## 過去の記録

過去の記録 ～10/22｜本日 10/23 | 今後の予定 10/24～

### 2017年07月07日(金)

#### 談話会・数理科学講演会

15:30-16:30 数理科学研究科棟(駒場) 002号室

Smith Normal Form and Combinatorics (English)

http://www-math.mit.edu/~rstan/

**Richard Stanley 氏**(MIT/University of Miami)Smith Normal Form and Combinatorics (English)

[ 講演概要 ]

Let $R$ be a commutative ring (with identity) and $A$ an $n \times n$ matrix over $R$. Suppose there exist $n \times n$ matrices $P,Q$ invertible over $R$ for which PAQ is a diagonal matrix $diag(e_1,...,e_r,0,...,0)$, where $e_i$ divides $e_{i+1}$ in $R$. We then call $PAQ$ a Smith normal form (SNF) of $A$. If $R$ is a PID then an SNF always exists and is unique up to multiplication by units. Moreover if $A$ is invertible then $\det A=ua_1\cdots a_n$, where $u$ is a unit, so SNF gives a

canonical factorization of $\det A$.

We will survey some connections between SNF and combinatorics. Topics will include (1) the general theory of SNF, (2) a close connection between SNF and chip firing in graphs, (3) the SNF of a random matrix of integers (joint work with Yinghui Wang), (4) SNF of special classes of matrices, including some arising in the theory of symmetric functions, hyperplane arrangements, and lattice paths.

[ 講演参考URL ]Let $R$ be a commutative ring (with identity) and $A$ an $n \times n$ matrix over $R$. Suppose there exist $n \times n$ matrices $P,Q$ invertible over $R$ for which PAQ is a diagonal matrix $diag(e_1,...,e_r,0,...,0)$, where $e_i$ divides $e_{i+1}$ in $R$. We then call $PAQ$ a Smith normal form (SNF) of $A$. If $R$ is a PID then an SNF always exists and is unique up to multiplication by units. Moreover if $A$ is invertible then $\det A=ua_1\cdots a_n$, where $u$ is a unit, so SNF gives a

canonical factorization of $\det A$.

We will survey some connections between SNF and combinatorics. Topics will include (1) the general theory of SNF, (2) a close connection between SNF and chip firing in graphs, (3) the SNF of a random matrix of integers (joint work with Yinghui Wang), (4) SNF of special classes of matrices, including some arising in the theory of symmetric functions, hyperplane arrangements, and lattice paths.

http://www-math.mit.edu/~rstan/

### 2017年07月04日(火)

#### 代数幾何学セミナー

15:30-17:00 数理科学研究科棟(駒場) 122号室

The space of rational curves and Manin's conjecture (English)

**谷本 祥 氏**(University of Copenhagen)The space of rational curves and Manin's conjecture (English)

[ 講演概要 ]

Manin's conjecture predicts the asymptotic formula for the counting function of rational points on a Fano variety after removing the exceptional thin set. There are many developments on birational geometry of exceptional sets using MMP, due to Lehmann, myself, Tschinkel, Hacon, and Jiang. Recently we found that the study of exceptional sets has applications to questions regarding the space of rational curves, i.e., its dimension and the number of components. I would like to explain these applications. This is joint work with Brian Lehmann.

Manin's conjecture predicts the asymptotic formula for the counting function of rational points on a Fano variety after removing the exceptional thin set. There are many developments on birational geometry of exceptional sets using MMP, due to Lehmann, myself, Tschinkel, Hacon, and Jiang. Recently we found that the study of exceptional sets has applications to questions regarding the space of rational curves, i.e., its dimension and the number of components. I would like to explain these applications. This is joint work with Brian Lehmann.

#### 数値解析セミナー

16:50-18:20 数理科学研究科棟(駒場) 002号室

Boundary conditions for Limited-Area Models (English)

**Ming-Cheng Shiue 氏**(National Chiao Tung University)Boundary conditions for Limited-Area Models (English)

[ 講演概要 ]

The problem of boundary conditions in a limited domain is recognized an important problem in geophysical fluid dynamics. This is due to that boundary conditions are proposed to have high resolution over a region of interest. The challenges for proposing later boundary conditions are of two types: on the computational side, if the proposed boundary conditions are not appropriate, it is well-known that the error from the lateral boundary can propagate into the computational domain and make a major effect on the numerical solution; on the mathematical side, the negative result of Oliger and Sundstrom that these equations including the inviscid primitive equations and shallow water equations in the multilayer case are not well-posed for any set of local boundary conditions.

In this talk, three-dimensional inviscid primitive equations and (one-layer and two-layer) shallow water equations which have been used in the limited-area numerical weather prediction modelings are considered. Our goals of this work are two folds: one is to propose boundary conditions which are physically suitable. That is, they let waves move freely out of the domain without producing spurious waves; the other is to numerically implement these boundary conditions by proposing suitable numerical methods. Numerical experiments are presented to demonstrate that these proposed boundary conditions and numerical schemes are suitable.

