## Seminar information archive

Seminar information archive ～02/15｜Today's seminar 02/16 | Future seminars 02/17～

### 2017/11/06

#### FMSP Lectures

17:00-18:00 Room #118 (Graduate School of Math. Sci. Bldg.)

Phaseless inverse problems for Maxwell equations (ENGLISH)

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Romanov2.pdf

**V. G. Romanov**(Sobolev Institute of Mathematics)Phaseless inverse problems for Maxwell equations (ENGLISH)

[ Abstract ]

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Romanov2.pdf

[ Reference URL ]http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Romanov2.pdf

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Romanov2.pdf

### 2017/11/02

#### Seminar on Probability and Statistics

14:00-15:10 Room #052 (Graduate School of Math. Sci. Bldg.)

Hermite processes and sheets

**Tudor Ciprian**(Université Lille 1)Hermite processes and sheets

[ Abstract ]

The Hermite process of order $\geq 1$ is a self-similar stochastic process with stationary increments living in the $q$th Wiener chaos. The class of Hermite processes includes the fractional Brownian motion (for $q=1$) and the Rosenblatt process (for $q=2$). We present the basic properties of these processes and we introduce their multiparameter version. We also discuss the behavior with respect to the self-similarity index and the possibility so solve stochastic equations with Hermite noise.

The Hermite process of order $\geq 1$ is a self-similar stochastic process with stationary increments living in the $q$th Wiener chaos. The class of Hermite processes includes the fractional Brownian motion (for $q=1$) and the Rosenblatt process (for $q=2$). We present the basic properties of these processes and we introduce their multiparameter version. We also discuss the behavior with respect to the self-similarity index and the possibility so solve stochastic equations with Hermite noise.

### 2017/11/01

#### Discrete mathematical modelling seminar

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

Discrete Painlevé equations associated with the E8 group (ENGLISH)

**Basile Grammaticos**(Université de Paris VII・XI)Discrete Painlevé equations associated with the E8 group (ENGLISH)

[ Abstract ]

I'll present a summary of the results of the Paris-Tokyo-Pondicherry group on equations associated with the affine Weyl group E8. I shall review the various parametrisations of the E8-related equations, introducing the trihomographic representation and the ancillary variable. Several examples of E8-associated equations will be given including what we believe is the simplest form for the generic elliptic discrete Painlevé equation.

I'll present a summary of the results of the Paris-Tokyo-Pondicherry group on equations associated with the affine Weyl group E8. I shall review the various parametrisations of the E8-related equations, introducing the trihomographic representation and the ancillary variable. Several examples of E8-associated equations will be given including what we believe is the simplest form for the generic elliptic discrete Painlevé equation.

### 2017/10/31

#### Tuesday Seminar on Topology

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

Nonamenable groups of piecewise projective homeomorphisms (ENGLISH)

**Yash Lodha**(École Polytechnique Fédérale de Lausanne)Nonamenable groups of piecewise projective homeomorphisms (ENGLISH)

[ Abstract ]

Groups of piecewise projective homeomorphisms provide elegant examples of groups that are non amenable, yet do not contain non abelian free subgroups. In this talk I will present a survey of these groups and discuss their striking properties. I will discuss properties such as (non)amenability, finiteness properties, normal subgroup structure, actions by various degrees of regularity and Tarski numbers.

Groups of piecewise projective homeomorphisms provide elegant examples of groups that are non amenable, yet do not contain non abelian free subgroups. In this talk I will present a survey of these groups and discuss their striking properties. I will discuss properties such as (non)amenability, finiteness properties, normal subgroup structure, actions by various degrees of regularity and Tarski numbers.

#### Algebraic Geometry Seminar

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

ACC for log canonical threshold polytopes (English)

**Zhan Li**(Beijing)ACC for log canonical threshold polytopes (English)

[ Abstract ]

We show that the log canonical threshold polytopes of varieties with log canonical singularities satisfy the ascending chain condition. This is a joint work with Jingjun Han and Lu Qi.

We show that the log canonical threshold polytopes of varieties with log canonical singularities satisfy the ascending chain condition. This is a joint work with Jingjun Han and Lu Qi.

#### Discrete mathematical modelling seminar

16:30-17:30 Room #126 (Graduate School of Math. Sci. Bldg.)

