# Curriculum Vitae

 Nationality Japan Date of birth 30th November 1962 Place of birth Tokyo, Japan Present Position Professor, Research Institute for Mathematical Sciences Kyoto University e-mail address nakajima@kurims.kyoto-u.ac.jp web page http://www.kurims.kyoto-u.ac.jp/~nakajima

## Education

 1981--1983 University of Tokyo General Education (Science - I) 1983--1985 University of Tokyo Department of Mathematics 1985--1987 Graduate School, University of Tokyo Department of Mathematics 1991 May 27 Ph. D. in Science Graduate School of Science, University of Tokyo

 1987--1992 University of Tokyo, Research Assistant Department of Mathematics 1992--1995 Tohoku University, Assistant Professor Mathematical Institute 1995--1997 University of Tokyo, Assistant Professor Department of Mathematical Science 1997--2000 Kyoto University, Assistant Professor Department of Mathematics 2000 Dec.-- Kyoto University, Professor -- 2008 Sep. Department of Mathematics 2008 Oct.-- Kyoto University, Professor Research Institute for Mathematical Sciences

## Ph.D. students supervised

 2005 Hiroshi Iritani Kyoto Univ. 2008 Daisuke Yamakawa Kyoto Univ. 2009 Kentaro Nagao Kyoto Univ. 2012 Yoshiyuki Kimura Kyoto Univ. 2015 Jaeyoo Choy Kyoto Univ. (JSPS RONPAKU program) 2016 Yuuya Takayama Kyoto Univ.

## Visiting positions

 1998-99 Institute for Advanced Study (Princeton, USA) Member School of Mathematics 2002 Centre for Advanced Study (Oslo, Norway) 2007-08 Institute for Advanced Study (Princeton, USA) Member School of Mathematics

## Awards

 1997 The geometry Prize of the Mathematical Society of Japan 2000 The spring Prize of the Mathematical Society of Japan 2003 American Mathematical Society, Cole Prize in Algebra 2006 JSPS prize 2014 Japan Academy Prize

## Talks

 2002 ICM 2002 Beijing Plenary address 2010 Hardy Lectureship, London Mathematical Society

## Current area of mathematical interest and research

Representation Theory, Geometry

## Editor

 2000 Oct.-- JAPANESE JOURNAL OF MATHEMATICS 2001 May--2010 Dec. Advances in Mathematics 2005 Jan.--2010 Dec. Transformation Groups 2005 Jan.-- 2009 Jan. Representation Theory 2005 May -- Duke Mathematical Journal 2011 Jan. -- Les Publications Math\'ematiques de l'IH\'ES

## Service

 2004 Apr. -- 2005 Mar. chairperson of Math. Dept. 2009 Apr. -- 2013 Mar. The chief of the library at RIMS 2010 Apr. -- 2011 Mar. chairperson of Graduate School, Mathematics Sciences Course

## Books

1. Hisenkeimondai to Fukusokika (in Japanese), Iwanami Shoten, 1999.
2. Lectures on Hilbert schemes of points on surfaces, AMS Univ. Lecture Series, 1999

