## Seminar information archive

Seminar information archive ～09/10｜Today's seminar 09/11 | Future seminars 09/12～

### 2018/03/30

#### Tuesday Seminar on Topology

15:00-16:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Graph Complexes and the Kashiwara-Vergne Lie algebra (ENGLISH)

**Matteo Felder**(University of Geneva)Graph Complexes and the Kashiwara-Vergne Lie algebra (ENGLISH)

[ Abstract ]

The Kashiwara-Vergne Lie algebra krv was introduced by A. Alekseev and C. Torossian. It describes the symmetries of the Kashiwara-Vergne problem in Lie theory. It has been shown to contain the Grothendieck-Teichmüller Lie algebra grt as a Lie subalgebra. Conjecturally, these two Lie algebras are expected to be isomorphic. An important theorem by T. Willwacher states that in degree zero the cohomology of M. Kontsevich's graph complex GC is isomorphic to grt. We will show how T. Willwacher's result induces a natural way to define a nested sequence of Lie subalgebras of krv whose intersection is grt. This infinite family therefore interpolates between the two Lie algebras. For this we will recall several techniques from the theory of graph complexes. If time permits, we will then sketch how one might generalize these notions to establish a "genus one" analogue of T. Willwacher theorem. More precisely, we will define a chain complex whose degree zero cohomology is given by a Lie subalgebra of the elliptic Grothendieck-Teichmüller Lie algebra introduced by B. Enriquez. The last part is joint work in progress with T. Willwacher.

The Kashiwara-Vergne Lie algebra krv was introduced by A. Alekseev and C. Torossian. It describes the symmetries of the Kashiwara-Vergne problem in Lie theory. It has been shown to contain the Grothendieck-Teichmüller Lie algebra grt as a Lie subalgebra. Conjecturally, these two Lie algebras are expected to be isomorphic. An important theorem by T. Willwacher states that in degree zero the cohomology of M. Kontsevich's graph complex GC is isomorphic to grt. We will show how T. Willwacher's result induces a natural way to define a nested sequence of Lie subalgebras of krv whose intersection is grt. This infinite family therefore interpolates between the two Lie algebras. For this we will recall several techniques from the theory of graph complexes. If time permits, we will then sketch how one might generalize these notions to establish a "genus one" analogue of T. Willwacher theorem. More precisely, we will define a chain complex whose degree zero cohomology is given by a Lie subalgebra of the elliptic Grothendieck-Teichmüller Lie algebra introduced by B. Enriquez. The last part is joint work in progress with T. Willwacher.

#### Tuesday Seminar on Topology

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

Goldman-Turaev formality in genus 0 from the KZ connection (ENGLISH)

**Florian Naef**(Massachusetts Institute of Technology)Goldman-Turaev formality in genus 0 from the KZ connection (ENGLISH)

[ Abstract ]

Using the intersection and self-intersection of loops on a surface one can define the Goldman-Turaev Lie bialgebra. By earlier joint work with A. Alekseev, N. Kawazumi and Y. Kuno it is known that the linearization problem of the Goldman-Turaev Lie bialgebra is closely related to the Kashiwara-Vergne problem and hence to Drinfeld associators. It turns out that in the case of a genus 0 surface and over the field of complex numbers there is a very direct and explicit proof of the formality of the Goldman-Turaev Lie bialgebra using the monodromy of the Knizhnik-Zamolodchikov connection. This is joint work with Anton Alekseev.

Using the intersection and self-intersection of loops on a surface one can define the Goldman-Turaev Lie bialgebra. By earlier joint work with A. Alekseev, N. Kawazumi and Y. Kuno it is known that the linearization problem of the Goldman-Turaev Lie bialgebra is closely related to the Kashiwara-Vergne problem and hence to Drinfeld associators. It turns out that in the case of a genus 0 surface and over the field of complex numbers there is a very direct and explicit proof of the formality of the Goldman-Turaev Lie bialgebra using the monodromy of the Knizhnik-Zamolodchikov connection. This is joint work with Anton Alekseev.

