## Seminar information archive

Seminar information archive ～10/21｜Today's seminar 10/22 | Future seminars 10/23～

#### PDE Real Analysis Seminar

10:30-11:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Optimal isoperimetric inequalities for surfaces in any codimension

in Cartan-Hadamard manifolds (English)

**Felix Schulze**(University College London)Optimal isoperimetric inequalities for surfaces in any codimension

in Cartan-Hadamard manifolds (English)

[ Abstract ]

Let $(M^n,g)$ be simply connected, complete, with non-positive sectional

curvatures, and $\Sigma$ a 2-dimensional surface in $M^n$. Let $S$ be an area

minimising 3-current such that $\partial S = \Sigma$. We use a weak mean

curvature flow, obtained via elliptic regularisation, starting from

$\Sigma$, to show that $S$ satisfies the optimal Euclidean isoperimetric

inequality: $|S| \leq 1/(6\sqrt{\pi}) |\Sigma|^{3/2}$. We also obtain the

optimal estimate in case the sectional curvatures of $M$ are bounded from

above by $\kappa < 0$ and characterise the case of equality. The proof

follows from an almost monotonicity of a suitable isoperimetric

difference along the approximating flows in one dimension higher.

Let $(M^n,g)$ be simply connected, complete, with non-positive sectional

curvatures, and $\Sigma$ a 2-dimensional surface in $M^n$. Let $S$ be an area

minimising 3-current such that $\partial S = \Sigma$. We use a weak mean

curvature flow, obtained via elliptic regularisation, starting from

$\Sigma$, to show that $S$ satisfies the optimal Euclidean isoperimetric

inequality: $|S| \leq 1/(6\sqrt{\pi}) |\Sigma|^{3/2}$. We also obtain the

optimal estimate in case the sectional curvatures of $M$ are bounded from

above by $\kappa < 0$ and characterise the case of equality. The proof

follows from an almost monotonicity of a suitable isoperimetric

difference along the approximating flows in one dimension higher.

#### Algebraic Geometry Seminar

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

Orbifold rational connectedness (English)

**Frédéric Campana**(Université de Lorraine/KIAS)Orbifold rational connectedness (English)

[ Abstract ]

The first step in the decomposition by canonical fibrations with fibres of `signed' canonical bundle of an arbitrary complex projective manifolds $X$ is its `rational quotient' (also called `MRC' fibration): it has rationally connected fibres and non-uniruled base. In general, the further steps (such as the Moishezon-Iitaka fibration) of this decomposition will require the consideration of 'orbifold base' of fibrations in order to deal with the multiple fibres (as seen already for elliptic surfaces). One thus needs to work in the larger category of (smooth) `orbifold pairs' $(X,D)$ to achieve this decomposition. The aim of the talk is thus to introduce the notions of Rational Connectedness and 'rational quotient' in this context, by means of suitable equivalent notions of negativity for the orbifold cotangent bundle (suitably defined. When $D$ is reduced, this is just the usual Log-version). The expected equivalence with connecting families of `orbifold rational curves' remains however presently open.

The first step in the decomposition by canonical fibrations with fibres of `signed' canonical bundle of an arbitrary complex projective manifolds $X$ is its `rational quotient' (also called `MRC' fibration): it has rationally connected fibres and non-uniruled base. In general, the further steps (such as the Moishezon-Iitaka fibration) of this decomposition will require the consideration of 'orbifold base' of fibrations in order to deal with the multiple fibres (as seen already for elliptic surfaces). One thus needs to work in the larger category of (smooth) `orbifold pairs' $(X,D)$ to achieve this decomposition. The aim of the talk is thus to introduce the notions of Rational Connectedness and 'rational quotient' in this context, by means of suitable equivalent notions of negativity for the orbifold cotangent bundle (suitably defined. When $D$ is reduced, this is just the usual Log-version). The expected equivalence with connecting families of `orbifold rational curves' remains however presently open.

