Higher Order Problems in Geometric Analysis
Lead Research Organisation:
University of Bath
Department Name: Mathematical Sciences
Abstract
Many problems in modern geometry are formulated in terms of nonlinear partial differential equations (PDEs), the analysis of which requires a high degree of expertise in the theory of PDEs as well as geometric insight. Since geometric principles or geometric constraints are often used in theoretical physics, engineering, or other sciences, the combination of analytic techniques with geometric ideas is also of tremendous interest outside of mathematics. Geometric analysis has thus always been interlinked with the theory of differential equations and with mathematical physics as well as geometry. But recently, interest in geometric PDEs has further increased through the work of Perelman, who solved a long-standing problem by proving the famous Poincare conjecture with a PDE-based approach, thereby increasing our understanding of three-dimensional spaces considerably.
The bulk of existing work in the area is concerned with equations of order 2, but there is increasing interest in higher order problems. Typically these require completely new methods, because much of the second order theory relies heavily on the maximum principle, which is not available for higher order equations. We propose to hold a workshop on `Higher Order Problems in Geometric Analysis', bringing together some of the leading experts on problems of this sort. We envisage a meeting that not only allows an exchange of the latest ideas within the geometric analysis community, but also generates interactions with geometers, applied mathematicians, or engineers. Furthermore, PhD students and other young researchers should have the opportunity to learn about questions, ideas, and techniques that they may rarely encounter otherwise.
The bulk of existing work in the area is concerned with equations of order 2, but there is increasing interest in higher order problems. Typically these require completely new methods, because much of the second order theory relies heavily on the maximum principle, which is not available for higher order equations. We propose to hold a workshop on `Higher Order Problems in Geometric Analysis', bringing together some of the leading experts on problems of this sort. We envisage a meeting that not only allows an exchange of the latest ideas within the geometric analysis community, but also generates interactions with geometers, applied mathematicians, or engineers. Furthermore, PhD students and other young researchers should have the opportunity to learn about questions, ideas, and techniques that they may rarely encounter otherwise.
Planned Impact
The direct impact of this workshop will mostly be academic. It is one of our objectives to encourage interaction between geometric analysts and researchers from different branches of mathematics (in particular, geometry, mathematical analysis, and numerical analysis) and other sciences.
We will achieve this by directly involving people with a variety of backgrounds. In particular, we will invite speakers from fields with links to geometric analysis. Furthermore, we will advertise the workshop widely. The advertisement will point out the possibility for PhD students to contribute a talk or poster.
The format of the workshop itself, with 2--3 talks in each morning or afternoon session and plenty of time for discussions, is designed to facilitate the exchange of ideas in an informal way. Similar events elsewhere, such as the workshops held regularly at the `Mathematisches Forschungsinstitut Oberwolfach' in Germany, have shown that the opportunities for discussions in a meeting of this sort are generally appreciated and used by the participants.
Economic and societal impacts are likely to be indirect, but there is a vast potential, as many problems in mathematical physics, materials science, or the life sciences have a geometric component. For example, the Willmore functional and related functionals have been used as models for the elastic energy of thin plates or thin rods, and also of DNA molecules. Methods from the theory of harmonic maps have been applied to models for liquid crystals, superconductors, and ferromagnetic materials. Geometric analysis also plays a significant role in the study of fundamental questions in cosmology. Many of the proposed participants (and the organisers) have worked on such problems in the past, and therefore it is likely that the ideas generated at this workshop will eventually have an impact on industrial and societal applications.
We will achieve this by directly involving people with a variety of backgrounds. In particular, we will invite speakers from fields with links to geometric analysis. Furthermore, we will advertise the workshop widely. The advertisement will point out the possibility for PhD students to contribute a talk or poster.
The format of the workshop itself, with 2--3 talks in each morning or afternoon session and plenty of time for discussions, is designed to facilitate the exchange of ideas in an informal way. Similar events elsewhere, such as the workshops held regularly at the `Mathematisches Forschungsinstitut Oberwolfach' in Germany, have shown that the opportunities for discussions in a meeting of this sort are generally appreciated and used by the participants.
Economic and societal impacts are likely to be indirect, but there is a vast potential, as many problems in mathematical physics, materials science, or the life sciences have a geometric component. For example, the Willmore functional and related functionals have been used as models for the elastic energy of thin plates or thin rods, and also of DNA molecules. Methods from the theory of harmonic maps have been applied to models for liquid crystals, superconductors, and ferromagnetic materials. Geometric analysis also plays a significant role in the study of fundamental questions in cosmology. Many of the proposed participants (and the organisers) have worked on such problems in the past, and therefore it is likely that the ideas generated at this workshop will eventually have an impact on industrial and societal applications.