### 2018/03/27

#### Infinite Analysis Seminar Tokyo

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

Schramm-Loewner evolutions and Liouville field theory (JAPANESE)

**Yoshiki Fukusumi**(The University of Tokyo, The Institute for Solid State Physics)Schramm-Loewner evolutions and Liouville field theory (JAPANESE)

[ Abstract ]

Schramm-Loewner evolutions (SLEs) are stochastic processes driven by Brownian motions which preserves conformal invariance. They describe the cluster boundaries associated with the minimal models of the conformal field theory, including the Ising model and the percolation as typical examples. The correlation functions of such models remarkably satisfy the martingale condition. We briefly review some known results. Then we analyse the time reversing procedure of Schramm Loewner evolutions and its relation to Liouville field theory or 2d pure gravity. We can get martingale observables by the calculation of the correlation functions of Liouville field theory without matter.

Schramm-Loewner evolutions (SLEs) are stochastic processes driven by Brownian motions which preserves conformal invariance. They describe the cluster boundaries associated with the minimal models of the conformal field theory, including the Ising model and the percolation as typical examples. The correlation functions of such models remarkably satisfy the martingale condition. We briefly review some known results. Then we analyse the time reversing procedure of Schramm Loewner evolutions and its relation to Liouville field theory or 2d pure gravity. We can get martingale observables by the calculation of the correlation functions of Liouville field theory without matter.

### 2018/03/26

#### FMSP Lectures

10:00-12:00 Room #002 (Graduate School of Math. Sci. Bldg.)

Geometric Recursion (ENGLISH)

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

**Jørgen Ellegaard Andersen**(Aarhus University)Geometric Recursion (ENGLISH)

[ Abstract ]

Geometric Recursion is a very general machinery for constructing mapping class group invariants objects associated to two dimensional surfaces. After presenting the general abstract definition we shall see how a number of constructions in low dimensional geometry and topology fits into this setting. These will include the Mirzakhani-McShane identies, mapping class group invariant closed forms on Teichmüller space (including the Weil-Petterson symplectic form) and the Goldman symplectic form on moduli spaces of flat connections for general compact simple Lie groups. We shall also discuss the process which establishes that any application of Topological Recursion can be lifted to a Geometric Recursion setting involving continuous functions on Teichmüller space, where the connection back to Topological Recursion is obtained by integration over the moduli space of curve. The work

presented is joint with G. Borot and N. Orantin.

[ Reference URL ]Geometric Recursion is a very general machinery for constructing mapping class group invariants objects associated to two dimensional surfaces. After presenting the general abstract definition we shall see how a number of constructions in low dimensional geometry and topology fits into this setting. These will include the Mirzakhani-McShane identies, mapping class group invariant closed forms on Teichmüller space (including the Weil-Petterson symplectic form) and the Goldman symplectic form on moduli spaces of flat connections for general compact simple Lie groups. We shall also discuss the process which establishes that any application of Topological Recursion can be lifted to a Geometric Recursion setting involving continuous functions on Teichmüller space, where the connection back to Topological Recursion is obtained by integration over the moduli space of curve. The work

presented is joint with G. Borot and N. Orantin.

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

### 2018/03/23

#### FMSP Lectures

10:00-12:00 Room #002 (Graduate School of Math. Sci. Bldg.)

Geometric Recursion (ENGLISH)

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

**Jørgen Ellegaard Andersen**(Aarhus University)Geometric Recursion (ENGLISH)

[ Abstract ]

Geometric Recursion is a very general machinery for constructing mapping class group invariants objects associated to two dimensional surfaces. After presenting the general abstract definition we shall see how a number of constructions in low dimensional geometry and topology fits into this setting. These will include the

Mirzakhani-McShane identies, mapping class group invariant closed forms on Teichmüller space (including the Weil-Petterson symplectic form) and the Goldman symplectic form on moduli spaces of flat connections for general compact simple Lie groups. We shall also discuss the process which establishes that any application of Topological Recursion can be lifted to a Geometric Recursion setting involving continuous functions on Teichmüller space, where the connection back to Topological Recursion is obtained by integration over the moduli space of curve. The work presented is joint with G. Borot and N. Orantin.

