過去の記録

過去の記録 ~10/22本日 10/23 | 今後の予定 10/24~

2017年01月19日(木)

統計数学セミナー

13:00-15:30   数理科学研究科棟(駒場) 052号室
Feng Chen 氏 (University of New South Wales)
Talk 1:Likelihood inference for a continuous time GARCH model
Talk 2:Nonparametric Estimation for Self-Exciting Point Processes: A Parsimonious Approach
[ 講演概要 ]
Talk 1:The continuous time GARCH (COGARCH) model of Kluppelberg, Lindner and Maller (2004) is a natural extension of the discrete time GARCH(1,1) model which preserves important features of the GARCH model in the discrete-time setting. For example, the COGARCH model is driven by a single source of noise as in the discrete time GARCH model, which is a Levy process in the COGARCH case, and both models can produced heavy tailed marginal returns even when the driving noise is light-tailed. However, calibrating the COGARCH model to data is a challenge, especially when observations of the COGARCH process are obtained at irregularly spaced time points. The method of moments has had some success in the case with regularly spaced data, yet it is not clear how to make it work in the more interesting case with irregularly spaced data. As a well-known method of estimation, the maximum likelihood method has not been developed for the COGARCH model, even in the quite simple case with the driving Levy process being compound Poisson, though a quasi-maximum likelihood (QML)method has been proposed. The challenge with the maximum likelihood method in this context is mainly due to the lack of a tractable form for the likelihood. In this talk, we propose a Monte Carlo method to approximate the likelihood of the compound Poisson driven COGARCH model. We evaluate the performance of the resulting maximum likelihood (ML) estimator using simulated data, and illustrate its application with high frequency exchange rate data. (Joint work with Damien Wee and William Dunsmuir).

Talk 2:There is ample evidence that in applications of self-exciting point process (SEPP) models, the intensity of background events is often far from constant. If a constant background is imposed, that assumption can reduce significantly the quality of statistical analysis, in problems as diverse as modelling the after-shocks of earthquakes and the study of ultra-high frequency financial data. Parametric models can be
used to alleviate this problem, but they run the risk of distorting inference by misspecifying the nature of the background intensity function. On the other hand, a purely nonparametric approach to analysis
leads to problems of identifiability; when a nonparametric approach is taken, not every aspect of the model can be identified from data recorded along a single observed sample path. In this paper we suggest overcoming this difficulty by using an approach based on the principle of parsimony, or Occam's razor. In particular, we suggest taking the point-process intensity to be either a constant or to have maximum differential entropy. Although seldom used for nonparametric function estimation in other settings, this approach is appropriate in the context of SEPP models. (Joint work with the late Peter Hall.)

2017年01月17日(火)

トポロジー火曜セミナー

17:30-18:30   数理科学研究科棟(駒場) 056号室
Tea: Common Room 17:00-17:30
杉山 聡 氏 (東京大学大学院数理科学研究科)
On an application of the Fukaya categories to the Koszul duality (JAPANESE)
[ 講演概要 ]
In this talk, we compute an A-Koszul dual of path algebras with relations over the directed An-type quivers via the Fukaya categories of exact Riemann surfaces.

The Koszul duality is originally a duality between certain quadratic algebras called Koszul algebras. In this talk, we are interested in the case when A is not a quadratic algebra, i.e. the case when A is defined as a quotient algebra of tensor algebra devided by higher degree relations.

The definition of Koszul duals for such algebras, A-Koszul duals, are given by some people, for example, D. M. Lu, J. H. Palmieri, Q. S. Wu, J. J. Zhang. However, the computation for a concrete examples is hard. In this talk, we use the Fukaya categories of exact Riemann surfaces to compute A-Koszul duals. Then, we understand the Koszul duality as a duality between higher products and relations.

2017年01月16日(月)

統計数学セミナー

16:50-18:00   数理科学研究科棟(駒場) 052号室
広瀬勇一 氏 (University of Wellington)
Profile likelihood approach to a large sample distribution of estimators in joint mixture model of survival and longitudinal ordered data
[ 講演概要 ]
We consider a semiparametric joint model that consists of item response and survival components, where these two components are linked through latent variables. We estimate the model parameters through a profile likelihood and the EM algorithm. We propose a method to derive an asymptotic variance of the estimators in this model.

