## Seminar information archive

Seminar information archive ～08/19｜Today's seminar 08/20 | Future seminars 08/21～

#### Operator Algebra Seminars

16:45-18:15 Room #122 (Graduate School of Math. Sci. Bldg.)

Unique prime factorization and bicentralizer problem for a class of type III factors

**Yusuke Isono**(RIMS, Kyoto Univ.)Unique prime factorization and bicentralizer problem for a class of type III factors

### 2015/05/19

#### PDE Real Analysis Seminar

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

Convex bodies and geometry of some associated Minkowski functionals (日本語)

**Sumio Yamada**(Gakushuin University)Convex bodies and geometry of some associated Minkowski functionals (日本語)

[ Abstract ]

In this talk, we will investigate the construction of so-called Hilbert metric, as well as Funk metric, defined on convex set from a new variational viewpoint. The local and global aspects of the geometry of the resulting Minkowski functionals will be contrasted. As an application, some remarks on the Perron-Frobenius theorem will be made. Part of the project is a joint work with Athanase Papadopoulos (Strasbourg).

In this talk, we will investigate the construction of so-called Hilbert metric, as well as Funk metric, defined on convex set from a new variational viewpoint. The local and global aspects of the geometry of the resulting Minkowski functionals will be contrasted. As an application, some remarks on the Perron-Frobenius theorem will be made. Part of the project is a joint work with Athanase Papadopoulos (Strasbourg).

#### Lie Groups and Representation Theory

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

RoCK blocks, wreath products and KLR algebras (English)

**Anton Evseev**(University of Birmingham)RoCK blocks, wreath products and KLR algebras (English)

[ Abstract ]

The so-called RoCK (or Rouquier) blocks play an important role in representation theory of symmetric groups over a finite field of characteristic $p$, as well as of Hecke algebras at roots of unity. Turner has conjectured that a certain idempotent truncation of a RoCK block is Morita equivalent to the principal block $B_0$ of the wreath product $S_p\wr S_d$ of symmetric groups, where $d$ is the "weight" of the block. The talk will outline a proof of this conjecture, which generalizes a result of Chuang-Kessar proved for $d < p$. The proof uses an isomorphism between a Hecke algebra at a root of unity and a cyclotomic Khovanov-Lauda-Rouquier algebra, the resulting grading on the Hecke algebra and the ideas behind a construction of R-matrices for modules over KLR algebras due to Kang-Kashiwara-Kim.

The so-called RoCK (or Rouquier) blocks play an important role in representation theory of symmetric groups over a finite field of characteristic $p$, as well as of Hecke algebras at roots of unity. Turner has conjectured that a certain idempotent truncation of a RoCK block is Morita equivalent to the principal block $B_0$ of the wreath product $S_p\wr S_d$ of symmetric groups, where $d$ is the "weight" of the block. The talk will outline a proof of this conjecture, which generalizes a result of Chuang-Kessar proved for $d < p$. The proof uses an isomorphism between a Hecke algebra at a root of unity and a cyclotomic Khovanov-Lauda-Rouquier algebra, the resulting grading on the Hecke algebra and the ideas behind a construction of R-matrices for modules over KLR algebras due to Kang-Kashiwara-Kim.

#### Tuesday Seminar on Topology

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

Quiver mutation loops and partition q-series (JAPANESE)

**Akishi Kato**(The University of Tokyo)Quiver mutation loops and partition q-series (JAPANESE)

[ Abstract ]

Quivers and their mutations are ubiquitous in mathematics and

mathematical physics; they play a key role in cluster algebras,

wall-crossing phenomena, gluing of ideal tetrahedra, etc.

Recently, we introduced a partition q-series for a quiver mutation loop

(a loop in a quiver exchange graph) using the idea of state sum of statistical

mechanics. The partition q-series enjoy some nice properties such

as pentagon move invariance. We also discuss their relation with combinatorial

Donaldson-Thomas invariants, as well as fermionic character formulas of

certain conformal field theories.

This is a joint work with Yuji Terashima.

Quivers and their mutations are ubiquitous in mathematics and

mathematical physics; they play a key role in cluster algebras,

wall-crossing phenomena, gluing of ideal tetrahedra, etc.

Recently, we introduced a partition q-series for a quiver mutation loop

(a loop in a quiver exchange graph) using the idea of state sum of statistical

mechanics. The partition q-series enjoy some nice properties such

as pentagon move invariance. We also discuss their relation with combinatorial

Donaldson-Thomas invariants, as well as fermionic character formulas of

certain conformal field theories.

