## Seminar information archive

Seminar information archive ～01/25｜Today's seminar 01/26 | Future seminars 01/27～

### 2020/01/22

#### FMSP Lectures

17:00-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Complex principal type operators in inverse conductivity problem (ENGLISH)

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

**Samuli Siltanen**(University of Helsinki)Complex principal type operators in inverse conductivity problem (ENGLISH)

[ Abstract ]

Stroke is a leading cause of death all around the world. There are two main types of stroke: ischemic (blood clot preventing blood flow to a part of the brain) and hemorrhagic (bleeding in the brain). The symptoms are the same, but treatments very different. A portable "stroke classifier" would be a life-saving equipment to have in ambulances, but so far it does not exist. Electrical Impedance Tomography (EIT) is a promising and harmless imaging method for stroke classification. In EIT one attempts to recover the electric conductivity inside a domain from electric boundary measurements. This is a nonlinear and ill-posed inverse problem. The so-called Complex Geometric Optics (CGO) solutions have proven to be a useful computational tool for reconstruction tasks in EIT. A new property of CGO solutions is presented, showing that a one-dimensional Fourier transform in the spectral variable provides a connection to parallel-beam Xray tomography of the conductivity. One of the consequences of this “nonlinear Fourier slice theorem” is a novel capability to recover inclusions within inclusions in EIT. In practical imaging, measurement noise causes strong blurring in the recovered profile functions. However, machine learning algorithms can be combined with the nonlinear PDE techniques in a fruitful way. As an example, simulated strokes are classified into hemorrhagic and ischemic using EIT measurements.

[ Reference URL ]Stroke is a leading cause of death all around the world. There are two main types of stroke: ischemic (blood clot preventing blood flow to a part of the brain) and hemorrhagic (bleeding in the brain). The symptoms are the same, but treatments very different. A portable "stroke classifier" would be a life-saving equipment to have in ambulances, but so far it does not exist. Electrical Impedance Tomography (EIT) is a promising and harmless imaging method for stroke classification. In EIT one attempts to recover the electric conductivity inside a domain from electric boundary measurements. This is a nonlinear and ill-posed inverse problem. The so-called Complex Geometric Optics (CGO) solutions have proven to be a useful computational tool for reconstruction tasks in EIT. A new property of CGO solutions is presented, showing that a one-dimensional Fourier transform in the spectral variable provides a connection to parallel-beam Xray tomography of the conductivity. One of the consequences of this “nonlinear Fourier slice theorem” is a novel capability to recover inclusions within inclusions in EIT. In practical imaging, measurement noise causes strong blurring in the recovered profile functions. However, machine learning algorithms can be combined with the nonlinear PDE techniques in a fruitful way. As an example, simulated strokes are classified into hemorrhagic and ischemic using EIT measurements.

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

### 2020/01/21

#### Algebraic Geometry Seminar

15:30-17:00 Room #118 (Graduate School of Math. Sci. Bldg.)

(Few) rational curves on K3 surfaces (English)

**Matthias Schütt**(Universität Hannover)(Few) rational curves on K3 surfaces (English)

[ Abstract ]

Rational curves play a fundamental role for the structure of a K3 surface. I will first review the general theory before focussing on the case of low degree curves where joint work with S. Rams (Krakow) extends bounds of Miyaoka and Degtyarev. Time permitting, I will also discuss the special case of smooth rational curves as well as applications to Enriques surfaces.

Rational curves play a fundamental role for the structure of a K3 surface. I will first review the general theory before focussing on the case of low degree curves where joint work with S. Rams (Krakow) extends bounds of Miyaoka and Degtyarev. Time permitting, I will also discuss the special case of smooth rational curves as well as applications to Enriques surfaces.

#### Lie Groups and Representation Theory

14:00-16:00 Room #126 (Graduate School of Math. Sci. Bldg.)

On Plancherel measure (English)

**Joseph Bernstein**(Tel Aviv University)On Plancherel measure (English)

### 2020/01/20

#### Numerical Analysis Seminar

16:50-18:20 Room #056 (Graduate School of Math. Sci. Bldg.)