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
Dinh Tuan Huynh 氏 (大阪大学)
A geometric second main theorem (ENGLISH)
[ 講演概要 ]
Using Ahlfors’ theory of covering surfaces, we establish a Cartan’s type Second Main Theorem in the complex projective plane with 1–truncated counting functions for entire holomorphic curves which cluster on an algebraic curve.

数値解析セミナー

16:50-18:20   数理科学研究科棟(駒場) 117号室
河原田秀夫 氏 (AMSOK, 千葉大学名誉教授)
炭酸カルシウムScale(湯あか)形成の抑止原理の解明 (日本語)
[ 講演概要 ]
Scaleとはボイラなどの内側にできる湯あかのことをいう。その成分は主としてカルシウムおよびマグネシウムの硫酸塩、炭酸塩、ケイ酸塩である。ボイラ内の対流の障害になるほか、熱伝導率が小さいので伝熱量が減るばかりでなく、伝熱面の過熱破損の原因となる。古くからそれらの弊害を除去すべく種々の装置が開発されてきた。しかし、現在に至ってもそれらの決定版と言われるのは見い出し難い状況にある。
最近、その表面にSiO2等の無機酸化物を含む球状(直径1cm程度)のセラミック球を金属銅、および金属銀の壁によって構成される円筒型の容器内に充填した装置が井川重信氏によって開発された(特許4660317号 登録日平成23年1月7日)。循環水中に上記装置を設置してセラミック球に接触させることにより、炭酸カルシウムのscale形成を抑止する。
しかしながら、上記装置がscale形成抑止に何故有効であるか、そのメカニズムは未だ解明されていない。我々はこの現象に対して基本的仮説を提示して、その原理の解明を試みる。そのカギとなるのは炭酸カルシウム結晶の表面自由エネルギーの制御にある。この仕組みの展開には数理モデルの構築とその数値解析が重要な役割を担っている。

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
桒田 和正 氏 (東京工業大学理学院)
Monotonicity and rigidity of the W-entropy on RCD (0,N) spaces (日本語)

2017年01月12日(木)

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
Emanuele Guidotti 氏 (Milan University)
yuimaGUI: a Graphical User Interface for the yuima Package
[ 講演概要 ]
The yuimaGUI package provides a user-friendly interface for yuima. It greatly simplifies tasks such as estimation and simulation of stochastic processes and it also includes additional tools. Some of them:
 data retrieval: stock prices and economic indicators
 time series clustering
 change point analysis
 lead-lag estimation
After a general overview of the whole interface, the yuimaGUI will be shown in real-time. All the settings and the inner workings will be discussed in detail. During this second part, you are kindly invited to ask questions whenever you feel that some problem may arise.

2017年01月11日(水)

代数学コロキウム

18:00-19:00   数理科学研究科棟(駒場) 056号室
Lei Fu 氏 (Tsinghua University)
Deformation and rigidity of $\ell$-adic sheaves (English)
[ 講演概要 ]
Let $X$ be a smooth connected algebraic curve over an algebraically closed field, let $S$ be a finite closed subset in $X$, and let $F_0$ be a lisse $\ell$-torsion sheaf on $X-S$. We study the deformation of $F_0$. The universal deformation space is a formal scheme. Its generic fiber has a rigid analytic space structure. By studying this rigid analytic space, we prove a conjecture of Katz which says that if a lisse $\overline{Q}_\ell$-sheaf $F$ is irreducible and physically rigid, then it is cohomologically rigid in the sense that $\chi(X,j_*End(F))=2$, where $j:X-S\to X$ is the open immersion.

(本講演は「東京北京パリ数論幾何セミナー」として, インターネットによる東大数理,Morningside Center of MathematicsとIHESの双方向同時中継で行います.)

2017年01月10日(火)

トポロジー火曜セミナー

17:00-18:00   数理科学研究科棟(駒場) 056号室
Tea: Common Room 16:30-17:00
斎藤 俊輔 氏 (東京大学大学院数理科学研究科)
Stability of anti-canonically balanced metrics (JAPANESE)
[ 講演概要 ]
Donaldson introduced "anti-canonically balanced metrics" on Fano manifolds, which is a finite dimensional analogue of Kähler-Einstein metrics. It is proved that anti-canonically balanced metrics are critical points of the quantized Ding functional.

