日時：月曜日，16:00～17:30
場所： 東京大学大学院数理科学研究科 数理科学研究科棟（駒場）１２６号室
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講師: Lu-Jing Huang 氏
時間: 16:00 - 17:30
場所：西早稲田キャンパスの51号館18階18-06号室
Title: The polynomial growth of effective resistances in one-dimensional critical long-range percolation
Abstract: We study the critical long-range percolation on $\mathbb{Z}$, where an edge connects $i$ and $j$ independently with probability $1-\exp\{-\beta\int_i^{i+1}\int_j^{j+1}|u-v|^{-2}\d u\d v\}$ for $|i-j|>1$ for some fixed $\beta>0$ and with probability 1 for $|i-j|=1$. Viewing this as a random electric network where each edge has a unit conductance, we show that the effective resistances from 0 to $[-n,n]^c$ and from the interval $[-n,n]$ to $[-2n,2n]^c$ (conditioned on no edge joining $[-n,n]$ and $[-2n,2n]^c$) both grow like $n^{\delta(\beta)}$ for some $\delta(\beta)\in (0,1)$.
講師: Hugo Da Cunha 氏 (Université Lyon 1)
時間: 16:00 - 17:30
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講師: Jessica Lin 氏 (McGill University)
時間: 16:00 - 17:30
場所: 東京大学大学院数理科学研究科 数理科学研究科棟（駒場）128号室
Title: Generalized Front Propagation for Stochastic Spatial Models
Abstract: In this talk, I will present a general framework which can be used to analyze the scaling limits of various stochastic spatial "population" models. Such models include ternary Branching Brownian motion subject to majority voting and several interacting particle systems motivated by biology. The approach is based on moment duality and a PDE methodology introduced by Barles and Souganidis, which can be used to study the asymptotic behaviour of rescaled reaction-diffusion equations. In the limit, the models exhibit phase separation with an evolving interface which is governed by a global-in-time, generalized notion of mean-curvature flow. This talk is based on joint work with Thomas Hughes (University of Bath).
講師: 名古路 浩辰 氏
時間: 16:00 - 17:30
場所: 東京大学大学院数理科学研究科 数理科学研究科棟（駒場）126号室
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講師: 仮確定
時間: 16:00 - 17:30
場所:
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講師: 角田 謙吉氏
時間: 17:00 - 18:30
場所：東京大学大学院数理科学研究科 数理科学研究科棟（駒場）126号室
Title: 粒子系に対する静的な揺らぎ
Abstract: 非平衡定常状態は数理物理の問題として様々な文脈の中で研究されている。非平衡定常状態とはカレントは0でないが時間に対して不変な状態であり、相互作用粒子系においては系の定常測度として定義される。非平衡定常状態の解析で難しい問題点として、その明示的な具体形が知られていないことや定常測度が粒子系に対して可逆でないことなどがあげられる。そのため非平衡定常状態の解析は容易ではないが、一般的な手法として対応するdynamicsに対する解析を用いる方法がある。本講演では粒子数密度に対する揺らぎの問題について焦点を当て、境界で流入・流出のある排他過程やGlauber+Kawasaki過程に対してその手法を解説する。
講師: Wai-Kit Lam 氏
時間: 16:00 - 17:30
場所：東京大学大学院数理科学研究科 数理科学研究科棟（駒場）126号室
Title:Disorder monomer-dimer model and maximum weight matching
Abstract: Given a finite graph, one puts i.i.d. weights on the edges and i.i.d. weights on the vertices. For a (partial) matching on this graph, define the weight of the matching by adding all the weights of the edges in the matching together with the weights of the unmatched vertices. One would like to understand how the maximum weight behaves as the size of the graph becomes large. The talk will be divided into two parts. In the first part, we consider the "positive temperature" case (a.k.a. the disorder monomer-dimer model). We show that the model exhibits correlation decay, and from this one can prove a Gaussian central limit theorem for the associated free energy. In the second part, we will focus on the "zero temperature" case, the maximum weight matching. We show that if the edge weights are exponentially distributed, and if the vertex weights are absent, then there is also correlation decay for a certain class of graphs. This correlation decay allows us to define the maximum weight matching on some infinite graphs and also prove limit theorems for the maximum weight matching. Joint work with Arnab Sen (Minnesota).
