## 過去の記録

過去の記録 ～11/11｜本日 11/12 | 今後の予定 11/13～

### 2018年11月28日(水)

#### 作用素環セミナー

16:45-18:15 数理科学研究科棟(駒場) 126号室

Localization of signature for singular fiber bundles

**山下真由子 氏**(東大数理)Localization of signature for singular fiber bundles

### 2018年11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

Frobenius summands and the finite F-representation type (TBA)

**原伸生 氏**(東京農工大)Frobenius summands and the finite F-representation type (TBA)

[ 講演概要 ]

We are motivated by a question arising from commutative algebra, asking what kind of

graded rings in positive characteristic p have finite F-representation type. In geometric

setting, this is related to the problem to looking out for Frobenius summands. Namely,

given aline bundle L on a projective variety X, we want to know how many and what

kind of indecomposable direct summands appear in the direct sum decomposition of

the iterated Frobenius push-forwards of L. We will consider the problem in the following

two cases, although the present situation in (2) is far from satisfactory.

(1) two-dimensional normal graded rings (a joint work with Ryo Ohkawa)

(2) the anti-canonical ring of a quintic del Pezzo surface

We are motivated by a question arising from commutative algebra, asking what kind of

graded rings in positive characteristic p have finite F-representation type. In geometric

setting, this is related to the problem to looking out for Frobenius summands. Namely,

given aline bundle L on a projective variety X, we want to know how many and what

kind of indecomposable direct summands appear in the direct sum decomposition of

the iterated Frobenius push-forwards of L. We will consider the problem in the following

two cases, although the present situation in (2) is far from satisfactory.

(1) two-dimensional normal graded rings (a joint work with Ryo Ohkawa)

(2) the anti-canonical ring of a quintic del Pezzo surface

#### トポロジー火曜セミナー

17:00-18:00 数理科学研究科棟(駒場) 056号室

Tea: Common Room 16:30-17:00

Fixed points for group actions on non-positively curved spaces (JAPANESE)

Tea: Common Room 16:30-17:00

**加藤 本子 氏**(東京大学大学院数理科学研究科)Fixed points for group actions on non-positively curved spaces (JAPANESE)

[ 講演概要 ]

In this talk, we introduce a fixed point property of groups which is a broad generalization of Serre's property FA, and give a criterion for groups to have such a property. We also apply the criterion to show that various generalizations of Thompson's group T have fixed points whenever they act on finite dimensional non-positively curved metric spaces, including CAT(0) spaces. Since Thompson's group T is known to have fixed point free actions on infinite dimensional CAT(0) spaces, it follows that there is a group which acts on infinite dimensional CAT(0) spaces without global fixed points, but not on finite dimensional ones.

In this talk, we introduce a fixed point property of groups which is a broad generalization of Serre's property FA, and give a criterion for groups to have such a property. We also apply the criterion to show that various generalizations of Thompson's group T have fixed points whenever they act on finite dimensional non-positively curved metric spaces, including CAT(0) spaces. Since Thompson's group T is known to have fixed point free actions on infinite dimensional CAT(0) spaces, it follows that there is a group which acts on infinite dimensional CAT(0) spaces without global fixed points, but not on finite dimensional ones.

### 2018年11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

DGA-Models of variations of mixed Hodge structures (JAPANESE)

**糟谷久矢 氏**(大阪大学)DGA-Models of variations of mixed Hodge structures (JAPANESE)

[ 講演概要 ]

Mixed Hodge structureは（Projectiveとは限らない)代数多様体のコホモロジー等に現れる非常に重要な構造です。Variations of mixed Hodge structures(VMHS)とは複素多様体をパラメーターとして複素幾何学的に良い振る舞いをしながら変化するMixed Hodge structureたちのことです。今回のお話ではこのVMHSの代数的なモデルについて考えてみたいと思いいます。具体的にはMorganの Mixed Hodge diagramと呼ばれるケーラー多様体のde Rham複体（あるいは対数的 de Rham複体）を積構造込みで模した代数的な対象に対して、”(Unipotent)VMHSのようなもの"を定義します。このVMHSのようなものは純粋に代数的に定義されたものであるため、本来のVMHSのようにベースとなる空間のパラメーターごとにMixed Hodge structureをとる（ファイバーをとる）ことを自然にはできません。本講演ではこの"VMHSのようなもの”からいかにファイバーを取るかということをメインテーマにしてお話ししたいと思います。さらに時間があれば、本結果の幾何学的応用についてもお話ししたいと思います。特に今回の結果によりMorganのMixed Hodge structureとHainのMixed Hodge structureの深い関係が見えることをお話ししたいと思います。

