## Seminar information archive

Seminar information archive ～05/28｜Today's seminar 05/29 | Future seminars 05/30～

#### thesis presentations

9:15-10:30 Room #122 (Graduate School of Math. Sci. Bldg.)

Sharp interface limit for the stochastic Allen-Cahn equation

(確率アレン・カーン方程式に対する鋭敏な界面極限）

(JAPANESE)

**李 嘉衣**(東京大学大学院数理科学研究科)Sharp interface limit for the stochastic Allen-Cahn equation

(確率アレン・カーン方程式に対する鋭敏な界面極限）

(JAPANESE)

#### thesis presentations

10:45-12:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Scaling limits in stochastic heat equation and stochastic chain model (確率熱方程式および確率鎖模型に対するスケール極限）

(JAPANESE)

**徐 路**(東京大学大学院数理科学研究科)Scaling limits in stochastic heat equation and stochastic chain model (確率熱方程式および確率鎖模型に対するスケール極限）

(JAPANESE)

#### thesis presentations

12:45-14:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Approximations of singular stochastic PDEs and their global well-posedness (特異な確率偏微分方程式の近似とその時間大域的適切性） (JAPANESE)

**星野 壮登**(東京大学大学院数理科学研究科)Approximations of singular stochastic PDEs and their global well-posedness (特異な確率偏微分方程式の近似とその時間大域的適切性） (JAPANESE)

#### thesis presentations

15:45-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Analysis for Viscosity Solutions with Special Emphasis on Anomalous Effects (不規則効果を強調した粘性解析） (JAPANESE)

**難波 時永**(東京大学大学院数理科学研究科)Analysis for Viscosity Solutions with Special Emphasis on Anomalous Effects (不規則効果を強調した粘性解析） (JAPANESE)

#### thesis presentations

17:15-18:30 Room #122 (Graduate School of Math. Sci. Bldg.)

Analysis of the minimal travelling wave speed via the methods of Young measures （進行波の最小速度のYoung 測度による解析） (JAPANESE)

**伊藤 涼**(東京大学大学院数理科学研究科)Analysis of the minimal travelling wave speed via the methods of Young measures （進行波の最小速度のYoung 測度による解析） (JAPANESE)

#### thesis presentations

9:15-10:30 Room #126 (Graduate School of Math. Sci. Bldg.)

Twist maps on quantized coordinate algebras (量子座標環における捻り写像） (JAPANESE)

**大矢 浩徳**(東京大学大学院数理科学研究科)Twist maps on quantized coordinate algebras (量子座標環における捻り写像） (JAPANESE)

#### thesis presentations

10:45-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

On simultaneous approximation to a pair of powers of a real number by rational numbers (実数の冪の対の有理数による同時近似について）

**Gantsooj Batzaya**(東京大学大学院数理科学研究科)On simultaneous approximation to a pair of powers of a real number by rational numbers (実数の冪の対の有理数による同時近似について）

#### thesis presentations

12:45-14:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Fukaya categories, surface Lefschetz fibrations, and Koszul duality theory (Fukaya 圏、曲面Lefschetz 束、Koszul 双対について） (JAPANESE)

**杉山 聡**(東京大学大学院数理科学研究科)Fukaya categories, surface Lefschetz fibrations, and Koszul duality theory (Fukaya 圏、曲面Lefschetz 束、Koszul 双対について） (JAPANESE)

#### thesis presentations

14:15-15:30 Room #126 (Graduate School of Math. Sci. Bldg.)

Partially ordered sets, order complexes and CAT(0) properties (半順序集合，順序複体，及びCAT(0) 性）

(JAPANESE)

**藤内 翔太**(東京大学大学院数理科学研究科)Partially ordered sets, order complexes and CAT(0) properties (半順序集合，順序複体，及びCAT(0) 性）

(JAPANESE)

### 2017/01/30

#### Tokyo Probability Seminar

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

**Jun Misumi**(Faculty of Science, Kochi University)### 2017/01/27

#### Algebraic Geometry Seminar

14:00-17:30 Room #122 (Graduate School of Math. Sci. Bldg.)