We first study the slope at infinity of the quantized Ding functional along Bergman geodesic rays. Then, we introduce a new algebro-geometric stability of Fano manifolds based on the slope formula, and show that the existence of anti-canonically balanced metrics implies our stability. The relationship between the stability and others is also discussed.

This talk is based on a joint work with R. Takahashi (Tohoku Univ).

トポロジー火曜セミナー

18:00-19:00   数理科学研究科棟(駒場) 056号室
林 晋 氏 (東京大学大学院数理科学研究科)
Topological Invariants and Corner States for Hamiltonians on a Three Dimensional Lattice (JAPANESE)
[ 講演概要 ]
In condensed matter physics, a correspondence between two topological invariants defined for a gapped Hamiltonian is well-known. One is defined for such a Hamiltonian on a lattice (bulk invariant), and the other is defined for its restriction onto a subsemigroup (edge invariant). The edge invariant is related to the wave functions localized near the edge. This correspondence is known as the bulk-edge correspondence. In this talk, we consider a variant of this correspondence. We consider a periodic Hamiltonian on a three dimensional lattice (bulk) and its restrictions onto two subsemigroups (edges) and their intersection (corner). We will show that, if our Hamiltonian is "gapped" in some sense, we can define a topological invariant for the bulk and edges. We will also define another topological invariant related to the wave functions localized near the corner. We will explain that there is a correspondence between these two topological invariants by using the six-term exact sequence associated to the quarter-plane Toeplitz extension obtained by E. Park.

2016年12月22日(木)

東京無限可積分系セミナー

14:00-17:30   数理科学研究科棟(駒場) 002号室
野崎雄太 氏 (東大数理) 14:00-15:30
種数 1 の曲面上のホモロジーコボルディズム (JAPANESE)
[ 講演概要 ]
種数 1 のファイバー結び目を含まないレンズ空間の存在が森元により示され,
その後 Baker によりそのようなレンズ空間が完全に決定された.
本講演では,ホモロジーコボルディズムの言葉で定式化される類似の問題に対し
て得られた結果を紹介する.
主結果の証明においては Chebotarev の密度定理と 2 次形式が重要な役割を果
たす.
土岡俊介 氏 (東大数理) 16:00-17:30
Schur分割定理の一般化について (JAPANESE)
[ 講演概要 ]
Rogers-Ramanujan(第1)恒等式は「隣接するパートの差が 2
以上である
ような n の分割は、各パートが mod 5で± 1であるようなnの分割と同数存在す
る」
という分割定理と同値であるが、Schurは1926年に後者の mod 6 版を発見した。
渡部正樹さんとの共同研究(arXiv:1609.01905)において、量子群の表現論を用
いて、
この定理を一般の奇数p¥geq 3に拡張したので報告する。p=3 の場合が Schur 分
割定理で、
p=5 の場合は、Andrews によって1970年代にRogers-Ramanujan 分割定理の
3パラメータ拡張に関連して予想され、1994年に Andrews-Bessenrodt-Olsson に
よって計算機を援用して証明された分割定理に対応する。

2016年12月20日(火)

PDE実解析研究会

10:30-11:30   数理科学研究科棟(駒場) 056号室
Nader Masmoudi 氏 (Courant Institute, NYU)
On the stability of the 3D Couette Flow (English)
[ 講演概要 ]
We will discuss the dynamics of small perturbations of the plane, periodic Couette flow in the 3D incompressible Navier-Stokes system at high Reynolds number. For sufficiently regular initial data, we determine the stability threshold for small perturbations and characterize the long time dynamics of solutions near this threshold. For rougher data, we obtain an estimate of the stability threshold which agrees closely with numerical experiments. The primary linear stability mechanism is an anisotropic enhanced dissipation resulting from the mixing caused by the large mean shear. There is also a linear inviscid damping similar to the one observed in 2D. The main linear instability is a non-normal instability known as the lift-up effect. There is clearly a competition between these linear effects. Understanding the variety of nonlinear resonances and devising the correct norms to estimate them form the core of the analysis we undertake. This is based on joint works with Jacob Bedrossian and Pierre Germain.