講師: Grégoire Allaire氏
時間: 16:00 - 17:30
場所：早稲田大学西早稲田キャンパス 62号館1階大会議室
Title: Long time homogenization of the wave equation in periodic media
Abstract: We report on a joint work with A. Lamacz-Keymling and J. Rauch. We study the homogenization of the wave equation in a periodic medium for long times of the order of any inverse power of the period. The unknown can be either a scalar or a vector field, while the coefficients can be purely periodic or locallyperiodic tensors. We obtain high order homogenized equations which include dispersive corrections that are crucial for long time accuracy. Our main tools are (i) a so-called "criminal ansatz", which generalizes to the hyperbolic setting an idea of Bakhavalov and Panasenko in the elliptic setting, (ii) an elimination process for the higher order time derivatives in the high order homogenization equation, (iii) a stability estimate for the corresponding homogenized solutions, based on frequency filtering(iv) an error estimate valid for any long times. The importance of considering high order homogenized equations to catch dispersive effects in the context of the wave equation was first recognized by Santosa and Symes and rigorously analyzed by Lamacz. Our work gives a systematic and complete analysis for all time scales and all high order corrective terms.
https://www.math.sci.waseda.ac.jp/math/activities/
講師: Franco Flandoli 氏
時間: 16:30 - 18:30
場所：西早稲田キャンパスの51号館18階18-06号室
Title: Reducing stochastic models in Climate, fluctuating Hydrodynamics and turbulent transport
Abstract: Many systems, taking the scaling limit in a parameter, converge to a deterministic limit equation. Examples are fast-slow systems in climate and their limit averaging equation; interacting particle systems and their mean field or hydrodynamic limit; stochastic turbulent transport and their diffusive limit, similar to diffusion limit in homogenisation. One could however stop before taking the limit and look for a simplified stochastic model, similar to the limit model but stochastic, which well represents the approximating problem without completely neglecting the fluctuations. This was the viewpoint in particular of Hasselmann proposal in climate modelling, here reviewed in comparison with Dean-Kawasaki approach to fluctuating hydrodynamics, with a detour on stochastic transport. The activities mentioned herein are performed in the framework of the European Research Council Project NoisyFluids 101053472.
講師: 長田博文氏
時間: 14:00 - 15:30
場所：東京大学大学院数理科学研究科 数理科学研究科棟（駒場）126号室
Title:クーロン点過程の対数微分に対する明示表現とその応用 (Explicit formula to logarithmic derivatives of Coulomb random point fields and their applications)
Abstract: Coulomb点過程とは、d次元Coulomb ポテンシャルで相互作用するd次元空間の無限粒子系である。対数微分とは、個々の粒子が、相互作用によって、他の（無限個の）粒子から受ける力を表すベクトル場である。各粒子は対数微分に従って運動する。一般に、対数微分が存在すれば、確率力学が存在することが共著者によって証明されている。本講演は、クーロン点過程の対数微分の存在を証明し、更に、明示表現を構築する。明示表現の応用として、対応する無限次元確率微分方程式のパスワイズ一意の強解の存在を証明する。これを、2次元以上のすべての次元の、すべての正の逆温度に対して行う。
Gibbs測度の理論は、1970年ごろ、DLR方程式を基に確立した。しかし、Ruelle族という、遠方での可積分性を持つ干渉ポテンシャルに適用範囲が限られていた。自然界の最も基本的なポテンシャルであるCoulombポテンシャルが、Gibbs測度の理論からずっと長い間、除外されてきた。本明示表現の応用として、Coulombポテンシャルを含む、強い遠距離相互作用を持つ点過程の広いクラスに対して有効な、干渉ポテンシャルと点過程を結び付ける方程式（定式化）を与える。これは、DLR方程式の役割を、CoulombやRieszポテンシャルという、遠距離強相互作用に対して果たすものである。
講師: Alejandro Ramirez氏
時間: 16:15 - 17:45
場所：東京大学大学院数理科学研究科 数理科学研究科棟（駒場）126号室
Title: GUE fluctuations near the time axis of the one-sided ballistic deposition model
Abstract: Ballistic deposition is a model of interface growth introduced by Vold in 1959, which has remained largely mathematically intractable. It is believed that it is in the KPZ universality class. We introduce the one-sided ballistic deposition model, in which vertically falling blocks can only stick to the top or the upper right corner of growing columns, but not to the upper left corners of growing columns as in ballistic deposition. We establish that strong KPZ universality holds near the time axis, proving that the fluctuations of the height function there are given by the Tracy-Widom GUE distribution. The proof is based on a graphical construction of the process in terms of a last passage percolation model. This is a joint work with Pablo Groisman, Santiago Saglietti and Sebastián Zaninovich.
講師: Ivan Corwin氏
時間: 10:00 - 11:30
場所：東京大学大学院数理科学研究科 数理科学研究科棟（駒場）126号室
Title: How Yang-Baxter unravels Kardar-Parisi-Zhang.