Mixed Hodge structureは（Projectiveとは限らない)代数多様体のコホモロジー等に現れる非常に重要な構造です。Variations of mixed Hodge structures(VMHS)とは複素多様体をパラメーターとして複素幾何学的に良い振る舞いをしながら変化するMixed Hodge structureたちのことです。今回のお話ではこのVMHSの代数的なモデルについて考えてみたいと思いいます。具体的にはMorganの Mixed Hodge diagramと呼ばれるケーラー多様体のde Rham複体（あるいは対数的 de Rham複体）を積構造込みで模した代数的な対象に対して、”(Unipotent)VMHSのようなもの"を定義します。このVMHSのようなものは純粋に代数的に定義されたものであるため、本来のVMHSのようにベースとなる空間のパラメーターごとにMixed Hodge structureをとる（ファイバーをとる）ことを自然にはできません。本講演ではこの"VMHSのようなもの”からいかにファイバーを取るかということをメインテーマにしてお話ししたいと思います。さらに時間があれば、本結果の幾何学的応用についてもお話ししたいと思います。特に今回の結果によりMorganのMixed Hodge structureとHainのMixed Hodge structureの深い関係が見えることをお話ししたいと思います。

### 2018年11月21日(水)

#### 代数学コロキウム

17:00-18:00 数理科学研究科棟(駒場) 056号室

Poncelet games, confinement of algebraic integers, and hyperbolic Ax-Schanuel (ENGLISH)

**Yves André 氏**(Université Pierre et Marie Curie)Poncelet games, confinement of algebraic integers, and hyperbolic Ax-Schanuel (ENGLISH)

[ 講演概要 ]

We shall theorize and exemplify the problem of torsion values of sections of abelian schemes. This « unlikely intersection problem », which arises in various diophantine and algebro-geometric contexts, can be reformulated in a non-trivial way in terms of Kodaira-Spencer maps. A key tool toward its general solution is then provided by recent theorems of Ax-Schanuel type (joint work with P. Corvaja, U. Zannier, and partly Z. Gao).

We shall theorize and exemplify the problem of torsion values of sections of abelian schemes. This « unlikely intersection problem », which arises in various diophantine and algebro-geometric contexts, can be reformulated in a non-trivial way in terms of Kodaira-Spencer maps. A key tool toward its general solution is then provided by recent theorems of Ax-Schanuel type (joint work with P. Corvaja, U. Zannier, and partly Z. Gao).

#### 作用素環セミナー

16:45-18:15 数理科学研究科棟(駒場) 126号室

Orbit equivalence classes for free actions of free products of infinite abelian groups

**守山貴顕 氏**(東大数理)Orbit equivalence classes for free actions of free products of infinite abelian groups

### 2018年11月20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

Artin-Mazur height, Yobuko height and

Hodge-Wittt cohomologies

**中島 幸喜 氏**(東京電機大)Artin-Mazur height, Yobuko height and

Hodge-Wittt cohomologies

[ 講演概要 ]

A few years ago Yobuko has introduced the notion of

a delicate invariant for a proper smooth scheme over a perfect field $k$

of finite characteristic. (We call this invariant Yobuko height.)

This generalize the notion of the F-splitness due to Mehta-Srinivas.

In this talk we give relations between Artin-Mazur heights

and Yobuko heights. We also give a finiteness result on

Hodge-Witt cohomologies of a proper smooth scheme $X$ over $k$

with finite Yobuko height. If time permits, we give a cofinite type result on

the $p$-primary torsion part of Chow group of of $X$

of codimension 2 if $\dim X=3$.

A few years ago Yobuko has introduced the notion of

a delicate invariant for a proper smooth scheme over a perfect field $k$

of finite characteristic. (We call this invariant Yobuko height.)

This generalize the notion of the F-splitness due to Mehta-Srinivas.

In this talk we give relations between Artin-Mazur heights

and Yobuko heights. We also give a finiteness result on

Hodge-Witt cohomologies of a proper smooth scheme $X$ over $k$

with finite Yobuko height. If time permits, we give a cofinite type result on

the $p$-primary torsion part of Chow group of of $X$

of codimension 2 if $\dim X=3$.

#### 離散数理モデリングセミナー

15:00-16:30 数理科学研究科棟(駒場) 002号室

情報幾何とその応用ー深層学習の解明に向けて

**甘利俊一 氏**(理化学研究所)情報幾何とその応用ー深層学習の解明に向けて

[ 講演概要 ]

情報幾何の基本的考えを説明した後で、深層学習にどう関係するのか、統計神経力学の立場から話をする。

情報幾何の基本的考えを説明した後で、深層学習にどう関係するのか、統計神経力学の立場から話をする。

#### 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

Torelli group, Johnson kernel and invariants of homology 3-spheres (JAPANESE)

Tea: Common Room 16:30-17:00

**逆井 卓也 氏**(東京大学大学院数理科学研究科)Torelli group, Johnson kernel and invariants of homology 3-spheres (JAPANESE)

[ 講演概要 ]

There are two filtrations of the Torelli group: One is the lower central series and the other is the Johnson filtration. They are closely related to Johnson homomorphisms as well as finite type invariants of homology 3-spheres. We compare the associated graded Lie algebras of the filtrations and report our explicit computational results. Then we discuss some applications of our computations. In particular, we give an explicit description of the rational abelianization of the Johnson kernel. This is a joint work with Shigeyuki Morita and Masaaki Suzuki.