The end of the World (ENGLISH)

**Basile Grammaticos**(Université de Paris VII・XI)The end of the World (ENGLISH)

[ Abstract ]

This is not a seminar on astrophysics or cosmology. I am not going to talk about something that will happen in billions of years. I will rather explain the menace to our civilisation and to the human species. Inspired from the works of several authors I will explain the existing risks. I will also present mathematical models which show that a general collapse is possible in the decades that follow.

This is not a seminar on astrophysics or cosmology. I am not going to talk about something that will happen in billions of years. I will rather explain the menace to our civilisation and to the human species. Inspired from the works of several authors I will explain the existing risks. I will also present mathematical models which show that a general collapse is possible in the decades that follow.

#### Discrete mathematical modelling seminar

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

A hierarchy of approximate regularity of functions (ENGLISH)

**Fon-Che Liu**(National Taiwan University)A hierarchy of approximate regularity of functions (ENGLISH)

[ Abstract ]

A hierarchy of a certain weakly sensed regularity of functions defined on subsets of Euclidean n-space which originated from the well-known Lusin theorem that characterizes measurable functions in terms of approximate continuity will be introduced. Its intimate relations with the ordinary hierarchy of regularity in terms of order of continuous differentiability will be exposed and explained.

A hierarchy of a certain weakly sensed regularity of functions defined on subsets of Euclidean n-space which originated from the well-known Lusin theorem that characterizes measurable functions in terms of approximate continuity will be introduced. Its intimate relations with the ordinary hierarchy of regularity in terms of order of continuous differentiability will be exposed and explained.

#### FMSP Lectures

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

Some Geometric Aspects in Inverse Problems (ENGLISH)

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

**V. G. Romanov**(Sobolev Institute of Mathematics)Some Geometric Aspects in Inverse Problems (ENGLISH)

[ Abstract ]

We consider inverse problems related to recovering coefficients in partial differential equations of the second order. It is supposed that some measurements of solutions to direct problems are produced on convenient sets. A study of some inverse problems for hyperbolic equations leads to geometric problems: recovering a function from its integrals along geodesic lines of the Riemannian metric or recovering the Riemannian metric inside a domain from given distances between arbitrary points of the domain boundary. Our main goal here is to demonstrate how such geometric problems arise for equations of parabolic and elliptic types.

[ Reference URL ]We consider inverse problems related to recovering coefficients in partial differential equations of the second order. It is supposed that some measurements of solutions to direct problems are produced on convenient sets. A study of some inverse problems for hyperbolic equations leads to geometric problems: recovering a function from its integrals along geodesic lines of the Riemannian metric or recovering the Riemannian metric inside a domain from given distances between arbitrary points of the domain boundary. Our main goal here is to demonstrate how such geometric problems arise for equations of parabolic and elliptic types.

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

### 2017/10/30

#### Tokyo Probability Seminar

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

Integration of controlled rough paths via fractional calculus (JAPANESE)

**Yu Ito**(Department of Mathematics, Faculty of Science, Kyoto Sangyo University)Integration of controlled rough paths via fractional calculus (JAPANESE)

#### Operator Algebra Seminars

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

The bicategory of $W^*$-bimodules

**Yusuke Sawada**(Nagoya Univ.)The bicategory of $W^*$-bimodules

#### Seminar on Geometric Complex Analysis

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

Odd dimensional complex analytic Kleinian groups

**Masahide Kato**(Sophia University)Odd dimensional complex analytic Kleinian groups

[ Abstract ]

In this talk, I would explain an idea to construct a higher dimensional analogue of the classical Kleinian group theory. For a group $G$ of a certain class of discrete subgroups of $\mathrm{PGL}(2n+2,\mathbf{C})$ which act on $\mathbf{P}^{2n+1}$, there is a canonical way to define the region of discontinuity, on which $G$ acts properly discontinuously. General principle in the discussion is to regard $\mathbf{P}^{n}$ in $\mathbf{P}^{2n+1}$ as a single point. We can consider the quotient space of the discontinuity region by the action of $G$. Though the Ahlfors finiteness theorem is hopeless because of a counter example, the Klein combination theorem and the handle attachment can be defined similarly. Any compact quotients which appear here are non-Kaehler. In the case $n=1$, we explain a new example of compact quotients which is related to a classical Kleinian group.

