Partial differential equations and more general relations which appear in Topology and Geometry are very often under-determined and have many solutions. This is in contrast with the PDE’s which arise in Physics where the solutions are few. This phenomenon was observed quite early in the work of Nash-Kuiper on C 1 -isometric embeddings (1954) and Smale-Hirsch theory of immersions (1965). Around 1969,Gromov introduced the theory of h-principle which brought all these work under the same framework. The theory of h-principle addresses the question of existence and homotopy classification of solutions of geometric partial differential relations.In the AIS, two important techniques of this theory - Holonomic approximation and convex integration - will be studied in details and several important applications will be seen, including Nash-Kuiper theorem and Smale-Hirsch theorem.Convex integration theory has some apparent similarity with Filippov’s relaxation theory for Differential inclusions (of first order) - though the techniques are completely different. Filippov’s theorem and its generalizations will be discussed in parallel with convex integration theory.
Basic Differential Topology and theory of vector bundles and fibre bundles will be reviewed during the first week. Preliminaries of PDE to develop the relaxation theory will be discussed in the third week.
Target Audience:
The School is meant for PhD students working in Geometry and Topology. Students working in Partial Differential Equations with strong interest in Topology and Geometry are also welcome. The participants must be familiar with basic Differential Topology (Chapter I and Chapter IV of Differential Topology by Guillemin and Pollack) and Algebraic topology (Basic knowledge of homotopy, fundamental groups,homology theory).
Names and affiliations of possible speakers.
(1) Adi Adimurthi, TIFR Centre for Applicable Mathematics
(2) Samik Basu, RKMVU, Belur
(3) Somnath Basu, IISER, Kolkata (not confirmed)
(4) Mahuya Datta, ISI Kolkata
(5) Dheeraj Kulkarni, ISI Kolkata
(6) Amiya Mukherjee, ISI, Kolkata
(7) Goutam Mukherjee, ISI, Kolkata
(8) Dishant Pancholi, IMSc, Chennai
(9) K. Sandeep, TIFR Centre for Applicable Mathematics
(10) Sushmita Venugopalan, CMI, Chennai (not confirmed)
Names and affiliations of the tutors.
(1) Aritra Bhowmick, ISI, Kolkata
(2) Sauvik Mukherjee, Presidency University
Speakers will conduct tutorials on the day of their lectures. Aritra Bhowmick will assist in the first week tutorials. Sauvik Mukherjee will assist in the second week.
Syllabus
Lectures in the first and the second week are of duration 90 minutes each. Lectures in the third week are of duration 1 hour.
A. Basic Differential Topology (8 lectures) - Samik Basu, Somnath Basu, Goutam Mukherjee
Smooth manifolds, tangent spaces, derivative of smooth maps. Immersions, submersions, transversal maps. Vector bundles - tangent, cotangent and exterior bundles. Vector fields and integration. Differentiation of forms. Lie derivative and Cartan formula. Existence of Riemannian metric.
B. Jet bundles, Whitney topology, Thom’s transversality theorem. (4 lectures) - Amiya Mukherjee
C. Introduction to h-principle, Holonomic approximation theorem and applications. Smale-Hisrsch theorem. (6 lectures) - Mahuya Datta, Dishant Pancholi
D. Preliminaries of Symplectic and Contact manifolds. Symplectic and Contact immersion theorems (6 lectures) - Dheeraj Kulkarni, Sushmita Venugopalan
E. Convex integration theory. Nash-Kuiper isometric C 1 -embedding theorem (6lectures) - Mahuya Datta
F. Filippov’s Relaxation theory and generalizations (12 lectures) - A. Adimurthi,K. Sandeep
Filippov’s theory–existence of solution for ODE Inclusion; Convex Analysis; Minimisation problems of convex functionals on Sobolev Spaces; Vector valued minimisations–Morrey’s theory– various notions of convexity; Differential
inclusions and Dacorogna-Marcellini’s results.
Topics (A), (B) will be covered in the first week, (C), (D) in the second week and (E), (F) in the third week.
Time Table | ||||||||
10 : 00 | 11 : 30 | 12 : 00 | 1 : 30 | 2 : 45 | 3 : 45 | 4 : 00 | 5 : 00 | |
22 May | A1 | T ea | A2 | Lunch | T utorial | T ea | T utorial | Snacks |
23 May | A3 | A4 | ||||||
24 May | A5 | B1 | ||||||
25 May | A6 | B2 | ||||||
26 May | A7 | B3 | ||||||
27 May | A8 | B4 | ||||||
29 May | C1 | T ea | D1 | Lunch | T utorial | T ea | T utorial | Snacks |
30 May | C2 | C3 | ||||||
31 May | C4 | C5 | ||||||
1 June | C6 | D2 | ||||||
2 June | D3 | D4 | ||||||
3 June | D5 | D6 |
10 : 00 | 11 : 00 | 11 : 30 | 12 : 30 | 1 : 30 | 2 : 45 | 3 : 45 | 4 : 00 | 5 : 00 | |
5 June | E1 | T ea | F1 | T utorial | Lunch | F2 | T ea | T utorial | Snacks |
6 June | E2 | F3 | F4 | ||||||
7 June | E3 | F5 | F6 | ||||||
8 June | E4 | F7 | F8 | ||||||
9 June | E5 | F9 | F10 | ||||||
10 June | E6 | F11 | F12 | ||||||