## Seminar information archive

Seminar information archive ～01/21｜Today's seminar 01/22 | Future seminars 01/23～

### 2018/01/17

#### Number Theory Seminar

18:00-19:00 Room #002 (Graduate School of Math. Sci. Bldg.)

On the vanishing of cohomology for certain Shimura varieties (ENGLISH)

**Ana Caraiani**(Imperial College)On the vanishing of cohomology for certain Shimura varieties (ENGLISH)

[ Abstract ]

I will prove that the compactly supported cohomology of certain unitary or symplectic Shimura varieties at level Gamma_1(p^\infty) vanishes above the middle degree. The key ingredients come from p-adic Hodge theory and studying the Bruhat decomposition on the Hodge-Tate flag variety. I will describe the steps in the proof using modular curves as a toy model. I will also mention an application to Galois representations for torsion classes in the cohomology of locally symmetric spaces for GL_n. This talk is based on joint work in preparation with D. Gulotta, C.Y. Hsu, C. Johansson, L. Mocz, E. Reineke, and S.C. Shih.

I will prove that the compactly supported cohomology of certain unitary or symplectic Shimura varieties at level Gamma_1(p^\infty) vanishes above the middle degree. The key ingredients come from p-adic Hodge theory and studying the Bruhat decomposition on the Hodge-Tate flag variety. I will describe the steps in the proof using modular curves as a toy model. I will also mention an application to Galois representations for torsion classes in the cohomology of locally symmetric spaces for GL_n. This talk is based on joint work in preparation with D. Gulotta, C.Y. Hsu, C. Johansson, L. Mocz, E. Reineke, and S.C. Shih.

#### Discrete mathematical modelling seminar

17:00-18:45 Room #056 (Graduate School of Math. Sci. Bldg.)

Quantum matter bounce with a dark energy expanding phase (ENGLISH)

Mass of the vacuum: a Newtonian perspective (ENGLISH)

**Samuel Colin**(CBPF, Rio de Janeiro, Brasil) 17:00-17:50Quantum matter bounce with a dark energy expanding phase (ENGLISH)

[ Abstract ]

The ``matter bounce'' is an alternative scenario to inflationary cosmology, according to which the universe undergoes a contraction, followed by an expansion, the bounce occurring when the quantum effects become important. In my talk, I will show that such a scenario can be unambiguously analyzed in the de Broglie-Bohm pilot-wave interpretation of quantum mechanics. More specifically, I will apply the pilot-wave theory to a Wheeler-DeWitt equation obtained from the quantization of a simple classical mini-superspace model, and show that there are numerical solutions describing bouncing universes with many desirable physical features. For example, one solution contains a dark energy phase during the expansion, without the need to postulate the existence of a cosmological constant in the classical action.

This work was done in collaboration with Nelson Pinto-Neto (CBPF, Rio de Janeiro, Brasil). Further details available at https://arxiv.org/abs/1706.03037.

The ``matter bounce'' is an alternative scenario to inflationary cosmology, according to which the universe undergoes a contraction, followed by an expansion, the bounce occurring when the quantum effects become important. In my talk, I will show that such a scenario can be unambiguously analyzed in the de Broglie-Bohm pilot-wave interpretation of quantum mechanics. More specifically, I will apply the pilot-wave theory to a Wheeler-DeWitt equation obtained from the quantization of a simple classical mini-superspace model, and show that there are numerical solutions describing bouncing universes with many desirable physical features. For example, one solution contains a dark energy phase during the expansion, without the need to postulate the existence of a cosmological constant in the classical action.

This work was done in collaboration with Nelson Pinto-Neto (CBPF, Rio de Janeiro, Brasil). Further details available at https://arxiv.org/abs/1706.03037.

**Thomas Durt**(Aix Marseille Université, Centrale Marseille, Institut Fresnel) 17:50-18:40Mass of the vacuum: a Newtonian perspective (ENGLISH)

[ Abstract ]

One could believe that special relativity forces us to totally renounce to the idea of an aether, but the aether reappears in general relativity which teaches us that space-time is structured by the local metrics. It also reappears in quantum field theory which teaches us that even at zero temperature space is filled by the quantum vacuum energy. Finally, aether reappears in modern astronomy where it was necessary to introduce ill-defined concepts such as dark matter and dark energy in order to explain apparent deviations from Newtonian dynamics (at the level of galactic rotation curves).

Newton dynamics being the unique limit of general relativistic dynamics in the classical regime, dark matter and dark energy can be seen as an ultimate, last chance strategy, aimed at reconciling the predictions of general relativity with astronomical data.

