Part II, 2020

Algebraic Geometry

• Paper 1, Section II, F

  Algebraic Geometry | Part II, 2020

  Let $k$ be an algebraically closed field of characteristic zero. Prove that an affine variety $V \subset \mathbb{A}_{k}^{n}$ is irreducible if and only if the associated ideal $I(V)$ of polynomials that vanish on $V$ is prime.

  Prove that the variety $\mathbb{V}\left(y^{2}-x^{3}\right) \subset \mathbb{A}_{k}^{2}$ is irreducible.

  State what it means for an affine variety over $k$ to be smooth and determine whether or not $\mathbb{V}\left(y^{2}-x^{3}\right)$ is smooth.

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• Paper 2, Section II, F

  Algebraic Geometry | Part II, 2020

  Let $k$ be an algebraically closed field of characteristic not equal to 2 and let $V \subset \mathbb{P}_{k}^{3}$ be a nonsingular quadric surface.

  (a) Prove that $V$ is birational to $\mathbb{P}_{k}^{2}$.

  (b) Prove that there exists a pair of disjoint lines on $V$.

  (c) Prove that the affine variety $W=\mathbb{V}(x y z-1) \subset \mathbb{A}_{k}^{3}$ does not contain any lines.

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• Paper 3, Section II, F

  Algebraic Geometry | Part II, 2020

  (i) Suppose $f(x, y)=0$ is an affine equation whose projective completion is a smooth projective curve. Give a basis for the vector space of holomorphic differential forms on this curve. [You are not required to prove your assertion.]

  Let $C \subset \mathbb{P}^{2}$ be the plane curve given by the vanishing of the polynomial

  $$X_{0}^{4}-X_{1}^{4}-X_{2}^{4}=0$$

  over the complex numbers.

  (ii) Prove that $C$ is nonsingular.

  (iii) Let $\ell$ be a line in $\mathbb{P}^{2}$ and define $D$ to be the divisor $\ell \cap C$. Prove that $D$ is a canonical divisor on $C$.

  (iv) Calculate the minimum degree $d$ such that there exists a non-constant map

  $$C \rightarrow \mathbb{P}^{1}$$

  of degree $d$.

  [You may use any results from the lectures provided that they are stated clearly.]

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• Paper 4, Section II, F

  Algebraic Geometry | Part II, 2020

  Let $P_{0}, \ldots, P_{n}$ be a basis for the homogeneous polynomials of degree $n$ in variables $Z_{0}$ and $Z_{1}$. Then the image of the $\operatorname{map} \mathbb{P}^{1} \rightarrow \mathbb{P}^{n}$ given by

  $$\left[Z_{0}, Z_{1}\right] \mapsto\left[P_{0}\left(Z_{0}, Z_{1}\right), \ldots, P_{n}\left(Z_{0}, Z_{1}\right)\right]$$

  is called a rational normal curve.

  Let $p_{1}, \ldots, p_{n+3}$ be a collection of points in general linear position in $\mathbb{P}^{n}$. Prove that there exists a unique rational normal curve in $\mathbb{P}^{n}$ passing through these points.

  Choose a basis of homogeneous polynomials of degree 3 as above, and give generators for the homogeneous ideal of the corresponding rational normal curve.

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Algebraic Topology

• Paper 1, Section II, $21 \mathbf{F}$

  Algebraic Topology | Part II, 2020

  Let $p: \mathbb{R}^{2} \rightarrow S^{1} \times S^{1}=: X$ be the map given by

  $$p\left(r_{1}, r_{2}\right)=\left(e^{2 \pi i r_{1}}, e^{2 \pi i r_{2}}\right)$$

  where $S^{1}$ is identified with the unit circle in $\mathbb{C}$. [You may take as given that $p$ is a covering map.]

  (a) Using the covering map $p$, show that $\pi_{1}\left(X, x_{0}\right)$ is isomorphic to $\mathbb{Z}^{2}$ as a group, where $x_{0}=(1,1) \in X$.

