## 過去の記録

### 2016年07月12日(火)

#### 代数幾何学セミナー

15:30-17:00   数理科学研究科棟(駒場) 122号室

Kenji Matsuki 氏 (Purdue/RIMS)
Hypersurfaces of maximal contact and jumping phenomenon in the problem of resolution of singularities in positive characteristic (English)
[ 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.
[ 講演参考URL ]
https://www.math.purdue.edu/people/bio/kmatsuki/home

#### トポロジー火曜セミナー

17:00-18:30   数理科学研究科棟(駒場) 056号室
Tea: Common Room 16:30-17:00
John Parker 氏 (Durham University)
Non-arithmetic lattices (ENGLISH)
[ 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.

#### PDE実解析研究会

10:20-11:00   数理科学研究科棟(駒場) 056号室

Piotr Rybka 氏 (University of Warsaw)
Special cases of the planar least gradient problem (English)
[ 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.

#### PDE実解析研究会

14:20-15:00   数理科学研究科棟(駒場) 056号室

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

#### 解析学火曜セミナー

16:50-18:20   数理科学研究科棟(駒場) 126号室
X. P. Wang 氏 (Université de Nantes, France)
Gevrey estimates of the resolvent and sub-exponential time-decay (English)
[ 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.

#### PDE実解析研究会

12:10-12:50   数理科学研究科棟(駒場) 056号室

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

#### PDE実解析研究会

11:20-12:00   数理科学研究科棟(駒場) 056号室

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

### 2016年07月11日(月)

#### 東京確率論セミナー

15:00-18:20   数理科学研究科棟(駒場) 128号室

Jin Feng 氏 (University of Kansas) 15:00-16:30
An introduction to Hamilton-Jacobi equation in the space of probability measures (English)
[ 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.

ある化学反応系のモデリングとそのシミュレーション解析

#### 作用素環セミナー

16:45-18:15   数理科学研究科棟(駒場) 118号室

Fraïssé Theory and Jiang-Su algebra

#### 数値解析セミナー

16:30-18:00   数理科学研究科棟(駒場) 056号室

Towards fast and reliable numerical computations of the stationary radiative transport equation (日本語)
[ 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).

### 2016年07月05日(火)

#### 代数幾何学セミナー

15:30-17:00   数理科学研究科棟(駒場) 122号室
Dulip Piyaratne 氏 (IPMU)
Generalized Bogomolov-Gieseker type inequality for Fano 3-folds (English)
[ 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.

#### Lie群論・表現論セミナー

17:00-18:00   数理科学研究科棟(駒場) 128号室

[ 講演概要 ]
SL(n,C)のあるタイプの冪等元を含む部分群が離散部分群になるための必要条件を与える不等式が、n=2の場合(もともとの清水の補題の場合に相当)を拡張する形で得られることと、対応する対称空間へのその群の作用が不等式によって影響を受けることについて去年７月のこのセミナーでお話しいたしました。これはその後の進展の報告です。

### 2016年07月04日(月)

#### 東京確率論セミナー

16:50-18:20   数理科学研究科棟(駒場) 128号室

Central limit theorems for non-symmetric random walks on nilpotent covering graphs
[ 講演概要 ]
ベキ零群を被覆変換群とするような有限グラフの被覆グラフのことをベキ零被覆グラフと呼ぶ。結晶格子(特に被覆変換群がアーベル群の場合)上のランダムウォークに関してはすでに様々なアプローチが図られ多くの結果が得られている。ベキ零被覆グラフ上のランダムウォークについては、対称な場合に幾つかの極限定理が知られているものの、非対称な場合にはあまり研究が進展していないように思われる。本講演では、ベキ零被覆グラフ上の非対称ランダムウォークを考察し、ある条件下で汎関数中心極限定理が成り立つことを報告する。本講演の内容は、石渡聡氏(山形大)および河備浩司氏(岡山大)との共同研究に基づく。