In our talk we shall describe a simple model, derived in the framework of Newtonian dynamics, aimed at explaining puzzling astronomical observations realized at the level of the solar system (Pioneer anomaly) and at the galactic scale (rotation curves), without adopting ad hoc hypotheses about the existence of dark matter and/or dark energy.

The basic idea is that Newtonian gravity is modified due to the presence of a (negative) density, everywhere in space, of mass-energy.

One could believe that special relativity forces us to totally renounce to the idea of an aether, but the aether reappears in general relativity which teaches us that space-time is structured by the local metrics. It also reappears in quantum field theory which teaches us that even at zero temperature space is filled by the quantum vacuum energy. Finally, aether reappears in modern astronomy where it was necessary to introduce ill-defined concepts such as dark matter and dark energy in order to explain apparent deviations from Newtonian dynamics (at the level of galactic rotation curves).

Newton dynamics being the unique limit of general relativistic dynamics in the classical regime, dark matter and dark energy can be seen as an ultimate, last chance strategy, aimed at reconciling the predictions of general relativity with astronomical data.

In our talk we shall describe a simple model, derived in the framework of Newtonian dynamics, aimed at explaining puzzling astronomical observations realized at the level of the solar system (Pioneer anomaly) and at the galactic scale (rotation curves), without adopting ad hoc hypotheses about the existence of dark matter and/or dark energy.

The basic idea is that Newtonian gravity is modified due to the presence of a (negative) density, everywhere in space, of mass-energy.

### 2018/01/16

#### Tuesday Seminar on Topology

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

On adequacy and the crossing number of satellite knots (JAPANESE)

**Jimenez Pascual Adrian**(The University of Tokyo)On adequacy and the crossing number of satellite knots (JAPANESE)

[ Abstract ]

It has always been difficult to prove results regarding the (minimal) crossing number of knots. In particular, apparently easy problems such as knowing the crossing number of the connected sum of knots, or bounding the crossing number of satellite knots have been conjectured through decades, yet still remain open. Focusing on this latter problem, in this talk I will prove that the crossing number of a satellite knot is bounded from below by the crossing number of its companion, when the companion is adequate.

It has always been difficult to prove results regarding the (minimal) crossing number of knots. In particular, apparently easy problems such as knowing the crossing number of the connected sum of knots, or bounding the crossing number of satellite knots have been conjectured through decades, yet still remain open. Focusing on this latter problem, in this talk I will prove that the crossing number of a satellite knot is bounded from below by the crossing number of its companion, when the companion is adequate.

#### Tuesday Seminar on Topology

18:00-19:00 Room #056 (Graduate School of Math. Sci. Bldg.)

A new relationship between the dilatation of pseudo-Anosov braids and fixed point theory (JAPANESE)

**Yumehito Kawashima**(The University of Tokyo)A new relationship between the dilatation of pseudo-Anosov braids and fixed point theory (JAPANESE)

[ Abstract ]

A relation between the dilatation of pseudo-Anosov braids and fixed point theory was studied by Ivanov. In this talk we reveal a new relationship between the above two subjects by showing a formula for the dilatation of pseudo-Anosov braids by means of the representations of braid groups due to B. Jiang and H. Zheng.

A relation between the dilatation of pseudo-Anosov braids and fixed point theory was studied by Ivanov. In this talk we reveal a new relationship between the above two subjects by showing a formula for the dilatation of pseudo-Anosov braids by means of the representations of braid groups due to B. Jiang and H. Zheng.

#### FMSP Lectures

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

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

[ Reference 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

[ Reference URL ]

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

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

[ Reference URL ]

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

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

[ Reference URL ]

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

### 2018/01/15

#### Seminar on Geometric Complex Analysis

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

Vanishing theorems of $L^2$-cohomology groups on Hessian manifolds

**Shinya Akagawa**(Osaka University)Vanishing theorems of $L^2$-cohomology groups on Hessian manifolds

[ Abstract ]

A Hessian manifold is a Riemannian manifold whose metric is locally given by the Hessian of a function with respect to flat coordinates. In this talk, we discuss vanishing theorems of $L^2$-cohomology groups on complete Hessian Manifolds coupled with flat line bundles. In particular, we obtain stronger vanishing theorems on regular convex cones with the Cheng-Yau metrics. Further we show that the Cheng-Yau metrics on regular convex cones give rise to harmonic maps to the positive symmetric matrices.