[ Reference URL ]Geometric Recursion is a very general machinery for constructing mapping class group invariants objects associated to two dimensional surfaces. After presenting the general abstract definition we shall see how a number of constructions in low dimensional geometry and topology fits into this setting. These will include the

Mirzakhani-McShane identies, mapping class group invariant closed forms on Teichmüller space (including the Weil-Petterson symplectic form) and the Goldman symplectic form on moduli spaces of flat connections for general compact simple Lie groups. We shall also discuss the process which establishes that any application of Topological Recursion can be lifted to a Geometric Recursion setting involving continuous functions on Teichmüller space, where the connection back to Topological Recursion is obtained by integration over the moduli space of curve. The work presented is joint with G. Borot and N. Orantin.

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

### 2018/03/19

#### Mathematical Biology Seminar

17:00-18:00 Room #509 (Graduate School of Math. Sci. Bldg.)

Large deviation theory for age-structured population dynamics

**Yuki Sugiyama**(Institute of Industrial Science, The University of Tokyo)Large deviation theory for age-structured population dynamics

[ Abstract ]

Control of population growth is ubiquitous problem in many fields. In the context of medical treatment, we attempt to diminish the growing speed of a cell population composed of cancer cells or pathogens by using antibiotics or some special therapies. In terms of evolutional biology, to survive in a fluctuating environment, cells maximize (optimize) their population growth by exploiting a risk hedge strategy for adaptation to the fluctuation. Recent development of experimental devices enables us to measure a big size lineage data that describes a growing cell population. In this study, by using these lineage data, we analyze a behavior of the population growth. Here, a structure of statistical mechanics using the large deviation theory plays an important role. As a results, we reveal that the population growth rate is given by the Legendre transform of the large deviation function for the semi-Markov process that describes a stochastic switch of cell types in the time evolution. Furthermore, by using this structure, we show that responses of the population growth rate with respect to an environmental change can be evaluated by statistics on a retrospective lineage path.

Control of population growth is ubiquitous problem in many fields. In the context of medical treatment, we attempt to diminish the growing speed of a cell population composed of cancer cells or pathogens by using antibiotics or some special therapies. In terms of evolutional biology, to survive in a fluctuating environment, cells maximize (optimize) their population growth by exploiting a risk hedge strategy for adaptation to the fluctuation. Recent development of experimental devices enables us to measure a big size lineage data that describes a growing cell population. In this study, by using these lineage data, we analyze a behavior of the population growth. Here, a structure of statistical mechanics using the large deviation theory plays an important role. As a results, we reveal that the population growth rate is given by the Legendre transform of the large deviation function for the semi-Markov process that describes a stochastic switch of cell types in the time evolution. Furthermore, by using this structure, we show that responses of the population growth rate with respect to an environmental change can be evaluated by statistics on a retrospective lineage path.

### 2018/03/15

#### Seminar on Probability and Statistics

16:00-17:10 Room #052 (Graduate School of Math. Sci. Bldg.)

On Hypotheses testing for discretely observed SDE (Joint work with Alessandro De Gregorio, University of Rome)

**Stefano Iacus**(University of Milan)On Hypotheses testing for discretely observed SDE (Joint work with Alessandro De Gregorio, University of Rome)

[ Abstract ]

In this talk we consider parametric hypotheses testing for discretely observed ergodic diffusion processes. We present the different test statistics proposed in literature and recall their asymptotic properties. We also compare the empirical performance of different tests in the case of small sample sizes.

In this talk we consider parametric hypotheses testing for discretely observed ergodic diffusion processes. We present the different test statistics proposed in literature and recall their asymptotic properties. We also compare the empirical performance of different tests in the case of small sample sizes.