The problem of boundary conditions in a limited domain is recognized an important problem in geophysical fluid dynamics. This is due to that boundary conditions are proposed to have high resolution over a region of interest. The challenges for proposing later boundary conditions are of two types: on the computational side, if the proposed boundary conditions are not appropriate, it is well-known that the error from the lateral boundary can propagate into the computational domain and make a major effect on the numerical solution; on the mathematical side, the negative result of Oliger and Sundstrom that these equations including the inviscid primitive equations and shallow water equations in the multilayer case are not well-posed for any set of local boundary conditions.

In this talk, three-dimensional inviscid primitive equations and (one-layer and two-layer) shallow water equations which have been used in the limited-area numerical weather prediction modelings are considered. Our goals of this work are two folds: one is to propose boundary conditions which are physically suitable. That is, they let waves move freely out of the domain without producing spurious waves; the other is to numerically implement these boundary conditions by proposing suitable numerical methods. Numerical experiments are presented to demonstrate that these proposed boundary conditions and numerical schemes are suitable.

#### トポロジー火曜セミナー

17:00-18:30 数理科学研究科棟(駒場) 056号室

Tea: Common Room 16:30-17:00

On link-homotopy for knotted surfaces in 4-space (ENGLISH)

Tea: Common Room 16:30-17:00

**Jean-Baptiste Meilhan 氏**(Université Grenoble Alpes)On link-homotopy for knotted surfaces in 4-space (ENGLISH)

[ 講演概要 ]

The purpose of this talk is to show how combinatorial objects (welded objects, which is a natural quotient of virtual knot theory) can be used to study knotted surfaces in 4-space.

We will first consider the case of 'ribbon' knotted surfaces, which are embedded surfaces bounding immersed 3-manifolds with only ribbon singularities. More precisely, we will consider ribbon knotted annuli ; these objects act naturally on the reduced free group, and we prove, using welded theory, that this action gives a classification up to link-homotopy, that is, up to continuous deformations leaving distinct component disjoint. This in turns implies a classification result for ribbon knotted tori.

Next, we will show how to extend this classification result beyond the ribbon case.

This is based on joint works with B. Audoux, P. Bellingeri and E. Wagner.

The purpose of this talk is to show how combinatorial objects (welded objects, which is a natural quotient of virtual knot theory) can be used to study knotted surfaces in 4-space.

We will first consider the case of 'ribbon' knotted surfaces, which are embedded surfaces bounding immersed 3-manifolds with only ribbon singularities. More precisely, we will consider ribbon knotted annuli ; these objects act naturally on the reduced free group, and we prove, using welded theory, that this action gives a classification up to link-homotopy, that is, up to continuous deformations leaving distinct component disjoint. This in turns implies a classification result for ribbon knotted tori.

Next, we will show how to extend this classification result beyond the ribbon case.

This is based on joint works with B. Audoux, P. Bellingeri and E. Wagner.

### 2017年07月03日(月)

#### 東京確率論セミナー

16:00-17:30 数理科学研究科棟(駒場) 126号室

Equilibrium fluctuation for a chain of anharmonic oscillators (JAPANESE)

**徐 路 氏**(九州大学 大学院数理学研究院)Equilibrium fluctuation for a chain of anharmonic oscillators (JAPANESE)

[ 講演概要 ]

A chain of oscillators is a particle system whose microscopic time evolution is given by Hamilton equations with various kinds of conservative noises. Mathematicians and physicians are interested in its macroscopic behaviors (ε → 0) under different space-time scales: ballistic (hyperbolic) (εx, εt), diffusive (εx, ε^2t) and superdiffusive (εx, ε^αt) for 1 < α < 2. In this talk, we consider a 1-dimensional chain of anharmonic oscillators perturbed by noises preserving the total momentum as well as the total energy. We present a result about the hyperbolic scaling limit of its equilibrium fluctuation as well as some further discussions. (A joint work with S. Olla, Université Paris-Dauphine)

A chain of oscillators is a particle system whose microscopic time evolution is given by Hamilton equations with various kinds of conservative noises. Mathematicians and physicians are interested in its macroscopic behaviors (ε → 0) under different space-time scales: ballistic (hyperbolic) (εx, εt), diffusive (εx, ε^2t) and superdiffusive (εx, ε^αt) for 1 < α < 2. In this talk, we consider a 1-dimensional chain of anharmonic oscillators perturbed by noises preserving the total momentum as well as the total energy. We present a result about the hyperbolic scaling limit of its equilibrium fluctuation as well as some further discussions. (A joint work with S. Olla, Université Paris-Dauphine)

#### 複素解析幾何セミナー

10:30-12:00 数理科学研究科棟(駒場) 128号室

Holomorphic isometric embeddings into Grassmannians of rank $2$

**長友 康行 氏**(明治大学)Holomorphic isometric embeddings into Grassmannians of rank $2$

[ 講演概要 ]

We suppose that Grassmannians are equipped with the standard Kähler metrics of Fubini-Study type. This means that the ${\it universal \ quotient}$ bundles over Grassmannians are provided with not only fibre metrics but also connections. Such connections are called the ${\it canonical}$ connection.