In this talk, I would explain an idea to construct a higher dimensional analogue of the classical Kleinian group theory. For a group $G$ of a certain class of discrete subgroups of $\mathrm{PGL}(2n+2,\mathbf{C})$ which act on $\mathbf{P}^{2n+1}$, there is a canonical way to define the region of discontinuity, on which $G$ acts properly discontinuously. General principle in the discussion is to regard $\mathbf{P}^{n}$ in $\mathbf{P}^{2n+1}$ as a single point. We can consider the quotient space of the discontinuity region by the action of $G$. Though the Ahlfors finiteness theorem is hopeless because of a counter example, the Klein combination theorem and the handle attachment can be defined similarly. Any compact quotients which appear here are non-Kaehler. In the case $n=1$, we explain a new example of compact quotients which is related to a classical Kleinian group.

#### Algebraic Geometry Seminar

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

Towards birational boundedness of elliptic Calabi-Yau varieties (English)

**Robeto Svaldi**(Cambridge)Towards birational boundedness of elliptic Calabi-Yau varieties (English)

[ Abstract ]

I will discuss new results towards the birational boundedness of

low-dimensional elliptic Calabi-Yau varieties, joint work with Gabriele

Di Certo.

Recent work in the minimal model program suggests that pairs with trivial log canonical

class should satisfy some boundedness properties.

I will show that 4-dimensional Calabi-Yau pairs which are not birational to a product are

indeed log birationally bounded. This implies birational boundedness of elliptically fibered

Calabi-Yau manifolds with a section, in dimension up to 5.

If time allows, I will also try to discuss a first approach towards boundedness of rationally

connected CY varieties in low dimension.

I will discuss new results towards the birational boundedness of

low-dimensional elliptic Calabi-Yau varieties, joint work with Gabriele

Di Certo.

Recent work in the minimal model program suggests that pairs with trivial log canonical

class should satisfy some boundedness properties.

I will show that 4-dimensional Calabi-Yau pairs which are not birational to a product are

indeed log birationally bounded. This implies birational boundedness of elliptically fibered

Calabi-Yau manifolds with a section, in dimension up to 5.

If time allows, I will also try to discuss a first approach towards boundedness of rationally

connected CY varieties in low dimension.

### 2017/10/25

#### Lectures

11:00-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

On Faltings' main comparison theorem in p-adic Hodge theory : the relative case (ENGLISH)

**Ahmed Abbes**(CNRS/IHES)On Faltings' main comparison theorem in p-adic Hodge theory : the relative case (ENGLISH)

[ Abstract ]

In the appendix of his 2002 Asterisque article, Faltings roughly sketched a proof of a relative version of his main comparison theorem in p-adic Hodge theory. I will briefly review the absolute case and then explain some of the key new inputs for the proof of the relative case (joint work with Michel Gros).

In the appendix of his 2002 Asterisque article, Faltings roughly sketched a proof of a relative version of his main comparison theorem in p-adic Hodge theory. I will briefly review the absolute case and then explain some of the key new inputs for the proof of the relative case (joint work with Michel Gros).

### 2017/10/24

#### Seminar on Mathematics for various disciplines

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

The initial value problem for the multidimensional system of gas dynamics may have infinitely many weak solutions (English)

**Christian Klingenberg**(Würzburg University)The initial value problem for the multidimensional system of gas dynamics may have infinitely many weak solutions (English)

[ Abstract ]

We consider the isentropic compressible Euler equations in two space dimensions together with particular initial data. This data consists of two constant states only, where one state lies on the lower and the other state on the upper half plane. The aim is to investigate if there exists a unique entropy solution or if the convex integration method produces infinitely many entropy solutions. In this lecture we will show that the solution of this Riemann problem for the 2-d isentropic Euler equations is non-unique (except if the solution is smooth). Next we are able to show that there exist Lipschitz data that may lead to infinitely many solutions even for the full system of Euler equations. This is joint work with Simon Markfelder.

We consider the isentropic compressible Euler equations in two space dimensions together with particular initial data. This data consists of two constant states only, where one state lies on the lower and the other state on the upper half plane. The aim is to investigate if there exists a unique entropy solution or if the convex integration method produces infinitely many entropy solutions. In this lecture we will show that the solution of this Riemann problem for the 2-d isentropic Euler equations is non-unique (except if the solution is smooth). Next we are able to show that there exist Lipschitz data that may lead to infinitely many solutions even for the full system of Euler equations. This is joint work with Simon Markfelder.

