## Seminar information archive

Seminar information archive ～02/25｜Today's seminar 02/26 | Future seminars 02/27～

### 2012/09/04

#### Tuesday Seminar on Topology

17:00-18:00 Room #002 (Graduate School of Math. Sci. Bldg.)

Poincare inequalities, rigid groups and applications (ENGLISH)

**Piotr Nowak**(the Institute of Mathematics, Polish Academy of Sciences)Poincare inequalities, rigid groups and applications (ENGLISH)

[ Abstract ]

Kazhdan’s property (T) for a group G can be expressed as a

fixed point property for affine isometric actions of G on a Hilbert

space. This definition generalizes naturally to other normed spaces. In

this talk we will focus on the spectral (aka geometric) method for

proving property (T), based on the work of Garland and studied earlier

by Pansu, Zuk, Ballmann-Swiatkowski, Dymara-Januszkiewicz

(“lambda_1>1/2” conditions) and we generalize it to to the setting of

all reflexive Banach spaces.

As applications we will show estimates of the conformal dimension of the

boundary of random hyperbolic groups in the Gromov density model and

present progress on Shalom’s conjecture on vanishing of 1-cohomology

with coefficients in uniformly bounded representations on Hilbert spaces.

Kazhdan’s property (T) for a group G can be expressed as a

fixed point property for affine isometric actions of G on a Hilbert

space. This definition generalizes naturally to other normed spaces. In

this talk we will focus on the spectral (aka geometric) method for

proving property (T), based on the work of Garland and studied earlier

by Pansu, Zuk, Ballmann-Swiatkowski, Dymara-Januszkiewicz

(“lambda_1>1/2” conditions) and we generalize it to to the setting of

all reflexive Banach spaces.

As applications we will show estimates of the conformal dimension of the

boundary of random hyperbolic groups in the Gromov density model and

present progress on Shalom’s conjecture on vanishing of 1-cohomology

with coefficients in uniformly bounded representations on Hilbert spaces.

### 2012/08/29

#### thesis presentations

10:30-11:45 Room #123 (Graduate School of Math. Sci. Bldg.)

Studies on the asymptotic invariants of cohomology groups and the positivity in complex geometry (JAPANESE)

**Shinichi MATSUMURA**(Graduate School of Mathematical Sciences the University of Tokyo)Studies on the asymptotic invariants of cohomology groups and the positivity in complex geometry (JAPANESE)

### 2012/08/20

#### thesis presentations

11:00-12:15 Room #128 (Graduate School of Math. Sci. Bldg.)

The variational formulation of the fully parabolic Keller-Segel system with degenerate diffusion (JAPANESE)

**Yoshifumi MIMURA**(Graduate School of Mathematical Sciences the University of Tokyo)The variational formulation of the fully parabolic Keller-Segel system with degenerate diffusion (JAPANESE)

### 2012/07/30

#### Algebraic Geometry Seminar

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

Log Bend-and-Break on Deligne-Mumford stacks (ENGLISH)

**Gianluca Pacienza**(Université de Strasbourg)Log Bend-and-Break on Deligne-Mumford stacks (ENGLISH)

[ Abstract ]

We prove a logarithmic Bend-and-Break lemma on a LCI Deligne-Mumford stacks with projective moduli space and integral boundary divisor. As a by-product we obtain a logarithmic version of the Miyaoka-Mori numerical criterion of uniruledness for DM stacks (under additional conditions on the boundary and on the non-schematic locus) and a Cone Theorem for Deligne-Mumford stacks with boundary. These results hold on an algebraically closed field of any characteristic. This is joint work with Michael McQuillan.

We prove a logarithmic Bend-and-Break lemma on a LCI Deligne-Mumford stacks with projective moduli space and integral boundary divisor. As a by-product we obtain a logarithmic version of the Miyaoka-Mori numerical criterion of uniruledness for DM stacks (under additional conditions on the boundary and on the non-schematic locus) and a Cone Theorem for Deligne-Mumford stacks with boundary. These results hold on an algebraically closed field of any characteristic. This is joint work with Michael McQuillan.

