## Seminar information archive

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

### 2016/05/09

#### Seminar on Geometric Complex Analysis

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

Nevanlinna type theorems for meromorphic functions on negatively curved Kähler manifolds (JAPANESE)

**Atsushi Atsuji**(Keio University)Nevanlinna type theorems for meromorphic functions on negatively curved Kähler manifolds (JAPANESE)

[ Abstract ]

We discuss a generalization of classical Nevanlinna theory to meromorphic functions on complete Kähler manifolds. Several generalization of domains of functions are known in Nevanlinna theory, especially the results due to W.Stoll are well-known. In general Kähler case the remainder term of the second main theorem of Nevanlinna theory usually takes a complicated form. It seems that we have to modify classical

methods in order to simplify the second main theorem. We will use heat diffusion to do that and show some defect relations. We would also like to give some Liouville type theorems for holomorphic maps by using similar heat diffusion methods.

We discuss a generalization of classical Nevanlinna theory to meromorphic functions on complete Kähler manifolds. Several generalization of domains of functions are known in Nevanlinna theory, especially the results due to W.Stoll are well-known. In general Kähler case the remainder term of the second main theorem of Nevanlinna theory usually takes a complicated form. It seems that we have to modify classical

methods in order to simplify the second main theorem. We will use heat diffusion to do that and show some defect relations. We would also like to give some Liouville type theorems for holomorphic maps by using similar heat diffusion methods.

#### Tokyo Probability Seminar

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

**Yosuke Kawamoto**(Graduate school of Mathematics, Kyushu university)#### Numerical Analysis Seminar

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

Potential theoretic approach to design of formulas for function approximation and numerical integration in weighted Hardy spaces

(日本語)

**Ken'ichiro Tanaka**(Musashino University)Potential theoretic approach to design of formulas for function approximation and numerical integration in weighted Hardy spaces

(日本語)

#### Operator Algebra Seminars

16:45-18:15 Room #118 (Graduate School of Math. Sci. Bldg.)

Surgery theory and discrete groups (English)

**Mikael Pichot**(McGill Univ./Univ．Tokyo)Surgery theory and discrete groups (English)

#### FMSP Lectures

15:00-17:00 Room #002 (Graduate School of Math. Sci. Bldg.)

Vertex Operator Algebras according to Newton (ENGLISH)

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

**Michael Tuite**(National University of Ireland, Galway)Vertex Operator Algebras according to Newton (ENGLISH)

[ Abstract ]

In this lecture I will give an introduction to Vertex Operator Algebras (VOAs) using elementary methods originally due to Isaac Newton. I will also discuss a class of exceptional VOAs including the Moonshine module which share a number of fundamental properties in common.

[ Reference URL ]In this lecture I will give an introduction to Vertex Operator Algebras (VOAs) using elementary methods originally due to Isaac Newton. I will also discuss a class of exceptional VOAs including the Moonshine module which share a number of fundamental properties in common.

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

### 2016/04/27

#### PDE Real Analysis Seminar

15:00-16:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Fourier transform versus Hilbert transform (English)

http://u.math.biu.ac.il/~liflyand/

**Elijah Liflyand**(Bar-Ilan University, Israel)Fourier transform versus Hilbert transform (English)

[ Abstract ]

We present several results in which the interplay between the Fourier transform and the Hilbert transform is of special form and importance.

1. In 50-s (Kahane, Izumi-Tsuchikura, Boas, etc.), the following problem in Fourier Analysis attracted much attention: Let $\{a_k\},$ $k=0,1,2...,$ be the sequence of the Fourier coefficients of the absolutely convergent sine (cosine) Fourier series of a function $f:\mathbb T=[-\pi,\pi)\to \mathbb C,$ that is $\sum |a_k|<\infty.$ Under which conditions on $\{a_k\}$ the re-expansion of $f(t)$ ($f(t)-f(0)$, respectively) in the cosine (sine) Fourier series will also be absolutely convergent?

We solve a similar problem for functions on the whole axis and their Fourier transforms. Generally, the re-expansion of a function with integrable cosine (sine) Fourier transform in the sine (cosine) Fourier transform is integrable if and only if not only the initial Fourier transform is integrable but also the Hilbert transform of the initial Fourier transform is integrable.

2. The following result is due to Hardy and Littlewood: If a (periodic) function $f$ and its conjugate $\widetilde f$ are both of bounded variation, their Fourier series converge absolutely.