A Hessian manifold is a Riemannian manifold whose metric is locally given by the Hessian of a function with respect to flat coordinates. In this talk, we discuss vanishing theorems of $L^2$-cohomology groups on complete Hessian Manifolds coupled with flat line bundles. In particular, we obtain stronger vanishing theorems on regular convex cones with the Cheng-Yau metrics. Further we show that the Cheng-Yau metrics on regular convex cones give rise to harmonic maps to the positive symmetric matrices.

### 2017/12/26

#### Algebraic Geometry Seminar

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

K-stability of log Fano hyperplane arrangements (English)

**Kento Fujita**(RIMS)K-stability of log Fano hyperplane arrangements (English)

[ Abstract ]

We completely determine which log Fano hyperplane arrangements are uniformly K-stable, K-stable, K-polystable, K-semistable or not.

We completely determine which log Fano hyperplane arrangements are uniformly K-stable, K-stable, K-polystable, K-semistable or not.

### 2017/12/25

#### Operator Algebra Seminars

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

Kazhdan's property (T) and semidefinite programming

**Narutaka Ozawa**(RIMS, Kyoto University)Kazhdan's property (T) and semidefinite programming

### 2017/12/21

#### Applied Analysis

16:00-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)

#### Mathematical Biology Seminar

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

Theoretical approaches to understand eco-evolutionary feedbacks

**Masato Yamamichi**(Department of General Systems Studies, The University of Tokyo)Theoretical approaches to understand eco-evolutionary feedbacks

### 2017/12/19

#### Numerical Analysis Seminar

16:50-18:20 Room #128 (Graduate School of Math. Sci. Bldg.)

#### Tuesday Seminar on Topology

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

Deformation of holomorphic quadratic differentials and its applications (JAPANESE)

**Hideki Miyachi**(Osaka university)Deformation of holomorphic quadratic differentials and its applications (JAPANESE)

[ Abstract ]

Quadratic differentials are standard and important objects in Teichmuller theory. The deformation space (moduli space) of the quadratic differentials is applied to many fields of mathematics. In this talk, I will develop the deformation of quadratic differentials. Indeed, following pioneer works by A. Douady, J. Hubbard, H. Masur and W. Veech, we describe the infinitesimal deformations in the odd (co)homology groups on the double covering spaces defined from the square roots of the quadratic differentials. We formulate the decomposition theorem for the infinitesimal deformations with keeping in mind of the induced deformation of the moduli of underlying complex structures. As applications, we obtain the Levi form of the Teichmuller distance, and an alternate proof of the Krushkal formula on the pluricomplex Green function on the Teichmuller space.

Quadratic differentials are standard and important objects in Teichmuller theory. The deformation space (moduli space) of the quadratic differentials is applied to many fields of mathematics. In this talk, I will develop the deformation of quadratic differentials. Indeed, following pioneer works by A. Douady, J. Hubbard, H. Masur and W. Veech, we describe the infinitesimal deformations in the odd (co)homology groups on the double covering spaces defined from the square roots of the quadratic differentials. We formulate the decomposition theorem for the infinitesimal deformations with keeping in mind of the induced deformation of the moduli of underlying complex structures. As applications, we obtain the Levi form of the Teichmuller distance, and an alternate proof of the Krushkal formula on the pluricomplex Green function on the Teichmuller space.

### 2017/12/18

#### Seminar on Geometric Complex Analysis

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

Gradient flow of the Ding functional

**Tomoyuki Hisamoto**(Nagoya University)Gradient flow of the Ding functional

[ Abstract ]

This is a joint work with T. Collins and R. Takahashi. We introduce the flow in the title to study the stability of a Fano manifold. The first result is the long-time existence of the flow. In the stable case it then converges to the Kähler-Einstein metric. In general the flow is expected to produce the optimally destabilizing degeneration of a Fano manifold. We confirm this expectation in the toric case.

This is a joint work with T. Collins and R. Takahashi. We introduce the flow in the title to study the stability of a Fano manifold. The first result is the long-time existence of the flow. In the stable case it then converges to the Kähler-Einstein metric. In general the flow is expected to produce the optimally destabilizing degeneration of a Fano manifold. We confirm this expectation in the toric case.

