[ 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 ]
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 ]
In ZFC, all the large cardinals are much bigger than $\omega_1$, the least uncountable cardinal,
while without assuming the Axiom of Choice, $\omega_1$ could have some large cardinal properties.
Jech and Takeuti independently proved that if the axiom system ZFC + There is a measurable cardinal is consistent,
then so is ZF + $\omega_1$ is a measurable cardinal.
Takeuti also proved that one can replace "measurable cardinal" above with "supercompact cardinal" as well as some other large cardinals.
Woodin proved that one can reduce the assumption, i.e., the consistency of ZFC + a supercompact cardinal,
to that of ZFC + There are proper class many Woodin cardinals which are limits of Woodin cardinals,
to obtain the consistency of ZF + $\omega_1$ is a supercompact cardinal.
Furthermore, the model he constructed also satisfies the Axiom of Determinacy (AD).
In this talk, after giving some background on the connections between large cardinals and determinacy, we discuss some consequences of the axiom system ZF + $\omega_1$ is a supercompact cardinal.
This is joint work with Nam Trang.

[ 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 ]
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 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 ]
5G時代に向けたモバイルブロードバンドの進展と今後の展望について、事例も踏まえながらお話をさせていただければと考えております。

[ 講演概要 ]
On a conformal 3D manifold with electromagnetic field, Einstein-Weyl equations are the counterpart of Einstein's classical field equations. In 1943, Elie Cartan showed, using abstract arguments, that the general solution depends on 4 functions of 2 variables. I will present families of explicit solutions depending on 9 functions of 1 variable, much beyond what was known before. Such solutions are generic in the sense that the Cotton tensor is nonzero. This is joint work with Pawel Nurowski.

[ 講演概要 ]
Morier-Genoud と Ovsienko によって連分数のある種の q-変形が導入された。このq-変形の最大の応用はそれを用いて向きづけられた有理絡み目の Jones 多項式がそれから直接求めることができることである。またこの連分数のq-変形は結び目理論への応用以外にも、2次無理数論、組み合わせ論への応用もあり、それについても紹介する。

一方、Lee-Schiffler の snake graph を用いる方法や Kogiso-Wakui による Conway-Coxeter frieze を持ちいる方法で Jones 多項式を計算するレシピが与えられている。そのことから、Morier-Genoud and Ovsienko の結果のそれらの観点からの別証明が考えられるが、それについて紹介し、さらに, Kogiso-Wakui の研究で用いた Ancestoral triangles の観点から連分数のq-変形をさらに一般化でき、連分数の cluster-variable 変形が出来ることを紹介する。

[ 講演概要 ]
The Navier-Stokes are used as a model for viscous incompressible fluids such as water. The question as to whether or not the equations in three dimensions form singularities is an open Millennium prize problem. In their celebrated paper in 1993, Constantin and Fefferman showed that (in the whole plane) if the vorticity is sufficiently well aligned in regions of high vorticity then the Navier-Stokes equations remain smooth. For the half-space it is commonly assumed that viscous fluids `stick' to the boundary, which generates vorticity at the boundary. In such a setting, it is open as to whether Constantin and Fefferman's result remains to be true. In my talk I will present recent results in this direction. Joint work with Christophe Prange (CNRS, Université de Bordeaux)

[ 講演概要 ]
In order to calculate a priori or a posteriori error estimates for solutions of an ill-posed operator equation with an injective operator we need to describe a set of approximate solutions that contains an exact solution. After that we have to calculate a diameter of this set or maximal distance from a fixed approximate solution to any element of this set. I will describe three approaches for constructing error estimates and also their practical applications.

[ 講演概要 ]
In 1950s, H. Fujita proposed a method to provide the upper and lower bounds in boundary value problems, which is based on the T*T theory of T. Kato about differential equations. Such a method can be regarded a different formulation of the hypercircle method from Prage-Synge's theorem.
Recently, the speaker extended Kato-Fujita's method to the case of the finite element solution of Poisson's equation and proposed a guaranteed point-wise error estimation. The newly proposed error estimation can be applied to problems defined over domains of general shapes along with general boundary conditions.

[ 講演概要 ]
Rumin 複体は接触多様体上に定まる，実数体の定数層の分解であり，実数体のde Rham 複体の部分複体である．本講演では，球面上のRumin ラプラシアンの固有値を書き下し，Rumin 複体の解析的捩率関数がRiemann のゼータ関数を用いて書き表されることを示す．特に，その関数の原点で消滅していることと，その解析的捩率が具体的に計算できることを紹介する．

[ 講演概要 ]
（株）エーアイスクエアでの活動を基に自然言語処理AIを題材として、事業化に際しての留意点と実際の商用化事例を紹介する。

[ 講演概要 ]
For l-adic sheaves on varieties over finite fields, the constant terms of the functional equations of the L-functions, called global epsilon factors, are important arithmetic invariants. When the varieties are curves, Deligne and Laumon show that they admit product formulae in terms of local epsilon factors.
In this talk, I will explain that, attaching some coefficients to irreducible components of singular supports, we can define refinements of characteristic cycles. We will see that, after taking modulo roots of unity, they give product formulae of global epsilon factors for higher dimensional varieties.
I will also explain that these results can be generalized to arbitrary perfect fields of any characteristic.

[ 講演概要 ]
I will talk about the underlying homotopical structures within field equations, which emerge in string theory as conformal invariance conditions for sigma models. I will show how these, often hidden, structures emerge from the homotopy Gerstenhaber algebra associated to vertex and Courant algebroids, thus making all such equations the natural objects within vertex algebra theory.

[ 講演概要 ]
In a celebrated series of papers, Mukai established structure theorems for polarized K3 surfaces of all genera g<21, with the exception of the case g=14. Using the identification between certain moduli spaces of polarized K3 surfaces of genera 14 and the moduli space of special cubic fourfolds of given discriminant, we discuss a novel approach to moduli spaces of K3 surfaces. As an application, we establish the rationality of the universal K3 surface of these genus 14,22. This is joint work with A. Verra.