Differential geometry has its origins in the monumental work of Gauss and Riemann in the 19th century. Its relevance for science has no better illustration than its use by Einstein as the mathematical foundation for his general theory of relativity. Over the course of the last 150 years, the subject has been invigorated by the adoption of global topological and analytical tools, and by its many applications in science and engineering, with general relativity, gauge field theory, and string theory deserving special mention. The pace of research activity and progress on fundamental problems in the field during the last decade continues to be breathtaking. The resolution of the Poincaré Conjecture and Thurston’s Geometrization Conjecture in low-dimensional topology was achieved by Perelman using Hamilton’s Ricci flow. In Kaehler geometry, the Fano case of Calabi’s problem about the existence of Kaehler-Einstein metrics was given a definitive answer by Chen-Donaldson-Sun and Tian. Recently in the area of global submanifold geometry, the Lawson conjecture was proved by Simon Brendle, and the Willmore conjecture was resolved jointly by F. Marques and A. Neves. These are only a few of a long list of deep results that were achieved in just the past decade.

 

The overall goal of the Program is to provide a forum for consolidating and extending these achievements and a platform for launching new directions of investigation, posing new problems, and developing new tools.

 

Although the program will include participation from researchers and graduate students working in many different areas within geometric analysis, there will be a focus on three main subthemes, to which we will denote individual workshops. These main subthemes are:

1. G2 manifolds and related topics

2. General relativity and the AdS/CFT correspondence

3. Mean curvature flow and the Ricci flow. 

Immediately preceding workshops for subthemes 1 and 3 there will be a two-day "minischool" during which guest lecturers will give short courses for postdocs, students, and non-expert researchers who are interested in the workshop subtheme, to provide them with the background they need to get the most out of the workshop talks.

Other activities are planned for the program, such including:

Graduate Courses. There will be two graduate courses at the Fields Institute during the 2017 Fall academic term. Both courses will be recorded. The first course is "Introduction to the mean curvature flow" by Robert Haslhofer; the second course will be taught jointly by F. Marques and A. Neves (see below for more details). Students from Fields Institute Principal Sponsoring Universities can get credit for taking these courses, however, arrangments must be made through the chair of students' home departments; the Fields Institute can help with providing attendance records and the administration of assignments and tests.  

Colloquia and Seminars. During the period of the thematic program, we will integrate the "Fields Institute Geometric Analysis Colloquium" with the activities of the program and have one speaker at least every two weeks during the September to December period. Moreover, graduate students and postdoctoral fellows associated with the thematic program will be encouraged to organize weekly working/learning seminars on topics related to the themes of the program. Finally, energetic visitors to the Institute during the program can also participate in the regular differential geometry seminars at the Universities of Toronto, McMaster, and Waterloo.

Public Outreach. Several public outreach activities are being planned.

GAP Meeting. There will be a 3 day minischool on geometric flows in complex geometry at the Fields Institute from Dec. 1-3, 2017. This separate event may be of interest to partcipants in this Thematic Program. The event webpage will be linked here once available.