  (b) Let $\mathrm{GL}_{2}(\mathbb{Z})$ denote the group of $2 \times 2$ matrices $A$ with integer entries such that $\operatorname{det} A=\pm 1$. If $A \in \mathrm{GL}_{2}(\mathbb{Z})$, we obtain a linear transformation $A: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$. Show that this linear transformation induces a homeomorphism $f_{A}: X \rightarrow X$ with $f_{A}\left(x_{0}\right)=x_{0}$ and such that $f_{A *}: \pi_{1}\left(X, x_{0}\right) \rightarrow \pi_{1}\left(X, x_{0}\right)$ agrees with $A$ as a map $\mathbb{Z}^{2} \rightarrow \mathbb{Z}^{2}$.

  (c) Let $p_{i}: \widehat{X}_{i} \rightarrow X$ for $i=1,2$ be connected covering maps of degree 2 . Show that there exist homeomorphisms $\phi: \widehat{X}_{1} \rightarrow \widehat{X}_{2}$ and $\psi: X \rightarrow X$ so that the diagram

  

  is commutative.

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• Paper 2, Section II, F

  Algebraic Topology | Part II, 2020

  (a) Let $f: X \rightarrow Y$ be a map of spaces. We define the mapping cylinder $M_{f}$ of $f$ to be the space

  $$(([0,1] \times X) \sqcup Y) / \sim$$

  with $(0, x) \sim f(x)$. Show carefully that the canonical inclusion $Y \hookrightarrow M_{f}$ is a homotopy equivalence.

  (b) Using the Seifert-van Kampen theorem, show that if $X$ is path-connected and $\alpha: S^{1} \rightarrow X$ is a map, and $x_{0}=\alpha\left(\theta_{0}\right)$ for some point $\theta_{0} \in S^{1}$, then

  $$\pi_{1}\left(X \cup_{\alpha} D^{2}, x_{0}\right) \cong \pi_{1}\left(X, x_{0}\right) /\langle\langle[\alpha]\rangle\rangle$$

  Use this fact to construct a connected space $X$ with

  $$\pi_{1}(X) \cong\left\langle a, b \mid a^{3}=b^{7}\right\rangle$$

  (c) Using a covering space of $S^{1} \vee S^{1}$, give explicit generators of a subgroup of $F_{2}$ isomorphic to $F_{3}$. Here $F_{n}$ denotes the free group on $n$ generators.

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• Paper 3, Section II, 20F

  Algebraic Topology | Part II, 2020

  Let $K$ be a simplicial complex with four vertices $v_{1}, \ldots, v_{4}$ with simplices $\left\langle v_{1}, v_{2}, v_{3}\right\rangle$, $\left\langle v_{1}, v_{4}\right\rangle$ and $\left\langle v_{2}, v_{4}\right\rangle$ and their faces.

  (a) Draw a picture of $|K|$, labelling the vertices.

  (b) Using the definition of homology, calculate $H_{n}(K)$ for all $n$.

  (c) Let $L$ be the subcomplex of $K$ consisting of the vertices $v_{1}, v_{2}, v_{4}$ and the 1 simplices $\left\langle v_{1}, v_{2}\right\rangle,\left\langle v_{1}, v_{4}\right\rangle,\left\langle v_{2}, v_{4}\right\rangle$. Let $i: L \rightarrow K$ be the inclusion. Construct a simplicial $\operatorname{map} j: K \rightarrow L$ such that the topological realisation $|j|$ of $j$ is a homotopy inverse to $|i|$. Construct an explicit chain homotopy $h: C_{\bullet}(K) \rightarrow C_{\bullet}(K)$ between $i_{\bullet} \circ j_{\bullet}$ and $\mathrm{id}_{C_{\bullet}(K)}$, and verify that $h$ is a chain homotopy.

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• Paper 4 , Section II, 21F

  Algebraic Topology | Part II, 2020

  In this question, you may assume all spaces involved are triangulable.

  (a) (i) State and prove the Mayer-Vietoris theorem. [You may assume the theorem that states that a short exact sequence of chain complexes gives rise to a long exact sequence of homology groups.]

  (ii) Use Mayer-Vietoris to calculate the homology groups of an oriented surface of genus $g$.

