Statements of Research accomplishment
I classify my major scientific works into several groups by their themes.
Early WorksI was guided by my supervisor Takushiro Ochiai to study nonlinear partial diﬀerentialequations (PDE) on manifolds, then I chose my research theme as analytic aspects of YangMills connections, which is an important examples of nonlinear PDE on manifolds.
In [1] I generalized Uhlenbeck’s convergence result of Yang-Mills connections on 4-manifolds
to higher dimensional manifolds. Namely a sequence of Yang-Mills connections on an ndimensional compact manifold with L2 curvature bounds, after taking a subsequence, converges outside a subset S with finite (n − 4)-dimenisonal Hausdorﬀ measure. The estimateof dimension of S is the best possible, as one can construct examples on product manifolds
X 4 × Y n−4 which converge outside finitely many points in X 4 -direction. [1] and [2] are mymaster thesis.
In [3] written by myself, and [4] written with Bando and Kasue, I proved a convergence
result for a sequence of compact 4-manifolds with Einstein metrics whose diameter, inverse ofvolume, and Euler number are uniformly bounded. Then it has a subsequence converging toan orbifold with Einstein metric outside finitely many points. Moreover if we rescale metrics
around a point where the convergence fails, we obtain a noncompact manifold with Einstein
metric, which is approximated by a quotient of the Euclidean space by a finite subgroup of theorthogonal group. Analytic aspects of Einstein metrics are very similar to those of Yang-Mills
connections, but we combine them with Gromov-Hausdoﬀ convergences, which were hot topics
in Riemannian geometry. I learned latter from Kenji Fukaya (who was an assistant professor
at University of Tokyo at that time), and later from the collaborator Kasue.
Noncompact spaces with Einstein metrics appearing as the limit in the rescaled sequence
above are called ALE (asymptotically locally Euclidean). The case when Einstein metrics
are hyper-Kählerian is the most important. In [8] I studied moduli spaces of anti-self-dual
connections (instantons) on such an ALE space. In particular, I showed that they are again
ALE if their dimension are 4.
Examples and classification of hyper-Kähler ALE spaces were obtained by Kronheimer
slightly earlier. I, together with him, continued study of moduli spaces of instantons on hyperKähler ALE spaces, and obtained their description in terms of matrices [10]. This description
is a generalization of a similar description of moduli spaces of instantons on R4 due to AtiyahDrinfeld-Hitchin-Manin, the famous ADHM description.
After [10], I changed my main interest from analytic aspects of Yang-Mills connections toalgebraic aspects of their moduli spaces.
Quiver varieties and representation theory
The ADHM description in [10] is understood in the framework of representation theoryof aﬃne ADE quivers. It turns out moduli spaces above, in the ADHM description, makesense for arbitrary quivers, hence I named the resulted spaces quiver varieties. Motivatedby earlier works of Ringel and Lusztig, I constructed irreducible integrable representationsof Kac-Moody Lie algebras on spaces of constructible functions on quiver varieties [14], andlater on homology groups of quiver varieties [21]. These constructions are very diﬀerent from1

usual geometric constructions (i.e., Borel-Weil theory) of representations of Lie algebras. It isimportant to consider various quiver varieties simultaneously, as a single quiver variety justcorresponds to a weight space of the representation. It is contradictory to a usual intuition
that Lie algebra representations appear as diﬀerential of Lie group representations, which arecontinuous symmetry, as my construction uses disconnected spaces essentially.
In [22], I obtained a similar construction of a representation of the Heisenberg algebra on
homology groups of Hilbert schemes of points on a complex algebraic surface. The same result
was obtained by Grojnowski around the same time. Hilbert schemes of points had been studied
by algebraic geometers, but this work gave a completely new way of looking at them.
I continued study of quiver varieties for several years. Papers [26,27,29,30,32,36] and [54]are about quiver varieties, but let me explain just two main papers. In [26] I constructed
representations of quantum loop algebras on equivariant K-theory of quiver varieties. Thisconstruction led me a definition of Kazhdan-Lusztig type polynomials by intersection cohomology groups of torus fixed point loci of quiver varieties. These polynomials were studied in[32], and in particular, I obtained their purely combinatorial characterization, which gave analgorithm for computation.
Computation of q-characters
The algorithm for analog of Kazhdan-Lusztig type polynomials is separated into two parts,and both are rather complicated, and handle huge data. Therefore even the first step, i.e.,
computation of t-analog of q-characters of ℓ-fundamental representations of quantum loop
algebras, requires use of supercomputer for the actual computation.
I started this project in 2002, but it finished in 2006 [47], as I waited a renovation of asupercomputer in Kyoto University so that it could handle enough data.
As a hope to get a better understanding of q-characters, I introduced a crystal structure, inthe sense of Kashiwara, on monomials appearing in q-characters [34]. This initial motivation
was not achieved, but the monomial realization of crystal bases has been studied in various
diﬀerent contexts, in my work with Hernandez [45], and others. Recent studies by KanakuboNakashima gave a unexpected link with cluster algebras.
Crystal bases of extremal weight modules
Equivariant K-groups of quiver varieties, as representations of quantum loop algebras,
have diﬀerent, completely algebraic realization found by Kashiwara. His construction simultaneously gave crystal bases, hence I was interested in Kashiwara’s approach. He posed aconjecture on their structure. I proved it for untwisted case in [33], and with Beck [38] infull generality. [38] also contained a proof of a conjecture by Lusztig on the cell structure ofquantum loop algebras. All techniques used in [33], [38] are purely algebraic unlike my otherworks.
Instanton counting and Donaldson invariants
Quiver varieties associated with Jordan quiver are original ADHM description, but I found
one new feature that they have natural resolution of singularities from general result for quivervarieties [15]. Later I realized that the resolutions are moduli spaces of framed sheaves on theprojective plane (later called Gieseker spaces by other people). In 2002 Nekrasov considered
the integration of the constant function 1 on the Gieseker spaces in the sense of equivariant cohomology groups, and conjectured that the leading part of the integration with respect2