On the number and boundedness of minimal models of general type (English)

Adjoint dimension of foliations (English)

https://www.dpmms.cam.ac.uk/~rs872/

**Luca Tasin**(Roma Tre University) 14:00-15:30On the number and boundedness of minimal models of general type (English)

[ Abstract ]

In this talk I will explain that the number of minimal models yields a constructible function on the base of any family of varieties of general type. From this it follows that the number of minimal models of a variety of general type can be bounded in terms of its volume. I will also show that in any dimension minimal models of general type and bounded volume form a bounded family. This is based on a joint work with D. Martinelli and S. Schreieder.

In this talk I will explain that the number of minimal models yields a constructible function on the base of any family of varieties of general type. From this it follows that the number of minimal models of a variety of general type can be bounded in terms of its volume. I will also show that in any dimension minimal models of general type and bounded volume form a bounded family. This is based on a joint work with D. Martinelli and S. Schreieder.

**Roberto Svaldi**(University of Cambridge/SISSA) 16:00-17:30Adjoint dimension of foliations (English)

[ Abstract ]

The classification of foliated surfaces by Brunella, McQuillan and Mendes carries many similarities with Enriques-Kodaira classification of surfaces but also many important differences. I will discuss an alternative classification scheme where the role of differential forms along the leaves is replaced by differential forms along the leaves with values in fractional powers of the conormal bundle of the foliation. In this alternative setup one obtains a classification of foliated surfaces closer to the usual Enriques-Kodaira classification. If time permits, I will show how to apply this alternative classification to describe the Zariski closure of the set foliations which admit rational first integral of bounded genus in families of foliated surfaces. Joint work with Jorge Vitorio Pereira.

[ Reference URL ]The classification of foliated surfaces by Brunella, McQuillan and Mendes carries many similarities with Enriques-Kodaira classification of surfaces but also many important differences. I will discuss an alternative classification scheme where the role of differential forms along the leaves is replaced by differential forms along the leaves with values in fractional powers of the conormal bundle of the foliation. In this alternative setup one obtains a classification of foliated surfaces closer to the usual Enriques-Kodaira classification. If time permits, I will show how to apply this alternative classification to describe the Zariski closure of the set foliations which admit rational first integral of bounded genus in families of foliated surfaces. Joint work with Jorge Vitorio Pereira.

https://www.dpmms.cam.ac.uk/~rs872/

### 2017/01/26

#### Seminar on Probability and Statistics

13:00-16:00 Room #052 (Graduate School of Math. Sci. Bldg.)

High-frequency financial data : trades and quotes databases, order flows and time resolution I, II, III

**Ioane Muni Toke**(Centrale Supelec Paris)High-frequency financial data : trades and quotes databases, order flows and time resolution I, II, III

[ Abstract ]

I present some of the challenges associated with preparing high-frequency trades and quotes databases for statistics purposes. In a first part, I investigate TRTH tick-by-tick data on three exchanges (Paris, London and Frankfurt) and on a five-year span. I analyse the performances of a procedure of reconstruction of orders flows. This turns out to be a forensic tool assessing the quality of the database: significant technical changes affecting the exchanges are tracked through the data. Moreover, the choices made when reconstructing order flows may have consequences on the quantitative models that are calibrated afterwards on such data. I also provide a refined look at the Lee–Ready procedure and its optimal lags. Findings are in line with both financial reasoning and the analysis of an illustrative Poisson model. In a second part, I investigate Nikkei-packaged Tokyo-traded ETF data. The application the order flow reconstruction procedure underlines the differences between the TRTH and Nikkei data. In a brief last part, we will discuss the time resolution of these databases and the potential problems arising when modelling a limit order book with simple point processes.