  (b) Let $S$ be an oriented surface of genus $g$, and let $D_{1}, \ldots, D_{n}$ be a collection of mutually disjoint closed subsets of $S$ with each $D_{i}$ homeomorphic to a two-dimensional disk. Let $D_{i}^{\circ}$ denote the interior of $D_{i}$, homeomorphic to an open two-dimensional disk, and let

  $$T:=S \backslash\left(D_{1}^{\circ} \cup \cdots \cup D_{n}^{\circ}\right)$$

  Show that

  $$H_{i}(T)= \begin{cases}\mathbb{Z} & i=0 \\ \mathbb{Z}^{2 g+n-1} & i=1 \\ 0 & \text { otherwise }\end{cases}$$

  (c) Let $T$ be the surface given in (b) when $S=S^{2}$ and $n=3$. Let $f: T \rightarrow S^{1} \times S^{1}$ be a map. Does there exist a map $g: S^{1} \times S^{1} \rightarrow T$ such that $f \circ g$ is homotopic to the identity map? Justify your answer.

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Analysis of Functions

• Paper 1, Section II, I

  Analysis of Functions | Part II, 2020

  Let $\mathbb{R}^{n}$ be equipped with the $\sigma$-algebra of Lebesgue measurable sets, and Lebesgue measure.

  (a) Given $f \in L^{\infty}\left(\mathbb{R}^{n}\right), g \in L^{1}\left(\mathbb{R}^{n}\right)$, define the convolution $f \star g$, and show that it is a bounded, continuous function. [You may use without proof continuity of translation on $L^{p}\left(\mathbb{R}^{n}\right)$ for $\left.1 \leqslant p<\infty .\right]$

  Suppose $A \subset \mathbb{R}^{n}$ is a measurable set with $0<|A|<\infty$ where $|A|$ denotes the Lebesgue measure of $A$. By considering the convolution of $f(x)=\mathbb{1}_{A}(x)$ and $g(x)=\mathbb{1}_{A}(-x)$, or otherwise, show that the set $A-A=\{x-y: x, y \in A\}$ contains an open neighbourhood of 0 . Does this still hold if $|A|=\infty$ ?

  (b) Suppose that $f: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ is a measurable function satisfying

  $$f(x+y)=f(x)+f(y), \quad \text { for all } x, y \in \mathbb{R}^{n}$$

  Let $B_{r}=\left\{y \in \mathbb{R}^{m}:|y|<r\right\}$. Show that for any $\epsilon>0$ :

  (i) $f^{-1}\left(B_{\epsilon}\right)-f^{-1}\left(B_{\epsilon}\right) \subset f^{-1}\left(B_{2 \epsilon}\right)$,

  (ii) $f^{-1}\left(B_{k \epsilon}\right)=k f^{-1}\left(B_{\epsilon}\right)$ for all $k \in \mathbb{N}$, where for $\lambda>0$ and $A \subset \mathbb{R}^{n}, \lambda A$ denotes the set $\{\lambda x: x \in A\}$.

  Show that $f$ is continuous at 0 and hence deduce that $f$ is continuous everywhere.

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• Paper 3, Section II, 22I

  Analysis of Functions | Part II, 2020

  Let $X$ be a Banach space.

  (a) Define the dual space $X^{\prime}$, giving an expression for $\|\Lambda\|_{X^{\prime}}$ for $\Lambda \in X^{\prime}$. If $Y=L^{p}\left(\mathbb{R}^{n}\right)$ for some $1 \leqslant p<\infty$, identify $Y^{\prime}$ giving an expression for a general element of $Y^{\prime}$. [You need not prove your assertion.]

  (b) For a sequence $\left(\Lambda_{i}\right)_{i=1}^{\infty}$ with $\Lambda_{i} \in X^{\prime}$, what is meant by: (i) $\Lambda_{i} \rightarrow \Lambda$, (ii) $\Lambda_{i} \rightarrow \Lambda$ (iii) $\Lambda_{i} \stackrel{*}{\rightarrow} \Lambda$ ? Show that (i) $\Longrightarrow$ (ii) $\Longrightarrow$ (iii). Find a sequence $\left(f_{i}\right)_{i=1}^{\infty}$ with $f_{i} \in$ $L^{\infty}(\mathbb{R})=\left(L^{1}(\mathbb{R})\right)^{\prime}$ such that, for some $f, g \in L^{\infty}\left(\mathbb{R}^{n}\right)$ :

  $$f_{i} \stackrel{*}{\rightarrow} f, \quad f_{i}^{2} \stackrel{*}{\rightarrow} g, \quad g \neq f^{2} .$$

