[ 講演概要 ]
I will explain a simplified proof of an optimal version of the　Ohsawa-Takegoshi L^2-extension theorem. In the proof, I use a method of Berndtsson-Lempert and skip some argument by the method of McNeal-Varolin. As an application, I will explain a result on extensions from possibly non-reduced varieties.

[ 講演概要 ]
We present a new time discretization method for strongly nonlinear parabolic systems. Our method is based on backward finite difference for the first derivative with second-order accuracy and the first-order linear discrete-time scheme for nonlinear systems which has been introduced by H. Murakawa. We propose a second-order stabilization method by combining these schemes.
Our error estimate requires testing the error equation by two test functions and showing $W^{1,\infty}$-boundedness which is proved by ($H^2$ or) $H^3$ energy estimate. We overcome the difficulty for establishing energy estimate by using the generating function technique which is popular in studying ordinary differential equations. Several numerical examples are provided to support the theoretical result.

[ 講演概要 ]
シンプレクティック・ベクトル空間の（コンパクト）部分集合に対して、シンプレクティック・ホモロジー（Floer ホモロジーの一種）を用いてそのシンプレクティック容量（capacity）を定義することができる。一般に、Floerホモロジーの定義には非線形偏微分方程式（いわゆるFloer方程式）の解の数え上げが関わるため、容量を定義から直接計算したり評価したりするのは難しい。この講演では（シンプレクティック・ベクトル空間をEuclid空間の余接空間とみなしたとき）fiberwiseに凸な集合のシンプレクティック・ホモロジーおよび容量をループ空間のホモロジーから計算する公式を示し、その応用を二つ与える。

[ 講演概要 ]
この講演では, アフィンA_1ディンキン図形に対応するADHM dataのモジュライを考える. これらは, 安定性条件の取り方に応じて,
(-2)曲線, あるいは群作用付きの平面上の枠付き連接層のモジュライとなる.
これら２種のモジュライ上の積分は, それぞれ組合せ論的な記述をもち, 特に(-2)曲線上ではNekrasov関数の広田微分がえられる.
これら２種のモジュライ上の積分が, ある場合に等しくなること, またそれに応じた関数等式について紹介する.
これは中島-吉岡による爆発公式の類似と考えられる.
また, Bershtein-ShchechkinによるPainleve tau functionの研究との関係についても触れたい.

[ 講演概要 ]
I would like to explain a classification result for p-divisible groups, which unifies many of the existing results in the literature. The main tool is the theory of prisms and prismatic cohomology recently developed by Bhatt and Scholze. This is joint work with Johannes Anschütz.

[ 講演概要 ]
株式会社ブロードバンドタワーのAI研究拠点での活動内容を通じて、AI研究に関する動向を紹介させていただきます。また、産業界におけるAI活用に対する課題と将来展望について事例をふまえお話をさせていただきます。

[ 講演概要 ]
本講演では，x軸上に2つの端点を持ち，その端点において異なる接触角を生成する曲線に対する表面拡散を考察する．上記の自由境界値問題は，曲線に対するある汎関数の形式的なH^{-1}勾配流として導出できる．この変分構造は，同様の接触角条件を課した面積保存型曲率流でも現れるため，解の漸近挙動も類似した構造を持つことが期待される．面積保存型曲率流においては，進行波解が安定性を持つことが知られているため，本講演では，表面拡散に対する進行波解の存在性，及びその形状を解析する．特に，面積保存型曲率流においては現れない構造である，角度条件に依存した進行波解の非一意性と非凸性に焦点を当てる．尚，本講演の内容は神戸大学の高坂良史氏との共同研究に基づく．

[ 講演概要 ]
機械学習が現在流行しているが、その手法にはさまざまなものがある。
本講演では、機械学習のある典型的な場合について、
その理論根拠となるはずの統計的学習理論の考え方を紹介する。
特に、深層学習（neural network）に関して、
その表現力とStone-Weierstrassの定理との関係、
一様大数の法則に関するいくつかの計算結果を紹介する。

講演者は機械学習の研究を始めたばかりの初心者であるが、機械学習の理論的研究に
は様々な数学の視点が有用ではないかと感じている。
機械学習への興味を持っていただければと思っている。

[ 講演概要 ]
Stein's method is a powerful tool to proving distributional approximations with error bounds. In this talk, we present two recent developments of Stein's method for high order approximations. (1) Together with Li Luo and Qi-Man Shao, we consider skewness correction in normal approximation. We prove a refined Cram\'er-type moderate deviation result for a class of statistics possessing a local structure. We discuss applications to $k$-runs, U-statistics and subgraph counts. (2) Together with Anton Braverman and Jim Dai, we derive and justify new diffusion approximations with state-dependent diffusion coefficients for stationary distributions of Markov chains. We discuss applications to the Erlang-C system, a hospital inpatient flow model and the auto-regressive model.

