## Seminar information archive

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

#### Lectures

16:00-17:30 Room #126 (Graduate School of Math. Sci. Bldg.)

Nori motives over function fields and period functions (ENGLISH)

**Javier Fresan**(Ecole Polytechnique)Nori motives over function fields and period functions (ENGLISH)

[ Abstract ]

Around twenty years ago, Nori introduced a tannakian category of mixed motives over a subfield of the complex numbers, thus giving the first unconditional construction of the motivic Galois group. In this series of lectures, I will first survey on Nori's theory and its relationship to other categories of motives. I will then explain how to extend his construction to functions fields and why the resulting tannakian group governs

algebraic relations between period functions.

This last part is based on an ongoing work with Peter Jossen.

Around twenty years ago, Nori introduced a tannakian category of mixed motives over a subfield of the complex numbers, thus giving the first unconditional construction of the motivic Galois group. In this series of lectures, I will first survey on Nori's theory and its relationship to other categories of motives. I will then explain how to extend his construction to functions fields and why the resulting tannakian group governs

algebraic relations between period functions.

This last part is based on an ongoing work with Peter Jossen.

### 2020/01/07

#### Tuesday Seminar on Topology

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

Magnitude homology of crushable spaces (JAPANESE)

**Yasuhiko Asao**(The University of Tokyo)Magnitude homology of crushable spaces (JAPANESE)

[ Abstract ]

The magnitude homology and the blurred magnitude homology are novel notions of homology theory for general metric spaces coined by Leinster et al. They are expected to be dealt with in the context of Topological Data Analysis since its original idea is based on a kind of "persistence of points clouds". However, little property of them has been revealed. In this talk, we see that the blurred magnitude homology is trivial when a metric space is contractible by a distance decreasing homotopy. We use techniques from singular homology theory.

The magnitude homology and the blurred magnitude homology are novel notions of homology theory for general metric spaces coined by Leinster et al. They are expected to be dealt with in the context of Topological Data Analysis since its original idea is based on a kind of "persistence of points clouds". However, little property of them has been revealed. In this talk, we see that the blurred magnitude homology is trivial when a metric space is contractible by a distance decreasing homotopy. We use techniques from singular homology theory.

#### Tuesday Seminar on Topology

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

Intersection number estimate of rational Lagrangian immersions in cotangent bundles via microlocal sheaf theory (JAPANESE)

**Tomohiro Asano**(The University of Tokyo)Intersection number estimate of rational Lagrangian immersions in cotangent bundles via microlocal sheaf theory (JAPANESE)

[ Abstract ]

Guillermou associated sheaves to exact Lagrangian submanifolds in cotangent bundles and proved topological properties of the Lagrangian submanifolds. In this talk, I will give an estimate on the displacement energy of rational Lagrangian immersions in cotangent bundles with intersection number estimates via microlocal sheaf theory. This result overlaps with results by Chekanov, Liu, and Akaho via Floer theory. This is joint work with Yuichi Ike.

Guillermou associated sheaves to exact Lagrangian submanifolds in cotangent bundles and proved topological properties of the Lagrangian submanifolds. In this talk, I will give an estimate on the displacement energy of rational Lagrangian immersions in cotangent bundles with intersection number estimates via microlocal sheaf theory. This result overlaps with results by Chekanov, Liu, and Akaho via Floer theory. This is joint work with Yuichi Ike.

### 2019/12/27

#### Seminar on Probability and Statistics

15:00-16:10 Room #126 (Graduate School of Math. Sci. Bldg.)

High order distributional approximations by Stein's method

**Xiao Fang**(Chinese University of Hong Kong)High order distributional approximations by Stein's method

[ Abstract ]

Stein's method is a powerful tool to proving distributional approximations with error bounds. In this talk, we present two recent developments of Stein's method for high order approximations. (1) Together with Li Luo and Qi-Man Shao, we consider skewness correction in normal approximation. We prove a refined Cram\'er-type moderate deviation result for a class of statistics possessing a local structure. We discuss applications to $k$-runs, U-statistics and subgraph counts. (2) Together with Anton Braverman and Jim Dai, we derive and justify new diffusion approximations with state-dependent diffusion coefficients for stationary distributions of Markov chains. We discuss applications to the Erlang-C system, a hospital inpatient flow model and the auto-regressive model.

