## Seminar information archive

Seminar information archive ～07/24｜Today's seminar 07/25 | Future seminars 07/26～

#### Operator Algebra Seminars

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

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

**Takaaki Moriyama**(the University of Tokyo)Orbit equivalence classes for free actions of free products of infinite abelian groups

### 2018/11/20

#### Algebraic Geometry Seminar

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

Artin-Mazur height, Yobuko height and

Hodge-Wittt cohomologies

**Nakkajima Yukiyoshi**(Tokyo Denki University)Artin-Mazur height, Yobuko height and

Hodge-Wittt cohomologies

[ Abstract ]

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 $\dim X=3$.

#### Discrete mathematical modelling seminar

15:00-16:30 Room #002 (Graduate School of Math. Sci. Bldg.)

#### Tuesday Seminar on Topology

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

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

**Takuya Sakasai**(The University of Tokyo)Torelli group, Johnson kernel and invariants of homology 3-spheres (JAPANESE)

[ Abstract ]

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.

### 2018/11/19

#### Tokyo Probability Seminar

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

Two-dimensional stochastic interface growth (ENGLISH)

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

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

[ Abstract ]

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.

[ Reference 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/

#### Seminar on Geometric Complex Analysis

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

BCOV invariants of Calabi-Yau varieties (ENGLISH)

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

[ Abstract ]

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.

#### Discrete mathematical modelling seminar

17:15-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Integrability for four-dimensional recurrence relations

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

[ Abstract ]

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.

### 2018/11/15

#### Applied Analysis

16:00-17:30 Room #118 (Graduate School of Math. Sci. Bldg.)

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

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

[ Abstract ]

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.

### 2018/11/14

#### Number Theory Seminar

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

A motivic construction of ramification filtrations (ENGLISH)

**Shuji Saito**(University of Tokyo)A motivic construction of ramification filtrations (ENGLISH)

[ Abstract ]

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.

### 2018/11/13

#### Algebraic Geometry Seminar

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

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

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

[ Abstract ]

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.

#### Tuesday Seminar on Topology

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

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

**Hidetoshi Masai**(Tokyo Institute of Technology)On continuity of drifts of the mapping class group (JAPANESE)

[ Abstract ]

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.

### 2018/11/12

#### Tokyo Probability Seminar

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

Random walk at weak and strong disorder (ENGLISH)

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

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

[ Abstract ]

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.

[ Reference 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/

### 2018/11/09

#### Seminar on Probability and Statistics

11:00-12:00 Room #123 (Graduate School of Math. Sci. Bldg.)

Market impact and option hedging in the presence of liquidity costs

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

[ Abstract ]

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).

### 2018/11/08

#### Tuesday Seminar on Topology

10:30-12:00 Room #056 (Graduate School of Math. Sci. Bldg.)

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

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

[ Abstract ]

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.

### 2018/11/06

#### Tuesday Seminar of Analysis

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

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

**SHIBATA Tetsutaro**(Hiroshima University)Global behavior of bifurcation curves and related topics (日本語)

[ Abstract ]

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.

#### Tuesday Seminar on Topology

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

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

**Shin-ichi Oguni**(Ehime University)Coarsely convex spaces and a coarse Cartan-Hadamard theorem (JAPANESE)

[ Abstract ]

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.

### 2018/11/05

#### Seminar on Geometric Complex Analysis

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

On the quasiconformal equivalence of Dynamical Cantor sets (JAPANESE)

**Hiroshige Shiga**(Tokyo Institute of Technology)On the quasiconformal equivalence of Dynamical Cantor sets (JAPANESE)

[ Abstract ]

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.

#### Numerical Analysis Seminar

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

Tosio Kato as an applied mathematician (Japanese)

**Hisashi Okamoto**(Gakushuin University)Tosio Kato as an applied mathematician (Japanese)

[ Abstract ]

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.

### 2018/11/02

#### Classical Analysis

17:00-18:30 Room #122 (Graduate School of Math. Sci. Bldg.)

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

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

[ Abstract ]

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.

### 2018/10/31

#### FMSP Lectures

15:00-16:30 Room #122 (Graduate School of Math. Sci. Bldg.)

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**(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)

[ Abstract ]

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.

[ Reference 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

#### Operator Algebra Seminars

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

Strong Tools in Free Probability Theory

**Tomohiro Hayase**(the University of Tokyo)Strong Tools in Free Probability Theory

### 2018/10/30

#### Tuesday Seminar of Analysis

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

Spectral structure of the Neumann-Poincaré operator in three dimensions: Willmore energy and surface geometry (日本語)

**MIYANISHI Yoshihisa**(Osaka University)Spectral structure of the Neumann-Poincaré operator in three dimensions: Willmore energy and surface geometry (日本語)

[ Abstract ]

The Neumann-Poincaré operator (abbreviated by NP) is a boundary integral operator naturally arising when solving classical boundary value problems using layer potentials. If the boundary of the domain, on which the NP operator is defined, is $C^{1, \alpha}$ smooth, then the NP operator is compact. Thus, the Fredholm integral equation, which appears when solving Dirichlet or Neumann problems, can be solved using the Fredholm index theory.

