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Mathematics at the Interface of PDEs, the Calculus of Variations, and Materials Science: A Celebration of Robert Kohn

From May 21 to May 23, experts on the calculus of variations and their applications to materials science will come together—as well as junior researchers and students in PDE—for an exciting hot topics workshop.

The workshop will cover an array of applied mathematics, including the effective behavior of composite materials, the role of nonconvexity in pattern formation and microstructure in materials, the importance of scaling laws in the analysis of energetic models and their gradient flows, the role of dimension reduction in the analysis of thin structures, and the rigorous and computational analysis of multiscale and stochastic dynamical systems. Additionally, the workshop will cover developments in the theory of homogenization and effective properties of composite materials. Another theme of the workshop will be the role of nonconvexity in understanding phase transitions, pattern formation, and formation of microstructure. All of these topics have a computational component which leads to an additional theme of the workshop: the computational analysis of multiscale problems, including recent efforts to carry out computations on models involving stochastic dynamics.

These fields have been profoundly influenced by the work of interdisciplinary mathematical scientist Robert V. Kohn, a professor of mathematics at the Courant Institute of Mathematical Sciences at New York University (NYU) since 1988. The workshop will celebrate his work on the occasion of his 60th birthday.

"It is no accident that we are organizing his birthday celebration at IMA. Kohn has been involved, either explicitly or implicitly, in all the IMA activities related to materials," explained workshop organizer Weinan E.

Workshop organizer Robert Lipton said that Kohn’s work provides unique and seminal contributions to the theory of extreme properties of composite media and to the theory of optimal structural design and phase transitions.

"His approach develops novel relaxation methods by identifying and exploiting the relationship between the mathematical notion of quasi-convexification and optimal bounds on energies coming from the homogenization theory," added Lipton.

Kohn is a world-renowned expert on nonlinear partial differential equations and on non-convex variational problems.

Weinan E. said that he has played a "pivotal" role in pushing math into material science, through his own work, his mentoring of students and postdocs, and leadership for the community.

His research focuses mainly on the analysis of mathematical models in materials science. He studies coarsening due to energy-driven motion, micromagnetics, pattern formation due to energy minimization, shape-memory materials, and surface energy as a selection mechanism. He received a bachelor’s degree in mathematics from Harvard University in 1974, a master’s degree in mathematics from the University of Warwick in 1975, and a doctoral degree in mathematics from Princeton University in 1979. He was a postdoctoral fellow at NYU’s Courant Institute of Mathematical Sciences from 1979 to 1981. He was promoted to professor of mathematics in 1988. Kohn's work was recognized with the Ralph E. Kleinman Award from the Society for Industrial and Applied Mathematics (SIAM) in 1999, the Keith Medal (Royal Society of Edinburgh) in 2007, and the Steele Prize for Seminal Contribution to Research (AMS) in 2014. He is also a fellow of the American Mathematical Society and SIAM. He has visited the IMA dozens of times over the course of his career, serving on the IMA’s Board of Governors and Community Relations Committee as well as a regular workshop speaker and organizer.