#### Tuesday Seminar on Topology

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

Approach from the submanifold theory to the Floer homology of Lagrangian intersections (JAPANESE)

**Reiko Miyaoka**(Tohoku University)Approach from the submanifold theory to the Floer homology of Lagrangian intersections (JAPANESE)

[ Abstract ]

The Gauss map images of isoparametric hypersurfaces in the spheres supply a rich family of minimal Lagrangian submanifolds of the complex hyperquadric Q_n(C). In simple cases, these are real forms of Q_n(C), and their Floer homology is known. In this talk, we consider the case when the number of distinct principal curvatures is 3,4,6, and report our results. This is a joint work with Hiroshi Iriyeh (Ibaraki U.), Hui Ma (Tsinghua U.) and Yoshihiro Ohnita (Osaka City U.).

The Gauss map images of isoparametric hypersurfaces in the spheres supply a rich family of minimal Lagrangian submanifolds of the complex hyperquadric Q_n(C). In simple cases, these are real forms of Q_n(C), and their Floer homology is known. In this talk, we consider the case when the number of distinct principal curvatures is 3,4,6, and report our results. This is a joint work with Hiroshi Iriyeh (Ibaraki U.), Hui Ma (Tsinghua U.) and Yoshihiro Ohnita (Osaka City U.).

#### Lie Groups and Representation Theory

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

Approach from the submanifold theory to the FLoer homology of Lagrangian intersections (JAPANESE)

**Reiko Miyaoka**(Tohoku University)Approach from the submanifold theory to the FLoer homology of Lagrangian intersections (JAPANESE)

[ Abstract ]

The Gauss map images of isoparametric hypersurfaces in the spheres supply a rich family of minimal Lagrangian submanifolds of the complex hyperquadric Q_n(C). In simple cases, these are real forms of Q_n(C), and their Floer homology is known. In this talk, we consider the case when the number of distinct principal curvatures is 3,4,6, and report our results, which do not directly follow from FOOO’s theory. This is a joint work with Hiroshi Iriyeh (Ibaraki U.), Hui Ma (Tsinghua U.) and Yoshihiro Ohnita (Osaka City U.).

The Gauss map images of isoparametric hypersurfaces in the spheres supply a rich family of minimal Lagrangian submanifolds of the complex hyperquadric Q_n(C). In simple cases, these are real forms of Q_n(C), and their Floer homology is known. In this talk, we consider the case when the number of distinct principal curvatures is 3,4,6, and report our results, which do not directly follow from FOOO’s theory. This is a joint work with Hiroshi Iriyeh (Ibaraki U.), Hui Ma (Tsinghua U.) and Yoshihiro Ohnita (Osaka City U.).

### 2017/10/23

#### Numerical Analysis Seminar

16:00-17:30 Room #056 (Graduate School of Math. Sci. Bldg.)

On the numerical discretization of the Euler equations with a gravitational force and applications in astrophysics (English)

**Christian Klingenberg**(Wuerzburg University, Germany)On the numerical discretization of the Euler equations with a gravitational force and applications in astrophysics (English)

[ Abstract ]

We consider astrophysical systems that are modeled by the multidimensional Euler equations with gravity.

First for the homogeneous Euler equations we look at flow in the low Mach number regime. Here for conventional finite volume discretizations one has excessive dissipation in this regime. We identify inconsistent scaling for low Mach numbers of the numerical fux function as the origin of this problem. Based on the Roe solver a technique that allows to correctly represent low Mach number flows with a discretization of the compressible Euler equations is proposed. We analyze properties of this scheme and demonstrate that its limit yields a discretization of the incompressible limit system.

Next for the Euler equations with gravity we seek well-balanced methods. We describe a numerical discretization of the compressible Euler equations with a gravitational potential. A pertinent feature of the solutions to these inhomogeneous equations is the special case of stationary solutions with zero velocity, described by a nonlinear PDE, whose solutions are called hydrostatic equilibria. We present well-balanced methods, for which we can ensure robustness, accuracy and stability, since it satisfies discrete entropy inequalities.

We will then present work in progress where we combine the two methods above.