We generalize the Hardy-Littlewood theorem (joint work with U. Stadtmüller) to the Fourier transform of a function on the real axis and its modified Hilbert transform. The initial Hardy-Littlewood theorem is a partial case of this extension, when the function is taken to be with compact support.

3. These and other problems are integrated parts of harmonic analysis of functions of bounded variation. We have found the maximal space for the integrability of the Fourier transform of a function of bounded variation. Along with those known earlier, various interesting new spaces appear in this study. Their inter-relations lead, in particular, to improvements of Hardy's inequality.

There are multidimensional generalizations of these results.

[ Reference URL ]We present several results in which the interplay between the Fourier transform and the Hilbert transform is of special form and importance.

1. In 50-s (Kahane, Izumi-Tsuchikura, Boas, etc.), the following problem in Fourier Analysis attracted much attention: Let $\{a_k\},$ $k=0,1,2...,$ be the sequence of the Fourier coefficients of the absolutely convergent sine (cosine) Fourier series of a function $f:\mathbb T=[-\pi,\pi)\to \mathbb C,$ that is $\sum |a_k|<\infty.$ Under which conditions on $\{a_k\}$ the re-expansion of $f(t)$ ($f(t)-f(0)$, respectively) in the cosine (sine) Fourier series will also be absolutely convergent?

We solve a similar problem for functions on the whole axis and their Fourier transforms. Generally, the re-expansion of a function with integrable cosine (sine) Fourier transform in the sine (cosine) Fourier transform is integrable if and only if not only the initial Fourier transform is integrable but also the Hilbert transform of the initial Fourier transform is integrable.

2. The following result is due to Hardy and Littlewood: If a (periodic) function $f$ and its conjugate $\widetilde f$ are both of bounded variation, their Fourier series converge absolutely.

We generalize the Hardy-Littlewood theorem (joint work with U. Stadtmüller) to the Fourier transform of a function on the real axis and its modified Hilbert transform. The initial Hardy-Littlewood theorem is a partial case of this extension, when the function is taken to be with compact support.

3. These and other problems are integrated parts of harmonic analysis of functions of bounded variation. We have found the maximal space for the integrability of the Fourier transform of a function of bounded variation. Along with those known earlier, various interesting new spaces appear in this study. Their inter-relations lead, in particular, to improvements of Hardy's inequality.

There are multidimensional generalizations of these results.

http://u.math.biu.ac.il/~liflyand/

#### Number Theory Seminar

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

On the endoscopic lifting of simple supercuspidal representations (Japanese)

**Masao Oi**(University of Tokyo)On the endoscopic lifting of simple supercuspidal representations (Japanese)

### 2016/04/26

#### Tuesday Seminar of Analysis

16:50-18:20 Room #126 (Graduate School of Math. Sci. Bldg.)

On microlocal analysis of Gauss-Manin connections for boundary singularities (Japanese)

**Saiei-Jaeyeong Matsubara-Heo**(Graduate School of Mathematical Sciences, the University of Tokyo)On microlocal analysis of Gauss-Manin connections for boundary singularities (Japanese)

#### Algebraic Geometry Seminar

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

A gentle introduction to K-stability and its recent development (Japanese)

https://sites.google.com/site/yujiodaka2013/

**Yuji Odaka**(Dept. of Math., Kyoto U.)A gentle introduction to K-stability and its recent development (Japanese)

[ Abstract ]

K安定性とは複素代数多様体上の「標準的な」ケーラー計量の存在問題に端を発する，代数幾何的な概念です．二木先生や満渕先生等の先駆的な仕事に感化されて導入され，特に近年ホットに研究され始めている一方，未だその大半はより微分幾何的な研究者の方々や背景の中でなされているように講演者には感じられます．

代数幾何的にもどのように面白いか，どういった意義があるかに私見で軽く触れた上で，その基礎付けをより拡張した枠組みで説明しつつ，最先端でどのようなことが問題になっているかをいくらか（私の力量と時間の許す限り）解説しつつ，文献をご紹介できればと思っています

[ Reference URL ]K安定性とは複素代数多様体上の「標準的な」ケーラー計量の存在問題に端を発する，代数幾何的な概念です．二木先生や満渕先生等の先駆的な仕事に感化されて導入され，特に近年ホットに研究され始めている一方，未だその大半はより微分幾何的な研究者の方々や背景の中でなされているように講演者には感じられます．

