[ 講演概要 ]
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.

[ 講演概要 ]
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.

[ 講演概要 ]
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.

[ 講演概要 ]
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.

[ 講演概要 ]
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.

[ 講演概要 ]
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.

[ 講演概要 ]
情報理論と人工知能の分野の中に、符号理論、圧縮センシング、ニューラルネットワーク
などの技術があり、ガロア体、代数曲線、確率を用いた尤度推定、多様体、L^1ノルム
正則化、微分方程式など様々な数学が用いられています。またこれらの技術はハードディスクや
携帯電話にも応用されています。本講演では、これらの技術から自分が取り組んできた研究に
ついていくつかをご紹介したいと思います。またニューラルネットワークの研究について、
脳神経科学との関わりについても少し触れてみたいと思います。

[ 講演概要 ]
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.

[ 講演概要 ]
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.

in

・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.

[ 講演概要 ]
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.

[ 講演概要 ]
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.

[ 講演概要 ]
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.

[ 講演概要 ]
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.

[ 講演概要 ]
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.

[ 講演概要 ]
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.

[ 講演概要 ]
I will discuss Hamilton-Jacobi equation in the space of probability measures.

Two types of applications motivate the issue: one is from the probabilistic large deviation study of weakly interacting particle systems in statistical mechanics, another is from an infinite particle version of the variational formulation of Newtonian mechanics.

In creating respective well-posedness theories, two mathematical observations played important roles: One, the free-particle flow picture naturally leads to the use of the optimal mass transportation calculus. Two, there is a hidden symmetry (particle permutation invariance) for elements in the space of probability measures. In fact, the space of probability measures in this context is best viewed as an infinite dimensional quotient space. Using a natural metric, we are lead to some fine aspects of the optimal transportation calculus that connect with the metric space analysis and probability.

Time permitting, I will discuss an open issue coming up from the study of the Gibbs-Non-Gibbs transitioning by the Dutch probability community.

The talk is based on my past works with the following collaborators: Markos Katsoulakis, Tom Kurtz, Truyen Nguyen, Andrzej Swiech and Luigi Ambrosio.

[ 講演概要 ]
The radiative transport equation (RTE) is a mathematical model of near-infrared light propagation in human tissue, and its analysis is required to develop a new noninvasive monitoring method of our body or brain activities. Since stationary RTE describes light intensity depending on a position and a direction, a discretization model of 3D-RTE is essentially a five dimensional problem. Therefore to establish a reliable and practical numerical method, both theoretical numerical analysis and computing techniques are required.

We firstly introduce huge-scale computation examples of RTE with bio-optical data. A high-accurate numerical cubature on the unit sphere and a hybrid parallel computing technique using GPGPU realize fast computation. Secondly we propose a semi-discrete upwind finite volume method to RTE. We also show its error estimate in two dimensions.

This talk is based on joint works with Prof. Y.Iso, Prof. N.Higashimori, and Prof. N.Oishi (Kyoto University).

[ 講演概要 ]
Construction of Bridgeland stability conditions on a given smooth projective 3-fold is an important problem. A conjectural construction for any 3-fold was introduced by Bayer, Macri and Toda, and the problem is reduced to proving so-called Bogomolov-Gieseker type inequality holds for certain stable objects in the derived category. It has been shown to hold for Fano 3-folds of Picard rank one due to the works of Macri, Schmidt and Li. However, Schmidt gave a counter-example for a Fano 3-fold of higher Picard rank. In this talk, I will explain how to modify the original conjectural inequality for general Fano 3-folds and why it holds.