The integral homology of a compact Hermitian symmetric spaces (CHSS) is generated by the homology classes of its Schubert varieties. Most Schubert varieties are singular. In 1961 Borel and Haefliger asked: when can the homology class [X] of a singular Schubert variety be represented by a smooth subvariety Y of the CHSS?

Remarkably, the subvarieties Y with [Y] = [X] are integrals of a (linear Pfaffian) differential system. I will discuss recent work with Dennis The in which we give a complete list of those Schubert varieties X for which there exists a first-order obstruction to the existence of a smooth Y. This extends (independent) work of M. Walters, R. Bryant and J. Hong.

The sine qua non of our analysis is a new characterization of the Schubert varieties by a non-negative integer and a marked Dynkin diagram. The description generalizes the well-known characterization of the smooth Schubert varieties by subdiagrams of the Dynkin diagram associated to the CHSS.

I will illustrate the talk with examples.

By marrying theory and empirical work, we can elucidate the patterns of key macroscopic measures within ecosystems, develop explanations of variation in those patterns, and develop predictive models of responses to changing environments. Beyond that, we need to bridge the gaps across scales, from the ecological to the evolutionary, from the physical and biological to the cultural and ethical. Ultimately, only by providing such linkages between the microscopic and the macroscopic can we further the science needed to achieve a sustainable future. This lecture will explore new approaches from evolutionary game theory, with application to a range of applications from marine and other systems

Compressive sampling (CoSa) is a new and fast-growing field which addresses the shortcomings of traditional signal acquisition. Many methods in CoSa have been developed to reconstruct a signal from few samples when the signal is sparse with respect to some orthonormal basis. This talk will introduce the field of CoSa and present new results in compressive sampling from undersampled data for which the signal is not sparse in an orthonormal basis, but rather in some arbitrary dictionary. We will highlight numerous applications to which this framework applies and interpret our results in these settings. Since the dictionary need not even be incoherent, this work bridges a gap in the literature by showing that signal recovery is feasible for truly redundant dictionaries. We show that the recovery can be accomplished by solving an l1-analysis optimization problem, and that the condition we impose on the measurement matrix which samples the signal is satisfied by many classes of random matrices. We will also show numerical results which highlight the potential of the l1-analysis problem.

Modular forms and their higher-dimensional analogues are a priori analytically defined objects which happen to have many interesting relations to other subjects (such as number theory). In this lecture, I will review how algebraic geometry of modular curves (in mixed characteristics) was used for studying an important class of modular forms, and explain how geometry of the so-called Shimura varieties can be used for an analogous theory in higher dimensions. If time permits, I will also explain some interesting new application of such a theory to the study of torsion in the singular cohomology of Shimura varieties.