First of all, we classify $\text{SU}(2)$ equivariant holomorphic embeddings of the complex projective line into complex Grassmannians of $2$-planes. To do so, we focus our attention on the pull-back connection of the canonical connection, which is an $\text{SU}(2)$ invariant connection by our hypothesis. We use ${\it extensions}$ of vector bundles to classify $\text{SU}(2)$ invariant connections on vector bundles of rank $2$ over the complex projective line. Since the extensions are in one-to-one correspondence with $H^1(\mathbf CP^1;\mathcal O(-2))$, the moduli space of non-trivial invariant connections modulo gauge transformations is identified with the quotient space of $H^1(\mathbf CP^1;\mathcal O(-2))$ by $S^1$-action. The positivity of the mean curvature of the pull-back connection implies that the moduli spaces of $\text{SU}(2)$ equivariant holomorphic embeddings are the open intervals $(0,l)$, where $l$ depends only on the ${\it degree}$ of the map.

Next, we describe moduli spaces of holomorphic isometric embeddings of the complex projective line into complex quadric hypersurfaces of the projective spaces. A harmonic map from a Riemannian manifold into a Grassmannian is characterized by the universal quotient bundle, a space of sections of the bundle and the Laplace operator. This characterization can be considered as a generalization of Theorem of Takahashi on minimal immersions into a sphere (J.Math.Soc.Japan 18 (1966)). Due to this, we can generalize do Carmo-Wallach theory. We apply a generalized do Carmo-Wallach theory to obtain the moduli spaces. This method also gives a description of the moduli space of Einstein-Hermitian harmonic maps with constant Kähler angles of the complex projective line into complex quadrics. It turns out that the moduli space is diffeomorphic to the moduli of holomorphic isometric embeddings of the same degree.

We suppose that Grassmannians are equipped with the standard Kähler metrics of Fubini-Study type. This means that the ${\it universal \ quotient}$ bundles over Grassmannians are provided with not only fibre metrics but also connections. Such connections are called the ${\it canonical}$ connection.

First of all, we classify $\text{SU}(2)$ equivariant holomorphic embeddings of the complex projective line into complex Grassmannians of $2$-planes. To do so, we focus our attention on the pull-back connection of the canonical connection, which is an $\text{SU}(2)$ invariant connection by our hypothesis. We use ${\it extensions}$ of vector bundles to classify $\text{SU}(2)$ invariant connections on vector bundles of rank $2$ over the complex projective line. Since the extensions are in one-to-one correspondence with $H^1(\mathbf CP^1;\mathcal O(-2))$, the moduli space of non-trivial invariant connections modulo gauge transformations is identified with the quotient space of $H^1(\mathbf CP^1;\mathcal O(-2))$ by $S^1$-action. The positivity of the mean curvature of the pull-back connection implies that the moduli spaces of $\text{SU}(2)$ equivariant holomorphic embeddings are the open intervals $(0,l)$, where $l$ depends only on the ${\it degree}$ of the map.

Next, we describe moduli spaces of holomorphic isometric embeddings of the complex projective line into complex quadric hypersurfaces of the projective spaces. A harmonic map from a Riemannian manifold into a Grassmannian is characterized by the universal quotient bundle, a space of sections of the bundle and the Laplace operator. This characterization can be considered as a generalization of Theorem of Takahashi on minimal immersions into a sphere (J.Math.Soc.Japan 18 (1966)). Due to this, we can generalize do Carmo-Wallach theory. We apply a generalized do Carmo-Wallach theory to obtain the moduli spaces. This method also gives a description of the moduli space of Einstein-Hermitian harmonic maps with constant Kähler angles of the complex projective line into complex quadrics. It turns out that the moduli space is diffeomorphic to the moduli of holomorphic isometric embeddings of the same degree.

#### 作用素環セミナー

16:45-18:15 数理科学研究科棟(駒場) 118号室

A State-Dependent Noncontextuality Inequality in Algebraic Quantum Theory

**北島雄一郎 氏**(日大生産工)A State-Dependent Noncontextuality Inequality in Algebraic Quantum Theory

### 2017年06月28日(水)

#### 数理人口学・数理生物学セミナー

14:55-15:45 数理科学研究科棟(駒場) 122号室

Stabilizing role of maturation delay on prey-predator dynamics (ENGLISH)

**Malay Banerjee 氏**(Department of Mathematics & Statistics，IIT Kanpur)Stabilizing role of maturation delay on prey-predator dynamics (ENGLISH)

[ 講演概要 ]

Discrete and continuous time delays are often introduced into mathematical models of interacting populations to take into account stage-structuring of one or more species. There are other aspects for the incorporation of time delays. In prey-predator models, maturation time delay is introduced to the growth equation of predators to implicitly model the stage-structure of predators. Most of the prey-predator models with maturation delay are known to exhibit regular and rregular, even chaotic, oscillations due to destabilization of coexistence steady-state when maturation time period is significantly large. However, such kind of instability can results in due to the introduction of maturation delay into predator’s growth equation with lack of ecological justification and inappropriate choice of the length of time delay. Recently we have worked on a class of delayed prey-predator models, where discrete time delay represents the maturation time for specialist predator implicitly, with ratio-dependent functional response [1] and Michaelis-Menten type

functional response [2]. We have established (i) the stabilizing role of maturation delay, (ii)extinction of predator for significantly long maturation period and (iii) suppression of Hopf bifurcation for large time delay, when the delayed model is constructed with appropriate biological rationale. Main objective of this talk is to discuss analytical results for the stable coexistence of both the species for a class of delayed prey-predator models with maturation delay for specialist predator. Analytical results will be illustrated with the help of numerical simulation results and appropriate bifurcation diagrams with time delay as bifurcation parameter. Main content of this talk is based upon the recent work with Prof. Y. Takeuchi [2].