Stein's method is a powerful tool to proving distributional approximations with error bounds. In this talk, we present two recent developments of Stein's method for high order approximations. (1) Together with Li Luo and Qi-Man Shao, we consider skewness correction in normal approximation. We prove a refined Cram\'er-type moderate deviation result for a class of statistics possessing a local structure. We discuss applications to $k$-runs, U-statistics and subgraph counts. (2) Together with Anton Braverman and Jim Dai, we derive and justify new diffusion approximations with state-dependent diffusion coefficients for stationary distributions of Markov chains. We discuss applications to the Erlang-C system, a hospital inpatient flow model and the auto-regressive model.

#### Seminar on Probability and Statistics

16:30-17:40 Room #126 (Graduate School of Math. Sci. Bldg.)

Heavy-Tailed Fractional Pearson Diffusions

**Nikolai Leonenko**(Cardiff University)Heavy-Tailed Fractional Pearson Diffusions

[ Abstract ]

We define fractional Pearson diffusions [5,7,8] by non-Markovian time change in the corresponding Pearson diffusions [1,2,3,4]. They are governed by the time-fractional diffusion equations with polynomial coefficients depending on the parameters of the corresponding Pearson distribution. We present the spectral representation of transition densities of fractional Pearson diffusions, which depend heavily on the structure of the spectrum of the infinitesimal generator of the corresponding non-fractional Pearson diffusion. Also, we present the strong solutions of the Cauchy problems associated with heavy-tailed fractional Pearson diffusions and the correlation structure of these diffusions [6] .

Continuous time random walks have random waiting times between particle jumps. We define the correlated continuous time random walks (CTRWs) that converge to fractional Pearson diffusions (fPDs) [9,10,11]. The jumps in these CTRWs are obtained from Markov chains through the Bernoulli urn-scheme model, Wright-Fisher model and Ehrenfest-Brillouin-type models. The jumps are correlated so that the limiting processes are not Lévy but diffusion processes with non-independent increments.

This is a joint work with M. Meerschaert (Michigan State University, USA), I. Papic (University of Osijek, Croatia), N.Suvak (University of Osijek, Croatia) and A. Sikorskii (Michigan State University and Arizona University, USA).

References:

[1] Avram, F., Leonenko, N.N and Suvak, N. (2013), On spectral analysis of heavy-tailed Kolmogorov-Pearson diffusion, Markov Processes and Related Fields, Volume 19, N 2 , 249-298

[2] Avram, F., Leonenko, N.N and Suvak, N., (2013), Spectral representation of transition density of Fisher-Snedecor diffusion, Stochastics, 85 (2013), no. 2, 346—369

[3] Bourguin, S., Campese, S., Leonenko, N. and Taqqu,M.S. (2019) Four moments theorems on Markov chaos, Annals of Probability, 47, N3, 1417–1446

[4] Kulik, A.M. and Leonenko, N.N. (2013) Ergodicity and mixing bounds for the Fisher-Snendecor diffusion, Bernoulli, Vol. 19, No. 5B, 2294-2329

[5] Leonenko, N.N., Meerschaert, M.M and Sikorskii, A. (2013) Fractional Pearson diffusions, Journal of Mathematical Analysis and Applications, vol. 403, 532-546

[6] Leonenko, N.N., Meerschaert, M.M and Sikorskii, A. (2013) Correlation Structure of Fractional Pearson diffusion, Computers and Mathematics with Applications, 66, 737-745

[7] Leonenko,N.N., Meerschaert,M.M., Schilling,R.L. and Sikorskii, A. (2014) Correlation Structure of Time-Changed Lévy Processes, Communications in Applied and Industrial Mathematics, Vol. 6 , No. 1, p. e-483 (22 pp.)