### 2016年06月28日(火)

#### トポロジー火曜セミナー

17:00-18:30   数理科学研究科棟(駒場) 056号室
Tea: Common Room 16:30-17:00

Strong algebraization of fixed point properties (JAPANESE)
[ 講演概要 ]
バナッハ空間（ないしは族）を固定したとき，有限生成群のそれ上の等長作用が常に大域的固定点を持つ，という性質を固定点性質と呼ぶ．ヒルベルト空間全体のなす族を考えたときの固定点性質は，「Kazhdan の性質(T)」と呼ばれる群の剛性と同値であることが知られている．

「有界生成」の成立がわからないような状況でもパッチワーク戦略を適用できうるようになった．系として，いろいろな離散群が強い固定点性質をもつことを示せ，しかも証明も非常にコンセプチュアルである．こうした応用面についても概観したい．

#### 解析学火曜セミナー

16:50-18:20   数理科学研究科棟(駒場) 126号室
Georgi Raikov 氏 (The Pontificia Universidad Católica de Chile)
Discrete spectrum of Schr\"odinger operators with oscillating decaying potentials (English)
[ 講演概要 ]
I will consider the Schr\"odinger operator $H_{\eta W} =-\Delta + \eta W$, self-adjoint in $L^2(\re^d)$, $d \geq 1$. Here $\eta$ is a non constant almost periodic function, while $W$ decays slowly and regularly at infinity. I will discuss the asymptotic behaviour of the discrete spectrum of $H_{\eta W}$ near the origin. Due to the irregular decay of $\eta W$, there exist some non semiclassical phenomena; in particular, $H_{\eta W}$ has less eigenvalues than suggested by the semiclassical intuition.

### 2016年06月27日(月)

#### 代数幾何学セミナー

15:30-17:00   数理科学研究科棟(駒場) 122号室
いつもと曜日が異なります。This week's seminar will be held on Monday, not on Tuesday.
Christopher Hacon 氏 (University of Utah)
Generic vanishing and birational geometry in char p>0 (ENGLISH)
[ 講演概要 ]
Many precise results on the birational geometry of irregular varieties have been obtained by combining the generic vanishing theorems of Green and Lazarsfeld with the Fourier-Mukai transform. In this talk we will discuss the failure of the generic vanishing theorems of Green and Lazarsfeld in positive characteristic. We will then explain a different approach to generic vanishing based on the theory of F-singularities that leads to concrete applications in birational geometry in positive characteristics
[ 講演参考URL ]
http://www.math.utah.edu/~hacon/

#### 複素解析幾何セミナー

10:30-12:00   数理科学研究科棟(駒場) 128号室

On a higher codimensional analogue of Ueda theory and its applications (JAPANESE)
[ 講演概要 ]
Let $Y$ be a compact complex manifold embedded in a complex manifold with unitary flat normal bundle. As a higher-codimensional generalization of Ueda's theory, we investigate the analytic structure of a neighborhood of $Y$. As an application, we give a criterion for the existence of a smooth Hermitian metric with semi-positive curvature on a nef line bundle.

### 2016年06月24日(金)

#### 幾何コロキウム

10:00-14:30   数理科学研究科棟(駒場) 118号室

[ 講演概要 ]

[ 講演概要 ]

#### 談話会・数理科学講演会

15:30-16:30   数理科学研究科棟(駒場) 123号室

[ 講演概要 ]
1980年代に確立した代数多様体に対する三次元極小モデル理論の高次元化および正標数体上の最近の進展の解説をし、自分の研究成果を交えて、その応用およびこれからの展望の話をする。あまり込み入った証明の話はしない予定である。

[ 講演参考URL ]
https://www.ms.u-tokyo.ac.jp/teacher/gongyo.html

### 2016年06月23日(木)

#### 古典解析セミナー

16:50-18:20   数理科学研究科棟(駒場) 118号室

Resurgence of formal series solutions of nonlinear differential and difference equations (JAPANESE)
[ 講演概要 ]
Resurgent analysis は1980年代に J. Ecalle により創始された. そこでは, alien derivatives 等の漸近解析における重要な概念が導入され, 近年数理物理学においても大きな注目を集めている. 本講演では Resurgent analysis の基本事項の概説から始め, 最近得られた非線形微(差)分方程式の形式解の resurgence に関する結果の紹介を行う.

