannian defined using Fontaine's ring B_dR, and related results on the Beilinson-Drinfeld Grassmannian over a self-product of Spa Q_p. These objects exist as diamonds, and in particular one can make sense of the fusion product in this situation; this is a priori surprising, as it entails colliding two distinct points of Spec Z. The focus of the talk will be on the geometry of the fusion product, and an analogue of the technically crucial ULA (Universally Locally Acyclic) condition that works in this non-algebraic setting.

In order to apply V. Lafforgue's ideas to the study of representations of p-adic groups, one needs a version of the geometric Satake equivalence in that setting. For the affine Grassmannian defined using the Witt vectors, this has been proven by Zhu. However, one actually needs a version for the affine Grassmannian defined using Fontaine's ring B_dR, and related results on the Beilinson-Drinfeld Grassmannian over a self-product of Spa Q_p. These objects exist as diamonds, and in particular one can make sense of the fusion product in this situation; this is a priori surprising, as it entails colliding two distinct points of Spec Z. The focus of the talk will be on the geometry of the fusion product, and an analogue of the technically crucial ULA (Universally Locally Acyclic) condition that works in this non-algebraic setting.

#### Numerical Analysis Seminar

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

Some issues in the Lagrange-Galerkin method and solutions: computability, dependence on the viscosity and inflow boundary conditions (日本語)

**Shinya Uchiumi**(Waseda University)Some issues in the Lagrange-Galerkin method and solutions: computability, dependence on the viscosity and inflow boundary conditions (日本語)

### 2017/04/17

#### Seminar on Geometric Complex Analysis

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

Dense holomorphic curves in spaces of holomorphic maps

**Yuta Kusakabe**(Osaka University)Dense holomorphic curves in spaces of holomorphic maps

[ Abstract ]

We study when there exists a dense holomorphic curve in a space of holomorphic maps from a Stein space. Our results state that for any bounded convex domain $\Omega \Subset \mathbb{C}^n$ and any connected complex manifold $Y$, the space $\mathcal{O}(\Omega,Y)$ contains a dense holomorphic disc, and that $Y$ is an Oka manifold if and only if for any Stein space $X$ there exists a dense entire curve in every path component of $\mathcal{O}(X,Y)$. The latter gives a new characterization of Oka manifolds. As an application of the former, we construct universal maps from bounded convex domains to any connected complex manifold.

We study when there exists a dense holomorphic curve in a space of holomorphic maps from a Stein space. Our results state that for any bounded convex domain $\Omega \Subset \mathbb{C}^n$ and any connected complex manifold $Y$, the space $\mathcal{O}(\Omega,Y)$ contains a dense holomorphic disc, and that $Y$ is an Oka manifold if and only if for any Stein space $X$ there exists a dense entire curve in every path component of $\mathcal{O}(X,Y)$. The latter gives a new characterization of Oka manifolds. As an application of the former, we construct universal maps from bounded convex domains to any connected complex manifold.

### 2017/04/24

#### Seminar on Geometric Complex Analysis

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

Lagrangian Mean Curvature Flows and Moment maps

**Hiroshi Konno**(Meiji University)Lagrangian Mean Curvature Flows and Moment maps

[ Abstract ]

In this talk, we construct various examples of Lagrangian mean curvature flows in Calabi-Yau manifolds, using moment maps for actions of abelian Lie groups on them. The examples include Lagrangian self-shrinkers and translating solitons in the Euclid spaces. We also construct Lagrangian mean curvature flows in non-flat Calabi-Yau manifolds. In particular, we describe Lagrangian mean curvature flows in 4-dimensional Ricci-flat ALE spaces in detail and investigate their singularities.

In this talk, we construct various examples of Lagrangian mean curvature flows in Calabi-Yau manifolds, using moment maps for actions of abelian Lie groups on them. The examples include Lagrangian self-shrinkers and translating solitons in the Euclid spaces. We also construct Lagrangian mean curvature flows in non-flat Calabi-Yau manifolds. In particular, we describe Lagrangian mean curvature flows in 4-dimensional Ricci-flat ALE spaces in detail and investigate their singularities.

### 2017/05/08

#### Seminar on Geometric Complex Analysis

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

TBA

**Taro Fujisawa**(Tokyo Denki University)TBA

[ Abstract ]

TBA

TBA

### 2017/05/22

#### Seminar on Geometric Complex Analysis

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

Complex K3 surfaces containing Levi-flat hypersurfaces

**Takayuki Koike**(Kyoto University)Complex K3 surfaces containing Levi-flat hypersurfaces

[ Abstract ]

We show the existence of a complex K3 surface $X$ which is not a Kummer surface and has a one-parameter family of Levi-flat hypersurfaces in which all the leaves are dense. We construct such $X$ by patching two open complex surfaces obtained as the complements of tubular neighborhoods of elliptic curves embedded in blow-ups of the projective planes at general nine points.

We show the existence of a complex K3 surface $X$ which is not a Kummer surface and has a one-parameter family of Levi-flat hypersurfaces in which all the leaves are dense. We construct such $X$ by patching two open complex surfaces obtained as the complements of tubular neighborhoods of elliptic curves embedded in blow-ups of the projective planes at general nine points.

### 2017/06/12

#### Seminar on Geometric Complex Analysis

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

TBA

**Yoshihiko Matsumoto**(Osaka University)TBA

[ Abstract ]

TBA

TBA

### 2017/06/26

#### Seminar on Geometric Complex Analysis

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

TBA

**Akito Futaki**(The University of Tokyo)TBA

[ Abstract ]

TBA

TBA

### 2017/10/17

#### PDE Real Analysis Seminar

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

**Hoài-Minh Nguyên**(École Polytechnique Fédérale de Lausanne)