There are two filtrations of the Torelli group: One is the lower central series and the other is the Johnson filtration. They are closely related to Johnson homomorphisms as well as finite type invariants of homology 3-spheres. We compare the associated graded Lie algebras of the filtrations and report our explicit computational results. Then we discuss some applications of our computations. In particular, we give an explicit description of the rational abelianization of the Johnson kernel. This is a joint work with Shigeyuki Morita and Masaaki Suzuki.

### 2018年11月19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) 128号室

Two-dimensional stochastic interface growth (ENGLISH)

http://math.univ-lyon1.fr/~toninelli/

**Fabio Toninelli 氏**(University Lyon 1)Two-dimensional stochastic interface growth (ENGLISH)

[ 講演概要 ]

I will discuss stochastic growth of two-dimensional, discrete interfaces, especially models in the so-called Anisotropic KPZ (AKPZ) class, that has the same large-scale behavior as the Stochastic Heat equation with additive noise. I will focus in particular on: 1) the relation between AKPZ exponents, convexity properties of the speed of growth and the preservation of the Gibbs property; and 2) the relation between singularities of the speed of growth and the occurrence of "smooth" (i.e. non-rough) stationary states.

[ 講演参考URL ]I will discuss stochastic growth of two-dimensional, discrete interfaces, especially models in the so-called Anisotropic KPZ (AKPZ) class, that has the same large-scale behavior as the Stochastic Heat equation with additive noise. I will focus in particular on: 1) the relation between AKPZ exponents, convexity properties of the speed of growth and the preservation of the Gibbs property; and 2) the relation between singularities of the speed of growth and the occurrence of "smooth" (i.e. non-rough) stationary states.

http://math.univ-lyon1.fr/~toninelli/

#### 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

BCOV invariants of Calabi-Yau varieties (ENGLISH)

**Gerard Freixas i Montplet 氏**(Centre National de la Recherche Scientifique)BCOV invariants of Calabi-Yau varieties (ENGLISH)

[ 講演概要 ]

The BCOV invariant of Calabi-Yau threefolds was introduced by Fang-Lu-Yoshikawa, themselves inspired by physicists Bershadsky-Cecotti-Ooguri-Vafa. It is a real number, obtained from a combination of holomorphic analytic torsion, and suitably normalized so that it only depends on the complex structure of the threefold. It is conjecturaly expected to encode genus 1 Gromov-Witten invariants of a mirror Calabi-Yau threefold. In order to confirm this prediction for a remarkable example, Fang-Lu-Yoshikawa studied the asymptotic behavior for degenerating families of Calabi-Yau threefolds acquiring at most ordinary double point (ODP) singularities. Their methods rely on former results by Yoshikawa on the singularities of Quillen metrics, together with more classical arguments in the theory of degenerations of Hodge structures and Hodge metrics. In this talk I will present joint work with Dennis Eriksson (Chalmers) and Christophe Mourougane (Rennes), where we extend the construction of the BCOV invariant to any dimension and we give precise asymptotic formulas for degenerating families of Calabi-Yau manifolds. Under several hypothesis on the geometry of the singularities acquired, our general formulas drastically simplify and prove some conjectures or predictions in the literature (Liu-Xia for semi-stable minimal families in dimension 3, Klemm-Pandharipande for ODP singularities in dimension 4, etc.). For this, we slightly improve Yoshikawa's results on the singularities of Quillen metrics, and we also provide a complement to Schmid's asymptotics of Hodge metrics when the monodromy transformations are non-unipotent.