### 2012/07/27

#### Seminar on Probability and Statistics

14:00-17:00 Room #006 (Graduate School of Math. Sci. Bldg.)

General approach to reinforcement learning based on statistical inference (JAPANESE)

[ Reference URL ]

http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2012/06.html

**UENO, Tsuyoshi**(Minato Discrete Structure Manipulation System Project, Japan Science and Technology Agency)General approach to reinforcement learning based on statistical inference (JAPANESE)

[ Reference URL ]

http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2012/06.html

### 2012/07/25

#### GCOE lecture series

16:30-18:00 Room #118 (Graduate School of Math. Sci. Bldg.)

A survey of recent results on the classification of C*-algebras (ENGLISH)

**George Elliott**(University of Toronto)A survey of recent results on the classification of C*-algebras (ENGLISH)

### 2012/07/24

#### Tuesday Seminar on Topology

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Orthospectra and identities (ENGLISH)

**Greg McShane**(Institut Fourier, Grenoble)Orthospectra and identities (ENGLISH)

[ Abstract ]

The orthospectra of a hyperbolic manifold with geodesic

boundary consists of the lengths of all geodesics perpendicular to the

boundary.

We discuss the properties of the orthospectra, asymptotics, multiplicity

and identities due to Basmajian, Bridgeman and Calegari. We will give

a proof that the identities of Bridgeman and Calegari are the same.

The orthospectra of a hyperbolic manifold with geodesic

boundary consists of the lengths of all geodesics perpendicular to the

boundary.

We discuss the properties of the orthospectra, asymptotics, multiplicity

and identities due to Basmajian, Bridgeman and Calegari. We will give

a proof that the identities of Bridgeman and Calegari are the same.

#### Lie Groups and Representation Theory

16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)

The Dynkin index and conformally invariant systems of Heisenberg parabolic type (ENGLISH)

**Toshihisa Kubo**(the University of Tokyo)The Dynkin index and conformally invariant systems of Heisenberg parabolic type (ENGLISH)

[ Abstract ]

Recently, Barchini-Kable-Zierau systematically constructed conformally invariant systems of differential operators using Heisenberg parabolic subalgebras. When they built such systems, two constants, which are defined as the constant of proportionality between two expressions,played an important role. In this talk we give concrete and uniform expressions for these constants. To do so the Dynkin index of a finite dimensional representation of a complex simple Lie algebra plays a key role.

Recently, Barchini-Kable-Zierau systematically constructed conformally invariant systems of differential operators using Heisenberg parabolic subalgebras. When they built such systems, two constants, which are defined as the constant of proportionality between two expressions,played an important role. In this talk we give concrete and uniform expressions for these constants. To do so the Dynkin index of a finite dimensional representation of a complex simple Lie algebra plays a key role.

### 2012/07/23

#### Algebraic Geometry Seminar

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

Derived category of smooth proper Deligne-Mumford stack with p_g>0 (JAPANESE)

**Shinnosuke Okawa**(University of Tokyo)Derived category of smooth proper Deligne-Mumford stack with p_g>0 (JAPANESE)

[ Abstract ]

Semiorthogonal decomposition (SOD) of the derived category of coherent sheaves reflects interesting geometry of varieties (more generally stacks), such as minimal model program. We show that the global sections of the canonical line bundle (if exists) give restrictions on the possible form of SODs. As a special case, we see that the global generation of the canonical line bundle implies the non-existence of SODs. (joint work with Kotaro Kawatani)

Semiorthogonal decomposition (SOD) of the derived category of coherent sheaves reflects interesting geometry of varieties (more generally stacks), such as minimal model program. We show that the global sections of the canonical line bundle (if exists) give restrictions on the possible form of SODs. As a special case, we see that the global generation of the canonical line bundle implies the non-existence of SODs. (joint work with Kotaro Kawatani)

#### Lectures

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

Combinatorial Ergodicity (ENGLISH)

**Thomas W. Roby**(University of Connecticut)Combinatorial Ergodicity (ENGLISH)

[ Abstract ]

Many cyclic actions $\\tau$ on a finite set $S$ of

combinatorial objects, along with many natural

statistics $\\phi$ on $S$, exhibit``combinatorial ergodicity'':

the average of $\\phi$ over each $\\tau$-orbit in $S$ is

the same as the average of $\\phi$ over the whole set $S$.