### 2018/03/13

#### Lectures

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

### 2018/03/12

#### Lie Groups and Representation Theory

15:00-16:30 Room #126 (Graduate School of Math. Sci. Bldg.)

Plethysms and Kronecker coefficients in geometric complexity theory

**Christian Ikenmeyer**(Max-Planck-Institut fur Informatik)Plethysms and Kronecker coefficients in geometric complexity theory

[ Abstract ]

Research on Kronecker coefficients and plethysms gained significant momentum when the topics were connected with geometric complexity theory, an approach towards computational complexity lower bounds via algebraic geometry and representation theory. This talk is about several recent results that were obtained with geometric complexity theory as motivation, namely the NP-hardness of deciding the positivity of Kronecker coefficients and an inequality between rectangular Kronecker coefficients and plethysm coefficients. While the proof of the former statement is mainly combinatorial, the proof of the latter statement interestingly uses insights from algebraic complexity theory. As far as we know algebraic complexity theory has never been used before to prove an inequality between representation theoretic multiplicities.

Research on Kronecker coefficients and plethysms gained significant momentum when the topics were connected with geometric complexity theory, an approach towards computational complexity lower bounds via algebraic geometry and representation theory. This talk is about several recent results that were obtained with geometric complexity theory as motivation, namely the NP-hardness of deciding the positivity of Kronecker coefficients and an inequality between rectangular Kronecker coefficients and plethysm coefficients. While the proof of the former statement is mainly combinatorial, the proof of the latter statement interestingly uses insights from algebraic complexity theory. As far as we know algebraic complexity theory has never been used before to prove an inequality between representation theoretic multiplicities.

### 2018/03/10

#### Colloquium

11:00-12:00 Room #大講義室 (Graduate School of Math. Sci. Bldg.)

(JAPANESE)

**Hitoshi ARAI**(Univ. Tokyo)(JAPANESE)

#### Colloquium

13:00-14:00 Room #大講義室 (Graduate School of Math. Sci. Bldg.)

(JAPANESE)

**Akito FUTAKI**(Univ. Tokyo)(JAPANESE)

#### Colloquium

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

(JAPANESE)

**Yujiro KAWAMATA**(Univ. Tokyo)(JAPANESE)

#### Colloquium

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

(JAPANESE)

**Hiroshi MATANO**(Univ. Tokyo)(JAPANESE)

### 2018/03/09

#### Lectures

13:30-14:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Sliced nearby cycles and duality, after W. Zheng (ENGLISH)

**Luc Illusie**Sliced nearby cycles and duality, after W. Zheng (ENGLISH)

[ Abstract ]

In the early 1980's Gabber proved duality for nearby cycles and, by a different method, Beilinson proved duality for vanishing cycles in the strictly local case (up to a twist of the inertia action on the tame part). Recently W. Zheng found a simple proof of a result, conjectured by Deligne, which implies them both, and extended it over finite dimensional excellent bases. I will explain the main ideas of his work, which relies on new developments, due to him, of Deligne's theory of fibered and oriented products.

In the early 1980's Gabber proved duality for nearby cycles and, by a different method, Beilinson proved duality for vanishing cycles in the strictly local case (up to a twist of the inertia action on the tame part). Recently W. Zheng found a simple proof of a result, conjectured by Deligne, which implies them both, and extended it over finite dimensional excellent bases. I will explain the main ideas of his work, which relies on new developments, due to him, of Deligne's theory of fibered and oriented products.

### 2018/03/02

#### Seminar on Probability and Statistics

15:00-16:10 Room #270 (Graduate School of Math. Sci. Bldg.)

"Estimating functions for SDE driven by stable Lévy processes"

Joint work with Emmanuelle Clément (Ecole Centrale)

**Arnaud Gloter**(Université d'Evry Val d'Essonne)"Estimating functions for SDE driven by stable Lévy processes"

Joint work with Emmanuelle Clément (Ecole Centrale)

[ Abstract ]

In this talk we will discuss about parametric inference for a stochastic differential equation driven by a pure-jump Lévy process, based on high frequency observations on a fixed time period. Assuming that the Lévy measure of the driving process behaves like that of an α-stable process around zero, we propose an estimating functions based method which leads to asymptotically efficient estimators for any value of α ∈ (0, 2).

