The theory of minimal surfaces arose historically from work of J. L. Lagrange and physical observations of J. Plateau almost 200 years ago. Rigorous mathematical theory was developed in the 20th century. In more recent times the theory has found important applications to diverse areas of geometry and relativity. In this talk, which is aimed at a general mathematical audience, we will introduce the subject and describe a few recent applications of the theory.

Traveling waves in actin have recently been reported in many cell types. Fish keratocyte cells, which usually exhibit rapid and steady motility, exhibit traveling waves of protrusion when plated on highly adhesive surfaces. We hypothesize that waving arises from a competition between actin polymerization and mature adhesions for VASP, a protein that associates with growing actin barbed ends. We developed a mathematical model of actin protrusion coupled with membrane tension, adhesions and VASP. The model is formulated as a system of partial differential equations with a nonlocal integral term and noise. Simulations of this model lead to a number of predictions, for example, that overexpression of VASP prevents waving, but depletion of VASP does not increase the fraction of cells that wave. The model also predicts that VASP exhibits a traveling wave whose peak is out of phase with the F-actin wave. Further experiments confirmed these predictions and provided quantitative data to estimate the model parameters. We thus conclude that the waves are the result of competition between actin and adhesions for VASP, rather than a regulatory biochemical oscillator or mechanical tag-of-war. We hypothesize that this waving behavior contributes to adaptation of cell motility mechanisms in perturbed environments.

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.