  (c) For $f \in C_{c}^{0}\left(\mathbb{R}^{n}\right)$, let $\Lambda: C_{c}^{0}\left(\mathbb{R}^{n}\right) \rightarrow \mathbb{C}$ be the map $\Lambda f=f(0)$. Show that $\Lambda$ may be extended to a continuous linear map $\tilde{\Lambda}: L^{\infty}\left(\mathbb{R}^{n}\right) \rightarrow \mathbb{C}$, and deduce that $\left(L^{\infty}\left(\mathbb{R}^{n}\right)\right)^{\prime} \neq L^{1}\left(\mathbb{R}^{n}\right)$. For which $1 \leqslant p \leqslant \infty$ is $L^{p}\left(\mathbb{R}^{n}\right)$ reflexive? [You may use without proof the Hahn-Banach theorem].

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• Paper 4, Section II, 23I

  Analysis of Functions | Part II, 2020

  (a) Define the Sobolev space $H^{s}\left(\mathbb{R}^{n}\right)$ for $s \in \mathbb{R}$.

  (b) Let $k$ be a non-negative integer and let $s>k+\frac{n}{2}$. Show that if $u \in H^{s}\left(\mathbb{R}^{n}\right)$ then there exists $u^{*} \in C^{k}\left(\mathbb{R}^{n}\right)$ with $u=u^{*}$ almost everywhere.

  (c) Show that if $f \in H^{s}\left(\mathbb{R}^{n}\right)$ for some $s \in \mathbb{R}$, there exists a unique $u \in H^{s+4}\left(\mathbb{R}^{n}\right)$ which solves:

  $$\Delta \Delta u+\Delta u+u=f$$

  in a distributional sense. Prove that there exists a constant $C>0$, independent of $f$, such that:

  $$\|u\|_{H^{s+4}} \leqslant C\|f\|_{H^{s}}$$

  For which $s$ will $u$ be a classical solution?

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Applications of Quantum Mechanics

• Paper 1, Section II, C

  Applications of Quantum Mechanics | Part II, 2020

  Consider the quantum mechanical scattering of a particle of mass $m$ in one dimension off a parity-symmetric potential, $V(x)=V(-x)$. State the constraints imposed by parity, unitarity and their combination on the components of the $S$-matrix in the parity basis,

  $$S=\left(\begin{array}{cc} S_{++} & S_{+-} \\ S_{-+} & S_{--} \end{array}\right)$$

  For the specific potential

  $$V=\hbar^{2} \frac{U_{0}}{2 m}\left[\delta_{D}(x+a)+\delta_{D}(x-a)\right]$$

  show that

  $$S_{--}=e^{-i 2 k a}\left[\frac{\left(2 k-i U_{0}\right) e^{i k a}+i U_{0} e^{-i k a}}{\left(2 k+i U_{0}\right) e^{-i k a}-i U_{0} e^{i k a}}\right]$$

  For $U_{0}<0$, derive the condition for the existence of an odd-parity bound state. For $U_{0}>0$ and to leading order in $U_{0} a \gg 1$, show that an odd-parity resonance exists and discuss how it evolves in time.

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• Paper 2, Section II, $35 \mathrm{C}$

  Applications of Quantum Mechanics | Part II, 2020

  a) Consider a particle moving in one dimension subject to a periodic potential, $V(x)=V(x+a)$. Define the Brillouin zone. State and prove Bloch's theorem.

  b) Consider now the following periodic potential

  $$V=V_{0}(\cos (x)-\cos (2 x))$$

  with positive constant $V_{0}$.

  i) For very small $V_{0}$, use the nearly-free electron model to compute explicitly the lowest-energy band gap to leading order in degenerate perturbation theory.

  ii) For very large $V_{0}$, the electron is localised very close to a minimum of the potential. Estimate the two lowest energies for such localised eigenstates and use the tight-binding model to estimate the lowest-energy band gap.