## Papers

1. Compactness of the moduli space of the Yang-Mills connections in higher dimensions, J. Math. Soc. Japan 40 (1988), 383--392.
Motivated by Professor Yukio Matsumoto's lectures on the famous paper by Donaldson, I applied the technique of Schoen's paper, which I had studied in seminars, to Yang-Mills connections. I submit the result as a report for Matsumoto's lectures. Then my supervisor Professor Takushiro Ochiai suggested me to write it as a research paper. This is the paper. It is natural that many analytical results are obtained in parallel for harmonic maps and Yang-Mills connections. This result was known in particular to Uhlenbeck, and in fact there was a written manuscript on it by her, as I knew when I met her later.
Later Tian wrote a paper analyzing structures of singular sets.
2. Removable singularities for Yang-Mills connections in higher dimensions, J. Fac. Sci. Univ. Tokyo 34 (1987), 299--207. Link to Univ. of Tokyo Repository.
3. Hausdorff convergence of Einstein 4-manifolds, J. Fac. Sci. Univ. Tokyo 35 (1988), 411--424. Link to Univ. of Tokyo Repository.
4. On a construction of coordinates at infinity on manifolds with fast curvature decay and maximal volume growth (with S.Bando and A.Kasue), Invent. Math. 97 (1989), 313--349.
5. Self-duality of ALE Ricci-flat $4$-manifolds and positive mass theorem, in Recent Topics in Differential and Analytic Geometry, Advanced Studies in Pure Math. 18-I (1990), 313--349.
6. Yang-Mills connections and Einstein-Hermitian metrics (with M.Itoh), in K\"ahler metrics and Moduli Spaces, Advanced Studies in Pure Math. 18-II (1990), 395--457.
7. Yau no trick (in Japanese), Suugaku 41 (1989), 253--258.
8. Moduli spaces of anti-self-dual connections on ALE gravitational instantons, Invent. Math. 102 (1990), 267--303.
9. Einstein-Hermitian connections on hyper-K\"ahler quotients (with T.Gocho), J. Math. Soc. Japan 44 (1992), 43--51.
10. Yang-Mills instantons on ALE gravitational instantons (with P.B.Kronheimer), Math. Ann. 288 (1990), 263--307.
11. Monopoles and Nahm's equations, in Einstein metrics and Yang-Mills connections, (1993) eds. Mabuchi, Mukai, Marcel Dekker, 193--211. pdf.
12. Einstein keiryou no syuusoku teiri to ALE kuukan (in Japanese), Suugaku 44 (1992), 133--146. Link to J-Stage.
13. Homology of moduli spaces of instantons on ALE spaces.I, J. of Differential Geometry, 40 (1994) 105--127. Link to JDG site.
citation search in SPIRES HEP
14. Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras, Duke Math. 76 (1994) 365--416.
15. Resolutions of moduli spaces of ideal instantons on $\mathbb R^4$, in Topology, Geometry and Field Theory, World Scientific (1994) 129--136.
16. Gauge theory on resolution of simple singularities and simple Lie algebras, Inter. Math. Res. Notices, 2 (1994) 61--74.
17. A convergence theorem for Einstein metrics and the ALE spaces (translation of 12), Amer. Math. Soc. Transl. 160 (1994) 79--94. preprint version. (Figures are the same as Japansese version [12], which can be found at J-Stage.)
18. Varieties associated with quivers, in Representation theory of algebras and related topics, CMS conference proceedings 19, AMS (1996) 139--157.
19. Hyper-K\"ahler structures on moduli spaces of parabolic Higgs bundles on Riemann surfaces, in Moduli of vector bundles, Lecture Notes in Pure and Appl. Math., 179, (1996), Marcel Dekker.
20. Gauge theory on resolution of simple singularities and affine Lie algebras, in Singularities and complex geometry (Beijing, 1994), 183--192, Amer. Math. Soc., Providence, RI, 1997
21. Quiver Varieties and Kac-Moody Algebras, Duke Math., 91, (1998), 515--560.
22. Heisenberg Algebra and Hilbert Schemes of Points on Projective Surfaces, Ann. of Math. 145, (1997) 379--388. Preprint version. alg-geom/9507012.
citation search in SPIRES HEP
23. Instantons and affine Lie algebras, in $S$-duality and mirror symmetry, Nucl. Phys. B (Proc. Suppl.) 46 (1996) 154--161. Preprint version. alg-geom/9510003.
citation search in SPIRES HEP
24. Kyokumenjyou no ten no Hilbert gaikei to Heisenberg daisuu (in Japanese), Suugaku 50, (1998) 385--398.
25. McKay correspondence and Hilbert schemes in dimension three (with Yukari Ito), Topology 39 (2000), 1155--1191. Link to Topology. Preprint version. math.AG/9803120.
citation search in SPIRES HEP
26. Quiver varieties and finite dimensional representations of quantum affine algebras, J. Amer. Math. Soc. 14 (2001), 145-238. Link to AMS; math.9912158.
27. Reflection functors for quiver varieties and Weyl group actions, Math. Ann. 327 (2003), 671--721. Link to Mathematische Annalen, (longer) preprint version.
28. Ebira tayoutai to Ryousi affine kan (Quiver varieties and quantum affine algebras) (in Japanese), Suugaku 52 (2000), 337--359.
