## Seminar information archive

Seminar information archive ～12/08｜Today's seminar 12/09 | Future seminars 12/10～

#### 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.

### 2017/10/11

#### 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).

#### Number Theory Seminar

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

Logarithmic resolution of singularities (ENGLISH)

**Michael Temkin**(The Hebrew University of Jerusalem)Logarithmic resolution of singularities (ENGLISH)

[ Abstract ]

The famous Hironaka's theorem asserts that any integral algebraic variety X of characteristic zero can be modified to a smooth variety X_res by a sequence of blowings up. Later it was shown that one can make this compatible with smooth morphisms Y --> X in the sense that Y_res --> Y is the pullback of X_res --> X. In a joint project with D. Abramovich and J. Wlodarczyk, we construct a new algorithm which is compatible with all log smooth morphisms (e.g. covers ramified along exceptional divisors). We expect that this algorithm will naturally extend to an algorithm of resolution of morphisms to log smooth ones. In particular, this should lead to functorial semistable reduction theorems. In my talk I will tell about main ideas of the classical algorithm and will then discuss logarithmic and stack-theoretic modifications we had to make in the new algorithm.

The famous Hironaka's theorem asserts that any integral algebraic variety X of characteristic zero can be modified to a smooth variety X_res by a sequence of blowings up. Later it was shown that one can make this compatible with smooth morphisms Y --> X in the sense that Y_res --> Y is the pullback of X_res --> X. In a joint project with D. Abramovich and J. Wlodarczyk, we construct a new algorithm which is compatible with all log smooth morphisms (e.g. covers ramified along exceptional divisors). We expect that this algorithm will naturally extend to an algorithm of resolution of morphisms to log smooth ones. In particular, this should lead to functorial semistable reduction theorems. In my talk I will tell about main ideas of the classical algorithm and will then discuss logarithmic and stack-theoretic modifications we had to make in the new algorithm.

### 2017/10/10

#### Tuesday Seminar on Topology

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

Poset-stratified spaces and some applications (JAPANESE)

**Shoji Yokura**(Kagoshima University)Poset-stratified spaces and some applications (JAPANESE)

[ Abstract ]

A poset-stratified space is a continuous map from a topological space to a poset with the Alexandroff topology. In this talk I will discuss some thoughts about poset-stratified spaces from a naive general-topological viewpoint, some applications such as hyperplane arrangements and poset-stratified space structures of hom-sets, and related topics such as characteristic classes of vector bundles, dependence of maps (by Borsuk) and dependence of cohomology classes (by Thom).

A poset-stratified space is a continuous map from a topological space to a poset with the Alexandroff topology. In this talk I will discuss some thoughts about poset-stratified spaces from a naive general-topological viewpoint, some applications such as hyperplane arrangements and poset-stratified space structures of hom-sets, and related topics such as characteristic classes of vector bundles, dependence of maps (by Borsuk) and dependence of cohomology classes (by Thom).

#### Numerical Analysis Seminar

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

Meshfree collocation methods for linear and fully nonlinear parabolic equations

**Yumiharu Nakano**(Tokyo Institute of Technology)Meshfree collocation methods for linear and fully nonlinear parabolic equations

#### Algebraic Geometry Seminar

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

Classification of Mukai pairs with corank 3 (English or Japanese)

**Akihiro Kanemitsu**(The University of Tokyo)Classification of Mukai pairs with corank 3 (English or Japanese)

[ Abstract ]

A Mukai pair $(X,E)$ is a pair of a Fano manifold $X$ and an ample vector bundle $E$ of rank $r$ on $X$ such that $c_1(X)=c_1(E)$. Study of such pairs was proposed by Mukai. It is known that, for a Mukai pair $(X,E)$, the rank $r$ of the bundle $E$ is at most $\dim X +1$, and Mukai conjectured the explicit

classification with $r \geq \dim X$. The above conjecture was solved independently by Fujita, Peternell and Ye-Zhang. Also the classification of Mukai pairs with $r= \dim X -1$ was given by Peternell-Szurek-Wi\'sniewski. In this talk I will give the classification of Mukai pairs with $r= \dim X -2$ and $\dim X \geq 5$.