We consider astrophysical systems that are modeled by the multidimensional Euler equations with gravity.

First for the homogeneous Euler equations we look at flow in the low Mach number regime. Here for conventional finite volume discretizations one has excessive dissipation in this regime. We identify inconsistent scaling for low Mach numbers of the numerical fux function as the origin of this problem. Based on the Roe solver a technique that allows to correctly represent low Mach number flows with a discretization of the compressible Euler equations is proposed. We analyze properties of this scheme and demonstrate that its limit yields a discretization of the incompressible limit system.

Next for the Euler equations with gravity we seek well-balanced methods. We describe a numerical discretization of the compressible Euler equations with a gravitational potential. A pertinent feature of the solutions to these inhomogeneous equations is the special case of stationary solutions with zero velocity, described by a nonlinear PDE, whose solutions are called hydrostatic equilibria. We present well-balanced methods, for which we can ensure robustness, accuracy and stability, since it satisfies discrete entropy inequalities.

We will then present work in progress where we combine the two methods above.

#### Seminar on Geometric Complex Analysis

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

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

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

[ Abstract ]

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

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

#### Tokyo Probability Seminar

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

(JAPANESE)

**Yoshihiro Abe**(Department of Mathematics, Gakushuin University)(JAPANESE)

### 2017/10/19

#### Colloquium of mathematical sciences and society

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

### 2017/10/17

#### PDE Real Analysis Seminar

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

Some perspectives on negative index materials (English)

**Hoài-Minh Nguyên**(École Polytechnique Fédérale de Lausanne)Some perspectives on negative index materials (English)

[ Abstract ]

Negative index materials (NIMs) are artificial structures whose refractive index has negative value over some frequency range. These materials were first investigated theoretically by Veselago in 1964. The existence of NIMs was confirmed experimentally by Shelby, Smith, and Schultz in 2001. New fabrication techniques now allow the construction of NIMs at scales that are interesting for applications. NIMs have attracted a lot of attention from the scientific community, not only because of potentially interesting applications, but also because of challenges in understanding their peculiar properties. Mathematically, the study of NIMs faces two difficulties. First, the equations describing the phenomenon have sign changing coefficients, hence the ellipticity and the compactness are lost in general. Second, the localized resonance, i.e., the field explodes in some regions and remains bounded in some others as the loss goes to 0, might appear. In this talk I will discuss various mathematics techniques used to understand various applications of NIMs such as cloaking and superlensing and to develop new designs for them.

Negative index materials (NIMs) are artificial structures whose refractive index has negative value over some frequency range. These materials were first investigated theoretically by Veselago in 1964. The existence of NIMs was confirmed experimentally by Shelby, Smith, and Schultz in 2001. New fabrication techniques now allow the construction of NIMs at scales that are interesting for applications. NIMs have attracted a lot of attention from the scientific community, not only because of potentially interesting applications, but also because of challenges in understanding their peculiar properties. Mathematically, the study of NIMs faces two difficulties. First, the equations describing the phenomenon have sign changing coefficients, hence the ellipticity and the compactness are lost in general. Second, the localized resonance, i.e., the field explodes in some regions and remains bounded in some others as the loss goes to 0, might appear. In this talk I will discuss various mathematics techniques used to understand various applications of NIMs such as cloaking and superlensing and to develop new designs for them.

#### Tuesday Seminar on Topology

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

Generalizations of twisted Alexander invariants and quandle cocycle invariants (JAPANESE)

**Atsushi Ishii**(University of Tsukuba)Generalizations of twisted Alexander invariants and quandle cocycle invariants (JAPANESE)

[ Abstract ]

We introduce augmented Alexander matrices, and construct link invariants. An augmented Alexander matrix is defined with an augmented Alexander pair, which gives an extension of a quandle. This framework gives the twisted Alexander invariant and the quandle cocycle invariant. This is a joint work with Kanako Oshiro.

We introduce augmented Alexander matrices, and construct link invariants. An augmented Alexander matrix is defined with an augmented Alexander pair, which gives an extension of a quandle. This framework gives the twisted Alexander invariant and the quandle cocycle invariant. This is a joint work with Kanako Oshiro.