The BCOV invariant of Calabi-Yau threefolds was introduced by Fang-Lu-Yoshikawa, themselves inspired by physicists Bershadsky-Cecotti-Ooguri-Vafa. It is a real number, obtained from a combination of holomorphic analytic torsion, and suitably normalized so that it only depends on the complex structure of the threefold. It is conjecturaly expected to encode genus 1 Gromov-Witten invariants of a mirror Calabi-Yau threefold. In order to confirm this prediction for a remarkable example, Fang-Lu-Yoshikawa studied the asymptotic behavior for degenerating families of Calabi-Yau threefolds acquiring at most ordinary double point (ODP) singularities. Their methods rely on former results by Yoshikawa on the singularities of Quillen metrics, together with more classical arguments in the theory of degenerations of Hodge structures and Hodge metrics. In this talk I will present joint work with Dennis Eriksson (Chalmers) and Christophe Mourougane (Rennes), where we extend the construction of the BCOV invariant to any dimension and we give precise asymptotic formulas for degenerating families of Calabi-Yau manifolds. Under several hypothesis on the geometry of the singularities acquired, our general formulas drastically simplify and prove some conjectures or predictions in the literature (Liu-Xia for semi-stable minimal families in dimension 3, Klemm-Pandharipande for ODP singularities in dimension 4, etc.). For this, we slightly improve Yoshikawa's results on the singularities of Quillen metrics, and we also provide a complement to Schmid's asymptotics of Hodge metrics when the monodromy transformations are non-unipotent.

#### 離散数理モデリングセミナー

17:15-18:30 数理科学研究科棟(駒場) 056号室

Integrability for four-dimensional recurrence relations

**Dinh T. Tran 氏**(School of Mathematics and Statistics, The University of Sydney)Integrability for four-dimensional recurrence relations

[ 講演概要 ]

In this talk, we study some fourth-order recurrence relations (or mappings). These recurrence relations were obtained by assuming that they possess two polynomial integrals with certain degree patterns.

For mappings with cubic growth, we will reduce them to three-dimensional ones using a procedure called deflation. These three-dimensional maps have two integrals; therefore, they are integrable in the sense of Liouville-Arnold. Furthermore, we can reduce the obtained three-dimensional maps to two-dimensional maps of Quispel-Roberts-Thompsons (QRT) type. On the other hand, for recurrences with quadratic growth and two integrals, we will show that they are integrable in the sense of Liouville-Arnold by providing their Poisson brackets. Non-degenerate Poisson brackets can be found by using the existence of discrete Lagrangians and a discrete analogue of the Ostrogradsky transformation.

This is joint work with G. Gubbiotti, N. Joshi, and C-M. Viallet.

In this talk, we study some fourth-order recurrence relations (or mappings). These recurrence relations were obtained by assuming that they possess two polynomial integrals with certain degree patterns.

For mappings with cubic growth, we will reduce them to three-dimensional ones using a procedure called deflation. These three-dimensional maps have two integrals; therefore, they are integrable in the sense of Liouville-Arnold. Furthermore, we can reduce the obtained three-dimensional maps to two-dimensional maps of Quispel-Roberts-Thompsons (QRT) type. On the other hand, for recurrences with quadratic growth and two integrals, we will show that they are integrable in the sense of Liouville-Arnold by providing their Poisson brackets. Non-degenerate Poisson brackets can be found by using the existence of discrete Lagrangians and a discrete analogue of the Ostrogradsky transformation.

This is joint work with G. Gubbiotti, N. Joshi, and C-M. Viallet.

### 2018年11月15日(木)

#### 応用解析セミナー

16:00-17:30 数理科学研究科棟(駒場) 118号室

Inhomogeneous Dirichlet-boundary value problem for one dimensional nonlinear Schr\"{o}dinger equations (Japanese)

**林 仲夫 氏**(大阪大学)Inhomogeneous Dirichlet-boundary value problem for one dimensional nonlinear Schr\"{o}dinger equations (Japanese)

[ 講演概要 ]

We consider the inhomogeneous Dirichlet-boundary value problem for the cubic nonlinear Schr\"{o}dinger equations on the half line. We present sufficient conditions of initial and boundary data which ensure asymptotic behavior of small solutions to equations by using the classical energy method and factorization techniques.

We consider the inhomogeneous Dirichlet-boundary value problem for the cubic nonlinear Schr\"{o}dinger equations on the half line. We present sufficient conditions of initial and boundary data which ensure asymptotic behavior of small solutions to equations by using the classical energy method and factorization techniques.

### 2018年11月14日(水)

#### 代数学コロキウム

18:00-19:00 数理科学研究科棟(駒場) 056号室

A motivic construction of ramification filtrations (ENGLISH)

**斎藤秀司 氏**(東京大学数理科学研究科)A motivic construction of ramification filtrations (ENGLISH)

[ 講演概要 ]

We give a new interpretation of Artin conductors of characters in the framework of theory of motives with modulus. It gives a unified way to understand Artin conductors of characters and irregularities of line bundle with integrable connections as well as overconvergent F-isocrystals of rank 1. It also gives rise to new conductors, for example, for G-torsors with G a finite flat group scheme, which specializes to the classical Artin conductor in case G = Z/nZ. We also give a motivic proof of a theorem of Kato and Matsuda on the determination of Artin conductors along divisors on smooth schemes by its restrictions to curves. Its proof is based on a motivic version of a theorem of Gabber-Katz. This is a joint work with Kay Rülling.