This is a joint work with Yuji Terashima.

### 2015/05/18

#### Seminar on Geometric Complex Analysis

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

On a global estimate of the Diederich–Fornaess index of Levi-flat real hypersurfaces (Japanese)

**Masanori Adachi**(Tokyo Univ. of Science)On a global estimate of the Diederich–Fornaess index of Levi-flat real hypersurfaces (Japanese)

[ Abstract ]

We give yet another proof for a global estimate of the Diederich-Fornaess index of relatively compact domains with Levi-flat boundary, namely, the index must be smaller than or equal to the reciprocal of the dimension of the ambient space. Although the Diederich-Fornaess index is originally defined for relatively compact domains in complex manifolds, our formulation reveals that it makes sense for abstract Levi-flat CR manifolds.

We give yet another proof for a global estimate of the Diederich-Fornaess index of relatively compact domains with Levi-flat boundary, namely, the index must be smaller than or equal to the reciprocal of the dimension of the ambient space. Although the Diederich-Fornaess index is originally defined for relatively compact domains in complex manifolds, our formulation reveals that it makes sense for abstract Levi-flat CR manifolds.

#### Algebraic Geometry Seminar

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

Twists and braids for general 3-fold flops (English)

http://db.ipmu.jp/member/personal/4007en.html

**Will Donovan**(IPMU)Twists and braids for general 3-fold flops (English)

[ Abstract ]

When a 3-fold contains a floppable rational curve, a theorem of Bridgeland provides a derived equivalence between the 3-fold and its flop. I will discuss recent joint work with Michael Wemyss, showing that these flop functors satisfy Coxeter-type braid relations. Using this result, we construct an action of a braid-type group on the derived category of the 3-fold. This group arises from the topology of a certain simplicial hyperplane arrangement, determined by the local geometry of the curve. I will give examples and explain key elements in the construction, including the noncommutative deformations of curves introduced in our previous work.

[ Reference URL ]When a 3-fold contains a floppable rational curve, a theorem of Bridgeland provides a derived equivalence between the 3-fold and its flop. I will discuss recent joint work with Michael Wemyss, showing that these flop functors satisfy Coxeter-type braid relations. Using this result, we construct an action of a braid-type group on the derived category of the 3-fold. This group arises from the topology of a certain simplicial hyperplane arrangement, determined by the local geometry of the curve. I will give examples and explain key elements in the construction, including the noncommutative deformations of curves introduced in our previous work.

http://db.ipmu.jp/member/personal/4007en.html

#### Numerical Analysis Seminar

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

Accurate matrix multiplication by error-free transformation (日本語)

**Katsuhisa Ozaki**(Shibaura Institute of Technology)Accurate matrix multiplication by error-free transformation (日本語)

#### Tokyo Probability Seminar

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

Central limit theorem for stochastic heat equations in random environments

**Lu Xu**(Graduate School of Mathematical Sciences, The University of Tokyo)Central limit theorem for stochastic heat equations in random environments

### 2015/05/14

#### Applied Analysis

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

Strong instability of standing waves for some nonlinear Schr\"odinger equations (Japanese)

**Masahito Ohta**(Tokyo University of Science)Strong instability of standing waves for some nonlinear Schr\"odinger equations (Japanese)

### 2015/05/13

#### Operator Algebra Seminars

16:45-18:15 Room #122 (Graduate School of Math. Sci. Bldg.)

Controlled topological phases and the bulk-edge correspondence for

topological insulators (English)

**Yosuke Kubota**(Univ. Tokyo)Controlled topological phases and the bulk-edge correspondence for

topological insulators (English)

### 2015/05/12

#### Tuesday Seminar of Analysis

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

Convergence of the Allen-Cahn equation with constraint to Brakke's mean curvature flow (Japanese)

**Keisuke Takasao**(Graduate School of Mathematical Sciences, the University of Tokyo)Convergence of the Allen-Cahn equation with constraint to Brakke's mean curvature flow (Japanese)

[ Abstract ]

In this talk we consider the Allen-Cahn equation with constraint. In 1994, Chen and Elliott studied the asymptotic behavior of the solution of the Allen-Cahn equation with constraint. They proved that the zero level set of the solution converges to the classical solution of the mean curvature flow under the suitable conditions on initial data. In 1993, Ilmanen proved the existence of the mean curvature flow via the Allen-Cahn equation without constraint in the sense of Brakke. We proved the same conclusion for the Allen-Cahn equation with constraint.

