This proposal seeks funding to support travel of American mathematicians to attend a conference and short courses at the Centre de Recerca Matematica (CRM) in Barcelona and a workshop at Univ. de Granada. The workshop on minimal surfaces will be held at Mathematics Institute of Univ. de Granada, June 17-27, 2013. The conference in conformal geometry will take place July 1-5, 2013 at CRM. The workshop seeks to review the recent advances in conformal geometry and minimal surfaces in order to encourage interactions of researchers in related areas. In between the two conferences will be the short courses for graduate students and postdocs on conformal geometry and minimal surfaces June 25 to June 28 at CRM. To promote the interests of young mathematicians in these topics, more than half of the funding from this proposal will go to support the travel of graduate students and young postdocs.
This proposed workshops and short courses are intended to stimulate advances and collaborations in the reseach topics of conformal geometry and minimal surfaces. Conformal geometry studies the structure of spaces in which there is a well defined notion of lengths, but the emphasis is on related measurements of length where the notion of angles is preserved. Similarly minimal surface theory studies the placement of surfaces in an ambient space in which there is a well defined notion of length and hence notion of area. Common to both disciplines is the use of differential equations to study related geometric invariants. In recent years there is extensive developments in analytic techniques in differential equations that have direct applications in both conformal geometry and minimal surfaces. These possible applications will in turn generated new advances the the theory of differential equations. The following urls give more detailed information: http://wdb.ugr.es/~geometry/seminar/en/eventos and www.crm.cat/en/Pages/default.aspx
The project was to support the travel of young American researchers to the program "Conformal Geometry and Geometric PDEs" held in the summer of 2013 at the Centre de Recerca Matematica at Barcelona Spain. Six postdocs and four graduate students were able to attend the program, supported by this grant. The program had dual purpose: to introduce to young researchers to the subject of Conformal Geometry/CR geometry and the theory of minimal surfaces all of which uses geometric partial differential equations as it main tool, and to encourage interaction among the participants across these fields.. There was one week of short courses given by the leaders of the field, and a one-week conference in which the lecturers discuss their work. The short courses were: Alice Chang, "Q-curvature and applications to problems in conformal geometry". Charles Fefferman, "Local conformal invariants". Bill Meeks, "Global theory of minimal surfaces". The conference comprise of twenty one leading experts discussing their current research. Spyros Alexakis spoke about compactness for family of minimal surfaces in hyperbolic 3-space. Sagun Chanillo spoke about the embedding problem for CR manifolds. Jeffrey Case spoke about weighted Yamabe equations. Jih-Hsin Cheng spoke about the positive mass theorem for CR manifolds. Manuel Fernandez spoke about Ricci solitons. Daniel Fox spoke about affine hyperspheres. Rod Gover spoke about projectively compact structures. Robin Graham spoke about conformal structures with G_2 holonomy. Matthew Gursky spoke about glueing Bach flat structures. Colin Guillarmou spoke about the spectral theory of the Dirichlet to Newmann operator. Kengo Hirachi spoke about the P-prime operator in CR geometry. Dimitry Jakobson spoke about nodal sets of solutions of Q-curvature equations. Robert Kusner spoke about moduli of CMC surfaces. Seong-Tag Kim spoke about compactness of Bach flat spaces. Andrea Malchiodi spoke about Liouville equation with sinularities. Luca Martinazzi spoke about supercritical equations and Trudinger's inequality. Pablo Mira spoke about constant mean curvature spheres in Sol_3. Juaquin Perez spoke about Calabi-Yau problem for embedded minimal surfaces. Jie Qing spoke about the geometric correspondence between hypersurfaces in hyperbolic space and the conformal metrics on the sphere at infinity. Fang Wang spoke about the Dirichlet to Neumann map for CCE spaces. Yi Wang spoke about isoperimtric inequality and the absolute Q-curvature integral. The short courses and the conference were well attended by more than eighty participants from the US, Canada, Japan, Australia, Brazil, Taiwan, France, Italy, Sweden and Spain. There were very successful interactions among the participants which develop into long term projects. The research team of Gursky/Malchiodi developed their ideas about a strong maximum principle for higher order equations, a key result that makes possible to apply the pde results to geometry. The reserach team of Case/Gover/Yang were able to map out their strategy to construct their new P-prime operators at the conference. There were a couple of informal talks in which the participants sketch out possible approaches to several problems of current interests.
