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Research

Research in Mathematical Sciences is conducted under the auspices of the Center for Applied Mathematics and Statistics (CAMS).

 

Research

Main Research Areas

CAMS members conduct research in a broad range of areas of applied mathematics, using, in particular, the techniques of scientific computation/numerical analysis, applied analysis and analysis of PDEs, mathematical modeling and asymptotics, dynamical systems theory, and statistics. The focus of CAMS research is roughly divided into five application areas (with significant overlap), listed below.

 
Mathematical Biology

Experimental, Computational and Mathematical Neuroscience

  • Neuronal dynamics, synaptic plasticity, neuronal networks
  • Electrical activity in excitable cells, calcium dynamics
  • Interaction between electrical dynamics and gene expression
  • Applications to visual cortex, central pattern generators, spreading cortical depression and circadian rhythms

Biological Fluid Dynamics

  • Immersed interface/continuum methods
  • Biomechanics and tissue engineering
  • Cell and vesicle dynamics

Developmental Biology

  • Morphogenetic patterning, embryogenesis
Calculated oxygen distributions in the tissue
surrounding three-dimensional arrays of capillaries
Members: Bose, Bukiet, Cummings, Diekman, Matveev, Muratov, Young (External Members: Booth, Bunker, Golowasch, Holzapfel, Huang, Miura, Nadim, Rotstein, Russell, Tao, Wylie)
Fluid Dynamics

Interfacial Flows

  • Fast/multiscale numerical methods
  • Analysis and simulation of jet breakup, bubble dynamics, surfactant effects
  • Modeling, analysis and simulation of thin films and wetting phenomena
  • Fluid-structure interaction
  • Electrohydrodynamics and electrokinetics
  • Geophysical fluid dynamics

Multiphase and Reactive Flows

  • Flame-flow interaction, droplet burning, ignition
  • Cryogenic fluids for rocket propulsion technology
  • Micro- and nano-fluidics
  • Complex fluids

Numerical Methods and PDE Analysis

  • Boundary integral, volume of fluid, level set methods
  • Convergence and error estimates for numerical methods
  • Fast/multiscale methods
  • Well-posedness for interfacial flows
  • Analysis of singularity formation

(see also Complex Flows and Soft Matter page)
Numerical simulation of the kinetics of
droplet formation in polymer-liquid crystal
mixtures
Members: Afkhami, Bechtold, Blackmore, Booty, Bukiet, Choi, Cummings, Hamfeldt, Jiang, Kondic, Luke, Muratov, Petropoulos, Shirokoff, Siegel, Young (External Members: Diez, Huang, Meegoda, Papageorgiou, Rosato, Vanden-Broeck, Wylie)
Linear and Nonlinear Waves

Underwater Acoustics and Ocean Waves

  • Inverse scattering, matched field processing, target detection and localization
  • Surface and internal waves
  • Data assimilation

Electromagnetics

  • Finite element and domain decomposition methods
  • Fast algorithms, fast multipole and related methods
  • Effective media, homogenization, dispersive media
  • Scattering from complex objects

Nonlinear Optics

  • Ultrafast optical communications
  • Solitons in optical fibers

Front Propagation

  • Traveling waves in reaction-diffusion systems
  • Population dynamics
  • Flame dynamics and detonation
Ambiguity surface generated by matched field
processing of an exhaustive search for unknown
source range and depth
Members: Bechtold, Boubendir, Choi, Goodman, Hamfeldt, Jiang, Michalopoulou, Moore, Muratov, Petropoulos, Shirokoff, Turc (External Members: Ahluwalia, Erneux, Kriegsmann, Papageorgiou, Russell, Vanden-Broeck)
Materials Science

Soft Matter

  • Molecular dynamics simulations of granular materials
  • Modeling and asymptotics for liquid crystal systems
  • Mesoscopic phases in polymer systems
  • Biomaterials

Energy Driven Pattern Formation

  • Analysis of variational problems with long-range forces
  • Modeling, analysis and simulation of magnetic systems
  • Numerical methods for interfacial phenomena in solids
  • Large deviations, rare events and nucleation phenomena

(see also Complex Flows and Soft Matter page)
Mesoscopic Model for Pattern Formation
in Materials
Members: Afkhami, Blackmore, Booty, Cummings, Kondic, Muratov, Shirokoff, Siegel, Young (External Members: Rosato, Wylie)
Statistics

Biostatistics

  • Multivariate methods
  • Clinical trials and Longitudinal Data Analysis
  • Microarray and Proteomics analysis
  • Generalized Additive Models and Semi-parametric methods

High dimensional data

  • Multiple testing and False Discovery Rate
  • Analysis of massive datasets
  • Variable selection

Spatial statistics

  • Analysis of spatial point patterns
  • Spatial anomaly detection

Survival Analysis

  • Copulas and competing risk models
  • Bootstrap of survival data
  • Empirical likelihood
  • High dimensional censored data analysis
  • Reliability models
The Mean Remaining Life function
MRL(x) ≡ E(X-x | X > x)
Members: Dhar, Fang, Guo, Loh, Subramanian, Wang (External Members: Johnson, Sverdlove)

Updated: 2/6/18