References:

[1] M. Sen, M. Banerjee, A. Morozov. (2014). Stage-structured ratio-dependent predatorprey models revisited: When should the maturation lag result in systems destabilization?, Ecological Complexity, 19(2), 23–34.

[2] M. Banerjee, Y. Takeuchi. (2017). Maturation delay for the predators can enhance stable coexistence for a class of prey-predator models, Journal of Theoretical Biology, 412, 154–171.

Discrete and continuous time delays are often introduced into mathematical models of interacting populations to take into account stage-structuring of one or more species. There are other aspects for the incorporation of time delays. In prey-predator models, maturation time delay is introduced to the growth equation of predators to implicitly model the stage-structure of predators. Most of the prey-predator models with maturation delay are known to exhibit regular and rregular, even chaotic, oscillations due to destabilization of coexistence steady-state when maturation time period is significantly large. However, such kind of instability can results in due to the introduction of maturation delay into predator’s growth equation with lack of ecological justification and inappropriate choice of the length of time delay. Recently we have worked on a class of delayed prey-predator models, where discrete time delay represents the maturation time for specialist predator implicitly, with ratio-dependent functional response [1] and Michaelis-Menten type

functional response [2]. We have established (i) the stabilizing role of maturation delay, (ii)extinction of predator for significantly long maturation period and (iii) suppression of Hopf bifurcation for large time delay, when the delayed model is constructed with appropriate biological rationale. Main objective of this talk is to discuss analytical results for the stable coexistence of both the species for a class of delayed prey-predator models with maturation delay for specialist predator. Analytical results will be illustrated with the help of numerical simulation results and appropriate bifurcation diagrams with time delay as bifurcation parameter. Main content of this talk is based upon the recent work with Prof. Y. Takeuchi [2].

References:

[1] M. Sen, M. Banerjee, A. Morozov. (2014). Stage-structured ratio-dependent predatorprey models revisited: When should the maturation lag result in systems destabilization?, Ecological Complexity, 19(2), 23–34.

[2] M. Banerjee, Y. Takeuchi. (2017). Maturation delay for the predators can enhance stable coexistence for a class of prey-predator models, Journal of Theoretical Biology, 412, 154–171.

#### 数理人口学・数理生物学セミナー

15:50-16:40 数理科学研究科棟(駒場) 122号室

Allee effect induced rich dynamics of a two prey one predator model where the predator is

generalist (ENGLISH)

**Moitri Sen 氏**(Department. of Mathematics, National Institute of Technology Patna)Allee effect induced rich dynamics of a two prey one predator model where the predator is

generalist (ENGLISH)

[ 講演概要 ]

One of the important ecological challenges is to capture the chaotic dynamics and understand the underlying regulating factors. Allee effect is one of the important factors in ecology and taking it into account can cause signicant changes to the system dynamics. In this work we propose a two prey-one predator model where the growth of both the prey population is governed by Allee effect, and the predator is generalist and hence survived on both the prey populations. We analyze the role of Allee eect on the chaotic dynamics of the system. Interestingly we have observed through a comprehensive bifurcation study that incorporation of Allee eect enriches the dynamics of the system. Specially after a certain threshold of the Allee eect, it has a very signicant eect on the chaotic dynamics of the system. In course of the bifurcation analysis we have explored all possible bifurca-tions such as namely the existence of transcritical bifurcation, saddle-node bifurcation, Hopf-bifurcation, Bogdanov-Takens bifurcation and Bautin bifurcation and period-doubling route to chaos respectively.

One of the important ecological challenges is to capture the chaotic dynamics and understand the underlying regulating factors. Allee effect is one of the important factors in ecology and taking it into account can cause signicant changes to the system dynamics. In this work we propose a two prey-one predator model where the growth of both the prey population is governed by Allee effect, and the predator is generalist and hence survived on both the prey populations. We analyze the role of Allee eect on the chaotic dynamics of the system. Interestingly we have observed through a comprehensive bifurcation study that incorporation of Allee eect enriches the dynamics of the system. Specially after a certain threshold of the Allee eect, it has a very signicant eect on the chaotic dynamics of the system. In course of the bifurcation analysis we have explored all possible bifurca-tions such as namely the existence of transcritical bifurcation, saddle-node bifurcation, Hopf-bifurcation, Bogdanov-Takens bifurcation and Bautin bifurcation and period-doubling route to chaos respectively.