29. $t$-analogue of the $q$-characters of finite dimensional representations of quantum affine algebras, in Physics and Combinatorics'', Proceedings of the Nagoya 2000 International Workshop, World Scientific, 2001, 195--218. Preprint version. math.0009231.
30. Quiver varieties and tensor products, Invent. Math., 146 (2001), 399--449. Link to Inventiones Mathematicae.; math.0103008.
31. Hilbert schemes of points on surfaces and Heisenberg algebras (translation of 24), Sugaku Expositions, 15 (2002), 207--222.
32. Quiver varieties and $t$--analogs of $q$--characters of quantum affine algebras, Ann. of Math. 160 (2004), 1057--1097 , PDF file from Ann. of Math.; math.QA/0105173.
33. Extremal weight modules of quantum affine algebras, Advanced Studies in Pure Math., 40 (2004), 343--369; math.QA/0204183.
34. $t$--analogs of $q$--characters of quantum affine algebras of type $A_n$, $D_n$, in Combinatorial and geometric representation theory (Seoul, 2001), 141--160, Contemp. Math., 325, Amer. Math. Soc., Providence, RI, 2003; math.QA/0204184.
35. $t$--analogs of $q$--characters of Kirillov-Reshetikhin modules of quantm affine algebras, Represent. Theory (elect.) 7 (2003), 259--274 , Link to Represent. Theory; math.QA/0204185.
36. Geometric construction of representations of affine algebras, in Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002), 423--438, Higher Ed. Press, Beijing, 2002; math.QA/0212401.
37. Convolution on homology groups of moduli spaces of sheaves on K3 surfaces, in Vector bundles and representation theory (Columbia, MO, 2002)'', 75--87, Contemp. Math. 322, 2003.
38. Crystal bases and two-sided cells of quantum affine algebras (with Jonathan Beck), Duke Math., 123 (2004), no. 2, 335--402 , Link to Duke Math.; math.QA/0212253.
39. Cells in quantum affine algebras, Algebra Colloquium 11 (2004), No. 1, 141--154. (Proceedings of the International Conference on Algebra, Suzhou 2002).
40. Instanton counting on blowup. I. $4$-dimensional pure gauge theory (with Kota Yoshioka), Invent. Math 162 (2005), no. 2, 313--355 , Link to Invent. Math; math.AG/0306198.
41. Lectures on instanton counting (with Kota Yoshioka), in Algebraic Structures and Moduli Spaces, CRM Proceedings \& Lecture Notes 38, AMS, 2004, 31--101; math.AG/0311058. (The preprint version was largely expanded in the printed version.)
42. Quiver varieties and quantum affine algebras (translation of 28), Sugaku Expositions, 19 (2006), no. 1, 53--78.
43. Crystal, canonical and PBW bases of quantum affine algebras, in Algebraic Groups and Homogeneous Spaces', Ed. V.B.Mehta, Narosa Publ House. 2007, 389--421. (Written in 2005 Apr.)
44. Instanton counting on blowup. II. $K$-theoretic partition funtion (with Kota Yoshioka), Transform. Groups 10 (2005), no. 3-4, 489--519 , Link to Transformation Group; math.AG/0505553.
45. Level $0$ monomial crystals (with David Hernandez), Nagoya Math. J. 184(2006), 85--153 , Link to Nagoya Math. J.; math.QA/0606174.
46. Instanton counting and Donaldson invariants (with Lothar G\"ottsche and Kota Yoshioka), J. of Differential Geometry, 80 (2008), 343--390; PDF from JDG site; preprint version math.AG/0606180.
47. $t$--analogs of $q$--characters of quantum affine algebras of type $E_6$, $E_7$, $E_8$, in Representation Theory of Algebraic Groups and Quantum Groups'', Progress in Mathematics, Vol. 284, 2011, 257--272, math.QA/0606637.
48. Instanton no kazoeage to Donaldson fuhen-ryo (Instanton counting and Donaldson invariants) (with Kota Yoshioka), (in Japanese), Suugaku, 59 (2007), 131--153.
49. Instanton counting and Donaldson invariants (with Kota Yoshioka), (translation of [48]) Sugaku Expositions, 23 (2010), no. 2, 189--212.
50. K-theoretic Donaldson invariants via instanton counting, (with Lothar G\"ottsche and Kota Yoshioka), Pure and Appl. Math. Quaterly, 5, No. 3 (2009) (Friedrich Hirzebruch special issue, part II), 1029--1111; Link to PAMQ site; prprint version math.AG/0611945.
51. Sheaves on ALE spaces and quiver varieties, Moscow Math. Journal, 7 (2007), No. 4, 699--722. Link to Online Journal, Errata
52. Perverse coherent sheaves on blow-up. I, a quiver description (with Kota Yoshioka), in Adv. Stu. in Pure Math. 61 (2011), 349--386. preprint version arXiv:0802.3120.
53. Perverse coherent sheaves on blow-up. II, wall-crossing and Betti numbers formula (with Kota Yoshioka), J. Algebraic Geom. 20 (2011), no. 1, 47--100. Link to Online Journal, arXiv:0806.0463.
54. Quiver varieties and branching, SIGMA, 5 (2009), 003, 37 pages. Link to Online Journal, arXiv:0809.2605.
55. Counting invariant of perverse coherent sheaves and its wall-crossing, (with Kentaro Nagao), IMRN, 2011, no. 17, 3885--3938. Link to Online Journal, arXiv:0809.2992.
56. Quiver varieties and cluster algebras, Kyoto J. Math. Volume 51, Number 1 (2011), 71-126. (Memorial Issue for the Late Professor Masayoshi Nagata). Link to online journal, arXiv:0905.0002.
