## Seminar information archive

Seminar information archive ～09/18｜Today's seminar 09/19 | Future seminars 09/20～

#### 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.

### 2017/12/04

#### Tokyo Probability Seminar

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

Some results for range of random walk on graph with spectral dimension two (JAPANESE)

**Kazuki Okamura**(Research Institute for Mathematical Sciences, Kyoto University)Some results for range of random walk on graph with spectral dimension two (JAPANESE)

[ Abstract ]

We consider the range of random walk on graphs with spectral dimension two. We show that a certain weak law of large numbers hold if a recurrent graph satisfies a uniform condition. We construct a recurrent graph such that the uniform condition holds but appropriately scaled expectations fluctuate. Our result is applicable to showing LILs for lamplighter random walks in the case that the spectral dimension of the underlying graph is two.

We consider the range of random walk on graphs with spectral dimension two. We show that a certain weak law of large numbers hold if a recurrent graph satisfies a uniform condition. We construct a recurrent graph such that the uniform condition holds but appropriately scaled expectations fluctuate. Our result is applicable to showing LILs for lamplighter random walks in the case that the spectral dimension of the underlying graph is two.

#### Operator Algebra Seminars

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

Subfactors and wiretapping channels

(English)

**Pieter Naaijkens**(UC Davis)Subfactors and wiretapping channels

(English)

### 2017/11/28

#### Tuesday Seminar on Topology

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

Diffeomorphism Groups of One-Manifolds (ENGLISH)

**Sang-hyun Kim**(Seoul National University)Diffeomorphism Groups of One-Manifolds (ENGLISH)

[ Abstract ]

Let a>=2 be a real number and k = [a]. We denote by Diff^a(S^1) the group of C^k diffeomorphisms such that the k--th derivatives are Hölder--continuous of exponent (a - k). For each real number a>=2, we prove that there exists a finitely generated group G < Diff^a(S^1) such that G admits no injective homomorphisms into Diff^b(S^1) for any b>a. This is joint work with Thomas Koberda.

Let a>=2 be a real number and k = [a]. We denote by Diff^a(S^1) the group of C^k diffeomorphisms such that the k--th derivatives are Hölder--continuous of exponent (a - k). For each real number a>=2, we prove that there exists a finitely generated group G < Diff^a(S^1) such that G admits no injective homomorphisms into Diff^b(S^1) for any b>a. This is joint work with Thomas Koberda.

#### Algebraic Geometry Seminar

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

Kodaira vanishing theorem for Witt canonical sheaves (English)

**Hiromu Tanaka**(Tokyo)Kodaira vanishing theorem for Witt canonical sheaves (English)

[ Abstract ]

We establish an analogue of the Kodaira vanishing theorem in terms of de Rham-Witt complex. More specifically, given a smooth projective variety over a perfect field of positive characteristic, we prove that the higher cohomologies vanish for the tensor product of the Witt canonical sheaf and the Teichmuller lift of an ample invertible sheaf.

We establish an analogue of the Kodaira vanishing theorem in terms of de Rham-Witt complex. More specifically, given a smooth projective variety over a perfect field of positive characteristic, we prove that the higher cohomologies vanish for the tensor product of the Witt canonical sheaf and the Teichmuller lift of an ample invertible sheaf.

#### Numerical Analysis Seminar

16:50-18:20 Room #117 (Graduate School of Math. Sci. Bldg.)

Hybrid discontinuous Galerkin methods for nearly incompressible elasticity problems

(Japanese)

**Daisuke Koyama**(The University of Electro-Communications)Hybrid discontinuous Galerkin methods for nearly incompressible elasticity problems

(Japanese)

[ Abstract ]

A Hybrid discontinuous Galerkin (HDG) method for linear elasticity problems has been introduced by Kikuchi et al. [Theor. Appl. Mech. Japan, vol.57, 395--404 (2009)], [RIMS Kokyuroku, vol.1971, 28--46 (2015)]. We consider to seek numerical solutions of the plane strain problem by the HDG method, especially in the case when materials are nearly incompressible, that is, when the first Lam\'e parameter $\lambda$ is large. In this talk, we consider two cases when the HDG method uses a lifting term and does not use it. When the lifting term is used, the method can be free of volumetric locking. On the other hand, when the lifting term is not used, we have to take an interior penalty parameter of order $\lambda$ as $\lambda$ tends to infinity, in order to guarantee the coercivity of the bilinear form. Taking such an interior penalty parameter causes volumetric locking phenomena. We thus conclude that the lifting term is essential for avoiding the volumetric locking in the HDG method.