代数幾何的にもどのように面白いか，どういった意義があるかに私見で軽く触れた上で，その基礎付けをより拡張した枠組みで説明しつつ，最先端でどのようなことが問題になっているかをいくらか（私の力量と時間の許す限り）解説しつつ，文献をご紹介できればと思っています

https://sites.google.com/site/yujiodaka2013/

#### Tuesday Seminar on Topology

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

Arithmetic topology on branched covers of 3-manifolds (JAPANESE)

**Jun Ueki**(The University of Tokyo)Arithmetic topology on branched covers of 3-manifolds (JAPANESE)

[ Abstract ]

The analogy between 3-dimensional topology and number theory was first pointed out by Mazur in the 1960s, and it has been studied systematically by Kapranov, Reznikov, Morishita, and others. In their analogies, for example, knots and 3-manifolds correspond to primes and number rings respectively. The study of these analogies is called arithmetic topology now.

In my talk, based on their dictionary of analogies, we study analogues of idelic class field theory, Iwasawa theory, and Galois deformation theory in the context of 3-dimensional topology, and establish various foundational analogies in arithmetic topology.

The analogy between 3-dimensional topology and number theory was first pointed out by Mazur in the 1960s, and it has been studied systematically by Kapranov, Reznikov, Morishita, and others. In their analogies, for example, knots and 3-manifolds correspond to primes and number rings respectively. The study of these analogies is called arithmetic topology now.

In my talk, based on their dictionary of analogies, we study analogues of idelic class field theory, Iwasawa theory, and Galois deformation theory in the context of 3-dimensional topology, and establish various foundational analogies in arithmetic topology.

#### Seminar on Probability and Statistics

16:10-17:10 Room #123 (Graduate School of Math. Sci. Bldg.)

LAMN property and optimal estimation for diffusion with non synchronous observations

**Teppei Ogihara**(Institute of Statistical Mathematics, JST PRESTO, JST CREST)LAMN property and optimal estimation for diffusion with non synchronous observations

[ Abstract ]

We study so-called local asymptotic mixed normality (LAMN) property for a statistical model generated by nonsynchronously observed diffusion processes using a Malliavin calculus technique. The LAMN property of the statistical model induces an asymptotic minimal variance of estimation errors for any estimators of the parameter. We also construct an optimal estimator which attains the best asymptotic variance.

We study so-called local asymptotic mixed normality (LAMN) property for a statistical model generated by nonsynchronously observed diffusion processes using a Malliavin calculus technique. The LAMN property of the statistical model induces an asymptotic minimal variance of estimation errors for any estimators of the parameter. We also construct an optimal estimator which attains the best asymptotic variance.

#### Seminar on Probability and Statistics

13:00-14:20 Room #123 (Graduate School of Math. Sci. Bldg.)

Stochastic heat equation with fractional noise 1

**Ciprian Tudor**(Université de Lille 1)Stochastic heat equation with fractional noise 1

[ Abstract ]

In the first part, we introduce the bifractional Brownian motion, which is a Gaussian process that generalizes the well- known fractional Brownian motion. We present the basic properties of this process and we also present its connection with the mild solution to the heat equation driven by a Gaussian noise that behaves as the Brownian motion in time.

In the first part, we introduce the bifractional Brownian motion, which is a Gaussian process that generalizes the well- known fractional Brownian motion. We present the basic properties of this process and we also present its connection with the mild solution to the heat equation driven by a Gaussian noise that behaves as the Brownian motion in time.

#### Seminar on Probability and Statistics

14:30-15:50 Room #123 (Graduate School of Math. Sci. Bldg.)

Stochastic heat equation with fractional noise 2

**Ciprian Tudor**(Université de Lille 1)Stochastic heat equation with fractional noise 2

[ Abstract ]

We will present recent result concerning the heat equation driven by q Gaussian noise which behaves as a fractional Brownian motion in time and has a correlated spatial structure. We give the basic results concerning the existence and the properties of the solution. We will also focus on the distribution of this Gaussian process and its connection with other fractional-type processes.

We will present recent result concerning the heat equation driven by q Gaussian noise which behaves as a fractional Brownian motion in time and has a correlated spatial structure. We give the basic results concerning the existence and the properties of the solution. We will also focus on the distribution of this Gaussian process and its connection with other fractional-type processes.

#### Mathematical Biology Seminar

15:00-16:00 Room #128演習室 (Graduate School of Math. Sci. Bldg.)

