## Seminar information archive

Seminar information archive ～02/01｜Today's seminar 02/02 | Future seminars 02/03～

#### FMSP Lectures

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

Discrete regularization of parameter identification problems (ENGLISH)

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

**Christian Clason**(University Duisburg-Essen)Discrete regularization of parameter identification problems (ENGLISH)

[ Abstract ]

This talk is concerned with parameter identification problems where a distributed parameter is known a priori to take on values from a given set. This property can be promoted with the aid of a convex regularization term in the Tikhonov functional. We discuss the properties of minimizers of this functional and their numerical computation using a semismooth Newton method.

[ Reference URL ]This talk is concerned with parameter identification problems where a distributed parameter is known a priori to take on values from a given set. This property can be promoted with the aid of a convex regularization term in the Tikhonov functional. We discuss the properties of minimizers of this functional and their numerical computation using a semismooth Newton method.

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

### 2016/08/10

#### FMSP Lectures

10:00-11:00 Room #370 (Graduate School of Math. Sci. Bldg.)

Global-local-integration-based kernel approximation methods: Technical arguments (ENGLISH)

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Hon2.pdf

**Benny Y C Hon**(City Univ. of Hong Kong)Global-local-integration-based kernel approximation methods: Technical arguments (ENGLISH)

[ Abstract ]

We discuss technical details of my talk on 8 Aug. and give also proofs of some main results.

[ Reference URL ]We discuss technical details of my talk on 8 Aug. and give also proofs of some main results.

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Hon2.pdf

#### Seminar on Probability and Statistics

13:00-14:30 Room #117 (Graduate School of Math. Sci. Bldg.)

**David Nualart**(Kansas University)### 2016/08/09

#### Seminar on Probability and Statistics

13:00-16:30 Room #117 (Graduate School of Math. Sci. Bldg.)

Malliavin calculus and normal approximations

http://www2.ms.u-tokyo.ac.jp/probstat/?page_id=180

**David Nualart**(Kansas University)Malliavin calculus and normal approximations

[ Abstract ]

The purpose of these lectures is to introduce some recent results on the application of Malliavin calculus combined with Stein's method to normal approximation. The Malliavin calculus is a differential calculus on the Wiener space. First, we will present some elements of Malliavin calculus, defining the basic differential operators: the derivative, its adjoint called the divergence operator and the generator of the Ornstein-Uhlenbeck semigroup. The behavior of these operators on the Wiener chaos expansion will be discussed. Then, we will introduce the Stein's method for normal approximation, which leads to general bounds for the Kolmogorov and total variation distances between the law of a Brownian functional and the standard normal distribution. In this context, the integration by parts formula of Malliavin calculus will allow us to express these bounds in terms of the Malliavin operators. We will present the application of this methodology to derive the Fourth Moment Theorem for a sequence of multiple stochastic integrals, and we will discuss some results on the uniform convergence of densities obtained using Malliavin calculus techniques. Finally, examples of functionals of Gaussian processes, such as the fractional Brownian motion, will be discussed.

[ Reference URL ]The purpose of these lectures is to introduce some recent results on the application of Malliavin calculus combined with Stein's method to normal approximation. The Malliavin calculus is a differential calculus on the Wiener space. First, we will present some elements of Malliavin calculus, defining the basic differential operators: the derivative, its adjoint called the divergence operator and the generator of the Ornstein-Uhlenbeck semigroup. The behavior of these operators on the Wiener chaos expansion will be discussed. Then, we will introduce the Stein's method for normal approximation, which leads to general bounds for the Kolmogorov and total variation distances between the law of a Brownian functional and the standard normal distribution. In this context, the integration by parts formula of Malliavin calculus will allow us to express these bounds in terms of the Malliavin operators. We will present the application of this methodology to derive the Fourth Moment Theorem for a sequence of multiple stochastic integrals, and we will discuss some results on the uniform convergence of densities obtained using Malliavin calculus techniques. Finally, examples of functionals of Gaussian processes, such as the fractional Brownian motion, will be discussed.

http://www2.ms.u-tokyo.ac.jp/probstat/?page_id=180

### 2016/08/08

#### FMSP Lectures

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

Global-local-integration-based kernel approximation methods (ENGLISH)

[ Reference URL ]

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

**Benny Y C Hon**(City Univ. of Hong Kong)Global-local-integration-based kernel approximation methods (ENGLISH)

[ Reference URL ]

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

#### FMSP Lectures

17:30-18:30 Room #128 (Graduate School of Math. Sci. Bldg.)