In this talk we consider the Allen-Cahn equation with constraint. In 1994, Chen and Elliott studied the asymptotic behavior of the solution of the Allen-Cahn equation with constraint. They proved that the zero level set of the solution converges to the classical solution of the mean curvature flow under the suitable conditions on initial data. In 1993, Ilmanen proved the existence of the mean curvature flow via the Allen-Cahn equation without constraint in the sense of Brakke. We proved the same conclusion for the Allen-Cahn equation with constraint.

#### Tuesday Seminar on Topology

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

Growth rate of the number of periodic points for generic dynamical systems (JAPANESE)

**Masayuki Asaoka**(Kyoto University)Growth rate of the number of periodic points for generic dynamical systems (JAPANESE)

[ Abstract ]

For any hyperbolic dynamical system, the number of periodic

points grows at most exponentially and the growth rate

reflects statistic property of the system. For dynamics far

from hyperbolicity, the situation is different. In 1999,

Kaloshin proved genericity of super-exponential growth in the

region where dense set of dynamical systems exhibits homoclinic

tangency (so called the Newhouse region).

How does the number of periodic points grow for generic

partially hyperbolic dynamical systems? Such systems are known

to be far from homoclinic tangency. Is the growth at most

exponential like hyperbolic system, or super-exponential by

a mechanism different from homoclinic tangency?

The speaker, Katsutoshi Shinohara, and Dimitry Turaev proved

super-exponential growth of the number of periodic points for

generic one-dimensional iterated function systems under some

reasonable conditions. Such systems are models of dynamics

of partially hyperbolic systems in neutral direction. So, we

expect genericity of super-exponential growth in a region of

partially hyperbolic systems.

In this talk, we start with a brief history of the problem on

growth rate of the number of periodic point and discuss two

mechanisms which lead to genericity of super-exponential growth,

Kaloshin's one and ours.

For any hyperbolic dynamical system, the number of periodic

points grows at most exponentially and the growth rate

reflects statistic property of the system. For dynamics far

from hyperbolicity, the situation is different. In 1999,

Kaloshin proved genericity of super-exponential growth in the

region where dense set of dynamical systems exhibits homoclinic

tangency (so called the Newhouse region).

How does the number of periodic points grow for generic

partially hyperbolic dynamical systems? Such systems are known

to be far from homoclinic tangency. Is the growth at most

exponential like hyperbolic system, or super-exponential by

a mechanism different from homoclinic tangency?

The speaker, Katsutoshi Shinohara, and Dimitry Turaev proved

super-exponential growth of the number of periodic points for

generic one-dimensional iterated function systems under some

reasonable conditions. Such systems are models of dynamics

of partially hyperbolic systems in neutral direction. So, we

expect genericity of super-exponential growth in a region of

partially hyperbolic systems.

In this talk, we start with a brief history of the problem on

growth rate of the number of periodic point and discuss two

mechanisms which lead to genericity of super-exponential growth,

Kaloshin's one and ours.

### 2015/05/11

#### Seminar on Geometric Complex Analysis

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

Integral Kahler Invariants and the Bergman kernel asymptotics for line bundles

**Kengo Hirachi**(The Univ. of Tokyo)Integral Kahler Invariants and the Bergman kernel asymptotics for line bundles

[ Abstract ]

On a compact Kahler manifold, one can define global invariants by integrating local invariants of the metric. Assume that a global invariant thus obtained depends only on the Kahler class. Then we show that the integrand can be decomposed into a Chern polynomial (the integrand of a Chern number) and divergences of one forms, which do not contribute to the integral. We apply this decomposition formula to describe the asymptotic expansion of the Bergman kernel for positive line bundles and to show that the CR Q-curvature on a Sasakian manifold is a divergence. This is a joint work with Spyros Alexakis (U Toronto).

On a compact Kahler manifold, one can define global invariants by integrating local invariants of the metric. Assume that a global invariant thus obtained depends only on the Kahler class. Then we show that the integrand can be decomposed into a Chern polynomial (the integrand of a Chern number) and divergences of one forms, which do not contribute to the integral. We apply this decomposition formula to describe the asymptotic expansion of the Bergman kernel for positive line bundles and to show that the CR Q-curvature on a Sasakian manifold is a divergence. This is a joint work with Spyros Alexakis (U Toronto).