### 2017/11/20

#### Seminar on Geometric Complex Analysis

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

Relative GIT stabilities of toric Fano manifolds in low dimensions

**Yasufumi Nitta**(Tokyo Institute of Technology)Relative GIT stabilities of toric Fano manifolds in low dimensions

[ Abstract ]

In 2000, Mabuchi extended the notion of Kaehler-Einstein metrics to Fano manifolds with non-vanishing Futaki invariant. Such a metric is called generalized Kaehler-Einstein metric or Mabuchi metric in the literature. Recently this metrics were rediscovered by Yao in the story of Donaldson's infinite dimensional moment map picture. Moreover, he introduced (uniform) relative Ding stability for toric Fano manifolds and showed that the existence of generalized Kaehler-Einstein metrics is equivalent to its uniform relative Ding stability. This equivalence is in the context of the Yau-Tian-Donaldson conjecture. In this talk, we focus on uniform relative Ding stability of toric Fano manifolds. More precisely, we determine all the uniformly relatively Ding stable toric Fano 3- and 4-folds as well as unstable ones. This talk is based on a joint work with Shunsuke Saito and Naoto Yotsutani.

In 2000, Mabuchi extended the notion of Kaehler-Einstein metrics to Fano manifolds with non-vanishing Futaki invariant. Such a metric is called generalized Kaehler-Einstein metric or Mabuchi metric in the literature. Recently this metrics were rediscovered by Yao in the story of Donaldson's infinite dimensional moment map picture. Moreover, he introduced (uniform) relative Ding stability for toric Fano manifolds and showed that the existence of generalized Kaehler-Einstein metrics is equivalent to its uniform relative Ding stability. This equivalence is in the context of the Yau-Tian-Donaldson conjecture. In this talk, we focus on uniform relative Ding stability of toric Fano manifolds. More precisely, we determine all the uniformly relatively Ding stable toric Fano 3- and 4-folds as well as unstable ones. This talk is based on a joint work with Shunsuke Saito and Naoto Yotsutani.

### 2017/11/16

#### Mathematical Biology Seminar

16:30-18:00 Room #123 (Graduate School of Math. Sci. Bldg.)

(JAPANESE)

**Jun Nakabayashi**(Yokohama City University)(JAPANESE)

#### Seminar on Probability and Statistics

13：00-16：00 Room #123 (Graduate School of Math. Sci. Bldg.)

### 2017/11/15

#### PDE Real Analysis Seminar

10:30-11:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Boundary value problems for parabolic equations with measurable coefficients (English)

**Kaj Nyström**(Uppsala University)Boundary value problems for parabolic equations with measurable coefficients (English)

[ Abstract ]

In recent joint works with P. Auscher and M. Egert we establish new results concerning boundary value problems in the upper half-space for second order parabolic equations (and systems) assuming only measurability and some transversal regularity in the coefficients of the elliptic part. To establish our results we introduce and develop a first order strategy by means of a parabolic Dirac operator at the boundary to obtain, in particular, Green's representation for solutions in natural classes involving square functions and non-tangential maximal functions, well-posedness results with data in $L^2$-Sobolev spaces together with invertibility of layer potentials, and perturbation results. In addition we solve the Kato square root problem for parabolic operators with coefficients of the elliptic part depending measurably on all variables. Using these results we are also able to solve the $L^p$-Dirichlet problem for parabolic equations with real, time-dependent, elliptic but non-symmetric coefficients. In this talk I will briefly describe some of these developments.

In recent joint works with P. Auscher and M. Egert we establish new results concerning boundary value problems in the upper half-space for second order parabolic equations (and systems) assuming only measurability and some transversal regularity in the coefficients of the elliptic part. To establish our results we introduce and develop a first order strategy by means of a parabolic Dirac operator at the boundary to obtain, in particular, Green's representation for solutions in natural classes involving square functions and non-tangential maximal functions, well-posedness results with data in $L^2$-Sobolev spaces together with invertibility of layer potentials, and perturbation results. In addition we solve the Kato square root problem for parabolic operators with coefficients of the elliptic part depending measurably on all variables. Using these results we are also able to solve the $L^p$-Dirichlet problem for parabolic equations with real, time-dependent, elliptic but non-symmetric coefficients. In this talk I will briefly describe some of these developments.

### 2017/11/14

#### Algebraic Geometry Seminar

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

A characterization of the birationality of 4-canonical maps of minimal 3-folds (English)

**Meng Chen**(Fudan)A characterization of the birationality of 4-canonical maps of minimal 3-folds (English)

[ Abstract ]

We explain the following theorem: For any minimal 3-fold X of general type with p_g>4, the 4-canonical map is non-birational if and only if X is birationally fibred by a pencil of (1,2) surfaces. The statement fails in the case of p_g=4.