I present some of the challenges associated with preparing high-frequency trades and quotes databases for statistics purposes. In a first part, I investigate TRTH tick-by-tick data on three exchanges (Paris, London and Frankfurt) and on a five-year span. I analyse the performances of a procedure of reconstruction of orders flows. This turns out to be a forensic tool assessing the quality of the database: significant technical changes affecting the exchanges are tracked through the data. Moreover, the choices made when reconstructing order flows may have consequences on the quantitative models that are calibrated afterwards on such data. I also provide a refined look at the Lee–Ready procedure and its optimal lags. Findings are in line with both financial reasoning and the analysis of an illustrative Poisson model. In a second part, I investigate Nikkei-packaged Tokyo-traded ETF data. The application the order flow reconstruction procedure underlines the differences between the TRTH and Nikkei data. In a brief last part, we will discuss the time resolution of these databases and the potential problems arising when modelling a limit order book with simple point processes.

### 2017/01/24

#### Tuesday Seminar on Topology

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

On the existence of infinitely many non-contractible periodic trajectories in Hamiltonian dynamics on closed symplectic manifolds (JAPANESE)

**Ryuma Orita**(The University of Tokyo)On the existence of infinitely many non-contractible periodic trajectories in Hamiltonian dynamics on closed symplectic manifolds (JAPANESE)

[ Abstract ]

We show that the presence of a non-contractible Hamiltonian one-periodic trajectory in a closed symplectic manifold yields the existence of infinitely many non-contractible periodic trajectories, provided that the symplectic form is aspherical and the fundamental group is virtually abelian. Moreover, we also show that a similar statement holds for closed monotone or negative monotone symplectic manifolds having virtually abelian fundamental groups. These results are certain generalizations of works by Ginzburg and Gurel who proved a similar statement holds for atoroidal or toroidally monotone closed symplectic manifolds. The proof is based on the machinery of filtered Floer--Novikov homology for non-contractible periodic trajectories.

We show that the presence of a non-contractible Hamiltonian one-periodic trajectory in a closed symplectic manifold yields the existence of infinitely many non-contractible periodic trajectories, provided that the symplectic form is aspherical and the fundamental group is virtually abelian. Moreover, we also show that a similar statement holds for closed monotone or negative monotone symplectic manifolds having virtually abelian fundamental groups. These results are certain generalizations of works by Ginzburg and Gurel who proved a similar statement holds for atoroidal or toroidally monotone closed symplectic manifolds. The proof is based on the machinery of filtered Floer--Novikov homology for non-contractible periodic trajectories.

#### Tuesday Seminar on Topology

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

Quantitative shadowing property, shadowable points, and local properties of topological dynamical systems (JAPANESE)

**Noriaki Kawaguchi**(The University of Tokyo)Quantitative shadowing property, shadowable points, and local properties of topological dynamical systems (JAPANESE)

[ Abstract ]

Shadowing property has been one of the key notions in topological hyperbolic dynamics, which is also common since C^0-generic homeomorphisms on a smooth closed manifold satisfy the property for instance. In this talk, the shadowing property in relation to other chaotic or non-chaotic properties of dynamical systems (entropy, sensitivity, equicontinuity, etc.) is discussed. Also, we introduce an idea of localizing and quantifying the shadowing property following the recent work of Morales, and present some of its consequences. The idea is shown to be effective for the description of local properties of dynamical systems.

Shadowing property has been one of the key notions in topological hyperbolic dynamics, which is also common since C^0-generic homeomorphisms on a smooth closed manifold satisfy the property for instance. In this talk, the shadowing property in relation to other chaotic or non-chaotic properties of dynamical systems (entropy, sensitivity, equicontinuity, etc.) is discussed. Also, we introduce an idea of localizing and quantifying the shadowing property following the recent work of Morales, and present some of its consequences. The idea is shown to be effective for the description of local properties of dynamical systems.