#### Operator Algebra Seminars

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

Isomorphism and Morita equivalence classes for crossed product of irrational rotation algebras by cyclic subgroups of $SL_2({\mathbb Z})$ (English)

**Zhuofeng He**(Univ. Tokyo)Isomorphism and Morita equivalence classes for crossed product of irrational rotation algebras by cyclic subgroups of $SL_2({\mathbb Z})$ (English)

### 2017/12/14

#### Algebraic Geometry Seminar

15:30-17:00 Room #123 (Graduate School of Math. Sci. Bldg.)

Algebraic curves and modular forms of low degree (English)

**Gerard van der Geer**(Universiteit van Amsterdam)Algebraic curves and modular forms of low degree (English)

[ Abstract ]

For genus 2 and 3 modular forms are intimately connected with the moduli of curves of genus 2 and 3. We give an explicit way to describe such modular forms for genus 2 and 3 using invariant theory and give some applications. This is based on joint work with Fabien Clery and Carel Faber.

For genus 2 and 3 modular forms are intimately connected with the moduli of curves of genus 2 and 3. We give an explicit way to describe such modular forms for genus 2 and 3 using invariant theory and give some applications. This is based on joint work with Fabien Clery and Carel Faber.

#### Applied Analysis

16:00-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)

Regularity for diffuse reflection boundary problem to the stationary linearized Boltzmann equation in a convex domain

(English)

**I-Kun, Chen**(Kyoto University)Regularity for diffuse reflection boundary problem to the stationary linearized Boltzmann equation in a convex domain

(English)

[ Abstract ]

We consider the diffuse reflection boundary problem for the linearized Boltzmann equation for hard sphere potential, cutoff hard potential, or Maxwellian molecular gases in a $C^2$ strictly convex bounded domain. We obtain a pointwise estimate for the derivative of the solution provided the boundary temperature is bounded differentiable and the solution is bounded. Velocity averaging effect for stationary solutions as well as observations in geometry are used in this research.

We consider the diffuse reflection boundary problem for the linearized Boltzmann equation for hard sphere potential, cutoff hard potential, or Maxwellian molecular gases in a $C^2$ strictly convex bounded domain. We obtain a pointwise estimate for the derivative of the solution provided the boundary temperature is bounded differentiable and the solution is bounded. Velocity averaging effect for stationary solutions as well as observations in geometry are used in this research.

#### Algebraic Geometry Seminar

10:30-12:00 Room #123 (Graduate School of Math. Sci. Bldg.)

Perfectoid test ideals (English)

**Linquan Ma**(University of Utah)Perfectoid test ideals (English)

[ Abstract ]

Inspired by the recent solution of the direct summand conjecture

of Andre and Bhatt, we introduce perfectoid multiplier/test ideals in mixed

characteristic. As an application, we obtain a uniform bound on the growth

of symbolic powers in regular local rings of mixed characteristic analogous

to results of Ein--Lazarsfeld--Smith and Hochster--Huneke in equal

characteristic. This is joint work with Karl Schwede.

Inspired by the recent solution of the direct summand conjecture

of Andre and Bhatt, we introduce perfectoid multiplier/test ideals in mixed

characteristic. As an application, we obtain a uniform bound on the growth

of symbolic powers in regular local rings of mixed characteristic analogous

to results of Ein--Lazarsfeld--Smith and Hochster--Huneke in equal

characteristic. This is joint work with Karl Schwede.

#### Mathematical Biology Seminar

13:00-16:40 Room #126 (Graduate School of Math. Sci. Bldg.)

Mathematical analysis for HBV model and HBV-HDV coinfection model (ENGLISH)

**Xu Yaya**15:40-16:10Mathematical analysis for HBV model and HBV-HDV coinfection model (ENGLISH)

[ Abstract ]

The hepatitis beta virus (HBV) and hepatitis delta viurs (HDV)

are two common forms of viral hepatitis. However HDV is dependent

on coinfection with HBV since replication of HDV requires the hepati-

tis B surface antigen (HBsAg) which can only been produced by HBV.

Here we start with analyzing HBV only model, the dynamics between

healthy cells, HBV infected cells and free HBV.We show that a postive

equilbrium exsits and it's globally asmptotically stable for R0 > 1, an

infection free equilibrium is globally asymptotically stable for R0 < 1.

Then we introduce HDV to form a coinfection model which contains

three more variables, HDV infected cells, coinfected cells and free HDV.

Additionally, we investigate two coinfection models, one without and

one with treatment by oral drugs which are valid for HBV only. We

consider several durgs with variable eciencies. As a result, compari-

son of model simulations indicate that treatment is necessary to taking

contiously for choric infection.