We explain the following theorem: For any minimal 3-fold X of general type with p_g>4, the 4-canonical map is non-birational if and only if X is birationally fibred by a pencil of (1,2) surfaces. The statement fails in the case of p_g=4.

#### Numerical Analysis Seminar

16:50-18:20 Room #002 (Graduate School of Math. Sci. Bldg.)

Energy-preserving numerical method based on the variational principle and application to unconstrained optimization problems (Japanese)

**Ai Ishikawa**(Kobe University)Energy-preserving numerical method based on the variational principle and application to unconstrained optimization problems (Japanese)

### 2017/11/13

#### Tokyo Probability Seminar

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

(JAPANESE)

**Masaki Wada**(Faculty of Human Development and Culture, Fukushima University)(JAPANESE)

#### Seminar on Geometric Complex Analysis

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

Relative Canonical Bundles for Families of Calabi-Yau Manifolds

**Georg Schumacher**(Philipps-Universität Marburg)Relative Canonical Bundles for Families of Calabi-Yau Manifolds

[ Abstract ]

We consider holomorphic families of Calabi-Yau manifolds (here being defined by the vanishing of the first real Chern class). We study induced hermitian metrics on the relative canonical bundle, which are related to the Weil-Petersson form on the base. Under a certain condition the total space possesses a Kähler form, whose restriction to fibers are equal to the Ricci flat metrics. Furthermore we prove an extension theorem for the Weil-Petersson form and give applications.

We consider holomorphic families of Calabi-Yau manifolds (here being defined by the vanishing of the first real Chern class). We study induced hermitian metrics on the relative canonical bundle, which are related to the Weil-Petersson form on the base. Under a certain condition the total space possesses a Kähler form, whose restriction to fibers are equal to the Ricci flat metrics. Furthermore we prove an extension theorem for the Weil-Petersson form and give applications.

#### Operator Algebra Seminars

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

Globularily generated double categories: On the problem of existence of internalizations for decorated bicategories (English)

**Juan Orendain**(UNAM)Globularily generated double categories: On the problem of existence of internalizations for decorated bicategories (English)

### 2017/11/10

#### Infinite Analysis Seminar Tokyo

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

Chern-Simons, gravity and integrable systems. (ENGLISH)

http://www.iip.ufrn.br/eventslecturer?inf==0EVRpXTR1TP

**Fabio Novaes**(International Institute of Physics (UFRN))Chern-Simons, gravity and integrable systems. (ENGLISH)

[ Abstract ]

It is known since the 80's that pure three-dimensional gravity is classically equivalent to a Chern-Simons theory with gauge group SL(2,R) x SL(2,R). For a given set of boundary conditions, the asymptotic classical phase space has a central extension in terms of two copies of Virasoro algebra. In particular, this gives a conformal field theory representation of black hole solutions in 3d gravity, also known as BTZ black holes. The BTZ black hole entropy can then be recovered using CFT. In this talk, we review this story and discuss recent results on how to relax the BTZ boundary conditions to obtain the KdV hierarchy at the boundary. More generally, this shows that Chern-Simons theory can represent virtually any integrable system at the boundary, given some consistency conditions. We also briefly discuss how this formulation can be useful to describe non-relativistic systems.

[ Reference URL ]It is known since the 80's that pure three-dimensional gravity is classically equivalent to a Chern-Simons theory with gauge group SL(2,R) x SL(2,R). For a given set of boundary conditions, the asymptotic classical phase space has a central extension in terms of two copies of Virasoro algebra. In particular, this gives a conformal field theory representation of black hole solutions in 3d gravity, also known as BTZ black holes. The BTZ black hole entropy can then be recovered using CFT. In this talk, we review this story and discuss recent results on how to relax the BTZ boundary conditions to obtain the KdV hierarchy at the boundary. More generally, this shows that Chern-Simons theory can represent virtually any integrable system at the boundary, given some consistency conditions. We also briefly discuss how this formulation can be useful to describe non-relativistic systems.

http://www.iip.ufrn.br/eventslecturer?inf==0EVRpXTR1TP

### 2017/11/09

#### Kavli IPMU Komaba Seminar

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

Various applications of supersymmetry in statistical physics (English)

**Edouard Brezin**(lpt ens, Paris)Various applications of supersymmetry in statistical physics (English)

[ Abstract ]

Supersymmetry is a fundamental concept in particle physics (although it has not been seen experimentally so far). But it is although a powerful tool in a number of problems arising in quantum mechanics and statistical physics. It has been widely used in the theory of disordered systems (Efetov et al.), it led to dimensional reduction for branched polymers (Parisi-Sourlas), for the susy classical gas (Brydges and Imbrie), for Landau levels with impurities. If has also many powerful applications in the theory of random matrices. I will briefly review some of these topics.

