2024年度
4月12日 高橋良輔（東北大学）
題目：J-equation and a Kobayashi-Hitchin-type correspondence on semistable vector bundles
要旨：We introduce the J-equation on higher rank holomorphic vector bundles with an application to the deformed Hermitian-Yang-Mills equation through the small volume limit. On semistable bundles over smooth projective surfaces, we provide a necessary and sufficient condition for the solvability of the J-equation in an asymptotic setup. Our result can be thought of as a perturbed version of the Kobayashi-Hitchin correspondence.
5月10日 阪本皓貴（東京大学）
題目: Harmonic measures in percolation clusters on hyperbolic groups
要旨: A random walk on a word hyperbolic group determines the hitting measure on the Gromov boundary under some mild assumptions. This measure is called the harmonic measure associated with the random walk.
For random walks with tranlation invariance (such as the simple random walk on the Cayley graph), it has been proved that the harmonic measures satisfy two important properties concerning Hausdorff dimensions, called exact dimensionality and Ledrappier-Young type formula.
We extend such results to the setting of random walks in random environments (RWRE) on hyperbolic groups. In particular, we can handle Bernoulli percolation clusters, which are very wild environments, by using cluster relations introduced by Gaboriau in the study of orbit equivalence relations.
5月31日 藤田玄（日本女子大学）
題目：“Remarks on toric geometry and probability density functions on a finite set”
要旨：
In information geometry a geometric structure called the dually flat structure is well studied. In fact, several important families of probability measures have dually flat structures. On the other hand a relation between Kähler geometry and dually flat structures is classically known as Dombrowski’s construction (or Hsu‘s theorem), which is one guiding principle of the study of dually flat structure.
In this talk we will focus on a relation between the toric Kähler geometry and the dually flat structure. A toric (symplectic) manifold is a symplectic manifold equipped with a maximal Hamiltonian torus action. It is well known that each toric manifold associates a convex polytope called the Delzant polytope, which characterizes the toric manifold. Any torus invariant Kähler structure on the toric manifold determines a dually flat structure on the Delzant polytope. Some Delzant polytope serves as a parameter space of probability density functions on a finite set called the mixture family. We give the condition for the Delzant polytope to become such a parameter space. We also explain a geometric description of the condition in terms of the symplectic reduction. If time permits, we will discuss topics relevant to Geometric Quantum Mechanics.
6月21日 伊敷喜斗（東京都立大学）
題目：Spaces of metrics are Baire
要旨： For a metrizable space, we consider the space of all metrics generating the same topology of the metrizable space, and this space of metrics is equipped with the supremum metric. In this talk, we first introduce the history of spaces of metrics. Next, we explain our main result: (1) for every metrizable space, that the space of metrics on the metrizable space is Baire.(2) the set of all complete metrics is comeager in the space of metrics. (3) the speaker also obtain non--Archimedean analogues of these results.