Finite dimensional representations of quantum affine algebras

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In the joint work with Jonathan Beck, I gave an explicit description of the crystal base of the upper triangular subalgebra of the quantum enveloping algebra U_q for arbitrary affine Lie algebras. Using this description, we proved that arbitrary extremal weight modules can be embedded into tensor products of level 0 fundamental representations. This result is an extension of a result obtained by Beck and me for a symmetric affine Lie algebra. Furthermore, we proved Lusztig's conjecture on cells of the crystal base of U_q. (The paper was published in Duke.Math.)I wrote the C program for the algorithm to compute q-characters of finite dimensional representation of quantum affine algebras. I performed the computation for the E_8 case, with which the previous program cannot deal. But the final answer was not obtained.On the other hand, I together with Kota Yoshioka, studied Nekrasov's partition, which is the generating function of the equivariant fundamental classes of moduli spaces of ins … More tantons on R^4, which are simplest examples of quiver varieties. It has an explicit combinatorial description in terms of Young tableaux, which we use to perform the experimental calculation via MAPLE with a super computer. Based on the calculation, we conjectured that the partition function satisfies the blowup formula, which determine the partition function recursively. We then proved this conjecture theoretically. As an application, we proved Nekrasov's conjecture affirmatively, i.e., the pole of the partition function with respect to the parameter ε_1, ε_2 is equal to the Seiberg-Witten prepotential. (The paper will appear in Invent.Math.)We gave a series of lectures on instanton counting and Seiberg-Witten prepotential, for which we published a lecture note. In it, we studied the genus 1 part of the partition function, and showed that it coincides with what was expected by physists.We further studied the relation between the partition function and Donaldson invariants with Lothar Goettsche. Donalson invariants are defined as integration of natural cohomology classes on moduli spaces of instantons on a 4-manifold. When b_+=1, the invariants depends on the choice of a Riemannian metric. The difference of invariants with respect to two metrics is given by the wall-crossing formula. We proved that the wall-crossing formula can be expressed by Nekrasov's partition function for rank 2 case. This is proved by studing the torus action on the moduli spaces on toric surfaces. Less