[8] Leonenko, N.N., Papic, I., Sikorskii, A. and Suvak, N. (2017) Heavy-tailed fractional Pearson diffusions, Stochastic Processes and their Applications, 127, N11, 3512-3535

[9] Leonenko, N.N., Papic, I., Sikorskii, A. and Suvak, N. (2018) Correlated continuous time random walks and fractional Pearson diffusions, Bernoulli, Vol. 24, No. 4B, 3603-3627

[10] Leonenko, N.N., Papic, I., Sikorskii, A. and Suvak, N. (2019) Ehrenfest-Brillouin-type correlated continuous time random walks and fractional Jacoby diffusion, Theory Probablity and Mathematical Statistics, Vol. 99,123-133.

[11] Leonenko, N.N., Papic, I., Sikorskii, A. and Suvak, N. (2019) Approximation of heavy-tailed fractional Pearson diffusions in Skorokhod topology, submitted

We define fractional Pearson diffusions [5,7,8] by non-Markovian time change in the corresponding Pearson diffusions [1,2,3,4]. They are governed by the time-fractional diffusion equations with polynomial coefficients depending on the parameters of the corresponding Pearson distribution. We present the spectral representation of transition densities of fractional Pearson diffusions, which depend heavily on the structure of the spectrum of the infinitesimal generator of the corresponding non-fractional Pearson diffusion. Also, we present the strong solutions of the Cauchy problems associated with heavy-tailed fractional Pearson diffusions and the correlation structure of these diffusions [6] .

Continuous time random walks have random waiting times between particle jumps. We define the correlated continuous time random walks (CTRWs) that converge to fractional Pearson diffusions (fPDs) [9,10,11]. The jumps in these CTRWs are obtained from Markov chains through the Bernoulli urn-scheme model, Wright-Fisher model and Ehrenfest-Brillouin-type models. The jumps are correlated so that the limiting processes are not Lévy but diffusion processes with non-independent increments.

This is a joint work with M. Meerschaert (Michigan State University, USA), I. Papic (University of Osijek, Croatia), N.Suvak (University of Osijek, Croatia) and A. Sikorskii (Michigan State University and Arizona University, USA).

References:

[1] Avram, F., Leonenko, N.N and Suvak, N. (2013), On spectral analysis of heavy-tailed Kolmogorov-Pearson diffusion, Markov Processes and Related Fields, Volume 19, N 2 , 249-298

[2] Avram, F., Leonenko, N.N and Suvak, N., (2013), Spectral representation of transition density of Fisher-Snedecor diffusion, Stochastics, 85 (2013), no. 2, 346—369

[3] Bourguin, S., Campese, S., Leonenko, N. and Taqqu,M.S. (2019) Four moments theorems on Markov chaos, Annals of Probability, 47, N3, 1417–1446

[4] Kulik, A.M. and Leonenko, N.N. (2013) Ergodicity and mixing bounds for the Fisher-Snendecor diffusion, Bernoulli, Vol. 19, No. 5B, 2294-2329

[5] Leonenko, N.N., Meerschaert, M.M and Sikorskii, A. (2013) Fractional Pearson diffusions, Journal of Mathematical Analysis and Applications, vol. 403, 532-546

[6] Leonenko, N.N., Meerschaert, M.M and Sikorskii, A. (2013) Correlation Structure of Fractional Pearson diffusion, Computers and Mathematics with Applications, 66, 737-745

[7] Leonenko,N.N., Meerschaert,M.M., Schilling,R.L. and Sikorskii, A. (2014) Correlation Structure of Time-Changed Lévy Processes, Communications in Applied and Industrial Mathematics, Vol. 6 , No. 1, p. e-483 (22 pp.)

[8] Leonenko, N.N., Papic, I., Sikorskii, A. and Suvak, N. (2017) Heavy-tailed fractional Pearson diffusions, Stochastic Processes and their Applications, 127, N11, 3512-3535

[9] Leonenko, N.N., Papic, I., Sikorskii, A. and Suvak, N. (2018) Correlated continuous time random walks and fractional Pearson diffusions, Bernoulli, Vol. 24, No. 4B, 3603-3627

[10] Leonenko, N.N., Papic, I., Sikorskii, A. and Suvak, N. (2019) Ehrenfest-Brillouin-type correlated continuous time random walks and fractional Jacoby diffusion, Theory Probablity and Mathematical Statistics, Vol. 99,123-133.