A Mukai pair $(X,E)$ is a pair of a Fano manifold $X$ and an ample vector bundle $E$ of rank $r$ on $X$ such that $c_1(X)=c_1(E)$. Study of such pairs was proposed by Mukai. It is known that, for a Mukai pair $(X,E)$, the rank $r$ of the bundle $E$ is at most $\dim X +1$, and Mukai conjectured the explicit

classification with $r \geq \dim X$. The above conjecture was solved independently by Fujita, Peternell and Ye-Zhang. Also the classification of Mukai pairs with $r= \dim X -1$ was given by Peternell-Szurek-Wi\'sniewski. In this talk I will give the classification of Mukai pairs with $r= \dim X -2$ and $\dim X \geq 5$.

### 2017/10/06

#### Colloquium

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

Singular Integrals and Real Variable Methods in Harmonic Analysis (JAPANESE)

[ Reference URL ]

http://lab.twcu.ac.jp/miyachi/English.html

**Akihiko Miyachi**(Tokyo Woman's Christian University)Singular Integrals and Real Variable Methods in Harmonic Analysis (JAPANESE)

[ Reference URL ]

http://lab.twcu.ac.jp/miyachi/English.html

### 2017/10/03

#### Tuesday Seminar on Topology

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

Transitional geometry (ENGLISH)

**Athanase Papadopoulos**(IRMA, Université de Strasbourg)Transitional geometry (ENGLISH)

[ Abstract ]

I will describe transitions, that is, paths between hyperbolic and spherical geometry, passing through the Euclidean. This is based on joint work with Norbert A’Campo and recent joint work with A’Campo and Yi Huang.

I will describe transitions, that is, paths between hyperbolic and spherical geometry, passing through the Euclidean. This is based on joint work with Norbert A’Campo and recent joint work with A’Campo and Yi Huang.

### 2017/10/02

#### Operator Algebra Seminars

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

Introduction to intermediate rank geometry (English)

**Mikael Pichot**(RIMS, Kyoto Univ./McGill Univ.)Introduction to intermediate rank geometry (English)

#### Seminar on Geometric Complex Analysis

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

The extension of holomorphic functions on a non-pluriharmonic locus

**Yusaku Tiba**(Ochanomizu University)The extension of holomorphic functions on a non-pluriharmonic locus

[ Abstract ]

Let $n \geq 4$ and let $\Omega$ be a bounded hyperconvex domain in $\mathbb{C}^{n}$. Let $\varphi$ be a negative exhaustive smooth plurisubharmonic function on $\Omega$. In this talk, we show that any holomorphic function defined on a connected open neighborhood of the support of $(i\partial \overline{\partial}\varphi)^{n-3}$ can be extended to the holomorphic function on $\Omega$.

Let $n \geq 4$ and let $\Omega$ be a bounded hyperconvex domain in $\mathbb{C}^{n}$. Let $\varphi$ be a negative exhaustive smooth plurisubharmonic function on $\Omega$. In this talk, we show that any holomorphic function defined on a connected open neighborhood of the support of $(i\partial \overline{\partial}\varphi)^{n-3}$ can be extended to the holomorphic function on $\Omega$.

### 2017/09/27

#### Number Theory Seminar

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

Height functions for motives, Hodge analogues, and Nevanlinna analogues (ENGLISH)

**Kazuya Kato**(University of Chicago)Height functions for motives, Hodge analogues, and Nevanlinna analogues (ENGLISH)

[ Abstract ]

We compare height functions for (1) points of an algebraic variety over a number field, (2) motives over a number field, (3) variations of Hodge structure with log degeneration on a projective smooth curve over the complex number field, (4) horizontal maps from the complex plane C to a toroidal partial compactification of the period domain. Usual Nevanlinna theory uses height functions for (5) holomorphic maps f from C to a compactification of an agebraic variety V and considers how often the values of f lie outside V. Vojta compares (1) and (5). In (4), V is replaced by a period domain. The comparisons of (1)--(4) provide many new questions to study.

