Organization
Graduate School of Mathematics Division of Mathematics Natural Mathematics Professor
Graduate School
Graduate School of Mathematics
Undergraduate School
School of Science Department of Mathematics
Title
Professor
ISHII Akira
Degree 1
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博士（理学） （ 2000.1 京都大学 ）
Committee Memberships 1
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日本数学会 代数学分科会運営委員会委員
2023.4
Committee type：Academic society
Papers 19
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Dimer models and group actions Open Access
Ishii, A; Nolla, A; Ueda, K
MATHEMATISCHE ZEITSCHRIFT Vol. 306 ( 1 ) 2024.1
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McKay correspondence for some finite reflection groups
Vol. 2023 page： 118 - 127 2024.1
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G-constellations and the maximal resolution of a quotient surface singularity Open Access
Ishii, A
HIROSHIMA MATHEMATICAL JOURNAL Vol. 50 ( 3 ) page： 375 - 398 2020.11
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Extended McKay correspondence for quotient surface singularities Open Access
Akira Ishii, Iku Nakamura
The Quarterly Journal of Mathematics Vol. 70 ( 2 ) page： 395 - 408 2019.6
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Dimer models and crepant resolutions Reviewed
Akira Ishii, Kazushi Ueda
Hokkaido Math. J. Vol. 45 ( 1 ) page： 1--42 2016
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Dimer models and the special McKay correspondence Reviewed Open Access
Akira Ishii, Kazushi Ueda
Geom. Topol. Vol. 19 ( 6 ) page： 3405--3466 2015
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The special McKay correspondence and exceptional collections Reviewed
Akira Ishii, Kazushi Ueda
Tohoku Math. J. Vol. 67 ( 4 ) page： 585--609 2015
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On G/N-Hilb of N-Hilb Reviewed Open Access
Akira Ishii, Álvaro Nolla de Celis, Yukari Ito
Kyoto J. Math. Vol. 53 ( 1 ) page： 91--130 2013
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A note on derived categories of Fermat varieties Reviewed
Akira Ishii, Kazushi Ueda
Derived categories in algebraic geometry, EMS Ser. Congr. Rep. page： 103--110 2012
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A note on consistency conditions on dimer models Reviewed Open Access
Akira Ishii, Kazushi Ueda
RIMS Kôkyûroku Bessatsu Vol. B24 page： 143--164 2011
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Stability conditions on An-singularities Reviewed Open Access
Akira Ishii, Kazushi Ueda, Hokuto Uehara
J. Differential Geom. Vol. 84 ( 1 ) page： 87--126 2010
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On moduli spaces of quiver representations associated with dimer models. Reviewed Open Access
Akira Ishii and Kazushi Ueda
RIMS Kôkyûroku Bessatsu Vol. B9 page： 127--141 2008
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Autoequivalences of derived categories on the minimal resolutions of An-singularities on surfaces. Reviewed Open Access
Akira Ishii, Hokuto Uehara
J. Differential Geom. Vol. 71 ( 3 ) page： 385--435 2005
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Representation moduli of the McKay quiver for finite abelian subgroups of SL(3,C). Reviewed
Akira Ishii
Clay Math. Proc. Vol. 3 page： 227--237 2004
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Flops of G-Hilb and equivalences of derived categories by variation of GIT quotient. Reviewed
Akira Ishii, Alastair Craw
Duke Math. J. Vol. 124 ( 2 ) page： 259--307 2004
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On the McKay correspondence for a finite small subgroup of GL(2,C) Reviewed
Akira Ishii
J. Reine Angew. Math. Vol. 549 page： 221--233 2002
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Versal deformation of reflexive modules over rational double points Reviewed
Akira Ishii
Math. Ann. Vol. 317 ( 2 ) page： 239--262 2000
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Semi-universal family of reflexive modules over a rational double point of type A Reviewed
Akira Ishii
Lecture Notes in Pure and Appl. Math. Vol. 179 page： 65--77 1996
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On the moduli of reflexive sheaves on a surface with rational double points Reviewed
Akira Ishii
Math. Ann. Vol. 294 ( 1 ) page： 125--150 1992
Books 1
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McKay correspondence, mutation and related topics
伊藤 由佳理, 石井 亮 (数学), 伊山 修（ Role： Edit）
Mathematical Society of Japan 2023 （ ISBN:9784864970983 ）
Presentations 6
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Derived McKay correspondence for real reflection groups of rank three Invited