We give a new interpretation of Artin conductors of characters in the framework of theory of motives with modulus. It gives a unified way to understand Artin conductors of characters and irregularities of line bundle with integrable connections as well as overconvergent F-isocrystals of rank 1. It also gives rise to new conductors, for example, for G-torsors with G a finite flat group scheme, which specializes to the classical Artin conductor in case G = Z/nZ. We also give a motivic proof of a theorem of Kato and Matsuda on the determination of Artin conductors along divisors on smooth schemes by its restrictions to curves. Its proof is based on a motivic version of a theorem of Gabber-Katz. This is a joint work with Kay Rülling.

### 2018年11月13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

Boundedness of varieties of Fano type with alpha-invariants and volumes bounded below (English)

**陳韋中 氏**(東大数理)Boundedness of varieties of Fano type with alpha-invariants and volumes bounded below (English)

[ 講演概要 ]

We show that fixed dimensional klt weak Fano pairs with alpha-invariants and volumes bounded away from 0 and the coefficients of the boundaries belong to the set of hyperstandard multiplicities Φ(R) associated to a fixed finite set R form a bounded family. We also show α(X, B)d−1vol(−(KX + B)) are bounded from above for all klt weak Fano pairs (X, B) of a fixed dimension d.

We show that fixed dimensional klt weak Fano pairs with alpha-invariants and volumes bounded away from 0 and the coefficients of the boundaries belong to the set of hyperstandard multiplicities Φ(R) associated to a fixed finite set R form a bounded family. We also show α(X, B)d−1vol(−(KX + B)) are bounded from above for all klt weak Fano pairs (X, B) of a fixed dimension d.

#### 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 continuity of drifts of the mapping class group (JAPANESE)

Tea: Common Room 16:30-17:00

**正井 秀俊 氏**(東京工業大学)On continuity of drifts of the mapping class group (JAPANESE)

[ 講演概要 ]

When a group is acting on a space isometrically, we may consider the "translation distance" of random walks, which is called the drift of the random walk. In this talk we consider mapping class group acting on the Teichmüller space. We first recall several characterizations of the drift. The drift is determined by the transition probability of the random walk. The goal of this talk is to show that the drift varies continuously with the transition probability measure.

When a group is acting on a space isometrically, we may consider the "translation distance" of random walks, which is called the drift of the random walk. In this talk we consider mapping class group acting on the Teichmüller space. We first recall several characterizations of the drift. The drift is determined by the transition probability of the random walk. The goal of this talk is to show that the drift varies continuously with the transition probability measure.

### 2018年11月12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) 128号室

Random walk at weak and strong disorder (ENGLISH)

http://www.mat.uc.cl/~aramirez/

**Alejandro Ramirez 氏**(Pontificia Universidad Catolica de Chile)Random walk at weak and strong disorder (ENGLISH)

[ 講演概要 ]

We consider random walks at low disorder on $\mathbb Z^d$. For dimensions $d\ge 4$, we exhibit a phase transition on the strength of the disorder expressed as an equality between the quenched and annealed rate functions. In dimension $d=2$ we exhibit a universal scaling limit to the stochastic heat equation. This talk is based on joint works with Bazaes, Mukherjee and Saglietti, and with Moreno and Quastel.

[ 講演参考URL ]We consider random walks at low disorder on $\mathbb Z^d$. For dimensions $d\ge 4$, we exhibit a phase transition on the strength of the disorder expressed as an equality between the quenched and annealed rate functions. In dimension $d=2$ we exhibit a universal scaling limit to the stochastic heat equation. This talk is based on joint works with Bazaes, Mukherjee and Saglietti, and with Moreno and Quastel.

http://www.mat.uc.cl/~aramirez/

### 2018年11月09日(金)

#### 統計数学セミナー

11:00-12:00 数理科学研究科棟(駒場) 123号室

Market impact and option hedging in the presence of liquidity costs

**Frédéric Abergel 氏**(CentraleSupélec)Market impact and option hedging in the presence of liquidity costs

[ 講演概要 ]

The phenomenon of market (or: price) impact is well-known among practicioners, and it has long been recognized as a key feature of trading in electronic markets. In the first part of this talk, I will present some new, recent results on market impact, especially for limit orders. I will then propose a theory for option hedging in the presence of liquidity costs.(Based on joint works with E. Saïd, G. Loeper).

The phenomenon of market (or: price) impact is well-known among practicioners, and it has long been recognized as a key feature of trading in electronic markets. In the first part of this talk, I will present some new, recent results on market impact, especially for limit orders. I will then propose a theory for option hedging in the presence of liquidity costs.(Based on joint works with E. Saïd, G. Loeper).