A Hybrid discontinuous Galerkin (HDG) method for linear elasticity problems has been introduced by Kikuchi et al. [Theor. Appl. Mech. Japan, vol.57, 395--404 (2009)], [RIMS Kokyuroku, vol.1971, 28--46 (2015)]. We consider to seek numerical solutions of the plane strain problem by the HDG method, especially in the case when materials are nearly incompressible, that is, when the first Lam\'e parameter $\lambda$ is large. In this talk, we consider two cases when the HDG method uses a lifting term and does not use it. When the lifting term is used, the method can be free of volumetric locking. On the other hand, when the lifting term is not used, we have to take an interior penalty parameter of order $\lambda$ as $\lambda$ tends to infinity, in order to guarantee the coercivity of the bilinear form. Taking such an interior penalty parameter causes volumetric locking phenomena. We thus conclude that the lifting term is essential for avoiding the volumetric locking in the HDG method.

### 2017/11/27

#### Tokyo Probability Seminar

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

Random Recursive Tree, Branching Markov Chains and Urn Models (ENGLISH)

**Antar Bandyopadhyay**(Indian Statistical Institute)Random Recursive Tree, Branching Markov Chains and Urn Models (ENGLISH)

[ Abstract ]

In this talk, we will establish a connection between random recursive tree, branching Markov chain and urn model. Exploring the connection further we will derive fairly general scaling limits for urn models with colors indexed by a Polish Space and show that several exiting results on classical/non-classical urn schemes can be easily derived out of such general asymptotic. We will further show that the connection can be used to derive exact asymptotic for the sizes of the connected components of a "random recursive forest", obtained by removing the root of a random recursive tree.

[This is a joint work with Debleena Thacker]

In this talk, we will establish a connection between random recursive tree, branching Markov chain and urn model. Exploring the connection further we will derive fairly general scaling limits for urn models with colors indexed by a Polish Space and show that several exiting results on classical/non-classical urn schemes can be easily derived out of such general asymptotic. We will further show that the connection can be used to derive exact asymptotic for the sizes of the connected components of a "random recursive forest", obtained by removing the root of a random recursive tree.

[This is a joint work with Debleena Thacker]

#### Seminar on Geometric Complex Analysis

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

On the proof of the optimal $L^2$ extension theorem by Berndtsson-Lempert and related results

**Genki Hosono**(The University of Tokyo)On the proof of the optimal $L^2$ extension theorem by Berndtsson-Lempert and related results

[ Abstract ]

We will present the recent progress on the Ohsawa-Takegoshi $L^2$ extension theorem. A version of the Ohsawa-Takegoshi $L^2$ extension with a optimal estimate has been proved by Blocki and Guan-Zhou. After that, by Berndtsson-Lempert, a new proof of the optimal $L^2$ extension theorem was given. In this talk, we will show an optimal $L^2$ extension theorem for jets of holomorphic functions by the Berndtsson-Lempert method. We will also explain the recent result about jet extensions by McNeal-Varolin. Their proof is also based on Berndtsson-Lempert, but there are some differences.

We will present the recent progress on the Ohsawa-Takegoshi $L^2$ extension theorem. A version of the Ohsawa-Takegoshi $L^2$ extension with a optimal estimate has been proved by Blocki and Guan-Zhou. After that, by Berndtsson-Lempert, a new proof of the optimal $L^2$ extension theorem was given. In this talk, we will show an optimal $L^2$ extension theorem for jets of holomorphic functions by the Berndtsson-Lempert method. We will also explain the recent result about jet extensions by McNeal-Varolin. Their proof is also based on Berndtsson-Lempert, but there are some differences.

### 2017/11/24

#### Colloquium

15:30-16:30 Room #002 (Graduate School of Math. Sci. Bldg.)

(JAPANESE)

**Yukari Ito**(IPMU, Nagoya University)(JAPANESE)

### 2017/11/21

#### Tuesday Seminar on Topology

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

The space of short ropes and the classifying space of the space of long knots (JAPANESE)

**Keiichi Sakai**(Shinshu University)The space of short ropes and the classifying space of the space of long knots (JAPANESE)

[ Abstract ]

We prove affirmatively the conjecture raised by J. Mostovoy; the space of short ropes is weakly homotopy equivalent to the classifying space of the topological monoid (or category) of long knots in R^3. We make use of techniques developed by S. Galatius and O. Randal-Williams to construct a manifold space model of the classifying space of the space of long knots, and we give an explicit map from the space of short ropes to the model in a geometric way. This is joint work with Syunji Moriya (Osaka Prefecture University).

