# Applied Math

## Applied Analysis

Developing analytical tools to understand complex physical and biological systems. The analytical methods range from dynamical systems tools, nonlinear asymptotic methods for deriving as well as solving complex physical models, bifurcation theory, and stochastic methods. There are strong ties to faculty in Mathematics including Hans Christianson, Jeremy Marzuola, Jason Metcalfe, Michael Taylor, and Mark Williams.

## Complex fluids and soft matter materials

Flows of complex fluids, with applications in materials science (nano-rod and platelet dispersions, liquid crystalline materials), solar energy, and biological media (ranging from individual cells to lung liquids). Research includes the development of models and analysis to describe: spatial and temporal scales that span molecular to processing scales; dynamics & structure competitions between soft molecular phases and flow; and effective material properties which result from molecular morphology created during processing. Faculty collaborate with the Superfine and Lopez groups in Physics and Astronomy, the Ashby, DeSimone, Rubinstein, Sheiko, and Samulski groups in Chemistry, several faculty (Boucher, Button, Hill, Tarran) in the Cystic Fibrosis Center, in mechanochemical modeling of cells with the Bloom group in Biology and the Jacobson group in Cell and Developmental Biology, and in antibody-pathogen diffusion-reaction processes in mucus with Sam Lai’s lab in Pharmacy.

## Computational and theoretical biophysics

Problems arising in various fields of tissue, cellular, and molecular biology. Goals include understanding the dynamics of complex biological systems and developing reliable mathematical models that capture the essential components of these systems. The problems range from modeling force generation in muscles to understanding the dynamics of biochemical regulatory networks. Collaborators include Tim Elston in Pharmacology, Ken Jacobson in Cell and Developmental Biology and Kerry Bloom in Biology.

## Computational methods

Bringing computational tools to the curriculum and to collaborations across the campus. Original research advances include development of methods to solve PDEs on complex surfaces, fast algorithms for electromagnetics, and multiscale, multiphysics algorithms for highly heterogeneous media.

## Experimental Fluid dynamics

Theory, modeling, simulation, and experiments in fluid dynamics. The Joint Fluids lab shared between the Departments of Mathematics and Marine Sciences engages undergrad and grad students in original fluid research, with publications by undergrads in leading fluid journals! Many faculty, postdocs, and grads work in mathematical fluid dynamics, with applications ranging from environmental, oceanic, atmospheric, and biological systems.

## Theoretical and Computational Fluid Dynamics

Our research in fluid dynamics spans the areas of swimming and flying, waves, stratified turbulent jets, Non-Newtonian fluids, fluid-structure interactions, cardiovascular flows, and geophysical fluid dynamics. These problems are attacked using a combination of analytical, computational, and comparisons to experiments in the Mathematics and Marine Sciences Joint Fluids Lab or with collaborators. Computations allow us to study full three-dimensional problems with moving boundaries and analytical methods allow us to gain fundamental understanding of the simplified systems. Collaborators at UNC include Brian White and Carol Arnosti in Marine Sciences and Casey Miller and Bill Gray in Environmental Sciences and Engineering.

- David Adalsteinsson
- Greg Forest
- Roberto Camassa
- Jingfang Huang
- Chris Jones
- Richard McLaughlin
- Laura Miller
- Sorin Mitran
- Peter Mucha

## Mathematical Biology

Research focuses on applying mathematical and computational tools to a variety of problems in the life sciences. Current work includes answering questions from the fields of animal swimming and flying, medical imaging, mechanotransduction, cell apoptosis, cell motility, heart development, and the biomechanics of lung surface liquids. Collaborators include the Hedrick, Kier, and Bloom labs in Biology, the Elston lab in Pharmacology, the Jacobson lab in Cell and Developmental Biology, and the Button, Hill and Tarran labs in the Cystic Fibrosis Center.

## Network Analysis

Involves a variety of problems in network analysis with a focus on developing and understanding tools for the study of various simulated and real-world network data. The field of networks spans across many academic disciplines, and our networks group collaborations have included coauthors in Biostatistics, Finance, Physics, Political Science, Psychology, Sociology, and Statistics.

## Nonlinear optics

Theory, modeling, and analysis of optical communications systems.

## Applied PDE

The study of fluids, Newtonian and non-Newtonian, of nonlinear optics, of transport phenomena in mathematical biology, and numerous other phenomena leads to partial differential equations. Members of the applied mathematics group use both computational methods and methods of applied analysis to attack these problems. Strong ties exist within Mathematics with Hans Christianson, Jason Metcalfe, Jeremy Marzuola, Michael Taylor, and Mark Williams.

- David Adalsteinsson
- Roberto Camassa
- Greg Forest
- Jingfang Huang
- Chris Jones
- Richard McLaughlin
- Laura Miller

## Stochastic Analysis

The application of analytical and computational tools to analyze systems under random excitation. Research areas include computational methods for stochastic differential equations, the advection-diffusion problems with Brownian motion, multiscale stochasticity, and stochastic processes on networks. Applications are widespread in biology, materials science, fluid mechanics, and complex dynamic networks.