Isogeometric Hierarchical Model Reduction: from analysis to patient-specific simulations (English)

**Yves A. B. C. Barbosa**(Politecnico di Milano)Isogeometric Hierarchical Model Reduction: from analysis to patient-specific simulations (English)

[ Abstract ]

In the field of hemodynamics, numerical models have evolved to account for the demands in speed and accuracy of modern diagnostic medicine. In this context, we studied in detail Hierarchical Model Reduction technique combined with Isogeometric Analysis (HigaMOD), a technique recently developed in [Perotto, Reali, Rusconi and Veneziani (2017)]. HigaMod is a reduction procedure used to downscale models when the phenomenon at hand presents a preferential direction of flow, e.g., when modelling the blood flow in arteries or the water flow in a channel network. The method showed a significant improvement in reducing the computational power and simulation time, while giving enough information to analyze the problem at hand.

Recently, we focused our work in solving the ADR problem and the Stokes problem in a patient-specific framework. Specifically, we evaluate the computational efficiency of HigaMod in simulating the blood flow in coronary arteries and cerebral arteries. The main goal is to assess the

mprovement that 1D enriched models can provide, with respect to traditional full models, when dealing with demanding 3D CFD simulations. The results obtained, even though preliminary, are promising [Brandes, Barbosa and Perotto (2019); Brandes, Barbosa, Perotto and Suito (2020)].

In the field of hemodynamics, numerical models have evolved to account for the demands in speed and accuracy of modern diagnostic medicine. In this context, we studied in detail Hierarchical Model Reduction technique combined with Isogeometric Analysis (HigaMOD), a technique recently developed in [Perotto, Reali, Rusconi and Veneziani (2017)]. HigaMod is a reduction procedure used to downscale models when the phenomenon at hand presents a preferential direction of flow, e.g., when modelling the blood flow in arteries or the water flow in a channel network. The method showed a significant improvement in reducing the computational power and simulation time, while giving enough information to analyze the problem at hand.

Recently, we focused our work in solving the ADR problem and the Stokes problem in a patient-specific framework. Specifically, we evaluate the computational efficiency of HigaMod in simulating the blood flow in coronary arteries and cerebral arteries. The main goal is to assess the

mprovement that 1D enriched models can provide, with respect to traditional full models, when dealing with demanding 3D CFD simulations. The results obtained, even though preliminary, are promising [Brandes, Barbosa and Perotto (2019); Brandes, Barbosa, Perotto and Suito (2020)].

#### Seminar on Geometric Complex Analysis

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

Diederich-Fornaess and Steinness indices for abstract CR manifolds

**Masanori Adachi**(Shizuoka Univ.)Diederich-Fornaess and Steinness indices for abstract CR manifolds

[ Abstract ]

The Diederich-Fornaes and Steinness indices are estimated for weakly pseudoconvex domains in complex manifolds in terms of the D'Angelo 1-form of the boundary CR manifolds. In particular, CR invariance of these indices is shown when the domain is Takeuchi 1-convex. This is a joint work with Jihun Yum (Pusan National University).

The Diederich-Fornaes and Steinness indices are estimated for weakly pseudoconvex domains in complex manifolds in terms of the D'Angelo 1-form of the boundary CR manifolds. In particular, CR invariance of these indices is shown when the domain is Takeuchi 1-convex. This is a joint work with Jihun Yum (Pusan National University).

### 2020/01/16

#### Information Mathematics Seminar

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

Foundation of Quantum Annealing (Japanese)

**Hirotaka Irie**(DENSO CORPORATION/RIKEN iTHEMS)Foundation of Quantum Annealing (Japanese)

[ Abstract ]

Explanation of Quantum Annealing

Explanation of Quantum Annealing

### 2020/01/14

#### Tuesday Seminar on Topology

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

SO(3)-invariant G

**Ryohei Chihara**(The University of Tokyo)SO(3)-invariant G

_{2}-geometry (JAPANESE)
[ Abstract ]

Berger's classification of holonomy groups of Riemannian manifolds includes exceptional cases of the Lie groups G

Berger's classification of holonomy groups of Riemannian manifolds includes exceptional cases of the Lie groups G

_{2}and Spin(7). Many authors have studied G_{2}- and Spin(7)-manifolds with torus symmetry. In this talk, we generalize the celebrated examples due to Bryant and Salamon and study G_{2}-manifolds with SO(3)-symmetry. Such torsion-free G_{2}-structures are described as a dynamical system of SU(3)-structures on an SO(3)-fibration over a 3-manifold. As a main result, we reduce this system into a constrained Hamiltonian dynamical system on the cotangent bundle over the space of all Riemannian metrics on the 3-manifold. The Hamiltonian function is very similar to that of the Hamiltonian formulation of general relativity.#### Tuesday Seminar on Topology

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

Algebraic entropy of sign-stable mutation loops (JAPANESE)

**Tsukasa Ishibashi**(The University of Tokyo)Algebraic entropy of sign-stable mutation loops (JAPANESE)

[ Abstract ]

Since its discovery, the cluster algebra has been developed with friutful connections with other branches of mathematics, unifying several combinatorial operations as well as their positivity notions. A mutation loop induces several dynamical systems via cluster transformations, and they form a group which can be seen as a combinatorial generalization of the mapping class groups of marked surfaces.