One example is the case where $S$ is the set of

length $n$ binary strings $a_{1}\\dots a_{n}$

with exactly $k$ 1's,

$\\tau$ is the map that cyclically rotates them,

and $\\phi$ is the number of \\textit{inversions}

(i.e, pairs $(a_{i},a_{j})=(1,0)$ for $iJ$ less than $j$).

This phenomenon was first noticed by Panyushev

in 2007 in the context of antichains in root posets;

Armstrong, Stump, and Thomas proved his

conjecture in 2011.

We describe a theoretical framework for results of this kind,

and discuss old and new results for products of two chains.

This is joint work with Jim Propp.

Many cyclic actions $\\tau$ on a finite set $S$ of

combinatorial objects, along with many natural

statistics $\\phi$ on $S$, exhibit``combinatorial ergodicity'':

the average of $\\phi$ over each $\\tau$-orbit in $S$ is

the same as the average of $\\phi$ over the whole set $S$.

One example is the case where $S$ is the set of

length $n$ binary strings $a_{1}\\dots a_{n}$

with exactly $k$ 1's,

$\\tau$ is the map that cyclically rotates them,

and $\\phi$ is the number of \\textit{inversions}

(i.e, pairs $(a_{i},a_{j})=(1,0)$ for $iJ$ less than $j$).

This phenomenon was first noticed by Panyushev

in 2007 in the context of antichains in root posets;

Armstrong, Stump, and Thomas proved his

conjecture in 2011.

We describe a theoretical framework for results of this kind,

and discuss old and new results for products of two chains.

This is joint work with Jim Propp.

### 2012/07/21

#### Monthly Seminar on Arithmetic of Automorphic Forms

13:30-16:00 Room #123 (Graduate School of Math. Sci. Bldg.)

On Fourier coefficients of Siegel-Eisenstein series of degree n. (JAPANESE)

Polylogarithms revisited from the viewpoint of the irrationality (JAPANESE)

**S. Takemori**(Kyoto Univ., School of Science) 13:30-14:30On Fourier coefficients of Siegel-Eisenstein series of degree n. (JAPANESE)

[ Abstract ]

We define an Siegel-Eisenstein series G_{k,\\chi} of degree n and talk about an explicit formula of the Fourier coefficients. This Eisenstein series is different from ordinarily defined Eisenstein series E_{k,\\chi}, but if \\chi satisfies a certain condition, we can obtain an explicit formula of Fourier coefficients of E_{k,\\chi}.

We define an Siegel-Eisenstein series G_{k,\\chi} of degree n and talk about an explicit formula of the Fourier coefficients. This Eisenstein series is different from ordinarily defined Eisenstein series E_{k,\\chi}, but if \\chi satisfies a certain condition, we can obtain an explicit formula of Fourier coefficients of E_{k,\\chi}.

**Noriko HIRATA-Kohno**(Nihon University) 15:00-16:00Polylogarithms revisited from the viewpoint of the irrationality (JAPANESE)

[ Abstract ]

In this report, we consider a polylogarithmic function to give a lower bound for the dimension of the linear space over the rationals spanned by $1$ and values of the function. Our proof uses Pad\\'e approximation and a criterion due to Yu. V. Nesterenko. We also describe what happens in the $p$-adic case and in the elliptic one.

In this report, we consider a polylogarithmic function to give a lower bound for the dimension of the linear space over the rationals spanned by $1$ and values of the function. Our proof uses Pad\\'e approximation and a criterion due to Yu. V. Nesterenko. We also describe what happens in the $p$-adic case and in the elliptic one.

### 2012/07/19

#### GCOE Seminars

17:00-18:00 Room #370 (Graduate School of Math. Sci. Bldg.)