57. Perverse coherent sheaves on blow-up. III. Blow-up formula from wall-crossing (with Kota Yoshioka), Kyoto J. Math. Volume 51, Number 2 (2011), 263--335. Link to online journal, arXiv:0911.1773.
58. Donaldson = Seiberg-Witten from Mochizuki's formula and instanton counting (with Lothar G\"ottsche and Kota Yoshioka), Publ. of RIMS, 47 (2011), No.1, 307--359. Link to Online Journal, arXiv:1001.5024.
After we had finished [46], [50], we glanced at Takuro Mochizuki's book and found that the same argument for a gauge theory with matter should show Witten's conjecture. It was at G\"ottsche's house at Trieste. Since we had developed the theory of perverse coherent sheaves on blowup to cover the gauge theory with matters, we started to consider. Before completing the calculation, I gave a talk at Univ. of Tokyo in December. We found that we overlooked resider at one of poles just one week before the talk. I reflected on my rush giving a talk before finishing the work, though Yoshioka warned me that it is not yet completed. Later we found that the pole plays an important role.
59. Handsaw quiver varieties and finite W-algebras, Moscow Mathematical Journal, 12 (2012), No.3, 633--666. Link to Online Journal, arXiv:1107.5073.
Since no volunteer gave talks in my postdoc seminars, I gave lectures on [BFFR] about finite W-algebras and Laumon spaces via handsaw quiver varieties. On the way, I found that I do not like the integral form in [BFFR]. Therefore I changed it and used the new one to prove character formulae for irreducible representations of W-algebras.
After posting the paper to arXiv, Losev pointed out that character formulas were proved earlier to him. So I refered his paper in ver.2. But my argument proved Kazhdan-Lusztig conjecture for type A independently, while Losev reduced the problem to KL conjecture. In this sense, the proof is completely different.
60. Quiver varieties and tensor products, II, in "Symmetries, Integrable Systems and Representations", Springer Proceedings in Mathematics \& Statistics Volume 40, 2013, pp 403--428. Link to Online Journal, arXiv:1207.0529.
The problem to understand tensor product in terms of quiver varieties has been with me since I had written [30]. In [30], I had used an algebraic definition to understand the tensor product when quiver is of finite type. The paper on stable envelop by Maulik-Okounkov gave a complete geometric understanding of the tensor product. In this paper I gave a framework to understand the stable envelop in perverse sheaves.
61. Refined Chern-Simons theory and Hilbert schemes of points on the plane, in 'Perspectives in Representation Theory', Contemporary Math., Volume 610, AMS, 2014, 305--331. preprint version, arXiv:1211.5821.
The result has been known for many years before I decided to write it.
62. Cluster algebras and singular supports of perverse sheaves, in "Advances in Representation Theory of Algebras", EMS Series of Congress Reports, 2014, 211--230. Link to Online Journal, arXiv:1301.5079.
63. A simple proof of the formula for the Betti numbers of the quasihomogeneous Hilbert schemes (with A. Buryak, B. L. Feigin), International Mathematics Research Notices, Volume 2015, Issue 13, 4708--4715, Link to Online Journal, preprint version, arXiv:1302.2789.
64. More lectures on Hilbert schemes of points on surfaces, preprint, arXiv:1401.6782.
65. Works of Yukinobu Toda, Suugaku, 66, No. 4, 414--421. (in Japanese)
66. Affine cellularity of quantum affine algebras, Journal of Algebra, Volume 441, 1 November 2015, Pages 601--608. published on line, preprint version, arXiv:1406.1298.
67. Instanton moduli spaces and W-algebras (with A. Braverman, M. Finkelberg), preprint, arXiv:1406.2381.
68. Towards a mathematical definition of Coulomb branches of $3$-dimensional $\mathcal N=4$ gauge theories, I, preprint, arXiv:1503.03676.
At the workshop "Warwick EPSRC Symposium: McKay correspondence, Orbifolds, Quivers'' on 2014 Sep., I heard a talk by Hanany on monopole formulas. Then I started to look for a definition of Coulomb branches to recover the formula. I first thought that the formula looks similar to the calculation of refined Donaldson-Thomas invariants of conifold by Nagao et al., hence I tried its generalization. But I realized that the use of the Chern-Simons functional gives rise a natural framework.
69. Towards a mathematical definition of Coulomb branches of $3$-dimensional $\mathcal N=4$ gauge theories, II (with A. Braverman, M. Finkelberg), preprint, arXiv:1601.03586.
70. Coulomb branches of $3d$ $\mathcal N=4$ quiver gauge theories and slices in the affine Grassmannian (with appendices by Alexander Braverman, Michael Finkelberg, Joel Kamnitzer, Ryosuke Kodera, Hiraku Nakajima, Ben Webster, and Alex Weekes), preprint arXiv:1604.03625.