#### Tokyo Probability Seminar

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

Phase transitions for controlled Markov chains on infinite graphs (JAPANESE)

**Naoyuki Ichihara**(College of Science and Engineering, Aoyama Gakuin University)Phase transitions for controlled Markov chains on infinite graphs (JAPANESE)

#### Algebraic Geometry Seminar

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

Deformations of weak Fano varieties (日本語 or English)

https://sites.google.com/site/tarosano222/

**Taro Sano**(Kyoto University)Deformations of weak Fano varieties (日本語 or English)

[ Abstract ]

A smooth projective variety often has obstructed deformations.

Nevertheless, important varieties such as Fano varieties and

Calabi-Yau varieties have unobstructed deformations.

In this talk, I explain about unobstructedness of deformations of weak

Fano varieties, in particular a weak Q-Fano 3-fold.

I also present several examples to show delicateness of this unobstructedness.

[ Reference URL ]A smooth projective variety often has obstructed deformations.

Nevertheless, important varieties such as Fano varieties and

Calabi-Yau varieties have unobstructed deformations.

In this talk, I explain about unobstructedness of deformations of weak

Fano varieties, in particular a weak Q-Fano 3-fold.

I also present several examples to show delicateness of this unobstructedness.

https://sites.google.com/site/tarosano222/

### 2015/05/08

#### Geometry Colloquium

10:00-11:30 Room #126 (Graduate School of Math. Sci. Bldg.)

On Perelman type functionals for the Ricci Yang-Mills flow (Japanese)

**Masashi Ishida**(Osaka University)On Perelman type functionals for the Ricci Yang-Mills flow (Japanese)

[ Abstract ]

In his works on the Ricci flow, Perelman introduced two functionals with monotonicity

formulas under the Ricci flow. The monotonicity formulas have many remarkable geometric applications. On the other hand, around 2007, Jeffrey Streets and Andrea Young independently and simultaneously introduced a new geometric flow which is called the Ricci Yang-Mills flow. The new flow can be regarded as the Ricci flow coupled with the Yang-Mills

heat flow. In this talk, we will introduce new functionals with monotonicity formulas under the Ricci Yang-Mills flow and discuss its applications.

In his works on the Ricci flow, Perelman introduced two functionals with monotonicity

formulas under the Ricci flow. The monotonicity formulas have many remarkable geometric applications. On the other hand, around 2007, Jeffrey Streets and Andrea Young independently and simultaneously introduced a new geometric flow which is called the Ricci Yang-Mills flow. The new flow can be regarded as the Ricci flow coupled with the Yang-Mills

heat flow. In this talk, we will introduce new functionals with monotonicity formulas under the Ricci Yang-Mills flow and discuss its applications.

### 2015/05/07

#### Tuesday Seminar on Topology

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

The group of parenthesized braids (ENGLISH)

**Patrick Dehornoy**(Univ. de Caen)The group of parenthesized braids (ENGLISH)

[ Abstract ]

We describe a group B obtained by gluing in a natural way two well-known

groups, namely Artin's braid group B_infty and Thompson's group F. The

elements of B correspond to braid diagrams in which the distances

between the strands are non uniform and some rescaling operators may

change these distances. The group B shares many properties with B_infty:

as the latter, it can be realized as a subgroup of a mapping class

group, namely that of a sphere with a Cantor set removed, and as a group

of automorphisms of a free group. Technically, the key point is the

existence of a self-distributive operation on B.

We describe a group B obtained by gluing in a natural way two well-known

groups, namely Artin's braid group B_infty and Thompson's group F. The

elements of B correspond to braid diagrams in which the distances

between the strands are non uniform and some rescaling operators may

change these distances. The group B shares many properties with B_infty:

as the latter, it can be realized as a subgroup of a mapping class

group, namely that of a sphere with a Cantor set removed, and as a group

of automorphisms of a free group. Technically, the key point is the

existence of a self-distributive operation on B.

### 2015/05/02

#### Harmonic Analysis Komaba Seminar

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

Two-weight Morrey norm inequality and the sequential testing

(日本語)

The topology of the dual space of ${\mathcal S}_0$

(日本語)

**Hitoshi Tanaka**(Univ Tokyo) 13:30-15:00Two-weight Morrey norm inequality and the sequential testing

(日本語)

[ Abstract ]

In this talk we extend Sawyer's two-weight theory to Morrey spaces and give a characterization of two-weight Morrey norm inequalities for the (general) Hardy-Littlewood maximal operators in terms of the sequential testing due to H\"{a}nninen, Hyt\"{o}nen and Li.

