ISL Geometric Measure Theory (2013)

Overview: To give a foundational course in Measure theory with geometric applications.

Schedule

Ln: Lecture n
Tn: Tutorial n
Tea Breaks: 10:30 - 11:00 and 15:30 - 16:00
Lunch Break: 13:15 - 14:30

Speakers and Syllabus

• C S Aravinda (CSA): Basic diophantine approximation: Theorems of Dirichlet, Hurwitz and Liouville, Liouville numbers, badly approximable numbers, Schmidt games, winning sets and Hausdorff dimensions of winning sets. Introduce the notion of ergodicity. (6 lectures)

• K Gowri Navada (GN): Continued fractions and a proof of Khinchine’s theorem (5 lectures)

• Shrihari Sridharan (SS): Brief discussion of limsup and liminf, Lebesgue and Hausdorff measures, Cantor sets, Borel-Cantelli lemma, computing measures and Hausdorff dimensions of Cantor sets. (5 lectures)

• Anandateertha Mangasuli (AM): Riesz representation theorem, Topological groups and Haar measures (briefly sketching the general construction but discussing a couple of simple examples explicitely). (3 lectures)

• Vikram Aithal (VA): Kakeya needle problem, Besicovich sets and Hausdorff dimension of Besicovich sets. (3 lectures)

• Jayadev Athreya (JA) and Anish Ghosh (AG): Will describe the notion of lattices in Euclidean space, and show how one can form a space of lattices in n-dimensional Euclidean space by considering the coset space Xn = SL(n, R)/SL(n, Z). Will describe and compute a finite invariant measure on this space, and show how to compute the volume using simple tools from number theory. Will discuss applications to the geometry of numbers,diophantine approximation, and counting problems like the Gauss circle problem. Will also discuss Khintchine’s theorem using Sullivan’s argument and the hyperbolic analogue of the lattice point counting problem. The second half of the course will describe how these problems connect to dynamical systems on the space Xn , and in particular discuss simple ex-amples of a phenomenon now know as the Dani correspondence. (5 lectures each)

• Kingshook Biswas (KB): Self-similar sets and computations of Hausdorff dimensions. (3 lectures)

 Course Associates:

• Vikram Aithal (VA) CBS, Mumbai

• Harish Chandra Pathak (HP) BHU, Varanasi

• Sanjay Pant (SP) DU and CEMS, Almora

Reference Books: The following books are the suggested references:

(1) Measure theory and Integration By G de Barra.
(2) Geometry of Fractal sets By K J Falconer.
(3) Continued Fractions (3rd Edition) By A I Khinchin.