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.

Since 2005, UCLA has run an internal NSF-funded summer REU program in applied mathematics for talented UCLA students and, more recently, students from other local colleges. The REU program has been very successful and is continuing to evolve into a better program. The unique feature of this program is that the undergraduate research projects are intrinscially tied into ongoing research carried out by the faculty and graduate students. I will discuss my involvement with the REU program for the last 3 years and present some of the projects I have mentored.The goal is to suggest a possible template for other schools to develop their own REU program in mathematics.