Delayed Models of Cancer Dynamics: Lessons Learned in Mathematical Modelling (ENGLISH)

https://web.viu.ca/idelsl/

**Lev Idels**(Vanvouver Island University)Delayed Models of Cancer Dynamics: Lessons Learned in Mathematical Modelling (ENGLISH)

[ Abstract ]

In general, delay differential equations provide a richer mathematical

framework (compared with ordinary differential equations) for the

analysis of biosystems dynamics. The inclusion of explicit time lags in

tumor growth models allows direct reference to experimentally measurable

and/or controllable cell growth characteristics. For three different

types of angiogenesis models with variable delays, we consider either

continuous or impulse therapy that eradicates tumor cells and suppresses

angiogenesis. It was shown that with the growth of delays, even

constant, the equilibrium can lose its stability, and sustainable

oscillation, as well as chaotic behavior, can be observed. The analysis

outlines the difficulties which occur in the case of unbounded growth

rates, such as classical Gompertz model, for small volumes of cancer

cells compared to available blood vessels. The Wheldon model (1975) of a

Chronic Myelogenous Leukemia (CML) dynamics is revisited in the light of

recent discovery that this model has a major drawback.

[ Reference URL ]In general, delay differential equations provide a richer mathematical

framework (compared with ordinary differential equations) for the

analysis of biosystems dynamics. The inclusion of explicit time lags in

tumor growth models allows direct reference to experimentally measurable

and/or controllable cell growth characteristics. For three different

types of angiogenesis models with variable delays, we consider either

continuous or impulse therapy that eradicates tumor cells and suppresses

angiogenesis. It was shown that with the growth of delays, even

constant, the equilibrium can lose its stability, and sustainable

oscillation, as well as chaotic behavior, can be observed. The analysis

outlines the difficulties which occur in the case of unbounded growth

rates, such as classical Gompertz model, for small volumes of cancer

cells compared to available blood vessels. The Wheldon model (1975) of a

Chronic Myelogenous Leukemia (CML) dynamics is revisited in the light of

recent discovery that this model has a major drawback.

https://web.viu.ca/idelsl/

### 2016/04/25

#### Operator Algebra Seminars

16:45-18:15 Room #118 (Graduate School of Math. Sci. Bldg.)

Graded twisting of quantum groups, actions, and categories

**Makoto Yamashita**(Ochanomizu Univ.)Graded twisting of quantum groups, actions, and categories

#### Seminar on Geometric Complex Analysis

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

The representative domain and its applications (JAPANESE)

**Atsushi Yamamori**(Academia Sinica)The representative domain and its applications (JAPANESE)

[ Abstract ]

Bergman introduced the notion of a representative domain to choose a nice holomorphic equivalence class of domains. In this talk, I will explain that the representative domain is also useful to obtain an analogue of Cartan's linearity theorem for some special class of domains.

Bergman introduced the notion of a representative domain to choose a nice holomorphic equivalence class of domains. In this talk, I will explain that the representative domain is also useful to obtain an analogue of Cartan's linearity theorem for some special class of domains.

#### Tokyo Probability Seminar

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

Concentration results for directed polymer with unbouded jumps

**Shuta Nakajima**(Research institute for mathematical sciences)Concentration results for directed polymer with unbouded jumps

### 2016/04/22

#### Seminar on Probability and Statistics

10:30-11:50 Room #002 (Graduate School of Math. Sci. Bldg.)

Stein method and Malliavin calculus : theory and some applications to limit theorems 1

**Ciprian Tudor**(Université de Lille 1)Stein method and Malliavin calculus : theory and some applications to limit theorems 1

[ Abstract ]

In this first part, we will present the basic ideas of the Stein method for the normal approximation. We will also describe its connection with the Malliavin calculus and the Fourth Moment Theorem.

In this first part, we will present the basic ideas of the Stein method for the normal approximation. We will also describe its connection with the Malliavin calculus and the Fourth Moment Theorem.

#### Seminar on Probability and Statistics

12:50-14:10 Room #002 (Graduate School of Math. Sci. Bldg.)

Stein method and Malliavin calculus : theory and some applications to limit theorems 2

**Ciprian Tudor**(Université de Lille 1)Stein method and Malliavin calculus : theory and some applications to limit theorems 2

[ Abstract ]

In the second presentation, we intend to do the following: to illustrate the application of the Stein method to the limit behavior of the quadratic variation of Gaussian processes and its connection to statistics. We also intend to present the extension of the method to other target distributions.

In the second presentation, we intend to do the following: to illustrate the application of the Stein method to the limit behavior of the quadratic variation of Gaussian processes and its connection to statistics. We also intend to present the extension of the method to other target distributions.