We introduce a new property of mutation loops called the sign stability, with a focus on an asymptotic behavior of the iteration of the tropicalized cluster X-transformation. A sign-stable mutation loop has a numerical invariant which we call the "cluster stretch factor", in analogy with the stretch factor of a pseudo-Anosov mapping class on a marked surface. We compute the algebraic entropies of the cluster A- and X-transformations induced by a sign-stable mutation loop, and conclude that these two coincide with the logarithm of the cluster stretch factor. This talk is based on a joint work with Shunsuke Kano.

Since its discovery, the cluster algebra has been developed with friutful connections with other branches of mathematics, unifying several combinatorial operations as well as their positivity notions. A mutation loop induces several dynamical systems via cluster transformations, and they form a group which can be seen as a combinatorial generalization of the mapping class groups of marked surfaces.

We introduce a new property of mutation loops called the sign stability, with a focus on an asymptotic behavior of the iteration of the tropicalized cluster X-transformation. A sign-stable mutation loop has a numerical invariant which we call the "cluster stretch factor", in analogy with the stretch factor of a pseudo-Anosov mapping class on a marked surface. We compute the algebraic entropies of the cluster A- and X-transformations induced by a sign-stable mutation loop, and conclude that these two coincide with the logarithm of the cluster stretch factor. This talk is based on a joint work with Shunsuke Kano.

#### Tuesday Seminar of Analysis

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

Scattering near a two-cluster threshold (English)

**Erik Skibsted**(Aarhus University)Scattering near a two-cluster threshold (English)

[ Abstract ]

For a one-body Schr\"odinger operator with an attractive slowly decaying potential the scattering matrix is well-defined at the energy zero, and the structure of its singularities is well-studied. The usual (non-relativistic) model for the Hydrogen atom is a particular example of such Schr\"odinger operator.

Less is known on scattering at a two-cluster threshold of an $N$-body Schr\"odinger operator for which the effective interaction between the two bound clusters is attractive Coulombic. An example of interest is scattering at a two-cluster threshold of a neutral atom/molecule. We present results of an ongoing joint work with X.P. Wang on the subject, including a version of the Sommerfeld uniqueness result and its applications.

We shall also present general results on spectral theory at a two-cluster threshold (not requiring the effective interaction to be attractive Coulombic). This includes a general structure theorem on the bound and resonance states at the threshold as well as a resolvent expansion in weighted spaces above the threshold (under more restrictive conditions). Applications to scattering theory will be indicated.

For a one-body Schr\"odinger operator with an attractive slowly decaying potential the scattering matrix is well-defined at the energy zero, and the structure of its singularities is well-studied. The usual (non-relativistic) model for the Hydrogen atom is a particular example of such Schr\"odinger operator.

Less is known on scattering at a two-cluster threshold of an $N$-body Schr\"odinger operator for which the effective interaction between the two bound clusters is attractive Coulombic. An example of interest is scattering at a two-cluster threshold of a neutral atom/molecule. We present results of an ongoing joint work with X.P. Wang on the subject, including a version of the Sommerfeld uniqueness result and its applications.

We shall also present general results on spectral theory at a two-cluster threshold (not requiring the effective interaction to be attractive Coulombic). This includes a general structure theorem on the bound and resonance states at the threshold as well as a resolvent expansion in weighted spaces above the threshold (under more restrictive conditions). Applications to scattering theory will be indicated.

### 2020/01/10

#### Lectures

10:00-11:30 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.

### 2020/01/09

#### Information Mathematics Seminar

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

Innovation and business administration of the manufacturing industry by AI/IoT (Japanese)

**Hiroshi Fujiwara**(BroadBand Tower, Inc.)Innovation and business administration of the manufacturing industry by AI/IoT (Japanese)

[ Abstract ]

Explanation on business administration of the manufacturing industry by AI/IoT

Explanation on business administration of the manufacturing industry by AI/IoT

#### Lectures

14:00-17:30 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.

#### 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.

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