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
Irene Pasquinelli 氏 (Durham University)
Deligne-Mostow lattices and cone metrics on the sphere (ENGLISH)
[ 講演概要 ]
Finding lattices in PU(n,1) has been one of the major challenges of the last decades. One way of constructing lattices is to give a fundamental domain for its action on the complex hyperbolic space.

One approach, successful for some lattices, consists of seeing the complex hyperbolic space as the configuration space of cone metrics on the sphere and of studying the action of some maps exchanging the cone points with same cone angle.

In this talk we will see how this construction of fundamental polyhedra can be extended to almost all Deligne-Mostow lattices with three folding symmetry.

2016年12月19日(月)

離散数理モデリングセミナー

17:30-18:30   数理科学研究科棟(駒場) 056号室
Anton Dzhamay 氏 (University of Northern Colorado)
Discrete Painlevé equations on the affine A3 surface (ENGLISH)
[ 講演概要 ]
We explain how to construct the birational representation of the extended affine Weyl symmetry group D5 and consider examples of discrete Painlevé equations that correspond to certain translation elements in this group. One of the examples is the famous q-PV equation of Jimbo-Sakai. Some other examples are conjugated to it via explicit change of variables and we explain how representing translation elements as words in the group allows us to see the corresponding change of coordinates explicitly. We also show a new example of a discrete Painlevé equation that is elementary (short translation), but at the same time is different from the q-PVI equation.

2016年12月17日(土)

離散数理モデリングセミナー

10:00-18:00   数理科学研究科棟(駒場) 056号室
Anton Dzhamay 氏 (University of Northern Colorado) 10:00-10:50
Factorization of Rational Mappings and Geometric Deautonomization (ENGLISH)
[ 講演概要 ]
This talk is the first of two talks describing the joint project with Tomoyuki Takenawa and Stefan Carstea on geometric deautonomization.
The goal of this project is to develop a systematic approach for deautonomizing discrete integrable mappings, such as the QRT mappings, to non-automonous mappings in the discrete Painlevé family, based on the action of the mapping on the Picard lattice of the surface and a choice of an elliptic fiber. In this talk we will explain the main ideas behind this approach and describe the technique that allows us to recover explicit formulas defining the mapping from the known action on the divisor group (the factorization technique). We illustrate our approach by reconstructing the famous example of the q-PVI equation of Jimbo-Sakai from a simple QRT mapping.
Tomoyuki Takenawa 氏 (Tokyo University of Marine Science and Technology) 11:00-11:50
From the QRT maps to elliptic difference Painlevé equations (ENGLISH)
[ 講演概要 ]
This talk is the second part of the joint project with Anton Dzhamay and Stefan Carstea on geometric deautonomization and focuses on the elliptic case and the special symmetry groups. It is well known that two-dimensional mappings preserving a rational elliptic fibration, like the Quispel-Roberts-Thompson mappings, can be deautonomized to discrete Painlevé equations. However, the dependence of this procedure on the choice of a particular elliptic fiber has not been sufficiently investigated.
In this talk we establish a way of performing the deautonomization for a pair of an autonomous mapping and a fiber. Especially, in the case where the fiber is smooth elliptic, imposing certain restrictions on such non autonomous mappings, we obtain new and simple elliptic difference Painlevé equations, including examples whose symmetry groups do not appear explicitly in Sakai's classification.
Hiroshi Kawakami 氏 (Aoyama Gakuin University) 13:30-14:20
The complete degeneration scheme of four-dimensional Painlevé-type equations (ENGLISH)
[ 講演概要 ]
In the joint work with H. Sakai and A. Nakamura, we constructed the degeneration scheme of four-dimensional Painlevé-type equations associated with unramified linear equations. In this talk I present the "complete" degeneration scheme of the four-dimensional Painlevé-type equations, which is constructed by means of the degeneration of HTL forms of associated linear equations.
Akane Nakamura 氏 (Josai University) 14:30-15:20
Degeneration of the Painlevé divisors (ENGLISH)
[ 講演概要 ]
There are three types of curves associated with 4-dimensional algebraically completely integrable systems, namely the spectral curve, the Painlevé divisors, and the separation curve. I am going to explain these three curves of genus two taking examples derived from the isospectral limit of the 4-dimensional Painlevé-type equations and study the Namikawa-Ueno type degeneration.
Teruhisa Tsuda 氏 (Hitotsubashi University) 16:00-16:50
Rational approximation and Schlesinger transformation (ENGLISH)
[ 講演概要 ]
We show how rational approximation problems for functions are related to the construction of Schlesinger transformations. Also we discuss their applications to the theory of isomonodromic deformations or Painlevé equations. This talk is based on a joint work with Toshiyuki Mano.
Takafumi Mase 氏 (the University of Tokyo) 17:00-17:50
Spaces of initial conditions for nonautonomous mappings of the plane (ENGLISH)
[ 講演概要 ]
Spaces of initial conditions are one of the most important and powerful tools to analyze mappings of the plane. In this talk, we study the basic properties of general nonautonomous equations that have spaces of initial conditions. We will consider the minimization of spaces of initial conditions for nonautonomous systems and we shall discuss a classification of nonautonomous integrable mappings of the plane with a space of initial conditions.