#### Seminar on Probability and Statistics

14:20-15:50 Room #002 (Graduate School of Math. Sci. Bldg.)

Equivalence between the convergence in total variation and that of the Stein factor to the invariant measures of diffusion processes

**Seiichiro Kusuoka**(Okayama University)Equivalence between the convergence in total variation and that of the Stein factor to the invariant measures of diffusion processes

[ Abstract ]

We consider the characterization of the convergence of distributions to a given distribution in a certain class by using Stein's equation and Malliavin calculus with respect to the invariant measures of one-dimensional diffusion processes. Precisely speaking, we obtain an estimate between the so-called Stein factor and the total variation norm, and the equivalence between the convergence of the distributions in total variation and that of the Stein factor. This talk is based on the joint work with C.A.Tudor (arXiv:1310.3785).

We consider the characterization of the convergence of distributions to a given distribution in a certain class by using Stein's equation and Malliavin calculus with respect to the invariant measures of one-dimensional diffusion processes. Precisely speaking, we obtain an estimate between the so-called Stein factor and the total variation norm, and the equivalence between the convergence of the distributions in total variation and that of the Stein factor. This talk is based on the joint work with C.A.Tudor (arXiv:1310.3785).

#### Seminar on Probability and Statistics

16:10-17:10 Room #002 (Graduate School of Math. Sci. Bldg.)

Asymptotic expansion and estimation of volatility

**Nakahiro Yoshida**(University of Tokyo, Institute of Statistical Mathematics, JST CREST)Asymptotic expansion and estimation of volatility

[ Abstract ]

Parametric estimation of volatility of an Ito process in a finite time horizon is discussed. Asymptotic expansion of the error distribution will be presented for the quasi likelihood estimators, i.e., quasi MLE, quasi Bayesian estimator and one-step quasi MLE. Statistics becomes non-ergodic, where the limit distribution is mixed normal. Asymptotic expansion is a basic tool in various areas in the traditional ergodic statistics such as higher order asymptotic decision theory, bootstrap and resampling plans, prediction theory, information criterion for model selection, information geometry, etc. Then a natural question is to obtain asymptotic expansion in the non-ergodic statistics. However, due to randomness of the characteristics of the limit, the classical martingale expansion or the mixing method cannot not apply. Recently a new martingale expansion was developed and applied to a quadratic form of the Ito process. The higher order terms are characterized by the adaptive random symbol and the anticipative random symbol. The Malliavin calculus is used for the description of the anticipative random symbols as well as for obtaining a decay of the characteristic functions. In this talk, the martingale expansion method and the quasi likelihood analysis with a polynomial type large deviation estimate of the quasi likelihood random field collaborate to derive expansions for the quasi likelihood estimators. Expansions of the realized volatility under microstructure noise, the power variation and the error of Euler-Maruyama scheme are recent applications. Further, some extension of martingale expansion to general martingales will be mentioned. References: SPA2013, arXiv:1212.5845, AISM2011, arXiv:1309.2071 (to appear in AAP), arXiv:1512.04716.

Parametric estimation of volatility of an Ito process in a finite time horizon is discussed. Asymptotic expansion of the error distribution will be presented for the quasi likelihood estimators, i.e., quasi MLE, quasi Bayesian estimator and one-step quasi MLE. Statistics becomes non-ergodic, where the limit distribution is mixed normal. Asymptotic expansion is a basic tool in various areas in the traditional ergodic statistics such as higher order asymptotic decision theory, bootstrap and resampling plans, prediction theory, information criterion for model selection, information geometry, etc. Then a natural question is to obtain asymptotic expansion in the non-ergodic statistics. However, due to randomness of the characteristics of the limit, the classical martingale expansion or the mixing method cannot not apply. Recently a new martingale expansion was developed and applied to a quadratic form of the Ito process. The higher order terms are characterized by the adaptive random symbol and the anticipative random symbol. The Malliavin calculus is used for the description of the anticipative random symbols as well as for obtaining a decay of the characteristic functions. In this talk, the martingale expansion method and the quasi likelihood analysis with a polynomial type large deviation estimate of the quasi likelihood random field collaborate to derive expansions for the quasi likelihood estimators. Expansions of the realized volatility under microstructure noise, the power variation and the error of Euler-Maruyama scheme are recent applications. Further, some extension of martingale expansion to general martingales will be mentioned. References: SPA2013, arXiv:1212.5845, AISM2011, arXiv:1309.2071 (to appear in AAP), arXiv:1512.04716.