### 2017/01/23

#### Seminar on Geometric Complex Analysis

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

A unified proof of Cousin I, II and d-bar equation on domains of holomorphy (JAPANESE)

**Junjiro Noguchi**(The University of Tokyo)A unified proof of Cousin I, II and d-bar equation on domains of holomorphy (JAPANESE)

[ Abstract ]

Oka's J\^oku-Ik\^o says that holomorphic functions on a complex submanifold of a polydisk extend holomorphically to the whole polydisk. By making use of Oka's J\^oku-Ik\^o we give a titled proof with introducing an argument that represents one of the three cases.

The proof is a modification of the cube dimension induction, used in the proof of Oka's Syzygy for coherent sheaves.

Oka's J\^oku-Ik\^o says that holomorphic functions on a complex submanifold of a polydisk extend holomorphically to the whole polydisk. By making use of Oka's J\^oku-Ik\^o we give a titled proof with introducing an argument that represents one of the three cases.

The proof is a modification of the cube dimension induction, used in the proof of Oka's Syzygy for coherent sheaves.

### 2017/01/19

#### Seminar on Probability and Statistics

13:00-15:30 Room #052 (Graduate School of Math. Sci. Bldg.)

Talk 1:Likelihood inference for a continuous time GARCH model

Talk 2:Nonparametric Estimation for Self-Exciting Point Processes: A Parsimonious Approach

**Feng Chen**(University of New South Wales)Talk 1:Likelihood inference for a continuous time GARCH model

Talk 2:Nonparametric Estimation for Self-Exciting Point Processes: A Parsimonious Approach

[ Abstract ]

Talk 1:The continuous time GARCH (COGARCH) model of Kluppelberg, Lindner and Maller (2004) is a natural extension of the discrete time GARCH(1,1) model which preserves important features of the GARCH model in the discrete-time setting. For example, the COGARCH model is driven by a single source of noise as in the discrete time GARCH model, which is a Levy process in the COGARCH case, and both models can produced heavy tailed marginal returns even when the driving noise is light-tailed. However, calibrating the COGARCH model to data is a challenge, especially when observations of the COGARCH process are obtained at irregularly spaced time points. The method of moments has had some success in the case with regularly spaced data, yet it is not clear how to make it work in the more interesting case with irregularly spaced data. As a well-known method of estimation, the maximum likelihood method has not been developed for the COGARCH model, even in the quite simple case with the driving Levy process being compound Poisson, though a quasi-maximum likelihood (QML)method has been proposed. The challenge with the maximum likelihood method in this context is mainly due to the lack of a tractable form for the likelihood. In this talk, we propose a Monte Carlo method to approximate the likelihood of the compound Poisson driven COGARCH model. We evaluate the performance of the resulting maximum likelihood (ML) estimator using simulated data, and illustrate its application with high frequency exchange rate data. (Joint work with Damien Wee and William Dunsmuir).

Talk 2:There is ample evidence that in applications of self-exciting point process (SEPP) models, the intensity of background events is often far from constant. If a constant background is imposed, that assumption can reduce significantly the quality of statistical analysis, in problems as diverse as modelling the after-shocks of earthquakes and the study of ultra-high frequency financial data. Parametric models can be

used to alleviate this problem, but they run the risk of distorting inference by misspecifying the nature of the background intensity function. On the other hand, a purely nonparametric approach to analysis

leads to problems of identifiability; when a nonparametric approach is taken, not every aspect of the model can be identified from data recorded along a single observed sample path. In this paper we suggest overcoming this difficulty by using an approach based on the principle of parsimony, or Occam's razor. In particular, we suggest taking the point-process intensity to be either a constant or to have maximum differential entropy. Although seldom used for nonparametric function estimation in other settings, this approach is appropriate in the context of SEPP models. (Joint work with the late Peter Hall.)

