Seminars: 2026/04/01--2027/04/01

距離空間がBusemannの凸性を満たすとはある意味で断面曲率が0以下であること, 測度距離空間がMCPを満たすとはある意味でリッチ曲率が下から抑えられていることをそれぞれ表す. この2つの条件を同時に満たす測度距離空間はどのような構造を持つか, が本研究のテーマである. 今回その位相的性質を調べるにあたって, sub-Finsler幾何学の技術が活躍する部分が現れたので, そこに重きをおいて話をする. 本研究は藤岡禎司 氏(福岡大)との共同研究に基づく.

In 2025, Zywina proved that there exist infinitely many elliptic curves of rank 2 over the rationals. It is striking that such a simple fact had not been known until recently. His proof requires the polynomial Szemerédi theorem for primes, proved by Tao and Ziegler in 2008. This theorem is used to keep the number of bad primes of the elliptic curves as small as possible. The elliptic curves he obtained have exactly five bad primes. In this talk, we show that for any given set of odd primes, there exist infinitely many elliptic curves of rank 2 over the rationals whose set of bad primes contains the given set. This result is obtained by applying Zywina’s method to elliptic curves parametrized by solutions to Pell equations. Since the solutions are determined by the fundamental unit, we can arrange that the set of bad primes contains the given set.

Optimal well-posedness of the cubic NLS system on the 1D torus アブストラクト：1次元トーラス上の三次非線形シュレーディンガー方程式は$L^2$で大域適切であることがよく知られている。本講演では、この結果を無限連立系に拡張する。これは作用素値の方程式と自然に同一視され、初期値は自己共役なシャッテン-$p$クラスの作用素で与えられる。本講演ではまず、無限連立系が$p=1$で大域適切、$p>1$では非適切となることを示す。次に、方程式に対してある意味での繰り込みを行うと、$1\le p \le 2$で大域適切、$p>2$では非適切となり、方程式の可解性が改善することを示す。本講演は、Andrew Rout 氏 (Politecnico di Milano) との共同研究に基づく。

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We consider expectations of the form $E[trh_1 (X^N_1) \dots trh_r(X^N_r)]$, where $X^N_i$ are self-adjoint polynomials in various independent classical random matrices and $h_i$ are smooth test function and obtain a large $N$ expansion of these quantities, building on the framework of polynomial approximation and Bernstein-type inequalities recently developed by Chen, Garza-Vargas, Tropp, and van Handel. As applications of the above, we prove the higher-order asymptotic vanishing of cumulants for smooth linear statistics, establish a Central Limit Theorem, and demonstrate the existence of formal asymptotic expansions for the free energy and observables of matrix integrals with smooth potentials. This talk is based on joint work with Benoît Collins

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TBA

TBA

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TBA

We study the complex-analytic geometry of semi-positive holomorphic line bundles on compact Kähler manifolds. In one of our main results, for a $\mathbb{Q}$-effective line bundle satisfying a natural torsion-type assumption, we show the equivalence between semi-positivity and semi-ampleness. More generally, for an effective nef divisor of numerical dimension one, we characterize the semi-positivity of the associated line bundle in terms of the existence of a certain type of pseudoflat fundamental system of neighborhoods of the support. Furthermore, for an effective semi-positive divisor, we prove a dichotomy: either the divisor is the pull-back of a $\mathbb{Q}$-divisor by a fibration onto a Riemann surface, or the Hartogs extension phenomenon holds on the complement of its support. Our proof is based on a pluripotential method that has previously been used for studying the boundaries of pseudoconvex domains, which allows us to investigate the complex-analytic structure of neighborhoods of the support of the divisor even when the manifold is non-compact.

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