Inverse boundary value problem for Schroedinger equation in two dimensions (ENGLISH)

**Oleg Emanouilov**(Colorado State Univ.)Inverse boundary value problem for Schroedinger equation in two dimensions (ENGLISH)

[ Abstract ]

We consider the Dirichlet-to-Neumann map for determining potential in two-dimensional Schroedinger equation. We relax the regularity condition on potentials and establish the uniqueness within L^p class with p > 2.

We consider the Dirichlet-to-Neumann map for determining potential in two-dimensional Schroedinger equation. We relax the regularity condition on potentials and establish the uniqueness within L^p class with p > 2.

### 2012/07/18

#### Number Theory Seminar

16:40-17:40 Room #056 (Graduate School of Math. Sci. Bldg.)

Voevodsky motives and a theorem of Gabber (ENGLISH)

**Shane Kelly**(Australian National University)Voevodsky motives and a theorem of Gabber (ENGLISH)

[ Abstract ]

The assumption that the base field satisfies resolution of singularities litters Voevodsky's work on motives. While we don't have resolution of singularities in positive characteristic p, there is a theorem of Gabber on alterations which may be used as a substitute if we are willing to work with Z[1/p] coefficients. We will discuss how this theorem of Gabber may be applied in the context of Voevodsky's work and mention some consequences.

The assumption that the base field satisfies resolution of singularities litters Voevodsky's work on motives. While we don't have resolution of singularities in positive characteristic p, there is a theorem of Gabber on alterations which may be used as a substitute if we are willing to work with Z[1/p] coefficients. We will discuss how this theorem of Gabber may be applied in the context of Voevodsky's work and mention some consequences.

#### Classical Analysis

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

Free divisors, holonomic systems and algebraic Painlev\\'{e} sixth solutions (ENGLISH)

**Jiro Sekiguchi**(Tokyo University of Agriculture and Technology)Free divisors, holonomic systems and algebraic Painlev\\'{e} sixth solutions (ENGLISH)

[ Abstract ]

In this talk, I will report an attempt to treat algebraic solutions of Painlev\\'{e} VI equation in a unified manner.

A classification of algebraic solutions of Painlev\\'{e} VI equation was accomplished by O. Lisovyy and Y. Tykhyy after efforts on the construction of such solutions by many authors, K. Iwasaki N. J. Hitchin, P. Boalch, B. Dubrovin, M. Mazzocco, A. V. Kitaev, R. Vidunas and others.

The outline of my approach is as follows.

Let $t$ be a variable and let $w$ be its algebraic function such that $w$ is a solution of Painlev\\'{e} sixth equation. Suppose that both $t$ and $w$ are rational functions of a parameter. Namely $(t,w)$ defines a rational curve.

(1) Find a polynomial $P(u)$ such that $t=\\frac{P(-u)}{P(u)}$.

(2) From $P(u)$, define a weighted homogeneous polynomial $f(x_1,x_2,x_3)=x_3f_1(x_1,x_2,x_3)$ of three variables $x_1,x_2,x_3$, where $(1,2,n)$ is the weight system of $(x_1,x_2,x_3)$ with $n=\\deg P(u)$. The hypersurface $D:f(x)=0$ is a free divisor in ${\\bf C}^3$. Note that $\\deg_{x_3}f_1=2$.

(3) Construct a holonomic system ${\\sl M}$ on ${\\bf C}^3$ of rank two with singularities along $D$.

(4) Construct an ordinary differential equation from the holonomic system ${\\sl M}$ with respect to $x_3$. This differential equation has three singular points $z_0,z_1,a_s$ in $x_3$-line.

(5) Putting $t=\\frac{z_1}{z_0},\\lambda=\\frac{a_s}{z_0}$, we conclude that $(t,\\lambda)$ is equivalent to the pair $(t,w)$.

Our study starts with showing the existence of $P(u)$ in (1). From the classification by Losovyy and Tykhyy, I find that the existence of $P(u)$ is guaranteed for Solutions III, IV, Solutions $k$ ($1\\le k\\le 21$, $k\\not= 4,13,14,20$) and Solution 30. We checked whether (1)-(5) are true or not in these cases separately and as a consequence (1)-(5) hold for the all these cases except Solutions 19, 21.

