Summer Graduate School

a Calabi-Yau manifold

This summer school will serve as an introduction to the SLMath program "Special geometric structures and analysis", which will take place in Fall 2024. There will be two mini-courses: one in Geometric Measure Theory and the other in Microlocal Analysis. The aim is to give the basic notions of two subjects also treated during the program.

School Structure

There will be two lectures and two problem sessions on most days.  Wednesday mornings will be devoted to a session of “Answers & Questions”.

Prerequisites

For both mini-courses, the undergraduate courses of Differential Geometry, Riemannian geometry, Topology and Analysis are pre-requisites. More precisely we expect a student to master the notions of
• differential manifolds (any dimension), tangent space, curvature tensor;
• riemannian metric, Levi-Civita connection, Ricci curvature;
• homotopy, first fundamental group, cohomology classes;

We suggest the following books:

References

[1] do Carmo, Manfredo P. Differential geometry of curves & surfaces. Dover Publications, Inc., Mineola, NY, 2016.
[2] Lee, Jeffrey. Manifolds and differential geometry. Graduate Studies in Mathematics, 107. American Mathematical Society, Providence.
[3] Munkres, James R. Topology. Prentice Hall, Inc., Upper Saddle River, NJ, 2000.
[4] Tao, Terence. Analysis II. Third edition. Texts and Readings in Mathematics, 38. Hindustan Book Agency, New Delhi.

To be more specific, we recommend the interested student to go over:
• Chapters 1, 2, 3, 7, 10 and 13 in [2]. We also suggest the problems 4, 7, 17, 18 at page 51-52, the problems 5, 12, 21 at page 123-125 and the problems 2, 9, 10 at page 634-635.
• Chapters 3, 4, 5 in [4]. We also suggest the exercises 4.6.8, 4.6.10, 4.6.16, 4.7.10, 5.5.3, 5.5.4.
(exercise numbering for Analysis II, editions 1, 3, & 4)

Geometric Measure Theory mini-course
• Lectures on geometric measure theory, by Leon Simon, chapters 4-8
• Geometric integration theory, by S. Krantz and H. Parks, chapter 8
• Measure theory and fine properties of functions, by L. Evans and R. Gariepy, chapters 2-3

Microlocal Analysis mini-course
• Lecture notes on pseudodifferential operators, by Richard Melrose
• Basics of the b-calculus, by Daniel Grieser
• Fredholm property of elliptic operators, by Semyon Dyatlov, chapter 15

Application Procedure

For eligibility and how to apply, see the Summer Graduate Schools homepage.