On the lifting of deterministic convergence results for inverse problems to the stochastic setting (ENGLISH)

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

**Daniel Gerth**(Tech. Univ. Chemnitz)On the lifting of deterministic convergence results for inverse problems to the stochastic setting (ENGLISH)

[ Abstract ]

In inverse problems, the inevitable measurement noise is modelled either by a deterministic worst-case model or a stochastic one.

The development of convergence theory in both approaches appears to be rather disconnected. In this talk we seek to bridge this gap and show how deterministic result can be transferred into the stochastic setting. The talk is split into two parts. In the first part, after briefly introducing "inverse problems" and the noise models, we examine the particular problem of sparsity-promoting regularization with a Besov-space penalty term to demonstrate the lifting technique. In the second part, we present a generalization of the technique that applies to a large group of regularization methods.

[ Reference URL ]In inverse problems, the inevitable measurement noise is modelled either by a deterministic worst-case model or a stochastic one.

The development of convergence theory in both approaches appears to be rather disconnected. In this talk we seek to bridge this gap and show how deterministic result can be transferred into the stochastic setting. The talk is split into two parts. In the first part, after briefly introducing "inverse problems" and the noise models, we examine the particular problem of sparsity-promoting regularization with a Besov-space penalty term to demonstrate the lifting technique. In the second part, we present a generalization of the technique that applies to a large group of regularization methods.

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

### 2016/08/06

#### Seminar on Probability and Statistics

10:00-17:10 Room #123 (Graduate School of Math. Sci. Bldg.)

Asymptotic expansion of variations

LAMN property and optimal estimation for diffusion with non synchronous observations

Approximation schemes for stochastic differential equations driven by a fractional Brownian motion

Parameter estimation for fractional Ornstein-Uhlenbeck processes

Stein's equations for invariant measures of diffusions processes and their applications via Malliavin calculus

Asymptotic expansion of a nonlinear oscillator with a jump diffusion

[ Reference URL ]

http://www2.ms.u-tokyo.ac.jp/probstat/?page_id=179

**Nakahiro Yoshida**(University of Tokyo, Institute of Statistical Mathematics, and JST CREST) 10:00-10:50Asymptotic expansion of variations

**Teppei Ogihara**(The Institute of Statistical Mathematics, JST PRESTO, and JST CREST) 11:00-11:50LAMN property and optimal estimation for diffusion with non synchronous observations

**David Nualart**(Kansas University) 13:10-14:00Approximation schemes for stochastic differential equations driven by a fractional Brownian motion

**David Nualart**(Kansas University) 14:10-15:00Parameter estimation for fractional Ornstein-Uhlenbeck processes

**Seiichiro Kusuoka**(Okayama University) 15:20-16:10Stein's equations for invariant measures of diffusions processes and their applications via Malliavin calculus

**Yasushi Ishikawa**(Ehime University) 16:20-17:10Asymptotic expansion of a nonlinear oscillator with a jump diffusion

[ Reference URL ]

http://www2.ms.u-tokyo.ac.jp/probstat/?page_id=179

### 2016/07/28

#### thesis presentations

15:00-16:15 Room #118 (Graduate School of Math. Sci. Bldg.)