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• Paper 3, Section II, C

  Applications of Quantum Mechanics | Part II, 2020

  (a) For the quantum scattering of a beam of particles in three dimensions off a spherically symmetric potential $V(r)$ that vanishes at large $r$, discuss the boundary conditions satisfied by the wavefunction $\psi$ and define the scattering amplitude $f(\theta)$. Assuming the asymptotic form

  $$\psi=\sum_{l=0}^{\infty} \frac{2 l+1}{2 i k}\left[(-1)^{l+1} \frac{e^{-i k r}}{r}+\left(1+2 i f_{l}\right) \frac{e^{i k r}}{r}\right] P_{l}(\cos \theta),$$

  state the constraints on $f_{l}$ imposed by the unitarity of the $S$-matrix and define the phase shifts $\delta_{l}$.

  (b) For $V_{0}>0$, consider the specific potential

  $$V(r)=\left\{\begin{array}{lc} \infty, & r \leqslant a \\ -V_{0}, & a<r \leqslant 2 a \\ 0, & r>2 a \end{array}\right.$$

  (i) Show that the s-wave phase shift $\delta_{0}$ obeys

  $$\tan \left(\delta_{0}\right)=\frac{k \cos (2 k a)-\kappa \cot (\kappa a) \sin (2 k a)}{k \sin (2 k a)+\kappa \cot (\kappa a) \cos (2 k a)},$$

  where $\kappa^{2}=k^{2}+2 m V_{0} / \hbar^{2}$.

  (ii) Compute the scattering length $a_{s}$ and find for which values of $\kappa$ it diverges. Discuss briefly the physical interpretation of the divergences. [Hint: you may find this trigonometric identity useful

  $$\left.\tan (A+B)=\frac{\tan A+\tan B}{1-\tan A \tan B} .\right]$$

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• Paper 4, Section II, $34 C$

  Applications of Quantum Mechanics | Part II, 2020

  (a) For a particle of charge $q$ moving in an electromagnetic field with vector potential $\boldsymbol{A}$ and scalar potential $\phi$, write down the classical Hamiltonian and the equations of motion.

  (b) Consider the vector and scalar potentials

  $$\boldsymbol{A}=\frac{B}{2}(-y, x, 0), \quad \phi=0$$

  (i) Solve the equations of motion. Define and compute the cyclotron frequency $\omega_{B}$.

  (ii) Write down the quantum Hamiltonian of the system in terms of the angular momentum operator

  $$L_{z}=x p_{y}-y p_{x}$$

  Show that the states

  $$\psi(x, y)=f(x+i y) e^{-\left(x^{2}+y^{2}\right) q B / 4 \hbar}$$

  for any function $f$, are energy eigenstates and compute their energy. Define Landau levels and discuss this result in relation to them.

  (iii) Show that for $f(w)=w^{M}$, the wavefunctions in ( $\dagger$ ) are eigenstates of angular momentum and compute the corresponding eigenvalue. These wavefunctions peak in a ring around the origin. Estimate its radius. Using these two facts or otherwise, estimate the degeneracy of Landau levels.

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Applied Probability

• Paper 1, Section II, 28K

  Applied Probability | Part II, 2020

  (a) What is meant by a birth process $N=(N(t): t \geqslant 0)$ with strictly positive rates $\lambda_{0}, \lambda_{1}, \ldots ?$ Explain what is meant by saying that $N$ is non-explosive.

  (b) Show that $N$ is non-explosive if and only if

  $$\sum_{n=0}^{\infty} \frac{1}{\lambda_{n}}=\infty$$

  (c) Suppose $N(0)=0$, and $\lambda_{n}=\alpha n+\beta$ where $\alpha, \beta>0$. Show that

  $$\mathbb{E}(N(t))=\frac{\beta}{\alpha}\left(e^{\alpha t}-1\right) .$$

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• Paper 2, Section II, 27K

  Applied Probability | Part II, 2020

  (i) Let $X$ be a Markov chain in continuous time on the integers $\mathbb{Z}$ with generator $\mathbf{G}=\left(g_{i, j}\right)$. Define the corresponding jump chain $Y$.

  Define the terms irreducibility and recurrence for $X$. If $X$ is irreducible, show that $X$ is recurrent if and only if $Y$ is recurrent.