The hepatitis beta virus (HBV) and hepatitis delta viurs (HDV)

are two common forms of viral hepatitis. However HDV is dependent

on coinfection with HBV since replication of HDV requires the hepati-

tis B surface antigen (HBsAg) which can only been produced by HBV.

Here we start with analyzing HBV only model, the dynamics between

healthy cells, HBV infected cells and free HBV.We show that a postive

equilbrium exsits and it's globally asmptotically stable for R0 > 1, an

infection free equilibrium is globally asymptotically stable for R0 < 1.

Then we introduce HDV to form a coinfection model which contains

three more variables, HDV infected cells, coinfected cells and free HDV.

Additionally, we investigate two coinfection models, one without and

one with treatment by oral drugs which are valid for HBV only. We

consider several durgs with variable eciencies. As a result, compari-

son of model simulations indicate that treatment is necessary to taking

contiously for choric infection.

### 2017/12/13

#### Number Theory Seminar

18:00-19:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Exponential motives (ENGLISH)

**Javier Fresán**(École polytechnique)Exponential motives (ENGLISH)

[ Abstract ]

What motives are to algebraic varieties, exponential motives are to pairs (X, f) consisting of an algebraic variety over some field k and a regular function f on X. In characteristic zero, one is naturally led to define the de Rham and rapid decay cohomology of such pairs when dealing with numbers like the special values of the gamma function or the Euler constant gamma which are not expected to be periods in the usual sense. Over finite fields, the étale and rigid cohomology groups of (X, f) play a pivotal role in the study of exponential sums.

Following ideas of Katz, Kontsevich, and Nori, we construct a Tannakian category of exponential motives when k is a subfield of the complex numbers. This allows one to attach to exponential periods a Galois group that conjecturally governs all algebraic relations among them. The category is equipped with a Hodge realisation functor with values in mixed Hodge modules over the affine line and, if k is a number field, with an étale realisation related to exponential sums. This is a joint work with Peter Jossen (ETH).

What motives are to algebraic varieties, exponential motives are to pairs (X, f) consisting of an algebraic variety over some field k and a regular function f on X. In characteristic zero, one is naturally led to define the de Rham and rapid decay cohomology of such pairs when dealing with numbers like the special values of the gamma function or the Euler constant gamma which are not expected to be periods in the usual sense. Over finite fields, the étale and rigid cohomology groups of (X, f) play a pivotal role in the study of exponential sums.

Following ideas of Katz, Kontsevich, and Nori, we construct a Tannakian category of exponential motives when k is a subfield of the complex numbers. This allows one to attach to exponential periods a Galois group that conjecturally governs all algebraic relations among them. The category is equipped with a Hodge realisation functor with values in mixed Hodge modules over the affine line and, if k is a number field, with an étale realisation related to exponential sums. This is a joint work with Peter Jossen (ETH).

#### FMSP Lectures

17:00-17:45 Room #470 (Graduate School of Math. Sci. Bldg.)

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

[ Abstract ]

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.

[ Reference 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

### 2017/12/12

#### PDE Real Analysis Seminar

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

Stochastic Three-Dimensional Navier-Stokes Equations + Waves: Averaging, Convergence, Regularity and Nonlinear Dynamics (English)

**Alex Mahalov**(Arizona State University)Stochastic Three-Dimensional Navier-Stokes Equations + Waves: Averaging, Convergence, Regularity and Nonlinear Dynamics (English)

[ Abstract ]

We establish multi-scale stochastic averaging, convergence and regularity theorems in a general framework by bootstrapping from global regularity of the averaged stochastic resonant equations. The averaged covariance operator couples stochastic and wave effects. We also present theoretical results for 3D nonlinear dynamics.

We establish multi-scale stochastic averaging, convergence and regularity theorems in a general framework by bootstrapping from global regularity of the averaged stochastic resonant equations. The averaged covariance operator couples stochastic and wave effects. We also present theoretical results for 3D nonlinear dynamics.

#### Tuesday Seminar on Topology

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

On the self-intersection of singular sets of maps and the signature defect (JAPANESE)

**Tatsuro Shimizu**(RIMS, Kyoto university)On the self-intersection of singular sets of maps and the signature defect (JAPANESE)

[ Abstract ]

Let $M$ be a closed oriented $n$-dimensional manifold. We give a geometric proof of that the $k$-times self-intersection of singular set of a Morin map from $M$ to $R^p$ coincides with the corank $k$ singular set of any generic map from $M$ to $R^{p+k-1}$ as homology classes with $Z/2$ coefficient ($n>p+k-2$). As an application we give a description of the signature defect of framed 3-manifold from the point of view of singular sets of maps.