Calabi-Yau 3-folds in Grassmannians and their I-functions （グラスマン多様体に含まれる3 次元カラビ・ヤウ多様体とそれらのI-関数）

(JAPANESE)

**井上 大輔**(東京大学大学院数理科学研究科)Calabi-Yau 3-folds in Grassmannians and their I-functions （グラスマン多様体に含まれる3 次元カラビ・ヤウ多様体とそれらのI-関数）

(JAPANESE)

### 2016/07/27

#### Mathematical Biology Seminar

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

The ecological dynamics of non-polio enteroviruses: Case studies from China and Japan (ENGLISH)

**Saki Takahashi**(Princeton University)The ecological dynamics of non-polio enteroviruses: Case studies from China and Japan (ENGLISH)

[ Abstract ]

As we approach global eradication of poliovirus (Enterovirus C species), its relatives are rapidly emerging as public health threats. One of these viruses, Enterovirus A71 (EV-A71), has been implicated in large outbreaks of hand, foot, and mouth disease (HFMD), a childhood illness that has had a substantial burden throughout East and Southeast Asia over the past fifteen years. HFMD is typically a self-limiting disease, but a small proportion of EV-A71 infections lead to the development of neurological and systemic complications that can be fatal. EV-A71 also exhibits puzzling spatial characteristics: the virus circulates at low levels worldwide, but has so far been endemic and associated with severe disease exclusively in Asia. In this talk, I will present findings from a recent study that we did to characterize the transmission dynamics of HFMD in China, where over one million cases are reported each year. I will then describe recent efforts to explain the observed multi-annual cyclicity of EV-A71 incidence in Japan and to probe the contributions of other serotypes to the observed burden of HFMD. In closing, I will discuss plans for unifying data and modeling to study this heterogeneity in the endemicity of EV-A71, as well as to broadly better understand the spatial and viral dynamics of this group of infections.

As we approach global eradication of poliovirus (Enterovirus C species), its relatives are rapidly emerging as public health threats. One of these viruses, Enterovirus A71 (EV-A71), has been implicated in large outbreaks of hand, foot, and mouth disease (HFMD), a childhood illness that has had a substantial burden throughout East and Southeast Asia over the past fifteen years. HFMD is typically a self-limiting disease, but a small proportion of EV-A71 infections lead to the development of neurological and systemic complications that can be fatal. EV-A71 also exhibits puzzling spatial characteristics: the virus circulates at low levels worldwide, but has so far been endemic and associated with severe disease exclusively in Asia. In this talk, I will present findings from a recent study that we did to characterize the transmission dynamics of HFMD in China, where over one million cases are reported each year. I will then describe recent efforts to explain the observed multi-annual cyclicity of EV-A71 incidence in Japan and to probe the contributions of other serotypes to the observed burden of HFMD. In closing, I will discuss plans for unifying data and modeling to study this heterogeneity in the endemicity of EV-A71, as well as to broadly better understand the spatial and viral dynamics of this group of infections.

### 2016/07/26

#### Seminar on Probability and Statistics

13:00-14:30 Room #052 (Graduate School of Math. Sci. Bldg.)

Multilevel Particle Filters

**Ajay Jasra**(National University of Singapore)Multilevel Particle Filters

[ Abstract ]

In this talk the filtering of partially observed diffusions,

with discrete-time observations, is considered.

It is assumed that only biased approximations of the diffusion can be

obtained, for choice of an accuracy parameter indexed by $l$.

A multilevel estimator is proposed, consisting of a telescopic sum of

increment estimators associated to the successive levels.

The work associated to $\cO(\varepsilon^2)$ mean-square error between

the multilevel estimator and average with respect to the filtering

distribution is shown to scale optimally, for example as

$\cO(\varepsilon^{-2})$ for optimal rates of convergence of the

underlying diffusion approximation.

The method is illustrated on several examples.

In this talk the filtering of partially observed diffusions,

with discrete-time observations, is considered.

It is assumed that only biased approximations of the diffusion can be

obtained, for choice of an accuracy parameter indexed by $l$.

A multilevel estimator is proposed, consisting of a telescopic sum of

increment estimators associated to the successive levels.

The work associated to $\cO(\varepsilon^2)$ mean-square error between

the multilevel estimator and average with respect to the filtering

distribution is shown to scale optimally, for example as

$\cO(\varepsilon^{-2})$ for optimal rates of convergence of the

underlying diffusion approximation.

The method is illustrated on several examples.