We also introduce the description of the K\"othe dual of Morrey type spaces generated by a basis of measurable functions.

The second topic is based on a joint work with Professors Sawano (Tokyo Metropolitan University) and Masty{\l}o (Adam Mickiewicz University and Institute of Mathematics).

In this talk we extend Sawyer's two-weight theory to Morrey spaces and give a characterization of two-weight Morrey norm inequalities for the (general) Hardy-Littlewood maximal operators in terms of the sequential testing due to H\"{a}nninen, Hyt\"{o}nen and Li.

We also introduce the description of the K\"othe dual of Morrey type spaces generated by a basis of measurable functions.

The second topic is based on a joint work with Professors Sawano (Tokyo Metropolitan University) and Masty{\l}o (Adam Mickiewicz University and Institute of Mathematics).

**Yoshihiro Sawano**(Tokyo Metropolitan University.) 15:30-17:00The topology of the dual space of ${\mathcal S}_0$

(日本語)

[ Abstract ]

Based on the notation of my Japanese book, I will consider the topology of ${\mathcal S}_0'$, the dual of ${\mathcal S}_0$.

In view of the linear isomorphism ${\mathcal S}_0' \sim {\mathcal S}/{\mathcal P}$, we can consider two different topologies;

1) the weak-* topology

and

2) the quotient topology in ${\mathcal S}/{\mathcal P}$.

We aim to show that these two topologies are the same. This will be an errortum of my Japanese book.

This work is done jointly with Takahiro Noi and Shohei Nakamura in Tokyo Metropolitan University.

Based on the notation of my Japanese book, I will consider the topology of ${\mathcal S}_0'$, the dual of ${\mathcal S}_0$.

In view of the linear isomorphism ${\mathcal S}_0' \sim {\mathcal S}/{\mathcal P}$, we can consider two different topologies;

1) the weak-* topology

and

2) the quotient topology in ${\mathcal S}/{\mathcal P}$.

We aim to show that these two topologies are the same. This will be an errortum of my Japanese book.

This work is done jointly with Takahiro Noi and Shohei Nakamura in Tokyo Metropolitan University.

### 2015/04/28

#### Tuesday Seminar on Topology

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

Verify hyperbolicity of 3-manifolds by computer and its applications. (JAPANESE)

**Hidetoshi Masai**(The University of Tokyo, JSPS)Verify hyperbolicity of 3-manifolds by computer and its applications. (JAPANESE)

[ Abstract ]

In this talk I will talk about the program called HIKMOT which

rigorously proves hyperbolicity of a given triangulated 3-manifold. To

prove hyperbolicity of a given triangulated 3-manifold, it suffices to

get a solution of Thurston's gluing equation. We use the notion called

interval arithmetic to overcome two types errors; round-off errors,

and truncated errors. I will also talk about its application to

exceptional surgeries along alternating knots. This talk is based on

joint work with N. Hoffman, K. Ichihara, M. Kashiwagi, S. Oishi, and

A. Takayasu.

In this talk I will talk about the program called HIKMOT which

rigorously proves hyperbolicity of a given triangulated 3-manifold. To

prove hyperbolicity of a given triangulated 3-manifold, it suffices to

get a solution of Thurston's gluing equation. We use the notion called

interval arithmetic to overcome two types errors; round-off errors,

and truncated errors. I will also talk about its application to

exceptional surgeries along alternating knots. This talk is based on

joint work with N. Hoffman, K. Ichihara, M. Kashiwagi, S. Oishi, and

A. Takayasu.

#### Lie Groups and Representation Theory

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

Restricting automorphic forms to geodesic cycles (English)

**Bent Orsted**(Aarhus University and the University of Tokyo)Restricting automorphic forms to geodesic cycles (English)

[ Abstract ]

We find estimates for the restriction of automorphic forms on hyperbolic manifolds to compact geodesic cycles in terms of their expansion into eigenfunctions of the Laplacian. Our method resembles earlier work on products of automorphic forms by Bernstein and Reznikov, and it uses Kobayashi's new symmetry-breaking kernels. This is joint work with Jan M\"o{}llers.

We find estimates for the restriction of automorphic forms on hyperbolic manifolds to compact geodesic cycles in terms of their expansion into eigenfunctions of the Laplacian. Our method resembles earlier work on products of automorphic forms by Bernstein and Reznikov, and it uses Kobayashi's new symmetry-breaking kernels. This is joint work with Jan M\"o{}llers.

