In three very interesting and suggestive papers, H. Carayol introduced new aspects of complex geometry and Hodge theory into the study of non-classical automorphic representations -- in particular, those involving the totally degenerate limits of discrete series. This talk is based on two joint projects which aim to put Carayol's work into a more general context, while hewing to his over-riding theme of producing arithmetic structures on the cohomology of non-algebraic generalizations of Shimura varieties.
Employment contracts are central to many current policy debates. New York City is experimenting with rewarding teachers based on value added in the hope that it will improve performance. Compensation practices in the financial sector are often viewed as a contributing factor to the financial crisis, resulting in increased regulation. At the same time, there are continued calls to reduce the public sector and rely more on market forces. In the 2012 Leon Levy Lecture, W. Bentley MacLeod, Sami Mnaymneh Professor of Economics and Professor of International and Public Affairs at Columbia University, discusses two approaches to compensation: “pay for performance” and “performance for pay.” When preconditions for market supply of goods and services are satisfied, then pay for performance is effective. But when performance is difficult to measure, there is a need to reward performance with pay. MacLeod illustrates these ideas with examples taken from the management of Roman villas from the time of Columella and Pliny the Younger, and explains why the lack of effective management may be a key factor in the poor performance of schools and financial markets.
The interface between water and vacuum (governed by the "water wave equation"), and the interface between oil and water in sand (governed by the "Muskat equation") can develop singularities in finite time. Joint work with A. Castro, D. Cordoba, F. Gancedo, J. Gomez and M. Lopez.
A basic fact of algebraic graph theory is that the number of connected components in an undirected graph is equal to the multiplicity of the eigenvalue zero in the Laplacian matrix of the graph. In particular, the graph is disconnected if and only if there are at least two eigenvalues equal to zero. Cheeger's inequality and its variants provide an approximate version of the latter fact; they state that a graph has a sparse (non-expanding) cut if and only if there are at least two eigenvalues that are close to zero.