Last Update 2018/08/08
My research interests are geometric representation theory, algebraic geometry, and mathematical physics. At the moment I am studying algebraic structures (called Hall algebras structure) on the homology (G-theory, etc) of moduli stacks parametrizing objects in 2-dimensional CY categories. In particular, I have already defined, together with O. Schiffmann, a cohomological Hall algebra for the moduli stack of Higgs sheaves (nilpotent or not, semistable or not) on a smooth projective curve. When the curve is the projective line, such algebras are related to the cohomology of moduli spaces of torsion-free sheaves on resolutions of A-type toric singularities and Nakajima cyclic quiver varieties. This yields to applications in physics, most notably in four-dimensional supersymmetric gauge theories on A-type ALE spaces and the AGT conjecture for these theories.
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