There are, so far, two classes of moduli examples which naturally carry symmetric obstruction theories: Moduli of stable objects in the abelian category of a) coherent sheaves on a Calabi-Yau threefold; and b) representations of a quiver with relations given by a superpotential. In this talk, we present one more such class motivated by the work of Diaconescu: Moduli of stable objects in the abelian category of coherent twisted quiver sheaves on a fixed projective smooth curve. It will arise as a curve counting on a holomorphic symplectic quotient described by a quiver and as an application of quasimap construction. The quasimap construction is joint with I. Ciocan-Fontanine and D. Maulik. When the symplectic quotients are the Hilbert schemes of points on the plane, the study was carried by D.E. Diaconescu.

In this talk, we consider mathematically and computationally optimal control problems for stochastic partial differential equations under the Neumann boundary condition. The control objective is to minimize the expectation of a cost functional, and the control is of the deterministic, boundary value type. Mathematically, we prove the existence of an optimal solution and a Lagrange multiplier; we represent the input data in their Karhunen-Loµeve (K-L) expansions and deduce the deterministic optimality system of equations. Computationally, we approximate the finite element solution of the optimality system and estimate its error through the discretizations of both the probability space and the spatial space.

I will discuss some computational problems that arise naturally and frequently in algebraic geometry. Esepcially I will explain how to formulate correct notion of limits of algebraic varieties, algorithms for computing the limits, and the cost of such computations. I will give various applications to several different areas, including a recent result in birational geometry of moduli space of stable curves.