Let $M$ be a closed oriented $n$-dimensional manifold. We give a geometric proof of that the $k$-times self-intersection of singular set of a Morin map from $M$ to $R^p$ coincides with the corank $k$ singular set of any generic map from $M$ to $R^{p+k-1}$ as homology classes with $Z/2$ coefficient ($n>p+k-2$). As an application we give a description of the signature defect of framed 3-manifold from the point of view of singular sets of maps.

### 2017/12/11

#### Seminar on Geometric Complex Analysis

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

Nishino's rigidity theorem and questions on locally pseudoconvex maps

**Takeo Ohsawa**(Nagoya University)Nishino's rigidity theorem and questions on locally pseudoconvex maps

[ Abstract ]

Nishino proved in 1969 that locally Stein maps with fibers $\cong \mathbb{C}$ are locally trivial. Yamaguchi gave an alternate proof of Nishino's theorem which later developed into a the theory of variations of the Bergman kernel. The proofs of Nishino and Yamaguchi will be reviewed and questions suggested by the result will be discussed. A new application of the $L^2$ extension theorem will be also presented in this context.

Nishino proved in 1969 that locally Stein maps with fibers $\cong \mathbb{C}$ are locally trivial. Yamaguchi gave an alternate proof of Nishino's theorem which later developed into a the theory of variations of the Bergman kernel. The proofs of Nishino and Yamaguchi will be reviewed and questions suggested by the result will be discussed. A new application of the $L^2$ extension theorem will be also presented in this context.

### 2017/12/05

#### Tuesday Seminar on Topology

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

Derivations and cohomologies of Lipschitz algebras (JAPANESE)

**Kazuhiro Kawamura**(University of Tsukuba)Derivations and cohomologies of Lipschitz algebras (JAPANESE)

[ Abstract ]

For a compact metric space M, Lip(M) denotes the Banach algebra of all complex-valued Lipschitz functions on M. Motivated by a classical work of de Leeuw, we define a compact, not necessarily metrizable, Hausdorff space \hat{M} so that each point of \hat{M} induces a derivation on Lip(M). To some extent, \hat{M} may be regarded as "the space of directions." We study, by an elementary method, the space of derivations and continuous Hochschild cohomologies (in the sense of B.E. Johnson and A.Y. Helemskii) of Lip(M) with coefficients C(\hat{M}) and C(M). The results so obtained show that the behavior of Lip(M) is (naturally) rather different than that of the algebra of smooth/C^1 functions on M.

For a compact metric space M, Lip(M) denotes the Banach algebra of all complex-valued Lipschitz functions on M. Motivated by a classical work of de Leeuw, we define a compact, not necessarily metrizable, Hausdorff space \hat{M} so that each point of \hat{M} induces a derivation on Lip(M). To some extent, \hat{M} may be regarded as "the space of directions." We study, by an elementary method, the space of derivations and continuous Hochschild cohomologies (in the sense of B.E. Johnson and A.Y. Helemskii) of Lip(M) with coefficients C(\hat{M}) and C(M). The results so obtained show that the behavior of Lip(M) is (naturally) rather different than that of the algebra of smooth/C^1 functions on M.

#### Algebraic Geometry Seminar

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

Ascending chain condition for F-pure thresholds on a fixed strongly F-regular germ (English or Japanese)

**Kenta Sato**(The University of Tokyo)Ascending chain condition for F-pure thresholds on a fixed strongly F-regular germ (English or Japanese)

[ Abstract ]

For a germ of a variety in positive characteristic and a non-zero ideal sheaf on the variety, we can define the F-pure threshold of the ideal by using Frobenius morphisms, which measures the singularities of the pair. In this talk, I will show that the set of all F-pure thresholds on a fixed strongly F-regular germ satisfies the ascending chain condition. This is a positive characteristic analogue of the "ascending chain condition for log canonical thresholds" in characteristic 0, which was recently proved by Hacon, McKernan, and Xu.

For a germ of a variety in positive characteristic and a non-zero ideal sheaf on the variety, we can define the F-pure threshold of the ideal by using Frobenius morphisms, which measures the singularities of the pair. In this talk, I will show that the set of all F-pure thresholds on a fixed strongly F-regular germ satisfies the ascending chain condition. This is a positive characteristic analogue of the "ascending chain condition for log canonical thresholds" in characteristic 0, which was recently proved by Hacon, McKernan, and Xu.

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