#### FMSPレクチャーズ

15:00-16:30   数理科学研究科棟(駒場) 056号室
Klaus Mainzer 氏 (Technische Universität München)
Complexity and Computability: Complex Dynamical Systems beyond Turing-Computability (ENGLISH)
[ 講演概要 ]
The computational theory of complexity is founded by digital computing (e.g. Turing machine) which cannot fully grasp continuous concepts of mathematics. The mathematical theory of complex dynamical systems (with interdisciplinary applications in natural and economic sciences) is based on continuous concepts. Further on, there is an outstanding tradition in mathematics since Newton, Leibniz, Euler et al. with real algorithms in, e.g., numerical analysis. How can the gap between the digital and continuous world be mathematically overcome? The talk aims at mathematical and philosophical foundations and interdisciplinary applications of complex dynamical systems beyond Turing-computability.
[ 講演参考URL ]
http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Mainzer.pdf

### 2016年06月21日(火)

#### 解析学火曜セミナー

16:50-18:20   数理科学研究科棟(駒場) 126号室

[ 講演概要 ]

#### トポロジー火曜セミナー

17:00-18:30   数理科学研究科棟(駒場) 056号室
Tea: Common Room 16:30-17:00

Spaces of chord diagrams of spherical curves (JAPANESE)
[ 講演概要 ]
In this talk, the speaker introduces a framework to obtain (possibly infinitely many) new topological invariants of spherical curves under local homotopy moves (several types of Reidemeister moves). They are defined by chord diagrams, each of which is a configurations of even paired points on a circle. We see that these invariants have useful properties.

#### 統計数学セミナー

13:00-15:00   数理科学研究科棟(駒場) 052号室
Lorenzo Mercuri 氏 (University of Milan)
New Classes and Methods in YUIMA package

[ 講演概要 ]
In this talk, we present three new classes recently introduced in YUIMA package.
These classes allow the user to manage three different problems:
・Construction of a multidimensional stochastic differential equation driven by a general multivariate Levy process. In particular we show how to define and then simulate a SDE driven by a multivariate Variance Gamma process.
・Definition and simulation of a functional of a general SDE.
・Definition and simulation of the integral of an object from the class yuima.model. In particular, we are able to evaluate Riemann Stieltjes integrals,deterministic integrals with random integrand and stochastic integrals.
Numerical examples are given in order to explain the new methods and classes.

### 2016年06月20日(月)

#### 代数幾何学セミナー

16:30-18:00   数理科学研究科棟(駒場) 123号室

De-Qi Zhang 氏 (National University of Singapore)
BUILDING BLOCKS OF POLARIZED ENDOMORPHISMS OF NORMAL PROJECTIVE VARIETIES (English)
[ 講演概要 ]
An endomorphism f of a normal projective variety X is polarized if f∗H ∼ qH for some ample Cartier divisor H and integer q > 1.

We first assert that a suitable maximal rationally connected fibration (MRC) can be made f-equivariant using a construction of N. Nakayama, that f descends to a polarized endomorphism of the base Y of this MRC and that this Y is a Q-abelian variety (quasi- ́etale quotient of an abelian variety). Next we show that we can run the minimal model program (MMP) f-equivariantly for mildly singular X and reach either a Q-abelian variety or a Fano variety of Picard number one.

As a consequence, the building blocks of polarized endomorphisms are those of Q- abelian varieties and those of Fano varieties of Picard number one.

Along the way, we show that f always descends to a polarized endomorphism of the Albanese variety Alb(X) of X, and that a power of f acts as a scalar on the Neron-Severi group of X (modulo torsion) when X is smooth and rationally connected.

Partial answers about X being of Calabi-Yau type or Fano type are also given with an extra primitivity assumption on f which seems necessary by an example.
This is a joint work with S. Meng.
[ 講演参考URL ]
http://www.math.nus.edu.sg/~matzdq/