Akira Ishii
2025.11.22 Kenta Hashizume, Akinari Hoshi, Hideo Kojima, Takuzo Okada, Kazuhiko Yamaki
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Derived McKay correspondence for real reflection groups of rank three Invited International conference
Akira Ishii
McKay correspondence, Dimer model and Derived category 2025.4.7 ukari Ito, Osamu Iyama, Yusuke Nakamura, Sho Tanimoto
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On the McKay correspondence for some reflection groups in dimension three Invited International conference
Akira Ishii
Perspectives in Tilting Theory and Related Topics 2025.2.17 Sota Asai, Aaron Chan, Osamu Iyama, Mayu Tsukamoto
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On the McKay correspondence for some reflection groups 1 Invited International conference
Akira Ishii
Japanese-European Symposium on Symplectic Varieties and Moduli Spaces 8th edition 2024.9.17
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On the McKay correspondence for some reflection groups 2 Invited International conference
Akira Ishii
Japanese-European Symposium on Symplectic Varieties and Moduli Spaces 8th edition 2024.9.19
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On the McKay correspondence for some reflection groups Invited International conference
Akira Ishii
Workshop on Tropical Geometry, Singularity theory, and Algebraic Geometry 2024.9.12 Yoshinori Gongyo, Yusuke Nakamura, Kohei Sato, JuAe Song
KAKENHI (Grants-in-Aid for Scientific Research) 4
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Grant number：25K06918 2025.4 - 2030.3
科学研究費助成事業 基盤研究(C)
石井 亮
Authorship：Principal investigator
Grant amount：\4680000 （ Direct Cost: \3600000 、 Indirect Cost：\1080000 ）
McKay 対応は，有限群G による商特異点の特異点解消の幾何学と，群Gの表現論を結びつけるものである．様々な状況で興味深い現象が観察されてきているが，本研究では，特に鏡映群や複素鏡映群と呼ばれる群による商ついて，判別式因子と呼ばれる超曲面を用いた商空間の幾何学的と群の表現の間にある関係を見出すことや，関連するモジュライ空間の記述を行うことなどを目標として研究を行う．
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McKay correspondence and derived category
Grant number：19K03444 2019.4 - 2025.3
Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)
Ishii Akira
Authorship：Principal investigator
Grant amount：\4290000 （ Direct Cost: \3300000 、 Indirect Cost：\990000 ）
We published a paper on the moduli of G-constellations for a subgroup G of GL(2) and a paper on consistent dimer models with group actions. We studied exceptional collections on the Hirzebruch surface Σ2 with Okawa and Uehara, and classified exceptional collections up to spherical twists and mutations of exceptional collections. This result is submitted. With a graduate student Nimura, we studied the derived McKay correspondence for real reflection groups of rank 3 and verified a conjecture on the existence of a certain semiorthogonal decomposition.
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Existence of higher dimensional crepant resolutions and generlization of the McKay correspondence
Grant number：18K03209 2018.4 - 2024.3
Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)
ITO YUKARI
Authorship：Coinvestigator(s)
Ito characterizes the exceptional set of resolutions of quotient singularities of a finite group G and their corresponding irreducible representations when resolutions of singularities are obtained. Ishii investigates the Dimer models of quotient singularities and exceptional corrections on Hirzebruch surfaces. Iyama gave an invited lecture at the International Congress of Mathematicians (ICM) in 2018 and conducted research on triangulated categories, cluster categories, and AR theory. Additionally, international research meetings were held in 2018, 2020, and 2023, and in April 2023, a collection of papers related to this research project titled "McKay correspondence, tilting theory and related topics" was published as Advanced Studies in Pure Mathematics 88.
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Singularities and derived catetories
Grant number：15K04819 2015.4 - 2019.3
Ishii Akira
Authorship：Principal investigator
Grant amount：\4680000 （ Direct Cost: \3600000 、 Indirect Cost：\1080000 ）
(1) We studied some globalisation of the so-called special McKay correspondence for a two-dimensional quotient singularity with Iku Nakamura and obtained a description of it.
(2) For a two-dimensional quotient singularity, we characterized those resolutions dominated by the so-called maximal resolution as the moduli spaces of G-constellations.
(3) We studied dimer models with group actions with Alvaro Nolla and Kazushi Ueda and constructed non-commutative crepant resolutions of the quotient of an affine topic variety by the symmetry of a lattice polygon.