Yuuji Tanaka

Addresses: Department of Mathematics, Faculty of Science, Kyoto University
Kitashirakawa Oiwake-cho, Sakyo-ku, Kyoto 606-8502, Japan

e-mail: y-tanaka (at) math.kyoto-u.ac.jp

Research interests: Gauge Theory, Geometric Analysis, Algebraic Geometry

More specifically, my current research interests are around the Vafa-Witten equations and the Kapustin-Witten ones on four-manifolds. They originated in an attractive theory called N=4 super (!) Yang-Mills theory in (Theoretical) Particle Physics, or more broadly in Superstring Theory. I've reached these objects with huge surprise and excitement from mathematical studies in higher-dimensional gauge theories such as the Donaldson-Thomas invariants and Spin(7)-instantons. Now I'm working on these from both algebraic and analytic aspects. (article in Case Studies)

Publications

(MathSciNet, zbMATH, arXiv)

Journal articles:

J. Gross, D. Joyce, and Y. Tanaka, Universal structures in ℂ-linear enumerative invariant theories, SIGMA 18 (2022), 068, 61pp, Special Issue on Enumerative and Gauge-Theoretic Invariants in honor of Lothar Göttsche on the occasion of his 60th birthday. (journal, arXiv)

C.-C. Liu, S. Rayan, and Y. Tanaka, The Kapustin-Witten equations and nonabelian Hodge theory, Eur. J. Math. 8 (2022), 23-41. (journal, arXiv)

Y. Tanaka and R. P. Thomas, Vafa-Witten invariants for projective surfaces I: stable case, J. Algebraic Geom. 29 (2020), 603-668. (journal, arXiv)

D. Joyce, Y. Tanaka, and M. Upmeier, On orientations for gauge-theoretic moduli spaces, Adv. Math. 362 (2020), 106957, 64pp. (journal, arXiv)

Y. Tanaka, A perturbation and generic smoothness of the Vafa-Witten moduli spaces on closed symplectic four-manifolds, Glasg. Math. J. 61 (2019), 471-486. (journal, arXiv)

Y. Tanaka, On the singular sets of solutions to the Kapustin-Witten equations and the Vafa-Witten ones on compact Kahler surfaces, Geom. Dedicata 199 (2019), 177-187. (journal, arXiv)

Y. Tanaka and R. P. Thomas, Vafa-Witten invariants for projective surfaces II: semistable case, Pure Appl. Math. Q. 13 (2017), 517-562, Special Issue in Honor of Simon Donaldson. (journal, arXiv)

Y. Tanaka, Some boundedness properties of solutions to the Vafa-Witten equations on closed 4-manifolds, Q. J. Math. 68 (2017), 1203-1225. (journal, arXiv)

Y. Tanaka, On the moduli space of the Donaldson-Thomas instanton, Extracta Math. 31 (2016), 89-107. (journal, arXiv)

Y. Tanaka, Stable sheaves with twisted sections and the Vafa-Witten equations on smooth projective surfaces, Manuscripta Math. 146 (2015), 351-363. (journal, arXiv)

Y. Tanaka, A removal singularity theorem of the Donaldson-Thomas instantons on compact Kahler threefolds, J. Math. Anal. and Appl. 411 (2014), 422-428. (journal, arXiv)

Y. Tanaka, A weak compactness theorem of the Donaldson-Thomas instantons on compact Kahler threefolds, J. Math. Anal. and Appl. 408 (2013), 27-34. (journal, arXiv)

Y. Tanaka, A construction of Spin(7)-instantons, Ann. Global Anal. Geom. 42 (2012), 495-521. (journal, arXiv)

Preprints:

M. Galdeano, D. Platt, Y. Tanaka, and L. Wang, Spin(7)-instantons on Joyce's first examples of compact Spin(7)-manifolds, October 2023, 70 pages. (arXiv)

N. Kuhn, O. Leigh, and Y. Tanaka, The blow-up formula for the instanton part of the Vafa-Witten invariants on projective surfaces, May 2022, 20 pages. (arXiv)

N. Kuhn and Y. Tanaka, A blowup formula for virtual enumerative invariants on projective surfaces, July 2021, 117 pages. (arXiv)

Y. Tanaka, Seiberg-Witten type equations on compact symplectic 6-manifolds, September 2017, 26 pages. (arXiv)

Talks:

Workshop on Algebraic Geometry, Tsinghua Sanya International Mathematics Forum, Sanya, December 2023.

Geometry of moduli spaces of Higgs bundles, Korea Institute for Advanced Study, Seoul, September 2023.

Physics and Special Holonomy, KITP, Santa Barbara, CA, March 2023.

Gauge Theory, Moduli Spaces and Representation Theory, Kashiwa 2023, In honor of the 60th birthday of Hiraku Nakajima, Kavli IPMU, University of Tokyo, Kashiwa, February, 2023. (slides)

list (including the above)

Graduate workshop:

British Isles Graduate Workshop III "Gauge theory with a view to higher dimensions", Jersey, 9-15 June 2019.

TCC course:

Introduction to Gauge Theory, Taught Course Centre, Trinity 2019.

Conference:

Online Workshop on Mirror symmetry and Related Topics, Kyoto 2021, Kyoto, 13-15 December 2021. (online)

As of 23 December 2023