My general interests are in computational and applied mathematics with an emphasis on classical continuum mechanics. My current efforts are focussed on modeling the mechanical properties of DNA at various length scales. Keywords: modeling, numerical analysis, differential equations, integral equations, geometry of curves and surfaces.

Main Projects

Nystrom-type numerical methods for boundary integral equations. The method of integral equations has a long and rich history in both the analysis and numerical treatment of boundary value problems. On the analysis side, the method provides a classic approach to the study of existence and uniqueness questions. On the numerical side, the method provides an alternative formulation which can be efficiently discretized. In the basic approach, a boundary value problem described by a partial differential equation on a domain of interest is reduced to an integral equation on the bounding surface. The unknown scalar or vector field throughout the domain is represented in terms of one or more integral potentials which depend on an unknown surface density. The representation can take various different direct and indirect forms and the integral potentials usually involve kernels that are at least weakly singular. The reduction in dimension makes the method of integral equations extremely attractive, especially for problems on exterior domains. Indeed, the method has been used in many different modern applications in acoustics, electromagnetics, hydrodynamics, elastostatics, and molecular modeling. The objective of this research is to develop Nystrom-type numerical methods for various types of integral equations on surfaces in three-dimensions and characterize their properties.

Hydrodynamic transport and diffusion of rigid and flexible particles. A detailed understanding of diffusion and its interplay with convection is crucially important in the design and performance of microfluidic devices for the transport, mixing, separation and manipulation of particles. Such devices have many applications, ranging from drug design, delivery and detection in the pharmaceutical and biomedical industries, to rapid sequencing, screening and analysis in the biotechnological and chemical industries. Classic theories of translational and rotational diffusion of symmetric, rigid particles such as ellipsoids and cylinders have provided extremely useful models for the analysis of such devices. However, the design of next-generation devices will likely require more detailed models that can capture various coupling effects for non-symmetric and/or non-rigid particles. The objective of this research is to develop mathematical models to characterize various cross- and self-coupling effects between translational, rotational and internal motions in the hydrodynamic diffusion of rigid and flexible particles of arbitrary shape.

Sequence-dependent models for DNA structure prediction. Various sequence-dependent structural features of DNA in solution, for example its intrinsic curvature and flexibility, are thought to be critical for its packaging into the cell, recognition by other molecules, and conformational changes during biochemical processes. At the scale of tens to hundreds of basepairs, the modeling of such features is prohibitively expensive for all-atom molecular dynamics (MD) models, which are usually considered at shorter scales, and are below the resolution of homogeneous chain and rod models, which are usually considered at longer scales. A class of models that are particularly well-suited for intermediate scales are those based on interacting, three-dimensional rigid bodies. Such models are coarse-grained and hence simpler to understand than those of the atomistic type, and are more detailed and hence better adapted to represent local features than those of the homogeneous chain and rod types. Indeed, rigid body models offer a promising approach to understand various sequence-dependent structural features of DNA as have been investigated in various works. The objective of this research is to develop and parameterize a family of rigid body models that can capture important local and non-local sequence-dependent structural features of DNA in solution.

Global curvature and optimal packing of curves with finite thickness. An understanding of how material filaments may be optimally packed in confined geometries and how they may supercoil or wrap around themselves is of great interest in the study of macromolecules such as DNA and other systems in chemistry and biology. For example, evidence published in the journals Nature and Proteins suggest that structural motifs in double-helical DNA and alpha-helical proteins may be explained by optimal packing rules. Moreover, experimental data on knotted DNA molecules suggest that certain physical properties of DNA knots can be predicted from a corresponding optimally tight shape. The objective of this research is to exploit the novel concept of global curvature to develop concise variational formulations, establish existence and regularity results, derive necessary conditions, and design numerical methods for various different optimal packing problems for curves.

Structure-preserving numerical methods in classical and continuum mechanics. Differential equations in the engineering and physical sciences are often derived from global balance laws pertaining to mass, momentum and energy, and are often formulated in terms of variables that evolve on smooth constraint manifolds. However, balance laws are typically lost under discretization and numerical solutions often drift away from their underlying manifold. The objective of this research is to develop special discretization techniques for preserving balance laws and constraint manifolds for various types of ordinary and partial differential equations and characterize their stability and convergence properties.

Representative talks

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