### 2017年06月27日(火)

#### 代数幾何学セミナー

15:30-17:00 数理科学研究科棟(駒場) 122号室

Cylinders in del Pezzo fibrations (English )

**岸本 崇 氏**(埼玉大学)Cylinders in del Pezzo fibrations (English )

[ 講演概要 ]

The cylinder is, by definition, an algebraic variety of the form Z × A1 . Certainly it is geometrically a very simple object, but it plays often an important role to connect unipotent group actions on special kinds of affine algebraic varieties to projective geometry. From the point of view of birational geometry, it is essential to look into cylinders found on Mori fiber spaces. In this talk, we shall focus mainly on Mori fiber spaces of relative dimension two or three. One of main results asserts that a del Pezzo fibration π : V → W contains a cylinder respecting the structure of π (so-called a vertical cylinder) if and only if the degree deg π of π is greater than or equal to 5 and π admits a rational section. Especially, in case of dim V = 3, the existence of a vertical cylinder is equivalent to saying deg π ≧ 5 in consideration of Tsen’s theorem, nevertheless, it is worthwhile to note that the affine 3-space A3C is embedded into certains del Pezzo fibrations π : V → P1C of deg π ≦ 4 in a twisted way. This is a joint work with Adrien Dubouloz (Universit ́e de Bourgogne).

The cylinder is, by definition, an algebraic variety of the form Z × A1 . Certainly it is geometrically a very simple object, but it plays often an important role to connect unipotent group actions on special kinds of affine algebraic varieties to projective geometry. From the point of view of birational geometry, it is essential to look into cylinders found on Mori fiber spaces. In this talk, we shall focus mainly on Mori fiber spaces of relative dimension two or three. One of main results asserts that a del Pezzo fibration π : V → W contains a cylinder respecting the structure of π (so-called a vertical cylinder) if and only if the degree deg π of π is greater than or equal to 5 and π admits a rational section. Especially, in case of dim V = 3, the existence of a vertical cylinder is equivalent to saying deg π ≧ 5 in consideration of Tsen’s theorem, nevertheless, it is worthwhile to note that the affine 3-space A3C is embedded into certains del Pezzo fibrations π : V → P1C of deg π ≦ 4 in a twisted way. This is a joint work with Adrien Dubouloz (Universit ́e de Bourgogne).

#### トポロジー火曜セミナー

17:00-18:30 数理科学研究科棟(駒場) 056号室

Tea: Common Room 16:30-17:00, RIKEN iTHEMS と共催

Braids and hyperbolic 3-manifolds from simple mixing devices (JAPANESE)

Tea: Common Room 16:30-17:00, RIKEN iTHEMS と共催

**金 英子 氏**(大阪大学)Braids and hyperbolic 3-manifolds from simple mixing devices (JAPANESE)

[ 講演概要 ]

Taffy pullers are devices for pulling candy. One can build braids from the motion of rods for taffy pullers. According to a beautiful article ``A mathematical history of taffy pullers" by Jean-Luc Thiffeault, all taffy pullers (except the first one) give rise to pseudo-Anosov braids. This means that the devices mix candies effectively. Following a study of Thiffeault, I will discuss which pseudo-Anosov braid is realized by taffy pullers. I will explain an interesting connection between braids coming from taffy pullers. I also discuss the hyperbolic mapping tori obtained from taffy pullers. Intriguingly, the two most common taffy pullers give rise to the complements of the the minimally twisted 4-chain link and 5-chain link which are important examples for the study of cusped hyperbolic 3-manifolds with small volumes.

Reference: A mathematical history of taffy pullers, Jean-Luc Thiffeault, https://arxiv.org/pdf/1608.00152.pdf

Taffy pullers are devices for pulling candy. One can build braids from the motion of rods for taffy pullers. According to a beautiful article ``A mathematical history of taffy pullers" by Jean-Luc Thiffeault, all taffy pullers (except the first one) give rise to pseudo-Anosov braids. This means that the devices mix candies effectively. Following a study of Thiffeault, I will discuss which pseudo-Anosov braid is realized by taffy pullers. I will explain an interesting connection between braids coming from taffy pullers. I also discuss the hyperbolic mapping tori obtained from taffy pullers. Intriguingly, the two most common taffy pullers give rise to the complements of the the minimally twisted 4-chain link and 5-chain link which are important examples for the study of cusped hyperbolic 3-manifolds with small volumes.

Reference: A mathematical history of taffy pullers, Jean-Luc Thiffeault, https://arxiv.org/pdf/1608.00152.pdf

### 2017年06月26日(月)

#### 複素解析幾何セミナー

10:30-12:00 数理科学研究科棟(駒場) 128号室

Volume minimization and obstructions to geometric problems

**二木 昭人 氏**(東京大学)Volume minimization and obstructions to geometric problems

[ 講演概要 ]

We discuss on the volume minimization principle for conformally Kaehler Einstein-Maxwell metrics in the similar spirit as the Kaehler-Ricci solitons and Sasaki-Einstein metrics. This talk is base on a joint work with Hajime Ono.

We discuss on the volume minimization principle for conformally Kaehler Einstein-Maxwell metrics in the similar spirit as the Kaehler-Ricci solitons and Sasaki-Einstein metrics. This talk is base on a joint work with Hajime Ono.

#### 作用素環セミナー

16:45-18:15 数理科学研究科棟(駒場) 118号室

Dimension, comparison, and almost finiteness (English)

**David Kerr 氏**(Texas A & M Univ.)Dimension, comparison, and almost finiteness (English)

### 2017年06月20日(火)

#### トポロジー火曜セミナー

17:00-18:30 数理科学研究科棟(駒場) 056号室

Tea: Common Room 16:30-17:00

Introduction to the AJ conjecture (ENGLISH)

Tea: Common Room 16:30-17:00

**Anh Tran 氏**(The University of Texas at Dallas)Introduction to the AJ conjecture (ENGLISH)

[ 講演概要 ]

The AJ conjecture was proposed by Garoufalidis about 15 years ago. It predicts a strong connection between two important knot invariants derived from very different background, namely the colored Jones function (a quantum invariant) and the A-polynomial (a geometric invariant). The colored Jones function is a sequence of Laurent polynomials which is known to satisfy a linear q-difference equation. The AJ conjecture states that by writing the linear q-difference equation into an operator form and setting q=1, one gets the A-polynomial. In this talk, I will give an introduction to this conjecture.