In this talk we will discuss about parametric inference for a stochastic differential equation driven by a pure-jump Lévy process, based on high frequency observations on a fixed time period. Assuming that the Lévy measure of the driving process behaves like that of an α-stable process around zero, we propose an estimating functions based method which leads to asymptotically efficient estimators for any value of α ∈ (0, 2).

### 2018/02/23

#### Colloquium

15:30-16:30 Room #002 (Graduate School of Math. Sci. Bldg.)

(JAPANESE)

**Hiromu Tanaka**(Univ. Tokyo)(JAPANESE)

#### FMSP Lectures

13:30-15:00 Room #002 (Graduate School of Math. Sci. Bldg.)

The topology of singular points of real analytic curves (ENGLISH)

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

**Etienne Ghys**(ENS de Lyon)The topology of singular points of real analytic curves (ENGLISH)

[ Abstract ]

In the neighborhood of a singular point, a germ of real analytic curve in the plane consists of a finite number of branches. Each of these branches intersects a small circle around the singular point in two points. Therefore, the local topology is described by a chord diagram : an even number of points on a circle paired two by two. Not all chord diagrams come from a singular point. The main purpose of this mini course is to give an complete description of those ‘’analytic ? chord diagrams. On our way, we shall meet some interesting concepts from computer science, graph theory and operads.

[ Reference URL ]In the neighborhood of a singular point, a germ of real analytic curve in the plane consists of a finite number of branches. Each of these branches intersects a small circle around the singular point in two points. Therefore, the local topology is described by a chord diagram : an even number of points on a circle paired two by two. Not all chord diagrams come from a singular point. The main purpose of this mini course is to give an complete description of those ‘’analytic ? chord diagrams. On our way, we shall meet some interesting concepts from computer science, graph theory and operads.

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

### 2018/02/22

#### FMSP Lectures

15:00-16:30 Room #117 (Graduate School of Math. Sci. Bldg.)

The topology of singular points of real analytic curves (ENGLISH)

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

**Etienne Ghys**(ENS de Lyon)The topology of singular points of real analytic curves (ENGLISH)

[ Abstract ]

In the neighborhood of a singular point, a germ of real analytic curve in the plane consists of a finite number of branches. Each of these branches intersects a small circle around the singular point in two points. Therefore, the local topology is described by a chord diagram : an even number of points on a circle paired two by two. Not all chord diagrams come from a singular point. The main purpose of this mini course is to give an complete description of those ‘’analytic ? chord diagrams. On our way, we shall meet some interesting concepts from computer science, graph theory and operads.

[ Reference URL ]In the neighborhood of a singular point, a germ of real analytic curve in the plane consists of a finite number of branches. Each of these branches intersects a small circle around the singular point in two points. Therefore, the local topology is described by a chord diagram : an even number of points on a circle paired two by two. Not all chord diagrams come from a singular point. The main purpose of this mini course is to give an complete description of those ‘’analytic ? chord diagrams. On our way, we shall meet some interesting concepts from computer science, graph theory and operads.

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

### 2018/02/21

#### FMSP Lectures

15:00-16:30 Room #117 (Graduate School of Math. Sci. Bldg.)

The topology of singular points of real analytic curves (ENGLISH)

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

**Etienne Ghys**(ENS de Lyon)The topology of singular points of real analytic curves (ENGLISH)

[ Abstract ]

In the neighborhood of a singular point, a germ of real analytic curve in the plane consists of a finite number of branches. Each of these branches intersects a small circle around the singular point in two points. Therefore, the local topology is described by a chord diagram : an even number of points on a circle paired two by two. Not all chord diagrams come from a singular point. The main purpose of this mini course is to give an complete description of those ‘’analytic ? chord diagrams. On our way, we shall meet some interesting concepts from computer science, graph theory and operads.