[ 講演概要 ]
We define fractional Pearson diffusions [5,7,8] by non-Markovian time change in the corresponding Pearson diffusions [1,2,3,4]. They are governed by the time-fractional diffusion equations with polynomial coefficients depending on the parameters of the corresponding Pearson distribution. We present the spectral representation of transition densities of fractional Pearson diffusions, which depend heavily on the structure of the spectrum of the infinitesimal generator of the corresponding non-fractional Pearson diffusion. Also, we present the strong solutions of the Cauchy problems associated with heavy-tailed fractional Pearson diffusions and the correlation structure of these diffusions [6] .
Continuous time random walks have random waiting times between particle jumps. We define the correlated continuous time random walks (CTRWs) that converge to fractional Pearson diffusions (fPDs) [9,10,11]. The jumps in these CTRWs are obtained from Markov chains through the Bernoulli urn-scheme model, Wright-Fisher model and Ehrenfest-Brillouin-type models. The jumps are correlated so that the limiting processes are not Lévy but diffusion processes with non-independent increments.

This is a joint work with M. Meerschaert (Michigan State University, USA), I. Papic (University of Osijek, Croatia), N.Suvak (University of Osijek, Croatia) and A. Sikorskii (Michigan State University and Arizona University, USA).

References:
[1] Avram, F., Leonenko, N.N and Suvak, N. (2013), On spectral analysis of heavy-tailed Kolmogorov-Pearson diffusion, Markov Processes and Related Fields, Volume 19, N 2 , 249-298
[2] Avram, F., Leonenko, N.N and Suvak, N., (2013), Spectral representation of transition density of Fisher-Snedecor diffusion, Stochastics, 85 (2013), no. 2, 346—369
[3] Bourguin, S., Campese, S., Leonenko, N. and Taqqu,M.S. (2019) Four moments theorems on Markov chaos, Annals of Probability, 47, N3, 1417–1446
[4] Kulik, A.M. and Leonenko, N.N. (2013) Ergodicity and mixing bounds for the Fisher-Snendecor diffusion, Bernoulli, Vol. 19, No. 5B, 2294-2329
[5] Leonenko, N.N., Meerschaert, M.M and Sikorskii, A. (2013) Fractional Pearson diffusions, Journal of Mathematical Analysis and Applications, vol. 403, 532-546
[6] Leonenko, N.N., Meerschaert, M.M and Sikorskii, A. (2013) Correlation Structure of Fractional Pearson diffusion, Computers and Mathematics with Applications, 66, 737-745
[7] Leonenko,N.N., Meerschaert,M.M., Schilling,R.L. and Sikorskii, A. (2014) Correlation Structure of Time-Changed Lévy Processes, Communications in Applied and Industrial Mathematics, Vol. 6 , No. 1, p. e-483 (22 pp.)
[8] Leonenko, N.N., Papic, I., Sikorskii, A. and Suvak, N. (2017) Heavy-tailed fractional Pearson diffusions, Stochastic Processes and their Applications, 127, N11, 3512-3535
[9] Leonenko, N.N., Papic, I., Sikorskii, A. and Suvak, N. (2018) Correlated continuous time random walks and fractional Pearson diffusions, Bernoulli, Vol. 24, No. 4B, 3603-3627
[10] Leonenko, N.N., Papic, I., Sikorskii, A. and Suvak, N. (2019) Ehrenfest-Brillouin-type correlated continuous time random walks and fractional Jacoby diffusion, Theory Probablity and Mathematical Statistics, Vol. 99,123-133.
[11] Leonenko, N.N., Papic, I., Sikorskii, A. and Suvak, N. (2019) Approximation of heavy-tailed fractional Pearson diffusions in Skorokhod topology, submitted

[ 講演概要 ]
Magnitude homology and its variant blurred magnitude homology are novel notions of homology theory for general metric space coined by Leinster et al. They are expected to be dealt with in the context of Topological Data Analysis since its original idea is based on a kind of "persistence of points clouds". However, little property of them have been revealed. In this talk, we see that magnitude homology and blurred magnitude homology is trivial when a metric space is contractible by a distance decreasing homotopy. We use techniques from singular homology theory.

[ 講演概要 ]
Guillermou associated sheaves to exact Lagrangian submanifolds in cotangent bundles and proved topological properties of the Lagrangian submanifolds. In this talk, I will give an estimate on the displacement energy of rational Lagrangian immersions in cotangent bundles with intersection number estimates via microlocal sheaf theory. This result overlaps with results by Chekanov, Liu, and Akaho via Floer theory. This is joint work with Yuichi Ike.

[ 講演概要 ]
TBA