The AJ conjecture was proposed by Garoufalidis about 15 years ago. It predicts a strong connection between two important knot invariants derived from very different background, namely the colored Jones function (a quantum invariant) and the A-polynomial (a geometric invariant). The colored Jones function is a sequence of Laurent polynomials which is known to satisfy a linear q-difference equation. The AJ conjecture states that by writing the linear q-difference equation into an operator form and setting q=1, one gets the A-polynomial. In this talk, I will give an introduction to this conjecture.

#### 談話会・数理科学講演会

15:30-16:30 数理科学研究科棟(駒場) 002号室

Some stochastic population models in a random environment (English)

http://www.ummisco.ird.fr/perso/bacaer/

**Nicolas Bacaër 氏**(研究開発研究所/東大数理)Some stochastic population models in a random environment (English)

[ 講演概要 ]

Two population models will be considered: an epidemic model [1] and a linear birth-and-death process [2]. The goal is to study the first non-zero eigenvalue, which is related to the speed of convergence towards extinction, using either WKB approximations or probabilistic arguments.

[1] "Le modèle stochastique SIS pour une épidémie dans un environnement aléatoire". Journal of Mathematical Biology (2016)

[2] "Sur les processus linéaires de naissance et de mort sous-critiques dans un environnement aléatoire". Journal of Mathematical Biology (2017)

[ 講演参考URL ]Two population models will be considered: an epidemic model [1] and a linear birth-and-death process [2]. The goal is to study the first non-zero eigenvalue, which is related to the speed of convergence towards extinction, using either WKB approximations or probabilistic arguments.

[1] "Le modèle stochastique SIS pour une épidémie dans un environnement aléatoire". Journal of Mathematical Biology (2016)

[2] "Sur les processus linéaires de naissance et de mort sous-critiques dans un environnement aléatoire". Journal of Mathematical Biology (2017)

http://www.ummisco.ird.fr/perso/bacaer/

### 2017年06月19日(月)

#### 複素解析幾何セミナー

10:30-12:00 数理科学研究科棟(駒場) 128号室

$Q$-prime curvature and Sasakian $\eta$-Einstein manifolds

**竹内 有哉 氏**(東京大学)$Q$-prime curvature and Sasakian $\eta$-Einstein manifolds

[ 講演概要 ]

The $Q$-prime curvature is defined for a pseudo-Einstein contact form on a strictly pseudoconvex CR manifold, and its integral, the total $Q$-prime curvature, defines a global CR invariant under some assumptions. In this talk, we will compute the $Q$-prime curvature for Sasakian $\eta$-Einstein manifolds. We will also study the first and the second variation of the total $Q$-prime curvature under deformations of real hypersurfaces at Sasakian $\eta$-Einstein manifolds.

The $Q$-prime curvature is defined for a pseudo-Einstein contact form on a strictly pseudoconvex CR manifold, and its integral, the total $Q$-prime curvature, defines a global CR invariant under some assumptions. In this talk, we will compute the $Q$-prime curvature for Sasakian $\eta$-Einstein manifolds. We will also study the first and the second variation of the total $Q$-prime curvature under deformations of real hypersurfaces at Sasakian $\eta$-Einstein manifolds.

#### 東京確率論セミナー

16:00-17:30 数理科学研究科棟(駒場) 126号室

Computation of first-order Greeks for barrier options using chain rules for Wiener path integrals (JAPANESE)

**石谷 謙介 氏**(首都大学東京 大学院理工学研究科)Computation of first-order Greeks for barrier options using chain rules for Wiener path integrals (JAPANESE)

[ 講演概要 ]

In this presentation, we present a new methodology to compute first-order Greeks for barrier options under the framework of path-dependent payoff functions with European, Lookback, or Asian type and with time-dependent trigger levels. In particular, we develop chain rules for Wiener path integrals between two curves that arise in the computation of first-order Greeks for barrier options. We also illustrate the effectiveness of our method through numerical examples.

In this presentation, we present a new methodology to compute first-order Greeks for barrier options under the framework of path-dependent payoff functions with European, Lookback, or Asian type and with time-dependent trigger levels. In particular, we develop chain rules for Wiener path integrals between two curves that arise in the computation of first-order Greeks for barrier options. We also illustrate the effectiveness of our method through numerical examples.