[11] Leonenko, N.N., Papic, I., Sikorskii, A. and Suvak, N. (2019) Approximation of heavy-tailed fractional Pearson diffusions in Skorokhod topology, submitted

### 2019/12/26

#### Operator Algebra Seminars

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

Categorical quantization of symmetric spaces and reflection equation

**Makoto Yamashita**(Oslo Univ.)Categorical quantization of symmetric spaces and reflection equation

### 2019/12/25

#### Operator Algebra Seminars

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

Connes fusion on the unit circle

(English)

**Bin Gui**(Rutgers Univ.)Connes fusion on the unit circle

(English)

### 2019/12/20

#### Colloquium

15:30-16:30 Room #056 (Graduate School of Math. Sci. Bldg.)

#### Logic

13:00-14:30 Room #156 (Graduate School of Math. Sci. Bldg.)

### 2019/12/19

#### Information Mathematics Seminar

16:50-18:35 Room #122 (Graduate School of Math. Sci. Bldg.)

Our activities in the research of artificial intelligence (Japanese)

**Shigeru Nemoto**(BroadBand Tower,inc. AI2 Open Innovation Lab.)Our activities in the research of artificial intelligence (Japanese)

[ Abstract ]

Explanation of activities in the research of artificial intelligence

Explanation of activities in the research of artificial intelligence

#### Applied Analysis

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

### 2019/12/18

#### Operator Algebra Seminars

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

The Hofstadter model, fractality, and topology

**Pasquale Marra**(Univ. Tokyo)The Hofstadter model, fractality, and topology

### 2019/12/17

#### Tuesday Seminar on Topology

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

Symplectic homology of fiberwise convex sets and homology of loop spaces (JAPANESE)

**Kei Irie**(The University of Tokyo)Symplectic homology of fiberwise convex sets and homology of loop spaces (JAPANESE)

[ Abstract ]

For any (compact) subset in the symplectic vector space, one can define its symplectic capacity by using symplectic homology, which is a version of Floer homology.

In general, it is very difficult to compute or estimate this capacity directly from its definition, since the definition of Floer homology involves counting solutions of nonlinear PDEs (so called Floer equations). In this talk, we consider the symplectic vector space as the cotangent bundle of the Euclidean space, and show a formula which computes symplectic homology and capacity of fiberwise convex sets from homology of loop spaces. We also explain two applications of this formula.

For any (compact) subset in the symplectic vector space, one can define its symplectic capacity by using symplectic homology, which is a version of Floer homology.

In general, it is very difficult to compute or estimate this capacity directly from its definition, since the definition of Floer homology involves counting solutions of nonlinear PDEs (so called Floer equations). In this talk, we consider the symplectic vector space as the cotangent bundle of the Euclidean space, and show a formula which computes symplectic homology and capacity of fiberwise convex sets from homology of loop spaces. We also explain two applications of this formula.

#### Infinite Analysis Seminar Tokyo

15:00-16:00 Room #駒場国際教育研究棟（旧６号館）108 (Graduate School of Math. Sci. Bldg.)

(-2) blow-up formula (JAPANESE)

**Ryo Ohkawa**(Waseda University)(-2) blow-up formula (JAPANESE)

[ Abstract ]

In this talk, we will consider the moduli of ADHM data

corresponding to the affine A_1 Dynkin diagram.

It is a moduli of framed sheaves on the (-2) curve or the projective

plane with a group action.

Each of these two types of moduli integrals has a combinatorial

description. In particular, the Hirota derivative of the Nekrasov

function can be obtained on the (-2) curve.

We introduce equalities among these two integrals and the

corresponding functional equations in some cases.

This is similar to the blow-up formula by Nakajima-Yoshioka.

