August 28 , 2008

Speaker: Andrei Zelevinsky (Northeastern University)
Title of talk: "Quivers with potentials and their representations"
Abstract: A quiver is a finite directed graph, that is, a finite set of vertices some of which are joined by arrows. A quiver representation assigns a finite-dimensional vector space to each vertex, and a linear map between the corresponding spaces to each arrow. The theory of quiver representations is a beautiful and well-developed field. A fundamental role in this theory is played by Bernstein-Gelfand-Ponomarev reflection functors associated to every source or sink of a quiver. In an ongoing joint work with Harm Derksen and Jerzy Weyman we extend these functors to arbitrary vertices. This construction is based on a framework of quivers with potentials; their representations are quiver representations satisfying relations of a special kind between the linear maps attached to arrows. The motivations for this work come from several sources: superpotentials in physics, Calabi-Yau algebras, cluster algebras. However no special knowledge will be assumed in any of these subjects, and the exposition should be accessible to graduate students.

September 18, 2008

Speaker: Xiao-Li Meng, Dept. of Statistics, Harvard University
Title of talk: Self-consistency: A General Recipe for Semi-parametric and Non-parametric Estimation
with Incomplete and Irregularly Spaced Data
Abstract: A common frustration in statistical estimation and, more generally, information recovery is when data are missing, incomplete or irregularly spaced (e.g., as with wavelets). Self-consistency is a general principle for handling such “bad” data problems in semi-parametric and non-parametric estimation and under an arbitrary loss function. It also provides a theoretical criterion to regulate and improve estimation procedures even when there is no missing data. Indeed, efficient estimation procedures, such as maximum likelihood estimation, are automatically self-consistent (asymptotically under square-loss). Conceptually, self-consistency is extremely appealing; it is essentially a mathematical formalization of the iteration of common-sense "trial-and-error" methods until no more improvement is possible.

Mathematically it is elegant, with one fixed-point equation to solve and a general projection theorem to establish its optimality. Practically, it is straightforward to program because it directly uses the regular/complete-data method for each iteration, much like the EM algorithm, which can be viewed as a version of self-consistency for maximum likelihood estimation. Its major disadvantage is that it can be computationally intensive. However, increasingly efficient (approximate) implementations are being discovered, such as for wavelet de-noising with hard and soft thresholding. This talk summarizes those these findings, based on joint work with Thomas Lee on wavelet applications and with Zhan Li on the theoretical foundation of the self-consistency principle.