### 2017年06月14日(水)

#### 代数学コロキウム

17:30-18:30 数理科学研究科棟(駒場) 056号室

Multiplicity one for the mod p cohomology of Shimura curves (ENGLISH)

[ 講演参考URL ]

http://www.ms.u-tokyo.ac.jp/~t-saito/title_Hu.pdf

**Yongquan Hu 氏**(Chinese Academy of Sciences, Morningside Center of Mathematics)Multiplicity one for the mod p cohomology of Shimura curves (ENGLISH)

[ 講演参考URL ]

http://www.ms.u-tokyo.ac.jp/~t-saito/title_Hu.pdf

### 2017年06月13日(火)

#### 数値解析セミナー

16:50-18:20 数理科学研究科棟(駒場) 002号室

Numerical analysis of viscoelastic fluid models (Japanese)

**野津裕史 氏**(金沢大学理工研究域)Numerical analysis of viscoelastic fluid models (Japanese)

[ 講演概要 ]

Numerical methods for viscoelastic fluid models are studied. In viscoelastic fluid models the stress tensor is often written as a sum of the viscous stress tensor depending linearly on the strain rate tensor and the extra stress tensor for the viscoelastic contribution. In order to describe the viscoelastic contribution another equation for the extra stress tensor is required. In the talk we mainly deal with the Oldroyd-B and the Peterlin models among several proposed viscoelastic fluid models, and present error estimates of finite element schemes based on the method of characteristics. The key issue in the estimates is the treatment of the divergence of the extra stress tensor appearing in the equation for the velocity and the pressure.

Numerical methods for viscoelastic fluid models are studied. In viscoelastic fluid models the stress tensor is often written as a sum of the viscous stress tensor depending linearly on the strain rate tensor and the extra stress tensor for the viscoelastic contribution. In order to describe the viscoelastic contribution another equation for the extra stress tensor is required. In the talk we mainly deal with the Oldroyd-B and the Peterlin models among several proposed viscoelastic fluid models, and present error estimates of finite element schemes based on the method of characteristics. The key issue in the estimates is the treatment of the divergence of the extra stress tensor appearing in the equation for the velocity and the pressure.

#### トポロジー火曜セミナー

17:00-18:30 数理科学研究科棟(駒場) 056号室

Tea: Common Room 16:30-17:00

Local criteria for non-embeddability of Levi-flat manifolds (JAPANESE)

Tea: Common Room 16:30-17:00

**小川 竜 氏**(東海大学)Local criteria for non-embeddability of Levi-flat manifolds (JAPANESE)

[ 講演概要 ]

In this talk, we consider the Levi-flat embedding problem. Barrett showed that a smooth Reeb foliation on S^3 cannot be realized as a Levi-flat hypersurface in any complex surfaces. To do this, he focused the relationship between the holonomy along the compact leaf and a system of its defining functions. We will show a new criterion for non-embeddability of Levi-flat manifolds. Our result is a higher dimensional analogue of Barrett's theorem. In particular, this enables us to weaken the compactness assumption. For this purpose, we pose a partial generalization of Ueda theory on the analytic neighborhood structure of complex hypersurfaces. This talk is based on a joint work with Takayuki Koike (Kyoto University).

In this talk, we consider the Levi-flat embedding problem. Barrett showed that a smooth Reeb foliation on S^3 cannot be realized as a Levi-flat hypersurface in any complex surfaces. To do this, he focused the relationship between the holonomy along the compact leaf and a system of its defining functions. We will show a new criterion for non-embeddability of Levi-flat manifolds. Our result is a higher dimensional analogue of Barrett's theorem. In particular, this enables us to weaken the compactness assumption. For this purpose, we pose a partial generalization of Ueda theory on the analytic neighborhood structure of complex hypersurfaces. This talk is based on a joint work with Takayuki Koike (Kyoto University).

### 2017年06月12日(月)

#### 複素解析幾何セミナー

10:30-12:00 数理科学研究科棟(駒場) 128号室

On Sp(2)-invariant asymptotically complex hyperbolic Einstein metrics on the 8-ball

**松本 佳彦 氏**(大阪大学)On Sp(2)-invariant asymptotically complex hyperbolic Einstein metrics on the 8-ball

[ 講演概要 ]

Following a pioneering work of Pedersen, Hitchin studied SU(2)-invariant asymptotically real/complex hyperbolic (often abbreviated as AH/ACH) solution to the Einstein equation on the 4-dimensional unit open ball. We discuss a similar problem on the 8-ball, on which the quaternionic unitary group Sp(2) acts naturally, focusing on ACH solutions rather than AH ones. The Einstein equation is treated as an asymptotic Dirichlet problem, and the Dirichlet data are Sp(2)-invariant “partially integrable” CR structures on the 7-sphere. A remarkable point is that most of such structures are actually non-integrable. I will present how we can practically compute the formal series expansion of the Einstein ACH metric corresponding to a given Dirichlet data, that is, an invariant partially integrable CR structure on the sphere.

Following a pioneering work of Pedersen, Hitchin studied SU(2)-invariant asymptotically real/complex hyperbolic (often abbreviated as AH/ACH) solution to the Einstein equation on the 4-dimensional unit open ball. We discuss a similar problem on the 8-ball, on which the quaternionic unitary group Sp(2) acts naturally, focusing on ACH solutions rather than AH ones. The Einstein equation is treated as an asymptotic Dirichlet problem, and the Dirichlet data are Sp(2)-invariant “partially integrable” CR structures on the 7-sphere. A remarkable point is that most of such structures are actually non-integrable. I will present how we can practically compute the formal series expansion of the Einstein ACH metric corresponding to a given Dirichlet data, that is, an invariant partially integrable CR structure on the sphere.