I would also like to talk about relationships with the study of the

Painleve tau function by Bershtein-Shchechkin.

In this talk, we will consider the moduli of ADHM data

corresponding to the affine A_1 Dynkin diagram.

It is a moduli of framed sheaves on the (-2) curve or the projective

plane with a group action.

Each of these two types of moduli integrals has a combinatorial

description. In particular, the Hirota derivative of the Nekrasov

function can be obtained on the (-2) curve.

We introduce equalities among these two integrals and the

corresponding functional equations in some cases.

This is similar to the blow-up formula by Nakajima-Yoshioka.

I would also like to talk about relationships with the study of the

Painleve tau function by Bershtein-Shchechkin.

### 2019/12/16

#### Seminar on Geometric Complex Analysis

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

A simplified proof of the optimal L^2 extension theorem and its application (Japanese)

**Genki Hosono**(Tohoku Univ.)A simplified proof of the optimal L^2 extension theorem and its application (Japanese)

[ Abstract ]

I will explain a simplified proof of an optimal version of the Ohsawa-Takegoshi L^2-extension theorem. In the proof, I use a method of Berndtsson-Lempert and skip some argument by the method of McNeal-Varolin. As an application, I will explain a result on extensions from possibly non-reduced varieties.

I will explain a simplified proof of an optimal version of the Ohsawa-Takegoshi L^2-extension theorem. In the proof, I use a method of Berndtsson-Lempert and skip some argument by the method of McNeal-Varolin. As an application, I will explain a result on extensions from possibly non-reduced varieties.

#### Numerical Analysis Seminar

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

A second-order stabilization method for linearizing and decoupling nonlinear parabolic systems (Japanese)

**Yuki Ueda**(The Hong Kong Polytechnic University)A second-order stabilization method for linearizing and decoupling nonlinear parabolic systems (Japanese)

[ Abstract ]

We present a new time discretization method for strongly nonlinear parabolic systems. Our method is based on backward finite difference for the first derivative with second-order accuracy and the first-order linear discrete-time scheme for nonlinear systems which has been introduced by H. Murakawa. We propose a second-order stabilization method by combining these schemes.

Our error estimate requires testing the error equation by two test functions and showing $W^{1,\infty}$-boundedness which is proved by ($H^2$ or) $H^3$ energy estimate. We overcome the difficulty for establishing energy estimate by using the generating function technique which is popular in studying ordinary differential equations. Several numerical examples are provided to support the theoretical result.

We present a new time discretization method for strongly nonlinear parabolic systems. Our method is based on backward finite difference for the first derivative with second-order accuracy and the first-order linear discrete-time scheme for nonlinear systems which has been introduced by H. Murakawa. We propose a second-order stabilization method by combining these schemes.

Our error estimate requires testing the error equation by two test functions and showing $W^{1,\infty}$-boundedness which is proved by ($H^2$ or) $H^3$ energy estimate. We overcome the difficulty for establishing energy estimate by using the generating function technique which is popular in studying ordinary differential equations. Several numerical examples are provided to support the theoretical result.

### 2019/12/12

#### Information Mathematics Seminar

16:50-18:35 Room #122 (Graduate School of Math. Sci. Bldg.)

５G Strategy (Japanese)

**Katsuya WATANABE**(Internet Research Institute, Inc.)５G Strategy (Japanese)

[ Abstract ]

Explanation of ５G Strategy

Explanation of ５G Strategy

### 2019/12/11

#### Seminar on Geometric Complex Analysis

16:00-17:00 Room #156 (Graduate School of Math. Sci. Bldg.)

Einstein-Weyl structures (English)

**Joel Merker**(Paris Sud)Einstein-Weyl structures (English)

[ Abstract ]

On a conformal 3D manifold with electromagnetic field, Einstein-Weyl equations are the counterpart of Einstein's classical field equations. In 1943, Elie Cartan showed, using abstract arguments, that the general solution depends on 4 functions of 2 variables. I will present families of explicit solutions depending on 9 functions of 1 variable, much beyond what was known before. Such solutions are generic in the sense that the Cotton tensor is nonzero. This is joint work with Pawel Nurowski.