Abstract: Over the past few decades, physicists and then mathematicians have sought to uncover the (conjecturally) universal long time and large space scaling limit for the so-called Kardar-Parisi-Zhang (KPZ) class of stochastically growing interfaces in (1+1)-dimensions. Progress has been marked by several breakthroughs, starting with the identification of a few free-fermionic integrable models in this class and their single-point limiting distributions, widening the field to include non-free-fermionic integrable representatives, evaluating their asymptotics distributions at various levels of generality, constructing the conjectural full space-time scaling limit, known as the directed landscape, and checking convergence to it for a few of the free-fermion representatives.
In this talk, I will describe a method that should prove convergence for all known integrable representatives of the KPZ class to this universal scaling limit. The method has been fully realized for the Asymmetric Simple Exclusion Process and the Stochastic Six Vertex Model. It relies on the Yang-Baxter equation as its only input and unravels the rich complexity of the KPZ class and its asymptotics from first principles. This is based on three works involving Amol Aggarwal, Alexei Borodin, Milind Hegde, Jiaoyang Huang and me.
2025年4月28日（月）
講師 檜垣 充朗氏
時間: 16:00 - 17:30
場所：東京大学大学院数理科学研究科 数理科学研究科棟（駒場）126号室
Title: ランダム粗面領域における粘性流体に対するナヴィエ壁法則
Abstract: 粗面を伴う領域における粘性流体運動の有効近似を得る経験的な手法として、工学分野では壁法則が知られてきた [cf. Nikuradse 1933]。筒状粗面領域における定常層流に対しては、壁法則により、ナヴィエ滑り境界条件に従う速度場が得られる (ナヴィエ壁法則)。本講演では、これが実際に有効近似を与えることを数学的に厳密に証明する。より正確には、粗面領域全体の標本空間を考えた際に、ある種のエルゴード性の仮定の下で、最適な近似率が得られることを報告する。証明の鍵は、粗面付近の流体運動を記述する境界層の確定的/確率的評価である。ここで我々は楕円型方程式に対する定量的確率均質化のアイディアを用いる [cf. Armstrong-Smart, Armstrong-Kuusi-Mourrat, Gloria-Neukamm-Otto, Shen]。ただし、係数行列ではなく粗面領域の標本空間を考えていることに注意されたい。なお、上述のエルゴード性としては、確率変数に対する関数不等式 (対数ソボレフ不等式やスペクトルギャップ不等式など) の成立を採用する。本講演の内容は Jinping Zhuge 氏 (Morningside Center of Mathematics, China)、Yulong Lu 氏 (University of Minnesota, USA) との共同研究に基づく。
講師 蛯名 真久 氏
時間: 16:00 - 17:30
場所：東京大学大学院数理科学研究科 数理科学研究科棟（駒場）126号室
タイトル Malliavin-Stein approach to local limit theorems
アブストラクト
Malliavin-Stein's method is a fruitful combination of the Malliavin calculus and Stein's method. It provides a powerful probabilistic technique for establishing the quantitative central limit theorems, particularly for functionals of Gaussian processes.
In this talk, we will see how the theory of generalized functionals in the Malliavin calculus can be combined with Malliavin-Stein's method to obtain quantitative local central limit theorems. If time allows, we will also discuss some applications to Wiener chaos. Part of this talk is based on the ongoing joint research with Ivan Nourdin and Giovanni Peccati.
講師: 星野壮登 氏
時間: 16:00 - 17:30
場所: 東京大学大学院数理科学研究科 数理科学研究科棟（駒場）126号室
Title: On the proofs of BPHZ theorem and future progress
Abstract: Hairer’s theory of regularity structures (2014) provides a robust framework to guarantee the renormalizability of stochastic partial differential equations (SPDEs). This theory is established in several steps, among which the final and most technically involved step is the proof of the so-called "BPHZ theorem." There are two main approaches to this proof: a graph-theoretic approach developed by Chandra and Hairer (2016+), and a Malliavin calculus-based inductive approach introduced by Linares, Otto, Tempelmayr, and Tsatsoulis (2024). As for Gaussian noises, the latter is simpler and more inductive. While the language used by Otto and his coauthors is different from that of regularity structures, similar arguments have been formulated in the language of regularity structures by Hairer and Steele (2024) and Bailleul and Hoshino (2023+) by different approaches. In this talk, I will first give an overview of the theory of regularity structures, then compare the outlines of the proofs of BPHZ theorem. If time permits, I will also discuss some current researches and future problems.