Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Community
Ask the community for help and clear up your study doubts
Discover the best universities in your country according to Docsity users
Free resources
Download our free guides on studying techniques, anxiety management strategies, and thesis advice from Docsity tutors
The questions and instructions for part iii paper 7 of the mathematical tripos exam, focusing on the atiyah-singer index theorem. The exam includes essay questions, proofs, and problem-solving tasks related to topics such as supersymmetry, fredholm operators, riemannian geometry, and the harmonic oscillator. Students are required to demonstrate a deep understanding of these concepts and their applications.
Typology: Exams
1 / 4
This page cannot be seen from the preview
Don't miss anything!
Friday, 4 June, 2010 1:30 pm to 4:30 pm
Attempt question ONE and no more than TWO other questions. There are FIVE questions in total. The questions carry equal weight.
Cover sheet None Treasury Tag Script paper
You may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator.
Write an essay outlining the supersymmetric proof of the Atiyah–Singer index theorem for twisted Dirac operators on compact spin manifolds.
(a) Prove that Hs(Tn) ⊂ C(Tn) for s > n/.
(b) Prove the double commutant theorem for *–algebras on a finite–dimensional inner product space.
(c) Let f : (a, b) → Mn(R) be a differentiable function. Establish a formula for d dt ef^ (t)
in terms of df /dt.
(d) If ∇X is the Riemannian connection on forms on a compact Riemannian manifold, prove that the exterior derivative is given by d =
i e(ωi)∇Xi where (Xi) is locally a basis of vector fields and (ωi) is the dual basis of 1–forms.
(a) Prove that a bounded operator on a Hilbert space is Fredholm of index 0 if and only if it is the sum of an invertible operator and a compact operator.
(b) Let V be a finite–dimensional real inner product space with orthonormal basis (ei). Let C : V → Mn(C) be a real linear map such that C(v)∗^ = C(v) and C(a) C(b) + C(b) C(a) = 2(a, b)I for a, b ∈ V. If T is a skew–adjoint operator on V , prove that π(T ) = (^14)
C(T ei) C(ei) satisfies π(T )∗^ = −π(T ) and [π(T ), C(v)] = C(T v) for all v ∈ V.
(c) Let M be compact manifold and p ∈ C∞(X, Mn(C)) a self–adjoint projection. Prove that pdp = dp(1 − p). Deduce that, if ω = p(dp)^2 , then tr ωk^ and det f (ω) are closed if f (t) is a polynomial with f (0) = 1.
(d) Let D be the Dirac operator acting on a Clifford bundle E = E+ ⊕E− over a compact Riemannian manifold. Let D± = D|E±. Assuming any spectral properties that you may require, prove that ind D+ = Tr(eD−D+t) − Tr(eD+^ D−t).
Part III, Paper 7
(a) State and prove Gauss’ lemma.
(b) Let gij (x) be a smooth metric on the ball B = {x ∈ Rn^ : ‖x‖ < r} in normal coordinates. Let g(x)−^1 = (gij^ (x)) and let L = −
ij ∂xj^ g ij (^) ∂xi. Prove that there is a
unique formal power series in t, G(x, t) = 1 +
n> 1 Bn(x)t n, with Bn smooth, such that
F (x, t) = (4πt)−n/^2 e−‖x‖ (^2) / 4 t G(x, t)
satisfies ∂tF + LF = 0.
(c) Define the Laplacian ∆ acting on functions on an n–dimensional compact Riemannian manifold M. Assuming any facts about Sobolev spaces that you may require, prove that the eigenvalues 0 = λ 0 6 λ 1 6 · · · of ∆ satisfy λk > Ak^2 /n^ for some constant A > 0.
(d) Show that if ∆ is the Laplacian acting on functions on a compact Riemannian manifold M , then e−t∆^ (t > 0) is a trace–class operator on L^2 (M ) given by a smooth kernel Kt(x, y). Establish a formula for Tr e−∆t^ in terms of Kt(x, y).
Part III, Paper 7
