Report of the Reading Committee for Reza Pakzad

January 31, 2011

I. Research

Broadly speaking the research of Reza Pakzad is in the area of geometric analysis and calculus of variations with applications to harmonic mappings, nonlinear elasticity and mathematical models of growing tissues. The research is from the perspective of rigorous mathematics, but it is motivated by problems arising in applied sciences.

R. Pakzad’s research can be divided into two different, but related categories. The first one is the theory of harmonic mappings and approximation of Sobolev mappings between manifolds (publications [1]-[5]). The papers have been published in high quality journals with [5] published in Geometric and Functional Analysis, which is the leading journal in the area. This part of research is related to his Ph.D.

After Ph.D. R. Pakzad accepted a postdoctoral position at the Max Plank Institute for Mathematics in the Sciences and started to collaborate with S. Müller and M.Lewicka thus moving his research interests toward nonlinear elasticity, the area in which he has been working since then. More precisely his research is focused on the theory of thin shells and related rigidity problems in Differential Geometry. The theory of thin shells has been extensively developed from the engineering point of view, but the rigorous mathematical model has been missing. As a result many of the existing theories were inconsistent. The first rigorous mathematical model of nonlinear plate theory have been developed by G. Frieseke, R.D. James and S.Müller and one of the main contributions of R. Pakzad in this area (mostly in collaboration with M.Lewicka) is the extension of the theory to a more general case of thin shells. The theory of shells and plates leads to interesting rigidity questions in Differential Geometry. Roughly speaking one may ask to what extent the results of Differential Geometry, known to be true in the case of smooth mappings can be extended to the case of Sobolev mappings – the class of mappings which appears in a natural way in nonlinear elasticity. Also in this area R. Pakzad has many important results. In particular his paper [6] has been published in Journal of Differential Geometry, the most important journal in the area.

All together R. Pakzad wrote 15 papers, 8 of which have been written since his acceptance of the Assistant Professor position at the University of Pittsburgh. 5 other papers are in preparation. Most of the papers have been published in leading journals including Geometric and Functional Analysis, Journal of Differential Geometry, Communications in Partial Differential Equations, Ann. Scuola Norm. Sup. Pisa, and Arch. Rational Mech. Analysis, just to name a few.

His research has been supported at the University of Pittsburgh by two grants: NSF grant Calculus of variations on rigid elastic structures, (PI, 2009-2012, $126,828) and the University of Pittsburgh CRDF research award, Derivation of shell theories from the 3d nonlinear elasticity, (PI, 2008-2010, $14,000).

II Service and Mentoring

R. Pakzad organized a special session Analysis of weakly differentiable maps with constrains and applications, for the Sectional AMS Meeting (2009).

He also works with Zhuomin Liu, a graduate student at the University of Pittsburgh.

III Teaching

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IV Conclusion

The committee strongly and unanimously gives a positive recommendation for the third year review of Dr. Reza Pakzad.

Piotr Hajlasz (chair) Christopher J. Lennard William C. Troy

Mathematics University of Pittsburgh