In this talk, I will report an attempt to treat algebraic solutions of Painlev\\'{e} VI equation in a unified manner.

A classification of algebraic solutions of Painlev\\'{e} VI equation was accomplished by O. Lisovyy and Y. Tykhyy after efforts on the construction of such solutions by many authors, K. Iwasaki N. J. Hitchin, P. Boalch, B. Dubrovin, M. Mazzocco, A. V. Kitaev, R. Vidunas and others.

The outline of my approach is as follows.

Let $t$ be a variable and let $w$ be its algebraic function such that $w$ is a solution of Painlev\\'{e} sixth equation. Suppose that both $t$ and $w$ are rational functions of a parameter. Namely $(t,w)$ defines a rational curve.

(1) Find a polynomial $P(u)$ such that $t=\\frac{P(-u)}{P(u)}$.

(2) From $P(u)$, define a weighted homogeneous polynomial $f(x_1,x_2,x_3)=x_3f_1(x_1,x_2,x_3)$ of three variables $x_1,x_2,x_3$, where $(1,2,n)$ is the weight system of $(x_1,x_2,x_3)$ with $n=\\deg P(u)$. The hypersurface $D:f(x)=0$ is a free divisor in ${\\bf C}^3$. Note that $\\deg_{x_3}f_1=2$.

(3) Construct a holonomic system ${\\sl M}$ on ${\\bf C}^3$ of rank two with singularities along $D$.

(4) Construct an ordinary differential equation from the holonomic system ${\\sl M}$ with respect to $x_3$. This differential equation has three singular points $z_0,z_1,a_s$ in $x_3$-line.

(5) Putting $t=\\frac{z_1}{z_0},\\lambda=\\frac{a_s}{z_0}$, we conclude that $(t,\\lambda)$ is equivalent to the pair $(t,w)$.

Our study starts with showing the existence of $P(u)$ in (1). From the classification by Losovyy and Tykhyy, I find that the existence of $P(u)$ is guaranteed for Solutions III, IV, Solutions $k$ ($1\\le k\\le 21$, $k\\not= 4,13,14,20$) and Solution 30. We checked whether (1)-(5) are true or not in these cases separately and as a consequence (1)-(5) hold for the all these cases except Solutions 19, 21.

### 2012/07/17

#### Tuesday Seminar on Topology

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Contact structure of mixed links (JAPANESE)

**Mutsuo Oka**(Tokyo University of Science)Contact structure of mixed links (JAPANESE)

[ Abstract ]

A strongly non-degenerate mixed function has a Milnor open book

structures on a sufficiently small sphere. We introduce the notion of

{\\em a holomorphic-like} mixed function

and we will show that a link defined by such a mixed function has a

canonical contact structure.

Then we will show that this contact structure for a certain

holomorphic-like mixed function

is carried by the Milnor open book.

A strongly non-degenerate mixed function has a Milnor open book

structures on a sufficiently small sphere. We introduce the notion of

{\\em a holomorphic-like} mixed function

and we will show that a link defined by such a mixed function has a

canonical contact structure.

Then we will show that this contact structure for a certain

holomorphic-like mixed function

is carried by the Milnor open book.

#### GCOE lecture series

16:30-18:00 Room #118 (Graduate School of Math. Sci. Bldg.)