to equivariant variables is given by period of certain hyperelliptic curves, called the SeibergWitten prepotential. The Seiberg-Witten prepotential was found 1994 in the study of gauge
theory in 4-dimension, but its derivation remained mysterious for a while, as it is based on physical argument. Nekrasov’s conjecture provided us a mathematical approach, and it was provedin my joint work with Yoshioka [40], and independently by Nekrasov-Okounkov, BravermanEtingof by completely diﬀerent methods.
Nekrasov’s framework gave us a better approach to Donaldson invariants for 4-manifolds,
whose computation was usually very diﬃcult. In [46] Göttsche, Yoshioka and I gave a proofof wallcrossing formula of Donaldson invariants for a 4-manifold X with b+ = 1, when X is acomplex projective surface. This result was proved earlier by Göttsche, assumping KotschieckMorgan conjecture. Our proof is based on earlier works: wallcrossing formula is given bycertain integral over Hilbert schemes of points on X (Ellingsrud-Göttsche), and this integralis ‘universal’ in X (Ellingsrud-Göttsche-Lehn). These results, in particular, imply that itis enough to compute wallcrossing formula for projective toric surfaces. They are expressedby combinatorial formula by localization theorem of equivariant cohomology groups, but one
needs to set the equivariant variables zero. The expected result means that there is a myracleof cancellation of poles in the localization formula.
In [58], a joint work with Göttsche and Yoshioka, I proved the so called Witten conjecture,
relating Donaldson invariants and Seiberg-Witten invariants for a complex projective surface.
By a result of Mochizuki, these invariants are related via certain integral over Hilbert schemesof points. We then use a similar technique as in our proof of wallcrossing formula, we computethe integral.
Perverse coherent sheaves on blow-up
This is a side project of instanton counting with Yoshioka. Perverse coherent sheaves onb The
blow-up are certain sheaves which live between a complex surface X and its blow-up X.
definition is motivated by a work of Bridgeland, who considered similar sheaves for a 3-foldflop. We studied moduli spaces of perverse coherent sheaves, and showed that they connect ab by a sequence of wallcrossing
moduli space of stable torsion free sheaves on X and that on X
[53]. There is a quiver description when X is the aﬃne plane C2 [52]. Using a sequence ofc2 . This method
wallcrossing, we compare Nekrasov partition function on C2 and the blow-up C
is more powerful than an earlier approach used in [40], as we can handle the partition functions
with matters [57].
Instanton moduli spaces and W-algebras
Motivated by a work of physicists, Alday-Gaiotto-Tachikawa (AGT), I, together with
Braverman and Finkleberg [68], constructed a representation of the W-algebra on the direct sum of equivariant intersection cohomology groups of Uhlenbeck partial compactificationof instanton moduli spaces on R4 (Uhlenbeck spaces, in short) of type ADE, where the direct
sum is taken over all instanton numbers. This is based on an earlier work by Maulik-Okounkov,
which proves a similar result for type A case. More precisely they consider equivariant cohomology groups of Gieseker spaces, which are resolution of singularities of Uhlenbeck spaces.
Since Gieseker spaces are available only in type A, a general case requires a new technique: We
reduced general ADE to the A2 -case by studying hyperbolic restriction on Uhlenbeck spaces.

3A mathematical definition of Coulomb branches of 3-dimensional SUSY gauge
theories
This is a on-going project. For a given pair of a compact Lie group G and its quaternionic
representation M, physicists consider a SUSY (supersymmetric) gauge theory in dimension
3, and associate its Coulomb branch, which is a noncompact hyperKähler manifold possiblywith singularities. The physical definition involves quantum corrections, which are diﬃcult tojustify mathematically. In [68], I propose an approach to a rigorous mathematical definition, as
an aﬃne algebraic symplectic variety. This approach was worked out in detail in [69], writtenwith Braverman, Finkelberg when M is of a form N ⊕ N∗ . Then I determine Coulomb branchesof quiver gauge theories of type ADE with known spaces, the so-called (generalized) slices
in the aﬃne Grassmannian in [70] (with Braverman-Finkelberg). In appendix of [70] (withBraverman-Finkelberg-Kamnitzer-Kodera-Webster-Weekes) we study quantization of Coulomb
branches of quiver gauge theories, and relate them to shifted Yangian. In [72], together withTakayama, I determine Coulomb branches of quiver gauge theories of aﬃne type A. They
are Cherkis bow varieties. In order to show this statement, we study Cherkis bow varieties
as aﬃne algebraic varieties, using quiver type description. In [73], together with Kodera, I
study quantized Coulomb branches of Jordan quiver gauge theories, and identify them withcyclotomic rational Cherednik algebras.

4