## Editors

1. Bibunkikagaku no saisentan -- Surveys in Geometry, special edition, Baifuukan, 2005 (in Japanese)

## Others

1. Regularity of minimizing harmonic maps into certain Riemannian manifolds, Part 3 of the master thesis, Tokyo University, 1987. (Part 1 and 2 were published as [1], [2] respectively.)
2. Convergence of anti-self-dual metrics, unpublished preprint, 1992(?), pdf file
3. Instanons on ALE spaces and canonical bases, (in Japansese), Symposium on Representation Theory, Yamagata, Nov. 1992, pdf file or pdf with pictures
4. Morse theory on moduli spaces of instantons on ALE scalar-flat K\"ahler surfaces, unpublished preprint, 1992(?), pdf file
5. Jack polynomials and Hilbert schemes of points on surfaces, unpublished preprint, alg-geom/9610021.
6. Quiver varieties and finite dimensional representations of quantum affine algebras, (in Japanese), in Representations of Lie Groups and Noncommutative Harmonic Analysis', RIMS Kokyuroku 1124 (2000), 135--149. pdf
7. Lectures at the University of Hong Kong -- a Geometric Construction of Algebras, 54 pages , pdf file
8. Moduli of sheaves on blown-up surfaces, in Proceedings of RIMS Project 1999/2000, Algebraic Geometry and Integrable Systems related to String theory', RIMS Kokyuroku 1232 (2001), 29--33. postscript file, or link to Kyoto Univ. Research Information Repository
The detail of the proofs of results in Sections 1,2 can be found in [41]. The approach in Section 4 was wrong, and later corrected in [40] in the formulation by Nekrasov.
9. Quiver varieties and McKay correspondence -- Lectures at Hokkaido University, 2001 Dec. --, 49 pages , pdf file
10. Problems on Quiver varieties, The 50th Geometry Symposium, Hokkaido Univ. 2003 Aug. , pdf file
11. Appendix to `Absolutely indecomposable representations and Kac-Moody Lie algebras'' by W.Crawley-Boevey and M.Van den Bergh, Invent. Math. 155 (2004), 537--539. preprint version math.RA/0106009.
12. Book Review, N.Chriss and V.Ginzburg : Representation theory and complex geometry , Suugaku 54 (2002), 318--322.
13. Questions on provisional Coulomb branches of $3$-dimensional $\mathcal N = 4$ gauge theories, RIMS kokyuroku "Representation theory, harmonic analysis and differential equation", to appear. preprint version arXiv:1510.03908.