### 2016/07/25

#### Algebraic Geometry Seminar

13:30-15:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Birational rigidity of complete intersections (English)

**Fumiaki Suzuki**(Tokyo)Birational rigidity of complete intersections (English)

[ Abstract ]

A complete intersection defined by s hypersurfaces of degree d_1, ... ,d_s in a projective space P^N is Q-Fano, i.e. normal, Q-factorial, terminal and having an ample anti-canonical divisor, if d_1 + ... + d_s is at most N and it has only mild singularities. Then it is rationally-connected by the results of Kollar-Miyaoka-Mori, Zhang and Hacon-Mckernan. A natural question is to determine its rationality. If its dimension or degree is at most 2, then it is rational. How about the remaining cases?

When d_1 + ... + d_s = N, birational rigidity give one of the most effective ways to tackle this problem. We recall that a Q-Fano variety is birationally superrigid if any birational map to the source of another Mori fiber space is isomorphism. It implies that X is non-rational and Bir(X) = Aut(X). After the works of Iskovskih-Manin, Pukhlikov, Chelt'so and de Fernex-Ein-Mustata, de Fernex proved that every smooth hypersurface of degree N in P^N is birationally superrigid for N at least 4. He also proved birational superrigidity of a large class of singular hypersurfaces of this type.

In this talk, we would like to extend de Fernex's results to complete intersections. As a key step, we generalize Pukhlikov's multiplicity bounds of cycles in hypersurfaces to complete intersections.

A complete intersection defined by s hypersurfaces of degree d_1, ... ,d_s in a projective space P^N is Q-Fano, i.e. normal, Q-factorial, terminal and having an ample anti-canonical divisor, if d_1 + ... + d_s is at most N and it has only mild singularities. Then it is rationally-connected by the results of Kollar-Miyaoka-Mori, Zhang and Hacon-Mckernan. A natural question is to determine its rationality. If its dimension or degree is at most 2, then it is rational. How about the remaining cases?

When d_1 + ... + d_s = N, birational rigidity give one of the most effective ways to tackle this problem. We recall that a Q-Fano variety is birationally superrigid if any birational map to the source of another Mori fiber space is isomorphism. It implies that X is non-rational and Bir(X) = Aut(X). After the works of Iskovskih-Manin, Pukhlikov, Chelt'so and de Fernex-Ein-Mustata, de Fernex proved that every smooth hypersurface of degree N in P^N is birationally superrigid for N at least 4. He also proved birational superrigidity of a large class of singular hypersurfaces of this type.

In this talk, we would like to extend de Fernex's results to complete intersections. As a key step, we generalize Pukhlikov's multiplicity bounds of cycles in hypersurfaces to complete intersections.

#### Tokyo Probability Seminar

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

Intermittent property of parabolic stochastic partial differential equations

**Bin Xie**(Department of Mathematical Sciences, Faculty of Science, Shinshu University)Intermittent property of parabolic stochastic partial differential equations

#### Algebraic Geometry Seminar

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

On the geometry of thin exceptional sets in Manin’s conjecture

**Sho Tanimoto**(University of Copenhagen)On the geometry of thin exceptional sets in Manin’s conjecture

[ Abstract ]

Manin’s conjecture predicts the asymptotic formula for the counting function of rational points on a Fano variety X after removing the exceptional sets. The original conjecture, which removes a proper closed subset, is wrong due to covering families of subvarieties violating the compatibility of Manin’s conjecture, and its refinement, suggested by Emmanuel Peyre, removes a thin set instead of a closed set. In this talk, first I would like to explain that subvarieties which conjecturally have more points than X only form a thin set using the minimal model program and the boundedness of log Fano varieties. After that, I would like to discuss our conjecture on the birational boundedness of covers violating the compatibility of Manin’s conjecture, and present some results in dimension 2 and 3. This is joint work with Brian Lehmann.

Manin’s conjecture predicts the asymptotic formula for the counting function of rational points on a Fano variety X after removing the exceptional sets. The original conjecture, which removes a proper closed subset, is wrong due to covering families of subvarieties violating the compatibility of Manin’s conjecture, and its refinement, suggested by Emmanuel Peyre, removes a thin set instead of a closed set. In this talk, first I would like to explain that subvarieties which conjecturally have more points than X only form a thin set using the minimal model program and the boundedness of log Fano varieties. After that, I would like to discuss our conjecture on the birational boundedness of covers violating the compatibility of Manin’s conjecture, and present some results in dimension 2 and 3. This is joint work with Brian Lehmann.