2016年12月14日(水)

代数学コロキウム

18:00-19:00   数理科学研究科棟(駒場) 056号室
Luc Illusie 氏 (Université Paris-Sud)
On vanishing cycles and duality, after A. Beilinson (English)
[ 講演概要 ]
It was proved by Gabber in the early 1980's that $R\Psi$ commutes with duality, and that $R\Phi$ preserves perversity up to shift. It had been in the folklore since then that this last result was in fact a consequence of a finer one, namely the compatibility of $R\Phi$ with duality. In this talk I'll give a proof of this, using a method explained to me by A. Beilinson.

(本講演は「東京北京パリ数論幾何セミナー」として, インターネットによる東大数理,Morningside Center of MathematicsとIHESの双方向同時中継で行います.)

2016年12月13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
三松 佳彦 氏 (中央大学)
3 次元多様体上の平面場とそれに接する非圧縮流の漸近的絡み目 (JAPANESE)
[ 講演概要 ]
This is a report on a project in (a very slow) progress which aims to prove the tightness of contact structures associated with algebraic Anosov flows without using Bennequin's nor Gromov's results.

After introducing an interpretation of asymptotic linking pairing in terms of differential forms, we attach a subspaces of exact 2-forms to each plane field. We analyze this space in the case where the plane field is an algebraic Anosov foliation and explain what can be done
using results from foliated cohomology and frameworks for secondary characteristic classes. We also show some explicit computations.

To close the talk, a quantization phenomenon which happens when a foliation is deformed into a contact structure is explained and we state some perspectives on applying the results on foliations to the tightness.

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
Hans Christianson 氏 (North Carolina State University)
Distribution of eigenfunction mass on some really simple domains (English)
[ 講演概要 ]
Eigenfunctions are fundamental objects of study in spectral geometry and quantum chaos. On a domain or manifold, they determine the behaviour of solutions to many evolution type equations using, for example, separation of variables. Eigenfunctions are very sensitive to background geometry, so it is important to understand what the eigenfunctions look like: where are they large and where are they small? There are many different ways to measure what "large" and "small" mean. One can consider local $L^2$ distribution, local and global $L^p$ distribution, as well as restrictions and boundary values. I will give an overview of what is known, and then discuss some very recent works in progress demonstrating that complicated things can happen even in very simple geometric settings.

2016年12月12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
秋元琢磨 氏 (慶應義塾大学)
異常拡散におけるエルゴード性の破れとサンプル依存性
[ 講演概要 ]
エルゴード的な系では、単一の軌道に対して、時間平均により得られる観測量は時間無限大の極限で一定値に収束する。特に、平衡系では、この一定値は、初期状態には依存せず、平衡分布による平均値と一致する。したがって、同じ条件の下で観測すれば、観測結果は不変である。つまり、エルゴード的な系は再現性を持つ。本講演では、連続時間ランダムウォークにおける拡散性(平均2乗変位)は再現性を持たないが分布的な再現性を持つことを紹介する [1,2]。連続時間ランダムウォークは、ランダムなポテンシャルエネルギー空間上のランダムウォーク(トラップモデル)をアニールした(空間的な不均質性は考えず、ランダムポテンシャルが常に時間変化している)モデルである。本講演では、さらに、このトラップモデルに対して、系のサイズが有限であるとき、系はエルゴード的であり、時間平均で定義された平均2乗変位は再現性を持つが、ある温度(ガラス温度)以下では、たとえ系のサイズを大きくしても、同じ値には収束せず、不均質さのサンプルに強く依存する(サンプルのゆらぎに起因して拡散性が大きく変わる)ことも示す[3]。換言すれば、ガラス温度以下では、大数の法則が破れ、拡散性はサンプルに依存して本質的にランダムになる。