An introduction to C*-algebra classification theory (ENGLISH)

**George Elliott**(University of Toronto)An introduction to C*-algebra classification theory (ENGLISH)

#### Tuesday Seminar of Analysis

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

On non-radially symmetric solutions of the Liouville-Gel'fand equation on a two-dimensional annular domain (JAPANESE)

**Toru Kan**(Mathematical institute, Tohoku University)On non-radially symmetric solutions of the Liouville-Gel'fand equation on a two-dimensional annular domain (JAPANESE)

[ Abstract ]

指数関数を非線形項に持つ非線形楕円型方程式(Liouville-Gel'fand方程式)について考察する。特に2次元の円環領域では、この方程式の非球対称な解が球対称解から分岐する形で現れる。本講演では、この分岐解の分岐図上での大域的な構造に関して得られた結果を紹介する。

指数関数を非線形項に持つ非線形楕円型方程式(Liouville-Gel'fand方程式)について考察する。特に2次元の円環領域では、この方程式の非球対称な解が球対称解から分岐する形で現れる。本講演では、この分岐解の分岐図上での大域的な構造に関して得られた結果を紹介する。

#### Lie Groups and Representation Theory

17:00-18:30 Room #126 (Graduate School of Math. Sci. Bldg.)

Dirac induction for graded affine Hecke algebras (ENGLISH)

**Eric Opdam**(Universiteit van Amsterdam)Dirac induction for graded affine Hecke algebras (ENGLISH)

[ Abstract ]

In recent joint work with Dan Ciubotaru and Peter Trapa we

constructed a model for the discrete series representations of graded affine Hecke algebras as the index of a Dirac operator.

We discuss the K-theoretic meaning of this result, and the remarkable relation between elliptic character theory of a Weyl group and the ordinary character theory of its Pin cover.

In recent joint work with Dan Ciubotaru and Peter Trapa we

constructed a model for the discrete series representations of graded affine Hecke algebras as the index of a Dirac operator.

We discuss the K-theoretic meaning of this result, and the remarkable relation between elliptic character theory of a Weyl group and the ordinary character theory of its Pin cover.

### 2012/07/14

#### Harmonic Analysis Komaba Seminar

13:30-17:00 Room #128 (Graduate School of Math. Sci. Bldg.)

On various inequalities characterizing critical Sobolev-Lorentz spaces (JAPANESE)

Boundedness of operators on Hardy spaces with variable exponents

(JAPANESE)

**Hidemitsu Wadade**(Gifu University) 13:30-15:00On various inequalities characterizing critical Sobolev-Lorentz spaces (JAPANESE)

**Yoshihiro Sawano**(Tokyo Metropolitan University) 15:30-17:00Boundedness of operators on Hardy spaces with variable exponents

(JAPANESE)

[ Abstract ]

In this talk, as an off-spring, we will discuss the boundedness of various operators. Our plan of the talk is as follows:

First we recall the definition of Hardy spaces with variable exponents and then we describe the atomic decomposition.

Based upon the atomic decomposition, I define linear operators such as singular integral operators and commutators.

After the definition, I will state the boundedness results and outline the proof of the boundedness of these operators.

In this talk, as an off-spring, we will discuss the boundedness of various operators. Our plan of the talk is as follows:

First we recall the definition of Hardy spaces with variable exponents and then we describe the atomic decomposition.

Based upon the atomic decomposition, I define linear operators such as singular integral operators and commutators.

After the definition, I will state the boundedness results and outline the proof of the boundedness of these operators.

### 2012/07/12

#### Lectures

16:30-17:30 Room #370 (Graduate School of Math. Sci. Bldg.)

Real time tsunami parameters evaluation (ENGLISH)

**M. Lavrentiev**(Sobolev Institute of Mathematics)Real time tsunami parameters evaluation (ENGLISH)

[ Abstract ]

We would like to propose several improvements to the existing software tools for tsunami modeling. Combination of optimaly located system of sensors with advantages of modern hardware architectures will make it possible to deliver calculated parameters of tsunami wave in 12-15 minutes after seismic event.

We would like to propose several improvements to the existing software tools for tsunami modeling. Combination of optimaly located system of sensors with advantages of modern hardware architectures will make it possible to deliver calculated parameters of tsunami wave in 12-15 minutes after seismic event.