### 2018年11月08日(木)

#### トポロジー火曜セミナー

10:30-12:00 数理科学研究科棟(駒場) 056号室

開催日，時刻にご注意下さい

Deformations of diagonal representations of knot groups into $\mathrm{SL}(n,\mathbb{C})$ (ENGLISH)

開催日，時刻にご注意下さい

**Michael Heusener 氏**(Université Clermont Auvergne)Deformations of diagonal representations of knot groups into $\mathrm{SL}(n,\mathbb{C})$ (ENGLISH)

[ 講演概要 ]

This is joint work with Leila Ben Abdelghani, Monastir (Tunisia).

Given a manifold $M$, the variety of representations of $\pi_1(M)$ into $\mathrm{SL}(2,\mathbb{C})$ and the variety of characters of such representations both contain information of the topology of $M$. Since the foundational work of W.P. Thurston and Culler & Shalen, the varieties of $\mathrm{SL}(2,\mathbb{C})$-characters have been extensively studied. This is specially interesting for $3$-dimensional manifolds, where the fundamental group and the geometrical properties of the manifold are strongly related.

However, much less is known of the character varieties for other groups, notably for $\mathrm{SL}(n,\mathbb{C})$ with $n\geq 3$. The $\mathrm{SL}(n,\mathbb{C})$-character varieties for free groups have been studied by S. Lawton and P. Will, and the $\mathrm{SL}(3,\mathbb{C})$-character variety of torus knot groups has been determined by V. Munoz and J. Porti.

In this talk I will present some results concerning the deformations of diagonal representations of knot groups in basic notations and some recent results concerning the representation and character varieties of $3$-manifold groups and in particular knot groups. In particular, we are interested in the local structure of the $\mathrm{SL}(n,\mathbb{C})$-representation variety at the diagonal representation.

This is joint work with Leila Ben Abdelghani, Monastir (Tunisia).

Given a manifold $M$, the variety of representations of $\pi_1(M)$ into $\mathrm{SL}(2,\mathbb{C})$ and the variety of characters of such representations both contain information of the topology of $M$. Since the foundational work of W.P. Thurston and Culler & Shalen, the varieties of $\mathrm{SL}(2,\mathbb{C})$-characters have been extensively studied. This is specially interesting for $3$-dimensional manifolds, where the fundamental group and the geometrical properties of the manifold are strongly related.

However, much less is known of the character varieties for other groups, notably for $\mathrm{SL}(n,\mathbb{C})$ with $n\geq 3$. The $\mathrm{SL}(n,\mathbb{C})$-character varieties for free groups have been studied by S. Lawton and P. Will, and the $\mathrm{SL}(3,\mathbb{C})$-character variety of torus knot groups has been determined by V. Munoz and J. Porti.

In this talk I will present some results concerning the deformations of diagonal representations of knot groups in basic notations and some recent results concerning the representation and character varieties of $3$-manifold groups and in particular knot groups. In particular, we are interested in the local structure of the $\mathrm{SL}(n,\mathbb{C})$-representation variety at the diagonal representation.

### 2018年11月06日(火)

#### 解析学火曜セミナー

16:50-18:20 数理科学研究科棟(駒場) 128号室

Global behavior of bifurcation curves and related topics (日本語)

**柴田徹太郎 氏**(広島大学)Global behavior of bifurcation curves and related topics (日本語)

[ 講演概要 ]

In this talk, we consider the asymptotic behavior of bifurcation curves for ODE with oscillatory nonlinear term. First, we study the global and local behavior of oscillatory bifurcation curves. We also consider the bifurcation problems with nonlinear diffusion.

In this talk, we consider the asymptotic behavior of bifurcation curves for ODE with oscillatory nonlinear term. First, we study the global and local behavior of oscillatory bifurcation curves. We also consider the bifurcation problems with nonlinear diffusion.

#### トポロジー火曜セミナー

17:30-18:30 数理科学研究科棟(駒場) 056号室

Tea: Common Room 17:00-17:30

Coarsely convex spaces and a coarse Cartan-Hadamard theorem (JAPANESE)

Tea: Common Room 17:00-17:30

**尾國 新一 氏**(愛媛大学)Coarsely convex spaces and a coarse Cartan-Hadamard theorem (JAPANESE)

[ 講演概要 ]

A coarse version of negatively-curved spaces have been very well studied as Gromov hyperbolic spaces. Recently we introduced a coarse version of non-positively curved spaces, named them coarsely convex spaces and showed a coarse version of the Cartan-Hadamard theorem for such spaces in a joint-work with Tomohiro Fukaya (arXiv:1705.05588). Based on the work, I introduce coarsely convex spaces and explain a coarse Cartan-Hadamard theorem, ideas for proof and its applications to differential topology.

