AIS Analysis and Geometry (2013)

School Convener(s)

Speakers and Syllabus 

Suggested Books:

1. Morse theory, J. Milnor
2. Topology from a Differentiable standpoint, John Milnor.
3. Differential Topology, Morris Hirsch.
4. Non-Linear Functional Analysis by Klaus Deimling.
5. Degree Theory by N. G. Lloyd.

Syllabus Details:

1. Definition of a Manifold.
2. Definition of a Complex manifold.
3. Tangent and co-tangent bundle.
4. Maps between manifolds, smooth maps.
5. Vector fields on manifolds. Vanishing of vector fields.
6. Implicit function and inverse function theorems on manifolds.
7. Sard’s theorem (finite dimensional version).
8. Degree of a map between two manifolds. Definition of the degree of a map using the signature of the Jacobian. ( see Lloyd’s book)
9. Brouwer fixed point theorem.
10. Borsuk-Ulam theorem The book Non-Linear Functional Analysis by Deimling, Chap 1 is tailor made for degree theory
    as related to PDE, lots of applications of Borsuk-Ulam, Brouwer fixed point theorem and degree theory to PDE and ODE.
11. Morse Deformation Lemma.
12. Attaching Handles
13. Morse Inequalities
14. The notion of Linking number of curves, computing these linking numbers. Intersection number.
15. Applications of the Morse inequalities.

Suggested Books:

1. Non-Linear Functional Analysis by Klaus Deimling.
2. Primer of Non-Linear Analysis by Ambrosetti and Prodi.
3. Variational Methods by Michael Struwe.
   

Syllabus Details:

1. Implicit Function theorem in Banach spaces.
2. Degree theory in Infinite dimensions, Leray-Schauder fixed point theorem. Students should have seen the notion of a compact operator in Linear Functional Analysis.
3. Liapunov-Schmidt finite dimensional reduction, basic bifurcation theory with examples. ODE examples can be picked from Ambrosetti’s book. Topics 1-3 are to be found in the first 91 pages in the book by Ambrosetti and Prodi and also the book by Deimling can be used to supplement. Struwe’s book is a good source for the next topics on our list.
4. Direct method in the calculus of variations. First eigenvalue problem for the Laplacian on a bounded domain with zero Dirichlet data should be done as an application of the direct method.
5. Perron’s method, sub and super solutions.
6. Concentration compactness( very important topic should not be skipped).
7. Infinite dimensional Morse theory.
8. Palais-Smale condition, Morse deformation lemma in infinite dimensions.
9. Mountain Pass Theorem of Ambrosetti-Rabinowitz. Morse index again. Application to finding solutions to with p sub-critical. Needs cooperation with Harmonic Analysis as Rellich compactness has to be done.
10. Variational principles based on Borsuk-Ulam( Ljusternik-Schnirelman category).
11. Variational principles based on the linking of curves, applications to Hamiltonian systems.
12. Pohozhaev's identity. A simple application to Non-existence of solutions. Link to conformal Killing fields from the Differential Geometry course.
13. Time permitting, Legendre transform, dual variational approaches, Hamiltonian dynamics.
14. Further applications of the Morse index, to ODE and PDE, or to radial solutions of PDE.

Syllabus Details:

1. Interpolation theorems of Marcinkiewicz and Stein.
2. Vitali Covering and the Hardy-Littlewood maximal function.
3. Hardy-Littlewood-Sobolev fractional integration and related Sobolev inequalities.
4. Sharp constants in the Hardy-Littlewood-Sobolev inequalities. Explain Lieb-Loss-Carlen method of competing symmetries and re-arrangements of functions.
5. Borderline Sobolev inequalities, Moser-Trudinger inequality. Local Poincare-Sobolev inequalities.
6. Calderon-Zygmund lemma.
7. Calderon-Zygmund singular integral theory.
8. Non-convolution Calderon-Zygmund operators, Cotlar-Stein lemma.
9. Applications of the above to fundamental estimates of elliptic regularity of PDE.
10. Bounded Mean Oscillation BMO, John-Nirenberg inequality.
11. Holder spaces and Singular integrals on Holder spaces.
12. Application of above ______ Schauder theory of elliptic PDE.
13. Sobolev spaces, traces and extensions.
14. Rellich-Kondrachov lemma of compactness.
15. Boundary elliptic problems based on Sobolev spaces. Talk some on Schauder theory based on Holder spaces. Difference quotient and regularity estimates upto the boundary.
16. Maximum principle, Hopf Lemma, strong maximum principle.
17. Littlewood-Paley theory.
18. Restriction theorems for the Fourier transform to sphere and light cones.
19. Strichartz estimates for the wave eqn., Schrodinger eqn. based on the restriction theorem. This is the cornerstone of everything nowadays in general Relativity, wave eqn. etc.
20. Fefferman’s fundamental duality of H1 and BMO, applications to compensated compactness, mean curvature equation, Harmonic maps etc.

Suggested Books:

1. Differential Geometry of Curves and Surfaces, M. P. Do Carmo
2. Riemannian Geometry, by M. P. Do Carmo.
3. Spivak
4. Goldberg, Curvature and Homology.
   

Syllabus Details:

1. 1. First Fundamental form( hopefully basic stuff has been done by the topology professor, tangent bundle etc)
2. Connections, Parallel transport, Levi-Civita connection.
3. Cartan Moving frame formulation.
4. Geodesic equation.
5. Exp map, injectivity radius.
6. Normal coordinates.
7. Riemann tensor, Curvature, Weyl, Ricci, Scalar.
8. Riem=Weyl+traceless Ricci+ Scalar curvature
9. Conformal properties of Weyl tensor, Isothermal coordinates on surfaces Weyl tensor vanishing and isothermal coordinates in general dimension.
10. Bianchi identities
11. Gauss map for surfaces.
12. Second fundamental form, Gauss curvature, Gauss eqn., Codazzi eqn. Theorema Egregium.
13. Gauss-Bonnet formula, Euler characteristic.
14. Invariant expressions for the divergence of a vector field and the Laplacian on a Riemannian manifold.
15. Formula for the Gauss curvature.
16. Jacobi fields, Jacobi equations.
17. Rauch comparison theorem, Myers theorem.
18. Conformal Killing vector fields.
19. Transformation laws for Gauss and scalar curvature under Conformal changes of the metric.

Selected Applicants