#### Algebraic Geometry Seminar

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

Intersection of currents, dimension excess and complex dynamics (English)

**Tien Cuong Dinh**(Singapore)Intersection of currents, dimension excess and complex dynamics (English)

[ Abstract ]

I will discuss dynamical properties of Henon maps in higher dimension, in particular, the equidistribution property of periodic points. Positive closed currents can be seen as an analytic counterpart of effective algebraic cycles. I will explain how a non-generic intersection theory for these currents, possibly with dimension excess, comes into the picture. Other applications of the intersection theory will be also discussed. This is a joint work with Nessim Sibony.

I will discuss dynamical properties of Henon maps in higher dimension, in particular, the equidistribution property of periodic points. Positive closed currents can be seen as an analytic counterpart of effective algebraic cycles. I will explain how a non-generic intersection theory for these currents, possibly with dimension excess, comes into the picture. Other applications of the intersection theory will be also discussed. This is a joint work with Nessim Sibony.

#### Lectures

17:00-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Integer partitions and hook length formulas (ENGLISH)

www-irma.u-strasbg.fr/~guoniu/

**Guoniu Han**(Université de Strasbourg/CNRS)Integer partitions and hook length formulas (ENGLISH)

[ Abstract ]

Integer partitions were first studied by Euler.

The Ferrers diagram of an integer partition is a very useful tool for

visualizing partitions. A Ferrers diagram is turned into a Young tableau

by filling each cell with a unique integer satisfying some conditions.

The number of Young tableaux is given by the famous hook length formula,

discovered by Frame-Robinson-Thrall.

In this talk, we introduce the hook length expansion technique and

explain how to find old and new hook length formulas for integer

partitions. In particular, we derive an expansion formula for the

powers of the Euler Product in terms of hook lengths, which is also

discovered by Nekrasov-Okounkov and Westburg. We obtain an extension

by adding two more parameters. It appears to be a discrete

interpolation between the Macdonald identities and the generating

function for t-cores. Several other summations involving hook length,

in particular, the Okada-Panova formula, will also be discussed.

[ Reference URL ]Integer partitions were first studied by Euler.

The Ferrers diagram of an integer partition is a very useful tool for

visualizing partitions. A Ferrers diagram is turned into a Young tableau

by filling each cell with a unique integer satisfying some conditions.

The number of Young tableaux is given by the famous hook length formula,

discovered by Frame-Robinson-Thrall.

In this talk, we introduce the hook length expansion technique and

explain how to find old and new hook length formulas for integer

partitions. In particular, we derive an expansion formula for the

powers of the Euler Product in terms of hook lengths, which is also

discovered by Nekrasov-Okounkov and Westburg. We obtain an extension

by adding two more parameters. It appears to be a discrete

interpolation between the Macdonald identities and the generating

function for t-cores. Several other summations involving hook length,

in particular, the Okada-Panova formula, will also be discussed.

www-irma.u-strasbg.fr/~guoniu/

### 2017/10/16

#### Seminar on Geometric Complex Analysis

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

Characterizations of hyperbolically $k$-convex domains in terms of hyperbolic metric

**Toshiyuki Sugawa**(Tohoku University)Characterizations of hyperbolically $k$-convex domains in terms of hyperbolic metric

[ Abstract ]

It is known that a plane domain $X$ with hyperbolic metric $h_X=h_X(z)|dz|$ of constant curvature $-4$ is (Euclidean) convex if and only if $h_X(z)d_X(z)\ge 1/2$, where $d_X(z)$ denotes the Euclidean distance from a point $z$ in $X$ to the boundary of $X$. We will consider spherical and hyperbolic versions of this result. More generally, we consider hyperbolic $k$-convexity (in the sense of Mejia and Minda) in the same line. A key is to observe a detailed behaviour of the hyperbolic density $h_X(z)$ near the boundary.

It is known that a plane domain $X$ with hyperbolic metric $h_X=h_X(z)|dz|$ of constant curvature $-4$ is (Euclidean) convex if and only if $h_X(z)d_X(z)\ge 1/2$, where $d_X(z)$ denotes the Euclidean distance from a point $z$ in $X$ to the boundary of $X$. We will consider spherical and hyperbolic versions of this result. More generally, we consider hyperbolic $k$-convexity (in the sense of Mejia and Minda) in the same line. A key is to observe a detailed behaviour of the hyperbolic density $h_X(z)$ near the boundary.

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