On a conformal 3D manifold with electromagnetic field, Einstein-Weyl equations are the counterpart of Einstein's classical field equations. In 1943, Elie Cartan showed, using abstract arguments, that the general solution depends on 4 functions of 2 variables. I will present families of explicit solutions depending on 9 functions of 1 variable, much beyond what was known before. Such solutions are generic in the sense that the Cotton tensor is nonzero. This is joint work with Pawel Nurowski.

### 2019/12/10

#### Operator Algebra Seminars

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

On gauging symmetries of topological phases of matter (English)

**Zhenghan Wang**(Microsoft Station Q)On gauging symmetries of topological phases of matter (English)

#### Tuesday Seminar on Topology

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

q-Deformation of a continued fraction and its applications (JAPANESE)

**Takeyoshi Kogiso**(Josai University)q-Deformation of a continued fraction and its applications (JAPANESE)

[ Abstract ]

A kind of q-Deformation of continued fractions was introduced by Morier-Genoud and Ovsienko. The most important application of this q-deformation of regular (or negative) continued fraction expansion of rational number r/s is to calculate the Jones polynomial of the rational link of r/s. Moreover we can apply the q-deformation of this continued fraction to quadratic irrational number theory and combinatorics.

On the other hand, there exist another recipes for determing the Jones polynomials by using Lee-Schiffler's snake graph and by using Kogiso-Wakui's Conway-Coxeter frieze method. Therefore, another approach of the result due to Morier-Genoud and Ovsienko can be considered from the viewpoint of a Conway-Coxeter frieze and a snake graph. Furthermore, we can consider a cluster-variable transformation of continued fractions as a further generalization by using ancestoral triangles used in the Kogiso-Wakui.

A kind of q-Deformation of continued fractions was introduced by Morier-Genoud and Ovsienko. The most important application of this q-deformation of regular (or negative) continued fraction expansion of rational number r/s is to calculate the Jones polynomial of the rational link of r/s. Moreover we can apply the q-deformation of this continued fraction to quadratic irrational number theory and combinatorics.

On the other hand, there exist another recipes for determing the Jones polynomials by using Lee-Schiffler's snake graph and by using Kogiso-Wakui's Conway-Coxeter frieze method. Therefore, another approach of the result due to Morier-Genoud and Ovsienko can be considered from the viewpoint of a Conway-Coxeter frieze and a snake graph. Furthermore, we can consider a cluster-variable transformation of continued fractions as a further generalization by using ancestoral triangles used in the Kogiso-Wakui.

#### Tuesday Seminar of Analysis

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

Vorticity alignment vs vorticity creation at the boundary (English)

**Tobias Barker**(École Normale Supérieure)Vorticity alignment vs vorticity creation at the boundary (English)

[ Abstract ]

The Navier-Stokes are used as a model for viscous incompressible fluids such as water. The question as to whether or not the equations in three dimensions form singularities is an open Millennium prize problem. In their celebrated paper in 1993, Constantin and Fefferman showed that (in the whole plane) if the vorticity is sufficiently well aligned in regions of high vorticity then the Navier-Stokes equations remain smooth. For the half-space it is commonly assumed that viscous fluids `stick' to the boundary, which generates vorticity at the boundary. In such a setting, it is open as to whether Constantin and Fefferman's result remains to be true. In my talk I will present recent results in this direction. Joint work with Christophe Prange (CNRS, Université de Bordeaux)

The Navier-Stokes are used as a model for viscous incompressible fluids such as water. The question as to whether or not the equations in three dimensions form singularities is an open Millennium prize problem. In their celebrated paper in 1993, Constantin and Fefferman showed that (in the whole plane) if the vorticity is sufficiently well aligned in regions of high vorticity then the Navier-Stokes equations remain smooth. For the half-space it is commonly assumed that viscous fluids `stick' to the boundary, which generates vorticity at the boundary. In such a setting, it is open as to whether Constantin and Fefferman's result remains to be true. In my talk I will present recent results in this direction. Joint work with Christophe Prange (CNRS, Université de Bordeaux)

#### FMSP Lectures

17:00-18:00 Room #118 (Graduate School of Math. Sci. Bldg.)