  (ii) Suppose

  $$g_{i, i-1}=3^{|i|}, \quad g_{i, i}=-3^{|i|+1}, \quad g_{i, i+1}=2 \cdot 3^{|i|}, \quad i \in \mathbb{Z} .$$

  Show that $X$ is transient, find an invariant distribution, and show that $X$ is explosive. [Any general results may be used without proof but should be stated clearly.]

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• Paper 3, Section II, 27K

  Applied Probability | Part II, 2020

  Define a renewal-reward process, and state the renewal-reward theorem.

  A machine $M$ is repaired at time $t=0$. After any repair, it functions without intervention for a time that is exponentially distributed with parameter $\lambda$, at which point it breaks down (assume the usual independence). Following any repair at time $T$, say, it is inspected at times $T, T+m, T+2 m, \ldots$, and instantly repaired if found to be broken (the inspection schedule is then restarted). Find the long run proportion of time that $M$ is working. [You may express your answer in terms of an integral.]

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• Paper 4, Section II, K

  Applied Probability | Part II, 2020

  (i) Explain the notation $\mathrm{M}(\lambda) / \mathrm{M}(\mu) / 1$ in the context of queueing theory. [In the following, you may use without proof the fact that $\pi_{n}=(\lambda / \mu)^{n}$ is the invariant distribution of such a queue when $0<\lambda<\mu$.

  (ii) In a shop queue, some customers rejoin the queue after having been served. Let $\lambda, \beta \in(0, \infty)$ and $\delta \in(0,1)$. Consider a $\mathrm{M}(\lambda) / \mathrm{M}(\mu) / 1$ queue subject to the modification that, on completion of service, each customer leaves the shop with probability $\delta$, or rejoins the shop queue with probability $1-\delta$. Different customers behave independently of one another, and all service times are independent random variables.

  Find the distribution of the total time a given customer spends being served by the server. Hence show that equilibrium is possible if $\lambda<\delta \mu$, and find the invariant distribution of the queue-length in this case.

  (iii) Show that, in equilibrium, the departure process is Poissonian, whereas, assuming the rejoining customers go to the end of the queue, the process of customers arriving at the queue (including the rejoining ones) is not Poissonian.

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Asymptotic Methods

• Paper 2, Section II, D

  Asymptotic Methods | Part II, 2020

  (a) Let $\delta>0$ and $x_{0} \in \mathbb{R}$. Let $\left\{\phi_{n}(x)\right\}_{n=0}^{\infty}$ be a sequence of (real) functions that are nonzero for all $x$ with $0<\left|x-x_{0}\right|<\delta$, and let $\left\{a_{n}\right\}_{n=0}^{\infty}$ be a sequence of nonzero real numbers. For every $N=0,1,2, \ldots$, the function $f(x)$ satisfies

  $$f(x)-\sum_{n=0}^{N} a_{n} \phi_{n}(x)=o\left(\phi_{N}(x)\right), \quad \text { as } \quad x \rightarrow x_{0}$$

  (i) Show that $\phi_{n+1}(x)=o\left(\phi_{n}(x)\right)$, for all $n=0,1,2, \ldots$; i.e., $\left\{\phi_{n}(x)\right\}_{n=0}^{\infty}$ is an asymptotic sequence.

  (ii) Show that for any $N=0,1,2, \ldots$, the functions $\phi_{0}(x), \phi_{1}(x), \ldots, \phi_{N}(x)$ are linearly independent on their domain of definition.

  (b) Let

  $$I(\varepsilon)=\int_{0}^{\infty}(1+\varepsilon t)^{-2} e^{-(1+\varepsilon) t} d t, \quad \text { for } \varepsilon>0$$

  (i) Find an asymptotic expansion (not necessarily a power series) of $I(\varepsilon)$, as $\varepsilon \rightarrow 0^{+}$.

  (ii) Find the first four terms of the expansion of $I(\varepsilon)$ into an asymptotic power series of $\varepsilon$, that is, with error $o\left(\varepsilon^{3}\right)$ as $\varepsilon \rightarrow 0^{+}$.