[ Reference URL ]In the neighborhood of a singular point, a germ of real analytic curve in the plane consists of a finite number of branches. Each of these branches intersects a small circle around the singular point in two points. Therefore, the local topology is described by a chord diagram : an even number of points on a circle paired two by two. Not all chord diagrams come from a singular point. The main purpose of this mini course is to give an complete description of those ‘’analytic ? chord diagrams. On our way, we shall meet some interesting concepts from computer science, graph theory and operads.

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

#### Tuesday Seminar on Topology

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

The category of bottom tangles in handlebodies, and the Kontsevich integral (ENGLISH)

**Gwénaël Massuyeau**(Université de Bourgogne)The category of bottom tangles in handlebodies, and the Kontsevich integral (ENGLISH)

[ Abstract ]

Habiro introduced the category B of « bottom tangles in handlebodies », which encapsulates the set of knots in the 3-sphere as well as the mapping class groups of 3-dimensional handlebodies. There is a natural filtration on the category B defined using an appropriate generalization of Vassiliev invariants. In this talk, we will show that the completion of B with respect to the Vassiliev filtration is isomorphic to a certain category A which can be defined either in a combinatorial way using « Jacobi diagrams », or by a universal property via the notion of « Casimir Hopf algebra ». Such an isomorphism will be obtained by extending the Kontsevich integral (originally defined as a knot invariant) to a functor Z from B to A. This functor Z can be regarded as a refinement of the TQFT-like functor derived from the LMO invariant and, if time allows, we will evoke the topological interpretation of the « tree-level » of Z. (This is based on joint works with Kazuo Habiro.)

Habiro introduced the category B of « bottom tangles in handlebodies », which encapsulates the set of knots in the 3-sphere as well as the mapping class groups of 3-dimensional handlebodies. There is a natural filtration on the category B defined using an appropriate generalization of Vassiliev invariants. In this talk, we will show that the completion of B with respect to the Vassiliev filtration is isomorphic to a certain category A which can be defined either in a combinatorial way using « Jacobi diagrams », or by a universal property via the notion of « Casimir Hopf algebra ». Such an isomorphism will be obtained by extending the Kontsevich integral (originally defined as a knot invariant) to a functor Z from B to A. This functor Z can be regarded as a refinement of the TQFT-like functor derived from the LMO invariant and, if time allows, we will evoke the topological interpretation of the « tree-level » of Z. (This is based on joint works with Kazuo Habiro.)

### 2018/02/19

#### Numerical Analysis Seminar

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

Existence, multiplicity, and orbital stability for travelling waves in a nonlinearly supported beam (English)

**Michael Plum**(Karlsruhe Insitute of Technology)Existence, multiplicity, and orbital stability for travelling waves in a nonlinearly supported beam (English)

[ Abstract ]

For a nonlinear beam equation with exponential nonlinearity, we prove existence of at least 36 travelling wave solutions for the specific wave speed c=1.3. Our proof makes heavy use of computer assistance: starting from numerical approximations, we use a fixed point argument to prove existence of solutions "close to" the approximate ones. Furthermore we investigate the orbital stability of these solutions by making use of both analytical and computer-assisted techniques.

For a nonlinear beam equation with exponential nonlinearity, we prove existence of at least 36 travelling wave solutions for the specific wave speed c=1.3. Our proof makes heavy use of computer assistance: starting from numerical approximations, we use a fixed point argument to prove existence of solutions "close to" the approximate ones. Furthermore we investigate the orbital stability of these solutions by making use of both analytical and computer-assisted techniques.

#### Numerical Analysis Seminar

16:15-17:15 Room #056 (Graduate School of Math. Sci. Bldg.)