### 2016/07/22

#### Operator Algebra Seminars

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

Radius of comparison for $C^*$ crossed products by free minimal actions of amenable groups

**N. Christopher Phillips**(Univ. Oregon)Radius of comparison for $C^*$ crossed products by free minimal actions of amenable groups

### 2016/07/19

#### Tuesday Seminar on Topology

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

The geometry of the curve graphs and beyond (JAPANESE)

**Yohsuke Watanabe**(University of Hawaii)The geometry of the curve graphs and beyond (JAPANESE)

[ Abstract ]

The curve graphs are locally infinite. However, by using Masur-Minsky's tight geodesics, one could view them as locally finite graphs. Bell-Fujiwara used a special property of tight geodesics and showed that the asymptotic dimension of the curve graphs is finite. In this talk, I will introduce a new class of geodesics which also has the property. If time permits, I will explain how such geodesics can be adapted in Out(F_n) setting.

The curve graphs are locally infinite. However, by using Masur-Minsky's tight geodesics, one could view them as locally finite graphs. Bell-Fujiwara used a special property of tight geodesics and showed that the asymptotic dimension of the curve graphs is finite. In this talk, I will introduce a new class of geodesics which also has the property. If time permits, I will explain how such geodesics can be adapted in Out(F_n) setting.

#### thesis presentations

13:00-14:15 Room #126 (Graduate School of Math. Sci. Bldg.)

Cube invariance of higher Chow groups with modulus （モジュラス付き高次チャウ群のキューブ不変性)

(JAPANESE)

**宮﨑 弘安**(東京大学大学院数理科学研究科)Cube invariance of higher Chow groups with modulus （モジュラス付き高次チャウ群のキューブ不変性)

(JAPANESE)

### 2016/07/13

#### Colloquium of mathematical sciences and society

17:00-18:30 Room #002 (Graduate School of Math. Sci. Bldg.)

Research on information theory and artificial intelligence based on mathematics

(JAPANESE)

[ Reference URL ]

http://fmsp.ms.u-tokyo.ac.jp/FMSP_colloquium20160713.pdf

**Toshio Ito**(Fujitsu Laboratories LTD.)Research on information theory and artificial intelligence based on mathematics

(JAPANESE)

[ Reference URL ]

http://fmsp.ms.u-tokyo.ac.jp/FMSP_colloquium20160713.pdf

#### Mathematical Biology Seminar

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

Dual role of delay effect in a tumor immune system (ENGLISH)

**Yu Min**(College of Science and Engineering, Aoyama Gakuin University)Dual role of delay effect in a tumor immune system (ENGLISH)

[ Abstract ]

In this talk, a previous tumor immune interaction model is simplified by considering a relatively weak immune activation, which can still keep the essential dynamics properties. Since the immune activation process is not instantaneous, we incorporate one delay effect for the activation of the effector cells by helper T cells into the model. Furthermore, we investigate the stability and instability region of the tumor-presence equilibrium state of the delay-induced system with respect to two parameters, the activation rate of effector cells by helper T cells and the helper T cells stimulation rate by the presence of identified tumor antigens. We show the dual role of this delay that can induce stability switches exhibiting destabilization as well as stabilization of the tumor-presence equilibrium. Besides, our results show that the appropriate immune activation time plays a significant role in control of tumor growth.

In this talk, a previous tumor immune interaction model is simplified by considering a relatively weak immune activation, which can still keep the essential dynamics properties. Since the immune activation process is not instantaneous, we incorporate one delay effect for the activation of the effector cells by helper T cells into the model. Furthermore, we investigate the stability and instability region of the tumor-presence equilibrium state of the delay-induced system with respect to two parameters, the activation rate of effector cells by helper T cells and the helper T cells stimulation rate by the presence of identified tumor antigens. We show the dual role of this delay that can induce stability switches exhibiting destabilization as well as stabilization of the tumor-presence equilibrium. Besides, our results show that the appropriate immune activation time plays a significant role in control of tumor growth.