Talk 1:The continuous time GARCH (COGARCH) model of Kluppelberg, Lindner and Maller (2004) is a natural extension of the discrete time GARCH(1,1) model which preserves important features of the GARCH model in the discrete-time setting. For example, the COGARCH model is driven by a single source of noise as in the discrete time GARCH model, which is a Levy process in the COGARCH case, and both models can produced heavy tailed marginal returns even when the driving noise is light-tailed. However, calibrating the COGARCH model to data is a challenge, especially when observations of the COGARCH process are obtained at irregularly spaced time points. The method of moments has had some success in the case with regularly spaced data, yet it is not clear how to make it work in the more interesting case with irregularly spaced data. As a well-known method of estimation, the maximum likelihood method has not been developed for the COGARCH model, even in the quite simple case with the driving Levy process being compound Poisson, though a quasi-maximum likelihood (QML)method has been proposed. The challenge with the maximum likelihood method in this context is mainly due to the lack of a tractable form for the likelihood. In this talk, we propose a Monte Carlo method to approximate the likelihood of the compound Poisson driven COGARCH model. We evaluate the performance of the resulting maximum likelihood (ML) estimator using simulated data, and illustrate its application with high frequency exchange rate data. (Joint work with Damien Wee and William Dunsmuir).

Talk 2:There is ample evidence that in applications of self-exciting point process (SEPP) models, the intensity of background events is often far from constant. If a constant background is imposed, that assumption can reduce significantly the quality of statistical analysis, in problems as diverse as modelling the after-shocks of earthquakes and the study of ultra-high frequency financial data. Parametric models can be

used to alleviate this problem, but they run the risk of distorting inference by misspecifying the nature of the background intensity function. On the other hand, a purely nonparametric approach to analysis

leads to problems of identifiability; when a nonparametric approach is taken, not every aspect of the model can be identified from data recorded along a single observed sample path. In this paper we suggest overcoming this difficulty by using an approach based on the principle of parsimony, or Occam's razor. In particular, we suggest taking the point-process intensity to be either a constant or to have maximum differential entropy. Although seldom used for nonparametric function estimation in other settings, this approach is appropriate in the context of SEPP models. (Joint work with the late Peter Hall.)

### 2017/01/17

#### Tuesday Seminar on Topology

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

On an application of the Fukaya categories to the Koszul duality (JAPANESE)

**Satoshi Sugiyama**(The University of Tokyo)On an application of the Fukaya categories to the Koszul duality (JAPANESE)

[ Abstract ]

In this talk, we compute an A

The Koszul duality is originally a duality between certain quadratic algebras called Koszul algebras. In this talk, we are interested in the case when A is not a quadratic algebra, i.e. the case when A is defined as a quotient algebra of tensor algebra devided by higher degree relations.

The definition of Koszul duals for such algebras, A

In this talk, we compute an A

_{∞}-Koszul dual of path algebras with relations over the directed A_{n}-type quivers via the Fukaya categories of exact Riemann surfaces.The Koszul duality is originally a duality between certain quadratic algebras called Koszul algebras. In this talk, we are interested in the case when A is not a quadratic algebra, i.e. the case when A is defined as a quotient algebra of tensor algebra devided by higher degree relations.

The definition of Koszul duals for such algebras, A

_{∞}-Koszul duals, are given by some people, for example, D. M. Lu, J. H. Palmieri, Q. S. Wu, J. J. Zhang. However, the computation for a concrete examples is hard. In this talk, we use the Fukaya categories of exact Riemann surfaces to compute A_{∞}-Koszul duals. Then, we understand the Koszul duality as a duality between higher products and relations.### 2017/01/16

#### Seminar on Probability and Statistics

16:50-18:00 Room #052 (Graduate School of Math. Sci. Bldg.)

Profile likelihood approach to a large sample distribution of estimators in joint mixture model of survival and longitudinal ordered data

**広瀬勇一**(University of Wellington)Profile likelihood approach to a large sample distribution of estimators in joint mixture model of survival and longitudinal ordered data

[ Abstract ]

We consider a semiparametric joint model that consists of item response and survival components, where these two components are linked through latent variables. We estimate the model parameters through a profile likelihood and the EM algorithm. We propose a method to derive an asymptotic variance of the estimators in this model.