An approach to computer-assisted existence proofs for nonlinear space-time fractional parabolic problems (English)

**Kaori Nagatou**(Karlsruhe Insitute of Technology)An approach to computer-assisted existence proofs for nonlinear space-time fractional parabolic problems (English)

[ Abstract ]

We consider an initial boundary value problem for a space-time fractional parabolic equation, which includes the fractional Laplacian, i.e. a nonlocal operator. We treat a corresponding local problem which is obtained by the Caffarelli-Silvestre extension technique, and show how to enclose a solution of the extended problem by computer-assisted means.

We consider an initial boundary value problem for a space-time fractional parabolic equation, which includes the fractional Laplacian, i.e. a nonlocal operator. We treat a corresponding local problem which is obtained by the Caffarelli-Silvestre extension technique, and show how to enclose a solution of the extended problem by computer-assisted means.

### 2018/02/14

#### Operator Algebra Seminars

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

Index theory on the Mishchenko bundle (English)

**Valerio Proietti**(Copenhagen Univ.)Index theory on the Mishchenko bundle (English)

### 2018/02/06

#### Infinite Analysis Seminar Tokyo

15:00-17:30 Room #002 (Graduate School of Math. Sci. Bldg.)

Sine-square deformation of one-dimensional critical systems (ENGLISH)

Modular invariant representations of the $N=2$ vertex operator superalgebra (ENGLISH)

**Hosho Katsura**(Department of Physics, Graduate School of Science, The Univeristy of Tokyo ) 15:00-16:00Sine-square deformation of one-dimensional critical systems (ENGLISH)

[ Abstract ]

Sine-square deformation (SSD) is one example of smooth boundary conditions that have significantly smaller finite-size effects than open boundary conditions. In a one-dimensional system with SSD, the interaction strength varies smoothly from the center to the edges according to the sine-square function. This means that the Hamiltonian of the system is inhomogeneous, as it lacks translational symmetry. Nevertheless, previous studies have revealed that the SSD leaves the ground state of the uniform chain with periodic boundary conditions (PBC) almost unchanged for critical systems. In particular, I showed in [1,2,3] that the correspondence is exact for critical XY and quantum Ising chains. The same correspondence between SSD and PBC holds for Dirac fermions in 1+1 dimension and a family of more general conformal field theories. If time permits, I will also introduce more recent results [4,5] and discuss the excited states of the SSD systems.

[1] H. Katsura, J. Phys. A: Math. Theor. 44, 252001 (2011).

[2] H. Katsura, J. Phys. A: Math. Theor. 45, 115003 (2012).

[3] I. Maruyama, H. Katsura, T. Hikihara, Phys. Rev. B 84, 165132 (2011).

[4] K. Okunishi and H. Katsura, J. Phys. A: Math. Theor. 48, 445208 (2015).

[5] S. Tamura and H. Katsura, Prog. Theor. Exp. Phys 2017, 113A01 (2017).

Sine-square deformation (SSD) is one example of smooth boundary conditions that have significantly smaller finite-size effects than open boundary conditions. In a one-dimensional system with SSD, the interaction strength varies smoothly from the center to the edges according to the sine-square function. This means that the Hamiltonian of the system is inhomogeneous, as it lacks translational symmetry. Nevertheless, previous studies have revealed that the SSD leaves the ground state of the uniform chain with periodic boundary conditions (PBC) almost unchanged for critical systems. In particular, I showed in [1,2,3] that the correspondence is exact for critical XY and quantum Ising chains. The same correspondence between SSD and PBC holds for Dirac fermions in 1+1 dimension and a family of more general conformal field theories. If time permits, I will also introduce more recent results [4,5] and discuss the excited states of the SSD systems.

[1] H. Katsura, J. Phys. A: Math. Theor. 44, 252001 (2011).

[2] H. Katsura, J. Phys. A: Math. Theor. 45, 115003 (2012).

[3] I. Maruyama, H. Katsura, T. Hikihara, Phys. Rev. B 84, 165132 (2011).

[4] K. Okunishi and H. Katsura, J. Phys. A: Math. Theor. 48, 445208 (2015).