We compare height functions for (1) points of an algebraic variety over a number field, (2) motives over a number field, (3) variations of Hodge structure with log degeneration on a projective smooth curve over the complex number field, (4) horizontal maps from the complex plane C to a toroidal partial compactification of the period domain. Usual Nevanlinna theory uses height functions for (5) holomorphic maps f from C to a compactification of an agebraic variety V and considers how often the values of f lie outside V. Vojta compares (1) and (5). In (4), V is replaced by a period domain. The comparisons of (1)--(4) provide many new questions to study.

### 2017/09/26

#### Tuesday Seminar on Topology

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

Representations of Semisimple Lie Groups and Penrose Transform (JAPANESE)

**Hideko Sekiguchi**(The University of Tokyo)Representations of Semisimple Lie Groups and Penrose Transform (JAPANESE)

[ Abstract ]

The classical Penrose transform is generalized to an intertwining operator on Dolbeault cohomologies of complex homogeneous spaces $X$ of (real) semisimple Lie groups.

I plan to discuss a detailed analysis when $X$ is an indefinite Grassmann manifold.

To be more precise, we determine the image of the Penrose transform, from the Dolbeault cohomology group on the indefinite Grassmann manifold consisting of maximally positive $k$-planes in ${\mathbb{C}}^{p,q}$ ($1 \le k \le \min(p,q)$) to the space of holomorphic functions over the bounded symmetric domain.

Furthermore, we prove that there is a duality between Dolbeault cohomology groups in two indefinite Grassmann manifolds,

namely, that of positive $k$-planes and that of negative $k$-planes.

The classical Penrose transform is generalized to an intertwining operator on Dolbeault cohomologies of complex homogeneous spaces $X$ of (real) semisimple Lie groups.

I plan to discuss a detailed analysis when $X$ is an indefinite Grassmann manifold.

To be more precise, we determine the image of the Penrose transform, from the Dolbeault cohomology group on the indefinite Grassmann manifold consisting of maximally positive $k$-planes in ${\mathbb{C}}^{p,q}$ ($1 \le k \le \min(p,q)$) to the space of holomorphic functions over the bounded symmetric domain.

Furthermore, we prove that there is a duality between Dolbeault cohomology groups in two indefinite Grassmann manifolds,

namely, that of positive $k$-planes and that of negative $k$-planes.

#### Lie Groups and Representation Theory

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

Representations of Semisimple Lie Groups and Penrose Transform (Japanese)

**Hideko Sekiguchi**(The University of Tokyo)Representations of Semisimple Lie Groups and Penrose Transform (Japanese)

[ Abstract ]

The classical Penrose transform is generalized to an intertwining operator on Dolbeault cohomologies of complex homogeneous spaces $X$ of (real) semisimple Lie groups.

I plan to discuss a detailed analysis when $X$ is an indefinite Grassmann manifold.

To be more precise, we determine the image of the Penrose transform, from the Dolbeault cohomology group on the indefinite Grassmann manifold consisting of maximally positive $k$-planes in ${\mathbb{C}}^{p,q}$ ($1 \le k \le \min(p,q)$) to the space of holomorphic functions over the bounded symmetric domain.

Furthermore, we prove that there is a duality between Dolbeault cohomology groups in two indefinite Grassmann manifolds, namely, that of positive $k$-planes and that of negative $k$-planes.

The classical Penrose transform is generalized to an intertwining operator on Dolbeault cohomologies of complex homogeneous spaces $X$ of (real) semisimple Lie groups.

I plan to discuss a detailed analysis when $X$ is an indefinite Grassmann manifold.

