Students will be equipped with the tools and intuition needed to tackle complex nonlinear problems arising in many guises and various technical fields. Cross-disciplinary research and communication skills will be developed through intensive project based courses, in which small graduate student teams investigate a topic guided by faculty members with complementary perspectives. This training will give students a deep learning experience outside their Ph.D. thesis research. Internships will provide students with additional cross-disciplinary perspectives. Cross-departmental research seminars and student-run seminars, regional workshops, yearly retreats, and an active visitor program will contribute to generating a highly cooperative, diverse, cross-disciplinary research environment.
We highlight here some of these research efforts, and the ways in which they are interconnected. The initials bracketed by [...] refer to faculty participants listed in section A.
The richness of pattern formation phenomena has stimulated very active research on spatially extended dynamical systems. Classic examples occur in fluid convection [?], parametrically-excited water waves [?], Turing patterns [?] in chemical and biological systems. Much progress has been achieved in the understanding of localized structures [?] and the morphological evolution of thin films [?]. The relevance of pattern formation extends beyond these macroscopic systems to microscopic scales, e.g. ?? [?] and ? [?], as well as ?, e.g. the structure and stability of ? flows [?]. Pattern formation issues arise in studies of information processing within the brain [?], as well as in characterization of spatio-temporal chaos, patterns chaotic in time and space [?].
Dynamics of complex fluids is technologically important, involving processes with rich chaotic dynamics. The application of the general theory of mappings and flows to experiments on mixing of highly viscous fluids has been notably successful [?]. Current theoretical and experimental investigations aim at the understanding of transport in complex fluids [?]. Much of naturally occurring and industrial mixing takes place in shear flows of fluids of slight viscosity, in which transport is mediated by localized structures [?].
Dynamics of thin films and interfaces. GT research on the growth of solid films, contributing to our understanding of morphological instabilities of interfaces in crystal growth [?], and the closely related problems of the dynamics of ? [?], is a prime example of a successful cross-departmental collaboration based on co-advising of graduate students.
Ergodic theory, classical and quantum chaos. The pure mathematics research at Georgia Tech ranges from work on knots that occur as closed orbits of flows and periodic orbits of low dimensional systems [?], geometrical and dynamical problems connected with geodesics on general manifolds [?], homoclinic bifurcations and cascades of sinks [?]. The periodic orbit theory [?] applies these deep mathematical results to physical problems such as far-from-equilibrium transport, conductance of mesoscopic devices, and the semiclassical quantization of classically chaotic systems such as helium.
Coupled nonlinear oscillators play an important role in
noise
Closely related nonlinear structures are
Expertise gained from studies of pattern-forming systems [?]
will be crucial for the understanding and
control of spatial instabilities in
Theoretical and experimental neurocomputation. Neurons respond to stimulation through the firing of action potentials; increasing stimulation results in an increased firing rate until saturation is reached. The built-in nonlinear response has dramatic consequences for the behavior of large and highly interconnected neuronal arrays, from synchronized oscillations and chaotic behavior in recurrent networks [?] to the enhanced computational capabilities of feed-forward networks [?]. The nonlinear neuronal dynamics provides a computational basis for understanding of a broad repertoire of motor behaviors [?]. The theoretical work synthesizes techniques from statistical physics, information theory, control theory, nonlinear dynamics, and offers great opportunities for new collaborations [?].