### 2016/04/21

#### Geometry Colloquium

17:00-18:00 Room #123 (Graduate School of Math. Sci. Bldg.)

Spectral convergence under bounded Ricci curvature (Japanese)

**Shouhei Honda**(Tohoku University)Spectral convergence under bounded Ricci curvature (Japanese)

[ Abstract ]

For a noncollapsed Gromov-Hausdorff convergent sequence of Riemannian manifolds with a uniform bound of Ricci curvature, we establish two spectral convergence. One of them is on the Hodge Laplacian acting on differential one-forms. The other is on the connection Laplacian acting on tensor fields of every type, which include all differential forms. These are sharp generalizations of Cheeger-Colding's spectral convergence of the Laplacian acting on functions to the cases of tensor fields and differential forms. These spectral convergence have two direct corollaries. One of them is to give new bounds on such eigenvalues, in terms of bounds on volume, diameter and the Ricci curvature. The other is that we show the upper semicontinuity of the first Betti numbers with respect to the Gromov-Hausdorff topology, and give the equivalence between the continuity of them and the existence of a uniform spectral gap. On the other hand we also define measurable curvature tensors of the noncollapsed Gromov-Hausdorff limit space of a sequence of Riemannian manifolds with a uniform bound of Ricci curvature, which include Riemannian curvature tensor, the Ricci curvature, and the scalar curvature. As fundamental properties of our Ricci curvature, we show that the Ricci curvature coincides with the difference between the Hodge Laplacian and the connection Laplacian, and is compatible with Gigli's one and Lott's Ricci measure. Moreover we prove a lower bound of the Ricci curvature is compatible with a reduced Riemannian curvature dimension condition. We also give a positive answer to Lott's question on the behavior of the scalar curvature with respect to the Gromov-Hausdorff topology by using our scalar curvature. This talk is based on arXiv:1510.05349.

For a noncollapsed Gromov-Hausdorff convergent sequence of Riemannian manifolds with a uniform bound of Ricci curvature, we establish two spectral convergence. One of them is on the Hodge Laplacian acting on differential one-forms. The other is on the connection Laplacian acting on tensor fields of every type, which include all differential forms. These are sharp generalizations of Cheeger-Colding's spectral convergence of the Laplacian acting on functions to the cases of tensor fields and differential forms. These spectral convergence have two direct corollaries. One of them is to give new bounds on such eigenvalues, in terms of bounds on volume, diameter and the Ricci curvature. The other is that we show the upper semicontinuity of the first Betti numbers with respect to the Gromov-Hausdorff topology, and give the equivalence between the continuity of them and the existence of a uniform spectral gap. On the other hand we also define measurable curvature tensors of the noncollapsed Gromov-Hausdorff limit space of a sequence of Riemannian manifolds with a uniform bound of Ricci curvature, which include Riemannian curvature tensor, the Ricci curvature, and the scalar curvature. As fundamental properties of our Ricci curvature, we show that the Ricci curvature coincides with the difference between the Hodge Laplacian and the connection Laplacian, and is compatible with Gigli's one and Lott's Ricci measure. Moreover we prove a lower bound of the Ricci curvature is compatible with a reduced Riemannian curvature dimension condition. We also give a positive answer to Lott's question on the behavior of the scalar curvature with respect to the Gromov-Hausdorff topology by using our scalar curvature. This talk is based on arXiv:1510.05349.

#### FMSP Lectures

15:00-16:00, 16:10-17:10 Room #002 (Graduate School of Math. Sci. Bldg.)

Rational homotopy theory : Quillen and Sullivan approach.(2) (ENGLISH)

[ Reference URL ]

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

**Aniceto Murillo et al**(Universidad de Malaga)Rational homotopy theory : Quillen and Sullivan approach.(2) (ENGLISH)

[ Reference URL ]

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

### 2016/04/20

#### FMSP Lectures

15:00-16:00, 16:10-17:10 Room #002 (Graduate School of Math. Sci. Bldg.)

Rational homotopy theory : Quillen and Sullivan approach.(1) (ENGLISH)

[ Reference URL ]

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

**Aniceto Murillo et al**(Universidad de Malaga)Rational homotopy theory : Quillen and Sullivan approach.(1) (ENGLISH)

[ Reference URL ]

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

#### Number Theory Seminar

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

On periodicity of geodesic continued fractions (Japanese)

**Hoto Bekki**(University of Tokyo)On periodicity of geodesic continued fractions (Japanese)

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