Regarding spectral properties of the NP operator, the spectrum consists of eigenvalues converging to $0$ for $C^{1, \alpha}$ smooth boundaries. Our main purpose here is to deduce eigenvalue asymptotics of the NP operators in three dimensions. This formula is the so-called Weyl's law for eigenvalue problems of NP operators. Then we discuss relationships among the Weyl's law, the Euler characteristic and the Willmore energy on the boundary surface.

The Neumann-Poincaré operator (abbreviated by NP) is a boundary integral operator naturally arising when solving classical boundary value problems using layer potentials. If the boundary of the domain, on which the NP operator is defined, is $C^{1, \alpha}$ smooth, then the NP operator is compact. Thus, the Fredholm integral equation, which appears when solving Dirichlet or Neumann problems, can be solved using the Fredholm index theory.

Regarding spectral properties of the NP operator, the spectrum consists of eigenvalues converging to $0$ for $C^{1, \alpha}$ smooth boundaries. Our main purpose here is to deduce eigenvalue asymptotics of the NP operators in three dimensions. This formula is the so-called Weyl's law for eigenvalue problems of NP operators. Then we discuss relationships among the Weyl's law, the Euler characteristic and the Willmore energy on the boundary surface.

#### PDE Real Analysis Seminar

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

The least gradient problem in the plain (English)

**Piotr Rybka**(University of Warsaw)The least gradient problem in the plain (English)

[ Abstract ]

The least gradient problem arises in many application, e.g. in the free material design. We show existence of solutions in bounded, strictly convex planar regions, when the data are functions on bounded variation.

Our main goal is to show existence of solution in convex, but not necessarily strictly convex planar regions. In order to avoid technicalities we consider only continuous data, but BV data will do to. We formulate a set of admissibility conditions. We show that they are sufficient for existence.

This is a joint project with Wojciech Górny and Ahmad Sabra.

The least gradient problem arises in many application, e.g. in the free material design. We show existence of solutions in bounded, strictly convex planar regions, when the data are functions on bounded variation.

Our main goal is to show existence of solution in convex, but not necessarily strictly convex planar regions. In order to avoid technicalities we consider only continuous data, but BV data will do to. We formulate a set of admissibility conditions. We show that they are sufficient for existence.

This is a joint project with Wojciech Górny and Ahmad Sabra.

#### Tuesday Seminar on Topology

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

The quasiconformal equivalence of Riemann surfaces and a universality of Schottky spaces (JAPANESE)

**Hiroshige Shiga**(Tokyo Institute of Technology)The quasiconformal equivalence of Riemann surfaces and a universality of Schottky spaces (JAPANESE)

[ Abstract ]

In the theory of Teichmüller space of Riemann surfaces, we consider the set of Riemann surfaces which are quasiconformally equivalent. For topologically finite Riemann surfaces, it is quite easy to examine if they are quasiconformally equivalent or not. On the other hand, for Riemann surfaces of topologically infinite type, the situation is rather complicated.

In this talk, after constructing an example which shows the complexity of the problem, we give some geometric conditions for Riemann surfaces to be quasiconformally equivalent. Our argument enables us to see a universality of Schottky spaces.

In the theory of Teichmüller space of Riemann surfaces, we consider the set of Riemann surfaces which are quasiconformally equivalent. For topologically finite Riemann surfaces, it is quite easy to examine if they are quasiconformally equivalent or not. On the other hand, for Riemann surfaces of topologically infinite type, the situation is rather complicated.

In this talk, after constructing an example which shows the complexity of the problem, we give some geometric conditions for Riemann surfaces to be quasiconformally equivalent. Our argument enables us to see a universality of Schottky spaces.

#### Seminar on Probability and Statistics

15:30-16:40 Room #126 (Graduate School of Math. Sci. Bldg.)

Asymptotic expansion for random vectors

**Ciprian A. Tudor**(Université de Lille 1, Université de Panthéon-Sorbonne Paris 1)Asymptotic expansion for random vectors

[ Abstract ]

We develop the asymptotic expansion theory for vector-valued sequences $F_{N}$ of random variables. We find the second-order term in the expansion of the density of $F_{N}$, based on assumptions in terms of the convergence of the Stein-Malliavin matrix associated to the sequence $F_{N}$ . Our approach combines the classical Fourier approach and the recent theory on Stein method and Malliavin calculus. We find the second order term of the asymptotic expansion of the density of $F_{N}$ and we discuss the main ideas on higher order asymptotic expansion. We illustrate our results by several examples.

We develop the asymptotic expansion theory for vector-valued sequences $F_{N}$ of random variables. We find the second-order term in the expansion of the density of $F_{N}$, based on assumptions in terms of the convergence of the Stein-Malliavin matrix associated to the sequence $F_{N}$ . Our approach combines the classical Fourier approach and the recent theory on Stein method and Malliavin calculus. We find the second order term of the asymptotic expansion of the density of $F_{N}$ and we discuss the main ideas on higher order asymptotic expansion. We illustrate our results by several examples.

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