### 2012/07/11

#### Classical Analysis

14:00-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)

On a q-analog of Painlevé III (D_7^{(1)}) and its algebraic function solutions (Joint work with N. Nakazono) (JAPANESE)

First order systems of linear ordinary differential equations and

representations of quivers (ENGLISH)

**Seiji Nishioka**(Yamagata University) 14:00-15:30On a q-analog of Painlevé III (D_7^{(1)}) and its algebraic function solutions (Joint work with N. Nakazono) (JAPANESE)

**Kazuki Hiroe**(Kyoto University) 16:00-17:30First order systems of linear ordinary differential equations and

representations of quivers (ENGLISH)

### 2012/07/10

#### Tuesday Seminar of Analysis

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Hadamard variational formula for the Green function

of the Stokes equations with the boundary condition (JAPANESE)

**Erika Ushikoshi**(Mathematical Institute, Tohoku University)Hadamard variational formula for the Green function

of the Stokes equations with the boundary condition (JAPANESE)

#### Tuesday Seminar on Topology

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Topology in Gravitational Lensing (ENGLISH)

**Marcus Werner**(Kavli IPMU)Topology in Gravitational Lensing (ENGLISH)

[ Abstract ]

General relativity implies that light is deflected by masses

due to the curvature of spacetime. The ensuing gravitational

lensing effect is an important tool in modern astronomy, and

topology plays a significant role in its properties. In this

talk, I will review topological aspects of gravitational lensing

theory: the connection of image numbers with Morse theory; the

interpretation of certain invariant sums of the signed image

magnification in terms of Lefschetz fixed point theory; and,

finally, a new partially topological perspective on gravitational

light deflection that emerges from the concept of optical geometry

and applications of the Gauss-Bonnet theorem.

General relativity implies that light is deflected by masses

due to the curvature of spacetime. The ensuing gravitational

lensing effect is an important tool in modern astronomy, and

topology plays a significant role in its properties. In this

talk, I will review topological aspects of gravitational lensing

theory: the connection of image numbers with Morse theory; the

interpretation of certain invariant sums of the signed image

magnification in terms of Lefschetz fixed point theory; and,

finally, a new partially topological perspective on gravitational

light deflection that emerges from the concept of optical geometry

and applications of the Gauss-Bonnet theorem.

### 2012/07/09

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Volume of graded linear series and the existence problem of constant scalar curvature Kaehler metric (JAPANESE)

**Tomoyuki Hisamoto**(Univ. of Tokyo)Volume of graded linear series and the existence problem of constant scalar curvature Kaehler metric (JAPANESE)

[ Abstract ]

We describe the volume of a graded linear series by the Monge-Ampere mass of the associated equilibrium metric. We relate this formula to the question whether the weak geodesic ray associated to a test configuration of given polarized manifold recovers the Donaldson-Futaki invariant.

We describe the volume of a graded linear series by the Monge-Ampere mass of the associated equilibrium metric. We relate this formula to the question whether the weak geodesic ray associated to a test configuration of given polarized manifold recovers the Donaldson-Futaki invariant.

### 2012/07/06

#### Colloquium

16:30-17:30 Room #002 (Graduate School of Math. Sci. Bldg.)

Large Deviations of Random Graphs and Random Matrices (ENGLISH)

**S.R.Srinivasa Varadhan**(Courant Institute of Mathematical Sciences, New York University)Large Deviations of Random Graphs and Random Matrices (ENGLISH)

[ Abstract ]

A random graph with $n$ vertices is a random symmetric matrix of $0$'s and $1$'s and they share some common aspects in their large deviation behavior. For random matrices it is the question of having large eigenvalues. For random graphs it is having too many or too few subgraph counts, like the number of triangles etc. The question that we will try to answer is what would a random matrix or a random graph conditioned to exhibit such a large deviation look like. Since the randomness is of size $n^2$ large deviation rates of order $n^2$ are possible.

A random graph with $n$ vertices is a random symmetric matrix of $0$'s and $1$'s and they share some common aspects in their large deviation behavior. For random matrices it is the question of having large eigenvalues. For random graphs it is having too many or too few subgraph counts, like the number of triangles etc. The question that we will try to answer is what would a random matrix or a random graph conditioned to exhibit such a large deviation look like. Since the randomness is of size $n^2$ large deviation rates of order $n^2$ are possible.

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