参考文献
[1] Y. He, S. Burov, R. Metzler, and E. Barkai, Phys. Rev. Lett. 101, 058101 (2008).
[2] T. Miyaguchi and T. Akimoto, Phys. Rev. E 87, 032130 (2013).
[3] T. Akimoto, E. Barkai, and K. Saito, Phys. Rev. Lett. 117, 180602 (2016).

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
川上 裕 氏 (金沢大学)
完備極小曲面の研究の最近の進展について (JAPANESE)
[ 講演概要 ]
ユークリッド空間内の完備極小曲面の性質は様々な観点から研究されていますが、複素解析的視点からの研究は近年においてもますます進展しています。そこで本講演では、完備極小曲面の研究の最近の進展について、講演者の成果も交えてお話したいと思います。

作用素環セミナー

16:45-18:15   数理科学研究科棟(駒場) 126号室
瀬戸樹 氏 (名大多元数理)
Roeコサイクルと分割された多様体の指数定理, そして一般化へ (Japanese)

2016年12月07日(水)

談話会・数理科学講演会

15:30-16:30   数理科学研究科棟(駒場) 056号室
Uwe Jannsen 氏 (Regensburg/東大数理)
On a conjecture of Bloch and Kato, and a local analogue.
[ 講演概要 ]
In their seminal paper on Tamagawa Numbers of motives,
Bloch and Kato introduced a notion of motivic pairs, without
loss of generality over the rational numbers, which should
satisfy certain properties (P1) to (P4). The last property
postulates the existence of a Galois stable lattice T in the
associated adelic Galois representation V such that for each
prime p the fixed module of the inertia group of Q_p of
V/T is l-divisible for almost all primes l different from p.

I postulate an analogous local conjecture and show that it
implies the global conjecture.

2016年12月06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
Horia Cornean 氏 (オールボー大学、デンマーク)
On the trivialization of Bloch bundles and the construction of localized Wannier functions (English)
[ 講演概要 ]
We shall present an introductory lecture on the trivialization of Bloch bundles and its physical implications. Simply stated, the main question we want to answer is the following: given a rank $N\geq 1$ family of orthogonal projections $P(k)$ where $k\in \mathbb{R}^d$, $P(\cdot)$ is smooth and $\mathbb{Z}^d$-periodic, is it possible to construct an orthonormal basis of its range which consists of vectors which are both smooth and periodic in $k$? We shall explain in detail the connection with solid state physics. This is joint work with I. Herbst and G. Nenciu.

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
吉田 建一 氏 (東京大学大学院数理科学研究科)
Union of 3-punctured spheres in a hyperbolic 3-manifold (JAPANESE)
[ 講演概要 ]
An essential 3-punctured sphere in a hyperbolic 3-manifold is isotopic to a totally geodesic one. We will classify the topological types for components of union of the totally geodesic 3-punctured spheres in an orientable hyperbolic 3-manifold. There are special types each of which appears in precisely one manifold.

古典解析セミナー

16:45-18:15   数理科学研究科棟(駒場) 154号室
David Sauzin 氏 (CNRS)
Introduction to resurgence on the example of saddle-node singularities (ENGLISH)
[ 講演概要 ]
Divergent power series naturally appear when solving such an elementary differential equation as x^2 dy = (x+y) dx, which is the simplest example of saddle-node singularity. I will discuss the formal classification of saddle-node singularities and illustrate on that case Ecalle's resurgence theory, which allows one to analyse the divergence of the formal solutions. One can also deal with resonant saddle-node singularities with one more dimension, a situation which covers the local study at infinity of some Painlevé equations.

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