We prove affirmatively the conjecture raised by J. Mostovoy; the space of short ropes is weakly homotopy equivalent to the classifying space of the topological monoid (or category) of long knots in R^3. We make use of techniques developed by S. Galatius and O. Randal-Williams to construct a manifold space model of the classifying space of the space of long knots, and we give an explicit map from the space of short ropes to the model in a geometric way. This is joint work with Syunji Moriya (Osaka Prefecture University).

#### PDE Real Analysis Seminar

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

Optimal isoperimetric inequalities for surfaces in any codimension

in Cartan-Hadamard manifolds (English)

**Felix Schulze**(University College London)Optimal isoperimetric inequalities for surfaces in any codimension

in Cartan-Hadamard manifolds (English)

[ Abstract ]

Let $(M^n,g)$ be simply connected, complete, with non-positive sectional

curvatures, and $\Sigma$ a 2-dimensional surface in $M^n$. Let $S$ be an area

minimising 3-current such that $\partial S = \Sigma$. We use a weak mean

curvature flow, obtained via elliptic regularisation, starting from

$\Sigma$, to show that $S$ satisfies the optimal Euclidean isoperimetric

inequality: $|S| \leq 1/(6\sqrt{\pi}) |\Sigma|^{3/2}$. We also obtain the

optimal estimate in case the sectional curvatures of $M$ are bounded from

above by $\kappa < 0$ and characterise the case of equality. The proof

follows from an almost monotonicity of a suitable isoperimetric

difference along the approximating flows in one dimension higher.

Let $(M^n,g)$ be simply connected, complete, with non-positive sectional

curvatures, and $\Sigma$ a 2-dimensional surface in $M^n$. Let $S$ be an area

minimising 3-current such that $\partial S = \Sigma$. We use a weak mean

curvature flow, obtained via elliptic regularisation, starting from

$\Sigma$, to show that $S$ satisfies the optimal Euclidean isoperimetric

inequality: $|S| \leq 1/(6\sqrt{\pi}) |\Sigma|^{3/2}$. We also obtain the

optimal estimate in case the sectional curvatures of $M$ are bounded from

above by $\kappa < 0$ and characterise the case of equality. The proof

follows from an almost monotonicity of a suitable isoperimetric

difference along the approximating flows in one dimension higher.

#### Algebraic Geometry Seminar

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

Orbifold rational connectedness (English)

**Frédéric Campana**(Université de Lorraine/KIAS)Orbifold rational connectedness (English)

[ Abstract ]

The first step in the decomposition by canonical fibrations with fibres of `signed' canonical bundle of an arbitrary complex projective manifolds $X$ is its `rational quotient' (also called `MRC' fibration): it has rationally connected fibres and non-uniruled base. In general, the further steps (such as the Moishezon-Iitaka fibration) of this decomposition will require the consideration of 'orbifold base' of fibrations in order to deal with the multiple fibres (as seen already for elliptic surfaces). One thus needs to work in the larger category of (smooth) `orbifold pairs' $(X,D)$ to achieve this decomposition. The aim of the talk is thus to introduce the notions of Rational Connectedness and 'rational quotient' in this context, by means of suitable equivalent notions of negativity for the orbifold cotangent bundle (suitably defined. When $D$ is reduced, this is just the usual Log-version). The expected equivalence with connecting families of `orbifold rational curves' remains however presently open.

The first step in the decomposition by canonical fibrations with fibres of `signed' canonical bundle of an arbitrary complex projective manifolds $X$ is its `rational quotient' (also called `MRC' fibration): it has rationally connected fibres and non-uniruled base. In general, the further steps (such as the Moishezon-Iitaka fibration) of this decomposition will require the consideration of 'orbifold base' of fibrations in order to deal with the multiple fibres (as seen already for elliptic surfaces). One thus needs to work in the larger category of (smooth) `orbifold pairs' $(X,D)$ to achieve this decomposition. The aim of the talk is thus to introduce the notions of Rational Connectedness and 'rational quotient' in this context, by means of suitable equivalent notions of negativity for the orbifold cotangent bundle (suitably defined. When $D$ is reduced, this is just the usual Log-version). The expected equivalence with connecting families of `orbifold rational curves' remains however presently open.

