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Affiliation |
Faculty of Science Department of Mathematics |
Degree 【 display / non-display 】
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Doctor of Science ( 2017.03 Waseda University )
Research Interests 【 display / non-display 】
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Hyperbolic geometry
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低次元トポロジー
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Geometric group theory
From Graduate School 【 display / non-display 】
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Waseda University Doctor's Course Completed
2014.04 - 2017.03
External Career 【 display / non-display 】
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Josai University Faculty of Sciences Assistant Professor
2024.04
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Gakushuin University Faculty of Science Department of Mathematics
2021.04 - 2024.03
Country：Japan
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Waseda University Lecturer
2018.04 - 2021.03
Country：Japan
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Waseda University
2017.04 - 2018.03
Professional Memberships 【 display / non-display 】
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THE MATHEMATICAL SOCIETY OF JAPAN
2014.10
Papers 【 display / non-display 】
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Currents on cusped hyperbolic surfaces and denseness property Reviewed
Dounnu Sasaki
Groups, Geometry, and Dynamics 16 ( 3 ) 1077 - 1117 2022.10
Authorship：Lead author Language：English Publishing type：Research paper (scientific journal) Publisher：European Mathematical Society - EMS - Publishing House GmbH
DOI： 10.4171/ggd/688
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Subset currents on surfaces Reviewed
Dounnu Sasaki
Memoirs of the American Mathematical Society 278 ( 1368 ) 2022.07
Authorship：Lead author Language：English Publishing type：Research paper (scientific journal) Publisher：American Mathematical Society (AMS)
<p>Subset currents on hyperbolic groups were introduced by Kapovich and Nagnibeda as a generalization of geodesic currents on hyperbolic groups, which were introduced by Bonahon and have been successfully studied in the case of the fundamental group of a compact hyperbolic surface . Kapovich and Nagnibeda particularly studied subset currents on free groups. In this article, we develop the theory of subset currents on , which we call subset currents on . We prove that the space of subset currents on is a measure-theoretic completion of the set of conjugacy classes of non-trivial finitely generated subgroups of , each of which geometrically corresponds to a convex core of a covering space of . This result was proved by Kapovich-Nagnibeda in the case of free groups, and is also a generalization of Bonahon’s result on geodesic currents on hyperbolic groups. We will also generalize several other results of them. Especially, we extend the (geometric) intersection number of two closed geodesics on to the intersection number of two convex cores on and, in addition, to a continuous -bilinear functional on .</p>
DOI： 10.1090/memo/1368
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An intersection functional on the space of subset currents on a free group Reviewed
Dounnu Sasaki
GEOMETRIAE DEDICATA 174 ( 1 ) 311 - 338 2015.02
Authorship：Lead author Language：English Publishing type：Research paper (scientific journal) Publisher：SPRINGER
Kapovich and Nagnibeda introduced the space of subset currents on a free group of rank , which can be thought of as a measure-theoretic completion of the set of all conjugacy classes of finitely generated subgroups of . We define a product of two finitely generated subgroups and of by the sum of the reduced rank over all double cosets , and extend the product to a continuous symmetric -bilinear functional . We also give an answer to a question presented by Kapovich and Nagnibeda. The definition of originates in the Strengthened Hanna Neumann Conjecture, which has been proven independently by Friedman and Mineyev, and can be stated as follows: for any finitely generated subgroups and of the inequality holds. As a corollary to our theorem, this inequality is generalized to the inequality for subset currents.
Books and Other Publications 【 display / non-display 】
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数学セミナー 2020年４月号（証明が書けない／→まずは証明のお作法を身につけよう）
（証明が書けない／→まずは証明のお作法を身につけよう）
日本評論社 2020.03
Scientific Research Funds Acquisition Results 【 display / non-display 】
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Currents on hyperbolic surfaces and non-cocompact actions
Grant number：21J01271 2021.04 - 2024.03
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for JSPS Fellows Grant-in-Aid for JSPS Fellows
Grant amount：\3640000 （ Direct Cost: \2800000 、 Indirect Cost：\840000 ）
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Currents on cusped hyperbolic surfaces
Grant number：19K14539 2019.04 - 2022.03
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Early-Career Scientists
Sasaki Dounnu
Authorship：Principal investigator Grant type：Competitive
Grant amount：\2080000 （ Direct Cost: \1600000 、 Indirect Cost：\480000 ）
The space GC(S) of geodesic currents on a hyperbolic surface S can be considered as a completion of the set of weighted closed geodesics on S when S is compact, since the set of rational geodesic currents on S, which correspond to weighted closed geodesics, is a dense subset of GC(S). I proved that even when S is a cusped hyperbolic surface with finite area, GC(S) has the denseness property of rational geodesic currents, which correspond not only to weighted closed geodesics on S but also to weighted geodesics connecting two cusps. In addition, I proved that the space of subset currents on a cusped hyperbolic surface, which is a generalization of geodesic currents, also has the denseness property of rational subset currents.
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双曲群上のサブセットカレントの研究
Grant number：16J02814 2016.04 - 2018.03
日本学術振興会 科学研究費助成事業 特別研究員奨励費
佐々木 東容
Authorship：Principal investigator Grant type：Competitive
Grant amount：\1700000 （ Direct Cost: \1700000 ）
サブセットカレントはThurstonの導入した測度付き測地線層（measured geodesic lamination）の概念の一般化である測地カレントのさらなる一般化である．幾何学的対象を測度という解析対象と関連付けることによって，曲面上の幾何学的対象の不変量を解析的視点から分析することが可能となる．サブセットカレントは測度付き測地線層に比べて扱う対象が格段に多くなっている．
本研究者は今年度サブセットカレントの研究をコンパクト双曲得曲面から非コンパクトな双曲曲面にまで拡大することを目指した．目標であった，稠密性定理および線形連続拡張に関する明確な結果は得られなかったものの，コンパクト双曲曲面の場合の結果を一部拡張することには成功した．
まず，非コンパクトな双曲曲面についてもコンパクトな場合と同様にサブセットカレント空間は定義でき，位相空間の観点から見てほぼ同様の性質を持つことがわかった．また，曲面の有限生成部分群からサブセットカレントを定義することが可能であるという点ではコンパクトな場合と全く同様であった，しかしながら，決定的に異なる点として，普遍被覆の無限遠に放物型極限点が存在することが挙げられる．放物型極限点を2個以上の有限個用意することによって，サブセットカレントが定義できることがわかった．簡単な例としては，曲面上のカスプを両端にもつ無限測地線がサブセットカレントを対応付ける（厳密には測地カレント）．また，稠密性定理の観点からも考察が必要であると考えている．
線形連続拡張については現状ではいくつかの点で否定的見解が得られている．連続拡張をする上での障害が見つかっており，不変量の捉え方を抜本的に見直すことが求められる．特に交点数については無限に発散することを解消する手段が必要であるとみている．
以上のように，限定的ではあるが，着実な成果を挙げることができた．