In this talk, an efficient rearrangement algorithm is introduced to the minimization of the positive principal eigenvalue under the constraint that the absolute value of the weight is bounded and the total weight is a fixed negative constant. Biologically, this minimization problem is motivated by the question of determining the optimal spatial arrangement of favorable and unfavorable regions for a species to survive. The method proposed is based on Rayleigh quotient formulation of eigenvalues and rearrangement algorithms which can handle topology changes automatically. Using the efficient rearrangement strategy, the new proposed method is more efficient than classical level set approaches based on shape and/or topological derivatives. The optimal results are explored theoretically and numerically under different geometries and boundary conditions.

Modern material sciences focus on studies on the microscopic scale. This calls for mathematical understanding of electronic structure and atomistic models, and also their connections to continuum theories. In this talk, we will discuss some recent works where we develop and generalize ideas and tools from mathematical analysis of continuum theories to these microscopic models. We will focus on macroscopic limit and microstructure pattern formation of electronic structure models.

Mathematical models have become essential tools in the increasingly quantitative world of biology. In some cases, mathematics can reveal patterns in pre-existing static biological data. In other cases, mathematical models can interact dynamically with experimental biology by providing insight into observed phenomena as well as generating novel and non-intuitive hypotheses to motivate experimental design. In this talk, I will present my recent work in both of these realms of mathematical biology.

I developed a mathematical model to discover inversion structural variants in human populations from pre-existing SNP data. Inversion chromosomal variants have long been considered important in understanding speciation because large inversions create reproductive isolation by suppressing recombination between inverted and normal chromosomes. Recent studies have identified many polymorphic inversion variants in human populations. Many of these inversions appear to have functional consequences and have been associated with genetic disorders and complex diseases, such as asthma. In addition, there is evidence some inversions may be under positive selection. I created a probabilistic mixture to identify putative inversion polymorphisms from phased haplotype data. By examining characteristic differences in allele frequencies around candidate inversion breakpoints, I partition the population into "normal" and "inverted" chromosomes. Predictions from my model are supported by previously validated inversions and represent a rich new source of candidate inversion polymorphisms.

In collaboration with experimental yeast biologists, I developed and validated a new model of prion transmission. Prion proteins are responsible for severe neurodegenerative disorders, such as bovine spongiform encephalopathy in cattle ("mad cow" disease) and Creutzfeldt-Jakob disease in humans. These diseases arise when a protein adopts an abnormal folded state and persists when that form self-replicates. While prion diseases are progressive, evidence in yeast suggests that this process can be reversed and eliminated. To understand the mechanistic basis of this "curing", I developed a stochastic model of prion dynamics that suggested a new theory for prion transmission. Results from my model guided experimental design, leading to new and non-intuitive insights about propagation of the abnormal fold through a population.

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.