To be more precise, we determine the image of the Penrose transform, from the Dolbeault cohomology group on the indefinite Grassmann manifold consisting of maximally positive $k$-planes in ${\mathbb{C}}^{p,q}$ ($1 \le k \le \min(p,q)$) to the space of holomorphic functions over the bounded symmetric domain.

Furthermore, we prove that there is a duality between Dolbeault cohomology groups in two indefinite Grassmann manifolds, namely, that of positive $k$-planes and that of negative $k$-planes.

### 2017/09/25

#### Seminar on Geometric Complex Analysis

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

Asymptotics of $L^2$ and Quillen metrics in degenerations of Calabi-Yau varieties

**Christophe Mourougane**(Université de Rennes 1)Asymptotics of $L^2$ and Quillen metrics in degenerations of Calabi-Yau varieties

[ Abstract ]

It is a joint work with Dennis Eriksson and Gerard Freixas i Montplet.

Our first motivation is to give a metric analogue of Kodaira's canonical bundle formula for elliptic surfaces, in the case of families of Calabi-Yau varieties. We consider degenerations of complex projective Calabi-Yau varieties and study the singularities of $L^2$, Quillen and BCOV metrics on Hodge and determinant bundles. The dominant and subdominant terms in the expansions of the metrics close to non-smooth fibres are shown to be related to well-known topological invariants of singularities, such as limit Hodge structures, vanishing cycles and log-canonical thresholds.

It is a joint work with Dennis Eriksson and Gerard Freixas i Montplet.

Our first motivation is to give a metric analogue of Kodaira's canonical bundle formula for elliptic surfaces, in the case of families of Calabi-Yau varieties. We consider degenerations of complex projective Calabi-Yau varieties and study the singularities of $L^2$, Quillen and BCOV metrics on Hodge and determinant bundles. The dominant and subdominant terms in the expansions of the metrics close to non-smooth fibres are shown to be related to well-known topological invariants of singularities, such as limit Hodge structures, vanishing cycles and log-canonical thresholds.

### 2017/09/11

#### Lectures

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

3D field theories with Chern-Simons term for large N in the Weyl gauge (ENGLISH)

**Jean Zinn-Justin**(CEA Saclay)3D field theories with Chern-Simons term for large N in the Weyl gauge (ENGLISH)

[ Abstract ]

ADS/CFT correspondance has led to a number of conjectures concerning, conformal invariant, U(N) symmetric 3D field theories with Chern-Simons term for N large. An example is boson-fermion duality. This has prompted a number of calculations to shed extra light on the ADS/CFT correspondance.

We study here the example of gauge invariant fermion matter coupled to a Chern-Simons term. In contrast with previous calculations, which employ the light-cone gauge, we use the more conventional temporal gauge. We calculate several gauge invariant correlation functions. We consider general massive matter and determine the conditions for conformal invariance. We compare massless results with previous calculations, providing a check of gauge independence.

We examine also the possibility of spontaneous breaking of scale invariance and show that this requires the addition of an auxiliary scalar field.

Our method is based on field integral and steepest descent. The saddle point equations involve non-local fields and take the form of a set of integral equations that we solve exactly.

ADS/CFT correspondance has led to a number of conjectures concerning, conformal invariant, U(N) symmetric 3D field theories with Chern-Simons term for N large. An example is boson-fermion duality. This has prompted a number of calculations to shed extra light on the ADS/CFT correspondance.

We study here the example of gauge invariant fermion matter coupled to a Chern-Simons term. In contrast with previous calculations, which employ the light-cone gauge, we use the more conventional temporal gauge. We calculate several gauge invariant correlation functions. We consider general massive matter and determine the conditions for conformal invariance. We compare massless results with previous calculations, providing a check of gauge independence.

We examine also the possibility of spontaneous breaking of scale invariance and show that this requires the addition of an auxiliary scalar field.

Our method is based on field integral and steepest descent. The saddle point equations involve non-local fields and take the form of a set of integral equations that we solve exactly.

### 2017/08/30

#### thesis presentations

10:00-11:15 Room #128 (Graduate School of Math. Sci. Bldg.)

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