Supersymmetry is a fundamental concept in particle physics (although it has not been seen experimentally so far). But it is although a powerful tool in a number of problems arising in quantum mechanics and statistical physics. It has been widely used in the theory of disordered systems (Efetov et al.), it led to dimensional reduction for branched polymers (Parisi-Sourlas), for the susy classical gas (Brydges and Imbrie), for Landau levels with impurities. If has also many powerful applications in the theory of random matrices. I will briefly review some of these topics.

### 2017/11/08

#### Number Theory Seminar

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

Iwasawa theory and Bloch-Kato conjecture for modular forms (ENGLISH)

**Xin Wan**(Morningside Center for Mathematics)Iwasawa theory and Bloch-Kato conjecture for modular forms (ENGLISH)

[ Abstract ]

Bloch and Kato formulated conjectures relating sizes of p-adic Selmer groups with special values of L-functions. Iwasawa theory is a useful tool for studying these conjectures and BSD conjecture for elliptic curves. For example the Iwasawa main conjecture for modular forms formulated by Kato implies the Tamagawa number formula for modular forms of analytic rank 0.

In this talk I'll first briefly review the above theory. Then we will focus on a different Iwasawa theory approach for this problem. The starting point is a recent joint work with Jetchev and Skinner proving the BSD formula for elliptic curves of analytic rank 1. We will discuss how such results are generalized to modular forms. If time allowed we may also explain the possibility to use it to deduce Bloch-Kato conjectures in both analytic rank 0 and 1 cases. In certain aspects such approach should be more powerful than classical Iwasawa theory, and has some potential to attack cases with bad ramification at p.

Bloch and Kato formulated conjectures relating sizes of p-adic Selmer groups with special values of L-functions. Iwasawa theory is a useful tool for studying these conjectures and BSD conjecture for elliptic curves. For example the Iwasawa main conjecture for modular forms formulated by Kato implies the Tamagawa number formula for modular forms of analytic rank 0.

In this talk I'll first briefly review the above theory. Then we will focus on a different Iwasawa theory approach for this problem. The starting point is a recent joint work with Jetchev and Skinner proving the BSD formula for elliptic curves of analytic rank 1. We will discuss how such results are generalized to modular forms. If time allowed we may also explain the possibility to use it to deduce Bloch-Kato conjectures in both analytic rank 0 and 1 cases. In certain aspects such approach should be more powerful than classical Iwasawa theory, and has some potential to attack cases with bad ramification at p.

### 2017/11/07

#### Tuesday Seminar on Topology

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

On an explicit example of topologically protected corner states (JAPANESE)

**Shin Hayashi**(AIST-TohokuU MathAM-OIL)On an explicit example of topologically protected corner states (JAPANESE)

[ Abstract ]

In condensed matter physics, topologically protected (codimension-one) edge states are known to appear on the surface of some insulators reflecting some topology of its bulk. Such phenomena can be understood from the point of view of an index theory associated to the Toeplitz extension and are called the bulk-edge correspondence. In this talk, we consider instead the quarter-plane Toeplitz extension and index theory associated with it. As a result, we show that topologically protected (codimension-two) corner states appear reflecting some topology of the gapped bulk and two edges. Such new topological phases can be obtained by taking a ``product’’ of two classically known topological phases (2d type A and 1d type AIII topological phases). By using this construction, we obtain an example of a continuous family of bounded self-adjoint Fredholm quarter-plane Toeplitz operators whose spectral flow is nontrivial, which gives an explicit example of topologically protected corner states.

In condensed matter physics, topologically protected (codimension-one) edge states are known to appear on the surface of some insulators reflecting some topology of its bulk. Such phenomena can be understood from the point of view of an index theory associated to the Toeplitz extension and are called the bulk-edge correspondence. In this talk, we consider instead the quarter-plane Toeplitz extension and index theory associated with it. As a result, we show that topologically protected (codimension-two) corner states appear reflecting some topology of the gapped bulk and two edges. Such new topological phases can be obtained by taking a ``product’’ of two classically known topological phases (2d type A and 1d type AIII topological phases). By using this construction, we obtain an example of a continuous family of bounded self-adjoint Fredholm quarter-plane Toeplitz operators whose spectral flow is nontrivial, which gives an explicit example of topologically protected corner states.