### 2016/07/12

#### Algebraic Geometry Seminar

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

Hypersurfaces of maximal contact and jumping phenomenon in the problem of resolution of singularities in positive characteristic (English)

https://www.math.purdue.edu/people/bio/kmatsuki/home

**Kenji Matsuki**(Purdue/RIMS)Hypersurfaces of maximal contact and jumping phenomenon in the problem of resolution of singularities in positive characteristic (English)

[ Abstract ]

According to our approach for resolution of singularities in positive characteristic (called the Idealistic Filtration Program, alias the I.F.P.) the algorithm is divided into the following two steps:

Step 1. Reduction of the general case to the monomial case.

Step 2. Solution in the monomial case.

While we have established Step 1 in abritrary dimension, Step 2 becomes very subtle and difficult in positive characteristic. This is in clear contrast to the classical setting in characteristic zero, where the solution in the monomial case is quite easy.

The talk consists of the two parts.

・Part I [13:30--15:00]: This part is mainly for the students, who are not familiar with the classical results in characteristic zero. Through Hironaka's reformulation of the problem of resolution of singularities, we will see how the notion of a hypersurface of maximal contact provides an inductive structure on dimension to the problem, and hence leading to a solution. Since our I.F.P. is closely modelled upon the classical algorithm in characteristic zero, this part should also give some background material and motivation for our approach in positive characteristic.

・Part II [15:30--17:00]: This is the main body of my talk. I will proceed according to the following menu.

{\bf Framewrok of the I.F.P.}: First I will explain the framewrok of the I.F.P., which further extends Hironaka's refomulation. The biggest obstacle to establish Step 1 is the fact that, in positive characteristic, a smooth hypersurface of maximal contact does not exist in general. In order to overcome this obstacle, we introduce the notion of the Leading Generator System, which is the collection of multiple singular hypersurfaces of maximal contcat.

{\bf Monomial Case}: As metioned above, then the problem is reduced to the one in the monomial case.

・ {\bf Inductive scheme on the invariant \boldmath$\tau$}: We firstly observe that, by the inductive scheme on the invariant $\tau$, we have only to consider the case with $\tau = 1$, i.e., the case where there is only one single singular hypersurface of maximal contact.

・ {\bf Tight Monomail Case}: We secondly observe that, if we reach the so-called Tight Monomial Case, then we can easily solve the problem.

・ {\bf Introduction of the invariant `` \boldmath$\mathrm{inv}_{\mathrm{MON},real}$''}: Thus our final task is, after arriving at the monimial case with $\tau = 1$, to reach the Tight Monomial Case, which is characterized by $\mathrm{inv}_{\mathrm{MON},real} = 0$.

・ {\bf Moh-Hauser Jumping phenomenon}: The invariant $\mathrm{inv}_{\mathrm{MON},real}$ usually behaves well, i.e., decreases after each blow up. But under some circustances, it strictly increases. I will explain this well-known Moh-Jumping phenomenon by giving a simple example.

・ {\bf Eventual decrease of the jumping peaks}: At last, the problem boils down to analyzing and overcoming the Moh-Hauser Jumping phenomenon. For this purpose, we will present the conjecture of ``Eventual decrease of the jumping peaks'', which is affirmatively solved in dimension 3, and is the current focus of our research in dimension 4.

[ Reference URL ]According to our approach for resolution of singularities in positive characteristic (called the Idealistic Filtration Program, alias the I.F.P.) the algorithm is divided into the following two steps:

Step 1. Reduction of the general case to the monomial case.

Step 2. Solution in the monomial case.

While we have established Step 1 in abritrary dimension, Step 2 becomes very subtle and difficult in positive characteristic. This is in clear contrast to the classical setting in characteristic zero, where the solution in the monomial case is quite easy.

The talk consists of the two parts.

・Part I [13:30--15:00]: This part is mainly for the students, who are not familiar with the classical results in characteristic zero. Through Hironaka's reformulation of the problem of resolution of singularities, we will see how the notion of a hypersurface of maximal contact provides an inductive structure on dimension to the problem, and hence leading to a solution. Since our I.F.P. is closely modelled upon the classical algorithm in characteristic zero, this part should also give some background material and motivation for our approach in positive characteristic.