### 2015/04/27

#### Seminar on Geometric Complex Analysis

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

Variational formulas for canonical differentials and application (Japanese)

**Sachiko Hamano**(Fukushima Univ.)Variational formulas for canonical differentials and application (Japanese)

[ Abstract ]

We prove the variational formulas of the second order for $L_1$- and $L_0$-canonical differentials, which with the remarkable contrast are our first example in the case of the deforming non-planar open Riemann surface. As a direct application, we show the rigidity of the Euclidean radius of the moduli disk on open torus under pseudoconvexity. The main part of this talk is a joint work with Masakazu Shiba and Hiroshi Yamaguchi.

We prove the variational formulas of the second order for $L_1$- and $L_0$-canonical differentials, which with the remarkable contrast are our first example in the case of the deforming non-planar open Riemann surface. As a direct application, we show the rigidity of the Euclidean radius of the moduli disk on open torus under pseudoconvexity. The main part of this talk is a joint work with Masakazu Shiba and Hiroshi Yamaguchi.

#### Algebraic Geometry Seminar

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

Lagrangian embeddings of cubic fourfolds containing a plane (日本語)

**Genki Ouchi**(University of Tokyo/IPMU)Lagrangian embeddings of cubic fourfolds containing a plane (日本語)

#### Numerical Analysis Seminar

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

A method of verified computations for solutions to semilinear parabolic equations using an analytic semigroup (日本語)

**Akitoshi Takayasu**(Waseda University)A method of verified computations for solutions to semilinear parabolic equations using an analytic semigroup (日本語)

### 2015/04/24

#### Geometry Colloquium

10:00-11:30 Room #126 (Graduate School of Math. Sci. Bldg.)

On K-stability and the volume functions of Q-Fano varieties (JAPANESE)

**Kento Fujita**(Kyoto University)On K-stability and the volume functions of Q-Fano varieties (JAPANESE)

[ Abstract ]

For Fano manifolds X, it is known that X admits K\"ahler-Einstein metrics if and only if the polarized pair

(X, -K_X) is K-polystable. In this talk, I will introduce a new effective stability named "divisorial stability" for Fano manifolds, which is weaker than K-stability and stronger than slope stability along divisors. I will show that:

1. We can easily test divisorial stability via the volume functions.

2. There is a relationship between divisorial stability and the structure property of Okounkov bodies of anti-canonical divisors.

3. For toric Fano manifolds, the existence of K\"ahler-Einstein metrics is equivalent to divisorial semistability.

For Fano manifolds X, it is known that X admits K\"ahler-Einstein metrics if and only if the polarized pair

(X, -K_X) is K-polystable. In this talk, I will introduce a new effective stability named "divisorial stability" for Fano manifolds, which is weaker than K-stability and stronger than slope stability along divisors. I will show that:

1. We can easily test divisorial stability via the volume functions.

2. There is a relationship between divisorial stability and the structure property of Okounkov bodies of anti-canonical divisors.

3. For toric Fano manifolds, the existence of K\"ahler-Einstein metrics is equivalent to divisorial semistability.

#### Colloquium

16:50-17:50 Room #123 (Graduate School of Math. Sci. Bldg.)

Rigidity of conformal functionals on spheres (ENGLISH)

**Bent Oersted**(Aarhus University and University of Tokyo)Rigidity of conformal functionals on spheres (ENGLISH)

[ Abstract ]

On a compact smooth manifold one may construct a Riemannian metric in many different ways. Each metric gives rise to natural elliptic operators such as the Laplace-Beltrami operator and corresponding spectral invariants, e.g. the eigenvalues, the trace of the heat semigroup, and the zeta function. In

this lecture we shall consider such functionals on the space of metrics on the sphere, combining conformal differential geometry and representation theory of semisimple Lie groups to obtain results about local extremal properties of special functionals. This is based on joint work with Niels Martin Moeller.

On a compact smooth manifold one may construct a Riemannian metric in many different ways. Each metric gives rise to natural elliptic operators such as the Laplace-Beltrami operator and corresponding spectral invariants, e.g. the eigenvalues, the trace of the heat semigroup, and the zeta function. In

this lecture we shall consider such functionals on the space of metrics on the sphere, combining conformal differential geometry and representation theory of semisimple Lie groups to obtain results about local extremal properties of special functionals. This is based on joint work with Niels Martin Moeller.

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