#### Algebraic Geometry Seminar

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

Characterizations of projective space and Seshadri constants in arbitrary characteristic

**Takumi Murayama**(University of Michigan)Characterizations of projective space and Seshadri constants in arbitrary characteristic

[ Abstract ]

Mori and Mukai conjectured that projective space should be the only n-dimensional Fano variety whose anti-canonical bundle has degree at least n + 1 along every curve. While this conjecture has been proved in characteristic zero, it remains open in positive characteristic. We will present some progress in this direction by giving another characterization of projective space using Seshadri constants and the Frobenius morphism. The key ingredient is a positive-characteristic analogue of Demailly’s criterion for separation of higher-order jets by adjoint bundles, whose proof gives new results for adjoint bundles even in characteristic zero.

Mori and Mukai conjectured that projective space should be the only n-dimensional Fano variety whose anti-canonical bundle has degree at least n + 1 along every curve. While this conjecture has been proved in characteristic zero, it remains open in positive characteristic. We will present some progress in this direction by giving another characterization of projective space using Seshadri constants and the Frobenius morphism. The key ingredient is a positive-characteristic analogue of Demailly’s criterion for separation of higher-order jets by adjoint bundles, whose proof gives new results for adjoint bundles even in characteristic zero.

### 2017/11/06

#### FMSP Lectures

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

Phaseless inverse problems for Maxwell equations (ENGLISH)

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Romanov2.pdf

**V. G. Romanov**(Sobolev Institute of Mathematics)Phaseless inverse problems for Maxwell equations (ENGLISH)

[ Abstract ]

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Romanov2.pdf

[ Reference URL ]http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Romanov2.pdf

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Romanov2.pdf

### 2017/11/02

#### Seminar on Probability and Statistics

14:00-15:10 Room #052 (Graduate School of Math. Sci. Bldg.)

Hermite processes and sheets

**Tudor Ciprian**(Université Lille 1)Hermite processes and sheets

[ Abstract ]

The Hermite process of order $\geq 1$ is a self-similar stochastic process with stationary increments living in the $q$th Wiener chaos. The class of Hermite processes includes the fractional Brownian motion (for $q=1$) and the Rosenblatt process (for $q=2$). We present the basic properties of these processes and we introduce their multiparameter version. We also discuss the behavior with respect to the self-similarity index and the possibility so solve stochastic equations with Hermite noise.

The Hermite process of order $\geq 1$ is a self-similar stochastic process with stationary increments living in the $q$th Wiener chaos. The class of Hermite processes includes the fractional Brownian motion (for $q=1$) and the Rosenblatt process (for $q=2$). We present the basic properties of these processes and we introduce their multiparameter version. We also discuss the behavior with respect to the self-similarity index and the possibility so solve stochastic equations with Hermite noise.

### 2017/11/01

#### Discrete mathematical modelling seminar

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

Discrete Painlevé equations associated with the E8 group (ENGLISH)

**Basile Grammaticos**(Université de Paris VII・XI)Discrete Painlevé equations associated with the E8 group (ENGLISH)

[ Abstract ]

I'll present a summary of the results of the Paris-Tokyo-Pondicherry group on equations associated with the affine Weyl group E8. I shall review the various parametrisations of the E8-related equations, introducing the trihomographic representation and the ancillary variable. Several examples of E8-associated equations will be given including what we believe is the simplest form for the generic elliptic discrete Painlevé equation.

I'll present a summary of the results of the Paris-Tokyo-Pondicherry group on equations associated with the affine Weyl group E8. I shall review the various parametrisations of the E8-related equations, introducing the trihomographic representation and the ancillary variable. Several examples of E8-associated equations will be given including what we believe is the simplest form for the generic elliptic discrete Painlevé equation.

### 2017/10/31

#### Tuesday Seminar on Topology

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

Nonamenable groups of piecewise projective homeomorphisms (ENGLISH)

**Yash Lodha**(École Polytechnique Fédérale de Lausanne)Nonamenable groups of piecewise projective homeomorphisms (ENGLISH)

[ Abstract ]

Groups of piecewise projective homeomorphisms provide elegant examples of groups that are non amenable, yet do not contain non abelian free subgroups. In this talk I will present a survey of these groups and discuss their striking properties. I will discuss properties such as (non)amenability, finiteness properties, normal subgroup structure, actions by various degrees of regularity and Tarski numbers.