A coarse version of negatively-curved spaces have been very well studied as Gromov hyperbolic spaces. Recently we introduced a coarse version of non-positively curved spaces, named them coarsely convex spaces and showed a coarse version of the Cartan-Hadamard theorem for such spaces in a joint-work with Tomohiro Fukaya (arXiv:1705.05588). Based on the work, I introduce coarsely convex spaces and explain a coarse Cartan-Hadamard theorem, ideas for proof and its applications to differential topology.

### 2018年11月05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 the quasiconformal equivalence of Dynamical Cantor sets (JAPANESE)

**志賀啓成 氏**(東京工業大学)On the quasiconformal equivalence of Dynamical Cantor sets (JAPANESE)

[ 講演概要 ]

Let $E$ be a Cantor set in the Riemann sphere $\widehat{\mathbb C}$, that is, a totally disconnected perfect set in $\widehat{\mathbb C}$.

The standard middle one-thirds Cantor set $\mathcal{C}$ is a typical example.

We consider the complement $X_{E}:=\widehat{\mathbb C}\setminus E$ of the Cantor set $E$.

It is an open Riemann surface with uncountable many boundary components.

We are interested in the quasiconformal equivalence of such surfaces.

In this talk, we discuss the quasiconformal equivalence for the complements of Cantor sets given by dynamical systems.

Let $E$ be a Cantor set in the Riemann sphere $\widehat{\mathbb C}$, that is, a totally disconnected perfect set in $\widehat{\mathbb C}$.

The standard middle one-thirds Cantor set $\mathcal{C}$ is a typical example.

We consider the complement $X_{E}:=\widehat{\mathbb C}\setminus E$ of the Cantor set $E$.

It is an open Riemann surface with uncountable many boundary components.

We are interested in the quasiconformal equivalence of such surfaces.

In this talk, we discuss the quasiconformal equivalence for the complements of Cantor sets given by dynamical systems.

#### 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

Tosio Kato as an applied mathematician (Japanese)

**岡本久 氏**(学習院大学理学部)Tosio Kato as an applied mathematician (Japanese)

[ 講演概要 ]

Tosio Kato (1917-1999) is nowadays considered to be a rigorous analyst or theorist. Many people consider his contributions in quantum mechanics to be epoch-making, his work on nonlinear partial differential equations elegant and inspiring. However, around the time when he visited USA for the first time in 1954, he was studying problems of applied mathematics, too, notably numerical computation of eigenvalues. I wish to shed light on the historical background of his study of applied mathematics. This is a joint work with Prof. Hiroshi Fujita.

Tosio Kato (1917-1999) is nowadays considered to be a rigorous analyst or theorist. Many people consider his contributions in quantum mechanics to be epoch-making, his work on nonlinear partial differential equations elegant and inspiring. However, around the time when he visited USA for the first time in 1954, he was studying problems of applied mathematics, too, notably numerical computation of eigenvalues. I wish to shed light on the historical background of his study of applied mathematics. This is a joint work with Prof. Hiroshi Fujita.

### 2018年11月02日(金)

#### 古典解析セミナー

17:00-18:30 数理科学研究科棟(駒場) 122号室

On the inverse problem of the discrete calculus of variations (ENGLISH)

**Giorgio Gubbiotti 氏**(The University of Sydney)On the inverse problem of the discrete calculus of variations (ENGLISH)

[ 講演概要 ]

One of the most powerful tools in Mathematical Physics since Euler and Lagrange is the calculus of variations. The variational formulation of mechanics where the equations of motion arise as the minimum of an action functional (the so-called Hamilton's principle), is fundamental in the development of theoretical mechanics and its foundations are present in each textbook on this subject [1, 3, 6]. Beside this, the application of calculus of variations goes beyond mechanics as many important mathematical problems, e.g. the isoperimetrical problem and the catenary, can be formulated in terms of calculus of variations.

An important problem regarding the calculus of variations is to determine which system of differential equations are Euler-Lagrange equations for some variational problem. This problem has a long and interesting history, see e.g. [4]. The general case of this problem remains unsolved, whereas several important results for particular cases were presented during the 20th century.

In this talk we present some conditions on the existence of a Lagrangian in the discrete scalar setting. We will introduce a set of differential operators called annihilation operators. We will use these operators to

reduce the functional equation governing of existence of a Lagrangian for a scalar difference equation of arbitrary even order 2k, with k > 1 to the solution of a system of linear partial differential equations. Solving such equations one can either find the Lagrangian or conclude that it does not exist.

We comment the relationship of our solution of the inverse problem of the discrete calculus of variation with the one given in [2], where a result analogous to the homotopy formula [5] for the continuous case was proven.

References

[1] H. Goldstein, C. Poole, and J. Safko. Classical Mechanics. Pearson Education, 2002.

[2] P. E. Hydon and E. L. Mansfeld. A variational complex for difference equations. Found. Comp. Math., 4:187{217, 2004.