We consider a semiparametric joint model that consists of item response and survival components, where these two components are linked through latent variables. We estimate the model parameters through a profile likelihood and the EM algorithm. We propose a method to derive an asymptotic variance of the estimators in this model.

#### Seminar on Geometric Complex Analysis

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

A geometric second main theorem (ENGLISH)

**Dinh Tuan Huynh**(Osaka University)A geometric second main theorem (ENGLISH)

[ Abstract ]

Using Ahlfors’ theory of covering surfaces, we establish a Cartan’s type Second Main Theorem in the complex projective plane with 1–truncated counting functions for entire holomorphic curves which cluster on an algebraic curve.

Using Ahlfors’ theory of covering surfaces, we establish a Cartan’s type Second Main Theorem in the complex projective plane with 1–truncated counting functions for entire holomorphic curves which cluster on an algebraic curve.

#### Numerical Analysis Seminar

16:50-18:20 Room #117 (Graduate School of Math. Sci. Bldg.)

Mathematical model for the generation of calcium carbonate scale (日本語)

**Hideo Kawarada**(AMSOK and Chiba University)Mathematical model for the generation of calcium carbonate scale (日本語)

#### Tokyo Probability Seminar

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

Monotonicity and rigidity of the W-entropy on RCD (0,N) spaces (日本語)

**Kazumasa Kuwada**(School of science, Tokyo institute of technology)Monotonicity and rigidity of the W-entropy on RCD (0,N) spaces (日本語)

### 2017/01/12

#### Seminar on Probability and Statistics

13:00-15:00 Room #052 (Graduate School of Math. Sci. Bldg.)

yuimaGUI: a Graphical User Interface for the yuima Package

**Emanuele Guidotti**(Milan University)yuimaGUI: a Graphical User Interface for the yuima Package

[ Abstract ]

The yuimaGUI package provides a user-friendly interface for yuima. It greatly simplifies tasks such as estimation and simulation of stochastic processes and it also includes additional tools. Some of them:

data retrieval: stock prices and economic indicators

time series clustering

change point analysis

lead-lag estimation

After a general overview of the whole interface, the yuimaGUI will be shown in real-time. All the settings and the inner workings will be discussed in detail. During this second part, you are kindly invited to ask questions whenever you feel that some problem may arise.

The yuimaGUI package provides a user-friendly interface for yuima. It greatly simplifies tasks such as estimation and simulation of stochastic processes and it also includes additional tools. Some of them:

data retrieval: stock prices and economic indicators

time series clustering

change point analysis

lead-lag estimation

After a general overview of the whole interface, the yuimaGUI will be shown in real-time. All the settings and the inner workings will be discussed in detail. During this second part, you are kindly invited to ask questions whenever you feel that some problem may arise.

### 2017/01/11

#### Number Theory Seminar

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

Deformation and rigidity of $\ell$-adic sheaves (English)

**Lei Fu**(Tsinghua University)Deformation and rigidity of $\ell$-adic sheaves (English)

[ Abstract ]

Let $X$ be a smooth connected algebraic curve over an algebraically closed field, let $S$ be a finite closed subset in $X$, and let $F_0$ be a lisse $\ell$-torsion sheaf on $X-S$. We study the deformation of $F_0$. The universal deformation space is a formal scheme. Its generic fiber has a rigid analytic space structure. By studying this rigid analytic space, we prove a conjecture of Katz which says that if a lisse $\overline{Q}_\ell$-sheaf $F$ is irreducible and physically rigid, then it is cohomologically rigid in the sense that $\chi(X,j_*End(F))=2$, where $j:X-S\to X$ is the open immersion.