Groups of piecewise projective homeomorphisms provide elegant examples of groups that are non amenable, yet do not contain non abelian free subgroups. In this talk I will present a survey of these groups and discuss their striking properties. I will discuss properties such as (non)amenability, finiteness properties, normal subgroup structure, actions by various degrees of regularity and Tarski numbers.

#### Algebraic Geometry Seminar

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

ACC for log canonical threshold polytopes (English)

**Zhan Li**(Beijing)ACC for log canonical threshold polytopes (English)

[ Abstract ]

We show that the log canonical threshold polytopes of varieties with log canonical singularities satisfy the ascending chain condition. This is a joint work with Jingjun Han and Lu Qi.

We show that the log canonical threshold polytopes of varieties with log canonical singularities satisfy the ascending chain condition. This is a joint work with Jingjun Han and Lu Qi.

#### Discrete mathematical modelling seminar

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

The end of the World (ENGLISH)

**Basile Grammaticos**(Université de Paris VII・XI)The end of the World (ENGLISH)

[ Abstract ]

This is not a seminar on astrophysics or cosmology. I am not going to talk about something that will happen in billions of years. I will rather explain the menace to our civilisation and to the human species. Inspired from the works of several authors I will explain the existing risks. I will also present mathematical models which show that a general collapse is possible in the decades that follow.

This is not a seminar on astrophysics or cosmology. I am not going to talk about something that will happen in billions of years. I will rather explain the menace to our civilisation and to the human species. Inspired from the works of several authors I will explain the existing risks. I will also present mathematical models which show that a general collapse is possible in the decades that follow.

#### Discrete mathematical modelling seminar

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

A hierarchy of approximate regularity of functions (ENGLISH)

**Fon-Che Liu**(National Taiwan University)A hierarchy of approximate regularity of functions (ENGLISH)

[ Abstract ]

A hierarchy of a certain weakly sensed regularity of functions defined on subsets of Euclidean n-space which originated from the well-known Lusin theorem that characterizes measurable functions in terms of approximate continuity will be introduced. Its intimate relations with the ordinary hierarchy of regularity in terms of order of continuous differentiability will be exposed and explained.

A hierarchy of a certain weakly sensed regularity of functions defined on subsets of Euclidean n-space which originated from the well-known Lusin theorem that characterizes measurable functions in terms of approximate continuity will be introduced. Its intimate relations with the ordinary hierarchy of regularity in terms of order of continuous differentiability will be exposed and explained.

#### FMSP Lectures

16:00-17:00 Room #118 (Graduate School of Math. Sci. Bldg.)

Some Geometric Aspects in Inverse Problems (ENGLISH)

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

**V. G. Romanov**(Sobolev Institute of Mathematics)Some Geometric Aspects in Inverse Problems (ENGLISH)

[ Abstract ]

We consider inverse problems related to recovering coefficients in partial differential equations of the second order. It is supposed that some measurements of solutions to direct problems are produced on convenient sets. A study of some inverse problems for hyperbolic equations leads to geometric problems: recovering a function from its integrals along geodesic lines of the Riemannian metric or recovering the Riemannian metric inside a domain from given distances between arbitrary points of the domain boundary. Our main goal here is to demonstrate how such geometric problems arise for equations of parabolic and elliptic types.

[ Reference URL ]We consider inverse problems related to recovering coefficients in partial differential equations of the second order. It is supposed that some measurements of solutions to direct problems are produced on convenient sets. A study of some inverse problems for hyperbolic equations leads to geometric problems: recovering a function from its integrals along geodesic lines of the Riemannian metric or recovering the Riemannian metric inside a domain from given distances between arbitrary points of the domain boundary. Our main goal here is to demonstrate how such geometric problems arise for equations of parabolic and elliptic types.

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

### 2017/10/30

#### Tokyo Probability Seminar

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

Integration of controlled rough paths via fractional calculus (JAPANESE)

**Yu Ito**(Department of Mathematics, Faculty of Science, Kyoto Sangyo University)Integration of controlled rough paths via fractional calculus (JAPANESE)

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