・Part II [15:30--17:00]: This is the main body of my talk. I will proceed according to the following menu.

{\bf Framewrok of the I.F.P.}: First I will explain the framewrok of the I.F.P., which further extends Hironaka's refomulation. The biggest obstacle to establish Step 1 is the fact that, in positive characteristic, a smooth hypersurface of maximal contact does not exist in general. In order to overcome this obstacle, we introduce the notion of the Leading Generator System, which is the collection of multiple singular hypersurfaces of maximal contcat.

{\bf Monomial Case}: As metioned above, then the problem is reduced to the one in the monomial case.

・ {\bf Inductive scheme on the invariant \boldmath$\tau$}: We firstly observe that, by the inductive scheme on the invariant $\tau$, we have only to consider the case with $\tau = 1$, i.e., the case where there is only one single singular hypersurface of maximal contact.

・ {\bf Tight Monomail Case}: We secondly observe that, if we reach the so-called Tight Monomial Case, then we can easily solve the problem.

・ {\bf Introduction of the invariant `` \boldmath$\mathrm{inv}_{\mathrm{MON},real}$''}: Thus our final task is, after arriving at the monimial case with $\tau = 1$, to reach the Tight Monomial Case, which is characterized by $\mathrm{inv}_{\mathrm{MON},real} = 0$.

・ {\bf Moh-Hauser Jumping phenomenon}: The invariant $\mathrm{inv}_{\mathrm{MON},real}$ usually behaves well, i.e., decreases after each blow up. But under some circustances, it strictly increases. I will explain this well-known Moh-Jumping phenomenon by giving a simple example.

・ {\bf Eventual decrease of the jumping peaks}: At last, the problem boils down to analyzing and overcoming the Moh-Hauser Jumping phenomenon. For this purpose, we will present the conjecture of ``Eventual decrease of the jumping peaks'', which is affirmatively solved in dimension 3, and is the current focus of our research in dimension 4.

https://www.math.purdue.edu/people/bio/kmatsuki/home

#### Tuesday Seminar on Topology

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

Non-arithmetic lattices (ENGLISH)

**John Parker**(Durham University)Non-arithmetic lattices (ENGLISH)

[ Abstract ]

In this talk I will discuss arithmetic and non-arithmetic lattices and I will give a history of the problem of finding non-arithmetic lattices. I will also briefly describe the construction of new non-arithmetic lattices in SU(2,1) found in my joint workwith Martin Deraux and Julien Paupert.

In this talk I will discuss arithmetic and non-arithmetic lattices and I will give a history of the problem of finding non-arithmetic lattices. I will also briefly describe the construction of new non-arithmetic lattices in SU(2,1) found in my joint workwith Martin Deraux and Julien Paupert.

#### PDE Real Analysis Seminar

10:20-11:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Special cases of the planar least gradient problem (English)

**Piotr Rybka**(University of Warsaw)Special cases of the planar least gradient problem (English)

[ Abstract ]

We study the least gradient problem in two special cases:

(1) the natural boundary conditions are imposed on a part of the strictly convex domain while the Dirichlet data are given on the rest of the boundary; or

(2) the Dirichlet data are specified on the boundary of a rectangle. We show existence of solutions and study properties of solution for special cases of the data. We are particularly interested in uniqueness and continuity of solutions.

We study the least gradient problem in two special cases:

(1) the natural boundary conditions are imposed on a part of the strictly convex domain while the Dirichlet data are given on the rest of the boundary; or

(2) the Dirichlet data are specified on the boundary of a rectangle. We show existence of solutions and study properties of solution for special cases of the data. We are particularly interested in uniqueness and continuity of solutions.

#### PDE Real Analysis Seminar

14:20-15:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Global Strong $L^p$ Well-Posedness of the 3D Primitive Equations (English)

**Amru Hussein**(TU Darmstadt)Global Strong $L^p$ Well-Posedness of the 3D Primitive Equations (English)

[ Abstract ]

Primitive Equations are considered to be a fundamental model for geophysical flows. Here, the $L^p$ theory for the full primitive equations, i.e. the three dimensional primitive equations coupled to the equation for temperature and salinity, is developed. This set of equations is globally strongly well-posed for arbitrary large initial data lying in certain interpolation spaces, which are explicitly characterized as subspaces of $H^2/p$, $p$, $1 < p < \infty$, satisfying certain boundary conditions. Thus, the general $L^p$ setting admits rougher data than the usual $L^2$ theory with initial data in $H^1$.

