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.

A region of space is cloaked for a class of measurements if observers are not only unaware of its contents, but also unaware of the presence of the cloak using such measurements. One approach to cloaking is the change of variables scheme introduced by Greenleaf, Lassas, and Uhlmann for electrical impedance tomography and by Pendry, Schurig, and Smith for the Maxwell equations. They used a singular change of variables which blows up a point into the cloaked region. To avoid this singularity, various regularized schemes have been proposed. In this talk I present results related to cloaking via change of variables for the Helmholtz equation using the natural regularized scheme introduced by Kohn, Shen, Vogelius, and Weintein, where the authors used a transformation which blows up a small ball instead of a point into the cloaked region. I will discuss the degree of invisibility for a finite range or the full range of frequencies, and the possibility of achieving perfect cloaking. At the end of my talk, I will also discuss some results related to the wave equation in 3d.

Teichmueller curves are central objects in geometry and dynamics. They provide fertile connections between polygon billiards, flat surfaces and moduli spaces. A class of special Teichmueller curves come from a branched cover construction. Using them as examples, I will introduce an algebro-geometric technique to study Teichmueller curves. As applications, we prove Kontsevich-Zorich's conjecture on the non-varying property of Siegel-Veech constants and the sum of Lyapunov exponents for Abelian differentials in low genus. Moreover, we provide a novel approach to the Schottky problem of describing geometrically the locus of Jacobians among Abelian varieties. This talk will be accessible to a general audience.

Chronic wound healing is a staggering public health problem, affecting 6.5 million individuals annually in the U.S. Ischemia, caused primarily by peripheral artery diseases, represents a major complicating factor in the healing process. In this talk, I will present a mathematical model of chronic wounds that represents the wounded tissue as a quasi-stationary Maxwell material, and incorporates the major biological processes involved in the wound closure. The model was formulated in terms of a system of partial differential equations with the surface of the open wound as a free boundary. Simulations of the model demonstrate how oxygen deficiency caused by ischemia limit macrophage recruitment to the wound-site and impair wound closure. The results are in tight agreement with recent experimental findings in a porcine model. I will also show analytical results of the model on the large-time asymptotic behavior of the free boundary under different ischemic conditions of the wound.