#### 代数幾何学セミナー

17:00-18:30 数理科学研究科棟(駒場) 056号室

普段と曜日・部屋が異なります

Rational and irrational singular quartic threefolds (English)

普段と曜日・部屋が異なります

**Ivan Cheltsov 氏**(The University of Edinburgh)Rational and irrational singular quartic threefolds (English)

[ 講演概要 ]

Burkhardt and Igusa quartics admit a faithful action of the symmetric group of degree 6.

There are other quartic threefolds with this property. All of them are singular.

Beauville proved that all but four of them are irrational. Burkhardt and Igusa quartics are known to be rational.

Two constructions of Todd imply the rationality of the remaining two quartic threefolds.

In this talk, I will give an alternative proof of both these (irrationality and rationality) results.

This proof is based on explicit small resolutions of the so-called Coble fourfold.

This fourfold is the double cover of the four-dimensional projective space branched over Igusa quartic.

This is a joint work with Sasha Kuznetsov and Costya Shramov.

Burkhardt and Igusa quartics admit a faithful action of the symmetric group of degree 6.

There are other quartic threefolds with this property. All of them are singular.

Beauville proved that all but four of them are irrational. Burkhardt and Igusa quartics are known to be rational.

Two constructions of Todd imply the rationality of the remaining two quartic threefolds.

In this talk, I will give an alternative proof of both these (irrationality and rationality) results.

This proof is based on explicit small resolutions of the so-called Coble fourfold.

This fourfold is the double cover of the four-dimensional projective space branched over Igusa quartic.

This is a joint work with Sasha Kuznetsov and Costya Shramov.

### 2017年06月06日(火)

#### 代数幾何学セミナー

15:30-17:00 数理科学研究科棟(駒場) 122号室

Fano varieties: K-stability and boundedness (English)

https://sites.google.com/site/chenjiangmath/

**Chen Jiang 氏**(IPMU)Fano varieties: K-stability and boundedness (English)

[ 講演概要 ]

There are two interesting problems for Fano varieties, K-stability and boundedness.

Significant progress has been made for both problems recently.

In this talk, I will show the boundedness of K-semistable Fano varieties with anti-canonical degree bounded from below, by using methods from birational geometry.

[ 講演参考URL ]There are two interesting problems for Fano varieties, K-stability and boundedness.

Significant progress has been made for both problems recently.

In this talk, I will show the boundedness of K-semistable Fano varieties with anti-canonical degree bounded from below, by using methods from birational geometry.

https://sites.google.com/site/chenjiangmath/

#### トポロジー火曜セミナー

17:00-18:30 数理科学研究科棟(駒場) 056号室

Tea: Common Room 16:30-17:00

A formula for the action of Dehn twists on the HOMFLY-PT type skein algebra and its application (JAPANESE)

Tea: Common Room 16:30-17:00

**辻 俊輔 氏**(東京大学大学院数理科学研究科)A formula for the action of Dehn twists on the HOMFLY-PT type skein algebra and its application (JAPANESE)

[ 講演概要 ]

We give an explicit formula for the action of the Dehn twist along a simple closed curve of a surface on the completed HOMFLY-PT type skein modules of the surface in terms of the action of the completed HOMFLY-PT type skein algebra of the surface. As an application, using this formula, we construct an invariant for an integral homology 3-sphere which is an element of Q[ρ] [[h]].

We give an explicit formula for the action of the Dehn twist along a simple closed curve of a surface on the completed HOMFLY-PT type skein modules of the surface in terms of the action of the completed HOMFLY-PT type skein algebra of the surface. As an application, using this formula, we construct an invariant for an integral homology 3-sphere which is an element of Q[ρ] [[h]].

### 2017年06月01日(木)

#### 古典解析セミナー

16:30-18:00 数理科学研究科棟(駒場) 122号室

Homological and monodromy representations of framed braid groups

(JAPANESE)

**池田 曉志 氏**(東京大学 IPMU)Homological and monodromy representations of framed braid groups

(JAPANESE)

[ 講演概要 ]

KZ方程式は配置空間上の可積分な微分方程式であり,そのモノドロミー表現を考えることで組みひも群の様々な表現が得られることはよく知られている. 2008年に神保-名古屋-Sunによって合流型のKZ方程式が導入された. この話では, 合流型のKZ方程式のモノドロミー表現を考えることで,枠付組みひも群(リボンの絡み方を表す群)の表現が得られることを説明する.

また, 枠付組みひも群の表現を, ある空間のホモロジー群を用いて構成し, 合流KZ方程式のモノドロミー表現との関係について説明する.

KZ方程式は配置空間上の可積分な微分方程式であり,そのモノドロミー表現を考えることで組みひも群の様々な表現が得られることはよく知られている. 2008年に神保-名古屋-Sunによって合流型のKZ方程式が導入された. この話では, 合流型のKZ方程式のモノドロミー表現を考えることで,枠付組みひも群(リボンの絡み方を表す群)の表現が得られることを説明する.

また, 枠付組みひも群の表現を, ある空間のホモロジー群を用いて構成し, 合流KZ方程式のモノドロミー表現との関係について説明する.

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