Let $X$ be a smooth connected algebraic curve over an algebraically closed field, let $S$ be a finite closed subset in $X$, and let $F_0$ be a lisse $\ell$-torsion sheaf on $X-S$. We study the deformation of $F_0$. The universal deformation space is a formal scheme. Its generic fiber has a rigid analytic space structure. By studying this rigid analytic space, we prove a conjecture of Katz which says that if a lisse $\overline{Q}_\ell$-sheaf $F$ is irreducible and physically rigid, then it is cohomologically rigid in the sense that $\chi(X,j_*End(F))=2$, where $j:X-S\to X$ is the open immersion.

### 2017/01/10

#### Tuesday Seminar on Topology

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

Stability of anti-canonically balanced metrics (JAPANESE)

**Shunsuke Saito**(The University of Tokyo)Stability of anti-canonically balanced metrics (JAPANESE)

[ Abstract ]

Donaldson introduced "anti-canonically balanced metrics" on Fano manifolds, which is a finite dimensional analogue of Kähler-Einstein metrics. It is proved that anti-canonically balanced metrics are critical points of the quantized Ding functional.

We first study the slope at infinity of the quantized Ding functional along Bergman geodesic rays. Then, we introduce a new algebro-geometric stability of Fano manifolds based on the slope formula, and show that the existence of anti-canonically balanced metrics implies our stability. The relationship between the stability and others is also discussed.

This talk is based on a joint work with R. Takahashi (Tohoku Univ).

Donaldson introduced "anti-canonically balanced metrics" on Fano manifolds, which is a finite dimensional analogue of Kähler-Einstein metrics. It is proved that anti-canonically balanced metrics are critical points of the quantized Ding functional.

We first study the slope at infinity of the quantized Ding functional along Bergman geodesic rays. Then, we introduce a new algebro-geometric stability of Fano manifolds based on the slope formula, and show that the existence of anti-canonically balanced metrics implies our stability. The relationship between the stability and others is also discussed.

This talk is based on a joint work with R. Takahashi (Tohoku Univ).

#### Tuesday Seminar on Topology

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

Topological Invariants and Corner States for Hamiltonians on a Three Dimensional Lattice (JAPANESE)

**Shin Hayashi**(The University of Tokyo)Topological Invariants and Corner States for Hamiltonians on a Three Dimensional Lattice (JAPANESE)

[ Abstract ]

In condensed matter physics, a correspondence between two topological invariants defined for a gapped Hamiltonian is well-known. One is defined for such a Hamiltonian on a lattice (bulk invariant), and the other is defined for its restriction onto a subsemigroup (edge invariant). The edge invariant is related to the wave functions localized near the edge. This correspondence is known as the bulk-edge correspondence. In this talk, we consider a variant of this correspondence. We consider a periodic Hamiltonian on a three dimensional lattice (bulk) and its restrictions onto two subsemigroups (edges) and their intersection (corner). We will show that, if our Hamiltonian is "gapped" in some sense, we can define a topological invariant for the bulk and edges. We will also define another topological invariant related to the wave functions localized near the corner. We will explain that there is a correspondence between these two topological invariants by using the six-term exact sequence associated to the quarter-plane Toeplitz extension obtained by E. Park.

In condensed matter physics, a correspondence between two topological invariants defined for a gapped Hamiltonian is well-known. One is defined for such a Hamiltonian on a lattice (bulk invariant), and the other is defined for its restriction onto a subsemigroup (edge invariant). The edge invariant is related to the wave functions localized near the edge. This correspondence is known as the bulk-edge correspondence. In this talk, we consider a variant of this correspondence. We consider a periodic Hamiltonian on a three dimensional lattice (bulk) and its restrictions onto two subsemigroups (edges) and their intersection (corner). We will show that, if our Hamiltonian is "gapped" in some sense, we can define a topological invariant for the bulk and edges. We will also define another topological invariant related to the wave functions localized near the corner. We will explain that there is a correspondence between these two topological invariants by using the six-term exact sequence associated to the quarter-plane Toeplitz extension obtained by E. Park.

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