In this study, the linearized Stokes type problem plays a prominent role, and it turns out that it can be treated efficiently using perturbation methods for $H^\infty$-calculus.

Primitive Equations are considered to be a fundamental model for geophysical flows. Here, the $L^p$ theory for the full primitive equations, i.e. the three dimensional primitive equations coupled to the equation for temperature and salinity, is developed. This set of equations is globally strongly well-posed for arbitrary large initial data lying in certain interpolation spaces, which are explicitly characterized as subspaces of $H^2/p$, $p$, $1 < p < \infty$, satisfying certain boundary conditions. Thus, the general $L^p$ setting admits rougher data than the usual $L^2$ theory with initial data in $H^1$.

In this study, the linearized Stokes type problem plays a prominent role, and it turns out that it can be treated efficiently using perturbation methods for $H^\infty$-calculus.

#### Tuesday Seminar of Analysis

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

Gevrey estimates of the resolvent and sub-exponential time-decay (English)

**X. P. Wang**(Université de Nantes, France)Gevrey estimates of the resolvent and sub-exponential time-decay (English)

[ Abstract ]

For a class of non-selfadjoint Schrodinger operators satisfying some weighted coercive condition, we prove that the resolvent satisfies the Gevrey estimates at the threshold. As applications, we show that the heat and Schrodinger semigroups decay sub-exponentially in appropriately weighted spaces. We also study compactly supported perturbations of this class of operators where zero may be an embedded eigenvalue.

For a class of non-selfadjoint Schrodinger operators satisfying some weighted coercive condition, we prove that the resolvent satisfies the Gevrey estimates at the threshold. As applications, we show that the heat and Schrodinger semigroups decay sub-exponentially in appropriately weighted spaces. We also study compactly supported perturbations of this class of operators where zero may be an embedded eigenvalue.

#### PDE Real Analysis Seminar

12:10-12:50 Room #056 (Graduate School of Math. Sci. Bldg.)

The role of convection in some Keller-Segel models (English)

**Elio Espejo**(National University of Colombia)The role of convection in some Keller-Segel models (English)

[ Abstract ]

An interesting problem in reaction-diffusion equations is the understanding of the role of convection in phenomena like blow-up or convergence. I will discuss this problem through some Keller-Segel type models arising in mathematical biology and show some recent results.

An interesting problem in reaction-diffusion equations is the understanding of the role of convection in phenomena like blow-up or convergence. I will discuss this problem through some Keller-Segel type models arising in mathematical biology and show some recent results.

#### PDE Real Analysis Seminar

11:20-12:00 Room #056 (Graduate School of Math. Sci. Bldg.)

The total variation flow in $H^{−s}$ (English)

**Monika Muszkieta**(Wroclaw University of Science and Technology)The total variation flow in $H^{−s}$ (English)

[ Abstract ]

In the talk, we consider the total variation flow in the Sobolev space $H^{−s}$. We explain the motivation to study this problem in the context of image processing applications and provide its rigorous interpretation under periodic boundary conditions. Furthermore, we introduce a numerical scheme for an approximate solution to this flow which has been derived based on the primal-dual approach and discuses some issues concerning its convergence. We also show and compare results of numerical experiments obtained by application of this scheme for a simple initial data and different values of the index $s$.

This is a join work with Y. Giga.

In the talk, we consider the total variation flow in the Sobolev space $H^{−s}$. We explain the motivation to study this problem in the context of image processing applications and provide its rigorous interpretation under periodic boundary conditions. Furthermore, we introduce a numerical scheme for an approximate solution to this flow which has been derived based on the primal-dual approach and discuses some issues concerning its convergence. We also show and compare results of numerical experiments obtained by application of this scheme for a simple initial data and different values of the index $s$.

This is a join work with Y. Giga.

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