[3] L. D. Landau and E. M. Lifshitz. Mechanics. Course of Theoretical Physics. Elsevier Science, 1982.

[4] P. J. Olver. Applications of Lie Groups to Differential Equations. Springer-Verlag, Berlin, 1986.

[5] M. M. Vainberg. Variational methods for the study of nonlinear operators. Holden-Day, San Francisco, 1964.

[6] E. T. Whittaker. A Treatise on the Analytical Dynamics of Particles and Rigid Bodies. Cambridge University Press, Cambridge, 1999.

One of the most powerful tools in Mathematical Physics since Euler and Lagrange is the calculus of variations. The variational formulation of mechanics where the equations of motion arise as the minimum of an action functional (the so-called Hamilton's principle), is fundamental in the development of theoretical mechanics and its foundations are present in each textbook on this subject [1, 3, 6]. Beside this, the application of calculus of variations goes beyond mechanics as many important mathematical problems, e.g. the isoperimetrical problem and the catenary, can be formulated in terms of calculus of variations.

An important problem regarding the calculus of variations is to determine which system of differential equations are Euler-Lagrange equations for some variational problem. This problem has a long and interesting history, see e.g. [4]. The general case of this problem remains unsolved, whereas several important results for particular cases were presented during the 20th century.

In this talk we present some conditions on the existence of a Lagrangian in the discrete scalar setting. We will introduce a set of differential operators called annihilation operators. We will use these operators to

reduce the functional equation governing of existence of a Lagrangian for a scalar difference equation of arbitrary even order 2k, with k > 1 to the solution of a system of linear partial differential equations. Solving such equations one can either find the Lagrangian or conclude that it does not exist.

We comment the relationship of our solution of the inverse problem of the discrete calculus of variation with the one given in [2], where a result analogous to the homotopy formula [5] for the continuous case was proven.

References

[1] H. Goldstein, C. Poole, and J. Safko. Classical Mechanics. Pearson Education, 2002.

[2] P. E. Hydon and E. L. Mansfeld. A variational complex for difference equations. Found. Comp. Math., 4:187{217, 2004.

[3] L. D. Landau and E. M. Lifshitz. Mechanics. Course of Theoretical Physics. Elsevier Science, 1982.

[4] P. J. Olver. Applications of Lie Groups to Differential Equations. Springer-Verlag, Berlin, 1986.

[5] M. M. Vainberg. Variational methods for the study of nonlinear operators. Holden-Day, San Francisco, 1964.

[6] E. T. Whittaker. A Treatise on the Analytical Dynamics of Particles and Rigid Bodies. Cambridge University Press, Cambridge, 1999.

### 2018年10月31日(水)

#### FMSPレクチャーズ

15:00-16:30 数理科学研究科棟(駒場) 122号室

K-THEORY AND THE DIRAC OPERATOR (4/4)

Lecture 4. BEYOND ELLIPTICITY or K-HOMOLOGY AND INDEX THEORY ON CONTACT MANIFOLDS (ENGLISH)

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

**Paul Baum 氏**(The Pennsylvania State University)K-THEORY AND THE DIRAC OPERATOR (4/4)

Lecture 4. BEYOND ELLIPTICITY or K-HOMOLOGY AND INDEX THEORY ON CONTACT MANIFOLDS (ENGLISH)

[ 講演概要 ]

K-homology is the dual theory to K-theory. The BD (Baum-Douglas) isomorphism of Atiyah-Kasparov K-homology and K-cycle K-homology provides a framework within which the Atiyah-Singer index theorem can be extended to certain differential operators which are hypoelliptic but not elliptic. This talk will consider such a class of differential operators on compact contact manifolds. These operators have been studied by a number of mathematicians (e.g. C.Epstein and R.Melrose).

Operators with similar analytical properties have also been studied (e.g. by Alain Connes and Henri Moscovici --- also Michel Hilsum and Georges Skandalis). Working within the BD framework, the index problem will be solved for these differential operators on compact contact manifolds.

This is joint work with Erik van Erp.

[ 講演参考URL ]K-homology is the dual theory to K-theory. The BD (Baum-Douglas) isomorphism of Atiyah-Kasparov K-homology and K-cycle K-homology provides a framework within which the Atiyah-Singer index theorem can be extended to certain differential operators which are hypoelliptic but not elliptic. This talk will consider such a class of differential operators on compact contact manifolds. These operators have been studied by a number of mathematicians (e.g. C.Epstein and R.Melrose).

Operators with similar analytical properties have also been studied (e.g. by Alain Connes and Henri Moscovici --- also Michel Hilsum and Georges Skandalis). Working within the BD framework, the index problem will be solved for these differential operators on compact contact manifolds.

This is joint work with Erik van Erp.

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

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