### 2017/11/20

#### Seminar on Geometric Complex Analysis

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

Relative GIT stabilities of toric Fano manifolds in low dimensions

**Yasufumi Nitta**(Tokyo Institute of Technology)Relative GIT stabilities of toric Fano manifolds in low dimensions

[ Abstract ]

In 2000, Mabuchi extended the notion of Kaehler-Einstein metrics to Fano manifolds with non-vanishing Futaki invariant. Such a metric is called generalized Kaehler-Einstein metric or Mabuchi metric in the literature. Recently this metrics were rediscovered by Yao in the story of Donaldson's infinite dimensional moment map picture. Moreover, he introduced (uniform) relative Ding stability for toric Fano manifolds and showed that the existence of generalized Kaehler-Einstein metrics is equivalent to its uniform relative Ding stability. This equivalence is in the context of the Yau-Tian-Donaldson conjecture. In this talk, we focus on uniform relative Ding stability of toric Fano manifolds. More precisely, we determine all the uniformly relatively Ding stable toric Fano 3- and 4-folds as well as unstable ones. This talk is based on a joint work with Shunsuke Saito and Naoto Yotsutani.

In 2000, Mabuchi extended the notion of Kaehler-Einstein metrics to Fano manifolds with non-vanishing Futaki invariant. Such a metric is called generalized Kaehler-Einstein metric or Mabuchi metric in the literature. Recently this metrics were rediscovered by Yao in the story of Donaldson's infinite dimensional moment map picture. Moreover, he introduced (uniform) relative Ding stability for toric Fano manifolds and showed that the existence of generalized Kaehler-Einstein metrics is equivalent to its uniform relative Ding stability. This equivalence is in the context of the Yau-Tian-Donaldson conjecture. In this talk, we focus on uniform relative Ding stability of toric Fano manifolds. More precisely, we determine all the uniformly relatively Ding stable toric Fano 3- and 4-folds as well as unstable ones. This talk is based on a joint work with Shunsuke Saito and Naoto Yotsutani.

### 2017/11/16

#### Mathematical Biology Seminar

16:30-18:00 Room #123 (Graduate School of Math. Sci. Bldg.)

(JAPANESE)

**Jun Nakabayashi**(Yokohama City University)(JAPANESE)

#### Seminar on Probability and Statistics

13：00-16：00 Room #123 (Graduate School of Math. Sci. Bldg.)

### 2017/11/15

#### PDE Real Analysis Seminar

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

Boundary value problems for parabolic equations with measurable coefficients (English)

**Kaj Nyström**(Uppsala University)Boundary value problems for parabolic equations with measurable coefficients (English)

[ Abstract ]

In recent joint works with P. Auscher and M. Egert we establish new results concerning boundary value problems in the upper half-space for second order parabolic equations (and systems) assuming only measurability and some transversal regularity in the coefficients of the elliptic part. To establish our results we introduce and develop a first order strategy by means of a parabolic Dirac operator at the boundary to obtain, in particular, Green's representation for solutions in natural classes involving square functions and non-tangential maximal functions, well-posedness results with data in $L^2$-Sobolev spaces together with invertibility of layer potentials, and perturbation results. In addition we solve the Kato square root problem for parabolic operators with coefficients of the elliptic part depending measurably on all variables. Using these results we are also able to solve the $L^p$-Dirichlet problem for parabolic equations with real, time-dependent, elliptic but non-symmetric coefficients. In this talk I will briefly describe some of these developments.

In recent joint works with P. Auscher and M. Egert we establish new results concerning boundary value problems in the upper half-space for second order parabolic equations (and systems) assuming only measurability and some transversal regularity in the coefficients of the elliptic part. To establish our results we introduce and develop a first order strategy by means of a parabolic Dirac operator at the boundary to obtain, in particular, Green's representation for solutions in natural classes involving square functions and non-tangential maximal functions, well-posedness results with data in $L^2$-Sobolev spaces together with invertibility of layer potentials, and perturbation results. In addition we solve the Kato square root problem for parabolic operators with coefficients of the elliptic part depending measurably on all variables. Using these results we are also able to solve the $L^p$-Dirichlet problem for parabolic equations with real, time-dependent, elliptic but non-symmetric coefficients. In this talk I will briefly describe some of these developments.

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