A priori and a posteriori error estimation for solutions of ill-posed problems (ENGLISH)

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

**Anatoly G. Yagola**(Lomonosov Moscow State University)A priori and a posteriori error estimation for solutions of ill-posed problems (ENGLISH)

[ Abstract ]

In order to calculate a priori or a posteriori error estimates for solutions of an ill-posed operator equation with an injective operator we need to describe a set of approximate solutions that contains an exact solution. After that we have to calculate a diameter of this set or maximal distance from a fixed approximate solution to any element of this set. I will describe three approaches for constructing error estimates and also their practical applications.

[ Reference URL ]In order to calculate a priori or a posteriori error estimates for solutions of an ill-posed operator equation with an injective operator we need to describe a set of approximate solutions that contains an exact solution. After that we have to calculate a diameter of this set or maximal distance from a fixed approximate solution to any element of this set. I will describe three approaches for constructing error estimates and also their practical applications.

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

### 2019/12/09

#### Numerical Analysis Seminar

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

Point-wise error estimation for the finite element solution to Poisson's equation --- new approach based on Kato-Fujita's method (Japanese)

**Xuefeng Liu**(Niigata University)Point-wise error estimation for the finite element solution to Poisson's equation --- new approach based on Kato-Fujita's method (Japanese)

[ Abstract ]

In 1950s, H. Fujita proposed a method to provide the upper and lower bounds in boundary value problems, which is based on the T*T theory of T. Kato about differential equations. Such a method can be regarded a different formulation of the hypercircle method from Prage-Synge's theorem.

Recently, the speaker extended Kato-Fujita's method to the case of the finite element solution of Poisson's equation and proposed a guaranteed point-wise error estimation. The newly proposed error estimation can be applied to problems defined over domains of general shapes along with general boundary conditions.

In 1950s, H. Fujita proposed a method to provide the upper and lower bounds in boundary value problems, which is based on the T*T theory of T. Kato about differential equations. Such a method can be regarded a different formulation of the hypercircle method from Prage-Synge's theorem.

Recently, the speaker extended Kato-Fujita's method to the case of the finite element solution of Poisson's equation and proposed a guaranteed point-wise error estimation. The newly proposed error estimation can be applied to problems defined over domains of general shapes along with general boundary conditions.

#### Seminar on Geometric Complex Analysis

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

Analytic torsions associated with the Rumin complex on contact spheres (Japanese)

**Akira Kitaoka**(The Univ. of Tokyo)Analytic torsions associated with the Rumin complex on contact spheres (Japanese)

[ Abstract ]

The Rumin complex, which is defined on contact manifolds, is a resolution of the constant sheaf of $\mathbb{R}$ given by a subquotient of the de Rham complex. In this talk, we explicitly write down all eigenvalues of the Rumin Laplacian on the standard contact spheres, and express the analytic torsion functions associated with the Rumin complex in terms of the Riemann zeta function. In particular, we find that the functions vanish at the origin and determine the analytic torsions.

The Rumin complex, which is defined on contact manifolds, is a resolution of the constant sheaf of $\mathbb{R}$ given by a subquotient of the de Rham complex. In this talk, we explicitly write down all eigenvalues of the Rumin Laplacian on the standard contact spheres, and express the analytic torsion functions associated with the Rumin complex in terms of the Riemann zeta function. In particular, we find that the functions vanish at the origin and determine the analytic torsions.

### 2019/12/05

#### Information Mathematics Seminar

16:50-18:35 Room #118 (Graduate School of Math. Sci. Bldg.)

Business development of NLP (Japanese)

**Akihito Ogino**(A.I. Squared, Inc.)Business development of NLP (Japanese)

[ Abstract ]

Explanation of AI in Business development of NLP

Explanation of AI in Business development of NLP

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