[5] S. Tamura and H. Katsura, Prog. Theor. Exp. Phys 2017, 113A01 (2017).

**Ryo Sato**(Graduate School of Mathematical Sciences, The University of Tokyo) 16:30-17:30Modular invariant representations of the $N=2$ vertex operator superalgebra (ENGLISH)

[ Abstract ]

One of the most remarkable features in representation theory of a (``good'') vertex operator superalgebra (VOSA) is the modular invariance property of the characters. As an application of the property, M. Wakimoto and D. Adamovic proved that all the fusion rules for the simple $N=2$ VOSA of central charge $c_{p,1}=3(1-2/p)$ are computed from the modular $S$-matrix by the so-called Verlinde formula. In this talk, we present a new ``modular invariant'' family of irreducible highest weight modules over the simple $N=2$ VOSA of central charge $c_{p,p'}:=3(1-2p'/p)$. Here $(p,p')$ is a pair of coprime integers such that $p,p'>1$. In addition, we will discuss some generalization of the Verlinde formula in the spirit of Creutzig--Ridout.

One of the most remarkable features in representation theory of a (``good'') vertex operator superalgebra (VOSA) is the modular invariance property of the characters. As an application of the property, M. Wakimoto and D. Adamovic proved that all the fusion rules for the simple $N=2$ VOSA of central charge $c_{p,1}=3(1-2/p)$ are computed from the modular $S$-matrix by the so-called Verlinde formula. In this talk, we present a new ``modular invariant'' family of irreducible highest weight modules over the simple $N=2$ VOSA of central charge $c_{p,p'}:=3(1-2p'/p)$. Here $(p,p')$ is a pair of coprime integers such that $p,p'>1$. In addition, we will discuss some generalization of the Verlinde formula in the spirit of Creutzig--Ridout.

### 2018/02/04

#### Seminar on Probability and Statistics

12:00-18:00 Room #118 (Graduate School of Math. Sci. Bldg.)

Nonlinear Economic Time Series Models

Analytic, representation and statistical aspects related to fractional Gaussian processes.

**Yukai Yang**(Uppsala University) 12:30-15:00Nonlinear Economic Time Series Models

[ Abstract ]

The lecture goes through several chapters in the book “Modelling Nonlinear Economic Time Series” by Teräsvirta, Tjøstheim and Granger in 2010. The lecture serves as an introduction for the students and researchers who are interested in this area. It introduces a number of examples of families of nonlinear time series parametric models in economic theory. It also talks about testing linearity against parametric alternatives with the presence of a characterization of the identification problem in many situations. Different ways of solving the identification problem are presented and their merits and disadvantages are discussed.

The lecture goes through several chapters in the book “Modelling Nonlinear Economic Time Series” by Teräsvirta, Tjøstheim and Granger in 2010. The lecture serves as an introduction for the students and researchers who are interested in this area. It introduces a number of examples of families of nonlinear time series parametric models in economic theory. It also talks about testing linearity against parametric alternatives with the presence of a characterization of the identification problem in many situations. Different ways of solving the identification problem are presented and their merits and disadvantages are discussed.

**Yuliia Mishura**(The Taras Shevchenko National University of Kiev ) 15:30-18:00Analytic, representation and statistical aspects related to fractional Gaussian processes.

[ Abstract ]

We consider the properties of fractional Gaussian processes whose covariance function is situated between two self-similarities, or, in other words, these processes belong to the generalized quasi-helix, according to geometric terminology of Kahane. For such processes we consider the two-sided bounds for maximal functionals and the representation results. We consider stochastic differential equations involving fractional Brownian motion and present also several results on statistical estimations for them.

We consider the properties of fractional Gaussian processes whose covariance function is situated between two self-similarities, or, in other words, these processes belong to the generalized quasi-helix, according to geometric terminology of Kahane. For such processes we consider the two-sided bounds for maximal functionals and the representation results. We consider stochastic differential equations involving fractional Brownian motion and present also several results on statistical estimations for them.

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