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• Paper 3, Section II, D

  Asymptotic Methods | Part II, 2020

  (a) Find the leading order term of the asymptotic expansion, as $x \rightarrow \infty$, of the integral

  $$I(x)=\int_{0}^{3 \pi} e^{(t+x \cos t)} d t$$

  (b) Find the first two leading nonzero terms of the asymptotic expansion, as $x \rightarrow \infty$, of the integral

  $$J(x)=\int_{0}^{\pi}(1-\cos t) e^{-x \ln (1+t)} d t$$

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• Paper 4, Section II, A

  Asymptotic Methods | Part II, 2020

  Consider the differential equation

  $$\tag{†} y^{\prime \prime}-y^{\prime}-\frac{2(x+1)}{x^{2}} y=0$$

  (i) Classify what type of regularity/singularity equation $(†)$ has at $x=\infty$.

  (ii) Find a transformation that maps equation ($†$) to an equation of the form

  $$u^{\prime \prime}+q(x) u=0$$

  (iii) Find the leading-order term of the asymptotic expansions of the solutions of equation $(*)$, as $x \rightarrow \infty$, using the Liouville-Green method.

  (iv) Derive the leading-order term of the asymptotic expansion of the solutions $y$ of ($†$). Check that one of them is an exact solution for $(†)$.

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Automata and Formal Languages

• Paper 1, Section I, $4 \mathbf{F}$

  Automata and Formal Languages | Part II, 2020

  Define an alphabet $\Sigma$, a word over $\Sigma$ and a language over $\Sigma$.

  What is a regular expression $R$ and how does this give rise to a language $\mathcal{L}(R) ?$

  Given any alphabet $\Sigma$, show that there exist languages $L$ over $\Sigma$ which are not equal to $\mathcal{L}(R)$ for any regular expression $R$. [You are not required to exhibit a specific $L$.]

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• Paper 1, Section II, F

  Automata and Formal Languages | Part II, 2020

  (a) Define a register machine, a sequence of instructions for a register machine and a partial computable function. How do we encode a register machine?

  (b) What is a partial recursive function? Show that a partial computable function is partial recursive. [You may assume that for a given machine with a given number of inputs, the function outputting its state in terms of the inputs and the time $t$ is recursive.]

  (c) (i) Let $g: \mathbb{N} \rightarrow \mathbb{N}$ be the partial function defined as follows: if $n$ codes a register machine and the ensuing partial function $f_{n, 1}$ is defined at $n$, set $g(n)=f_{n, 1}(n)+1$. Otherwise set $g(n)=0$. Is $g$ a partial computable function?

  (ii) Let $h: \mathbb{N} \rightarrow \mathbb{N}$ be the partial function defined as follows: if $n$ codes a register machine and the ensuing partial function $f_{n, 1}$ is defined at $n$, set $h(n)=f_{n, 1}(n)+1$. Otherwise, set $h(n)=0$ if $n$ is odd and let $h(n)$ be undefined if $n$ is even. Is $h$ a partial computable function?

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• Paper 2, Section I, F

  Automata and Formal Languages | Part II, 2020

  Assuming the definition of a partial recursive function from $\mathbb{N}$ to $\mathbb{N}$, what is a recursive subset of $\mathbb{N}$ ? What is a recursively enumerable subset of $\mathbb{N}$ ?

  Show that a subset $E \subseteq \mathbb{N}$ is recursive if and only if $E$ and $\mathbb{N} \backslash E$ are recursively enumerable.

  Are the following subsets of $\mathbb{N}$ recursive?

  (i) $\mathbb{K}:=\left\{n \mid n\right.$ codes a program and $f_{n, 1}(n)$ halts at some stage $\}$.

  (ii) $\mathbb{K}_{100}:=\left\{n \mid n\right.$ codes a program and $f_{n, 1}(n)$ halts within 100 steps $\}$.

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• Paper 3, Section I, F

  Automata and Formal Languages | Part II, 2020

  Define a context-free grammar $G$, a sentence of $G$ and the language $\mathcal{L}(G)$ generated by $G$.

  For the alphabet $\Sigma=\{a, b\}$, which of the following languages over $\Sigma$ are contextfree? (i) $\left\{a^{2 m} b^{2 m} \mid m \geqslant 0\right\}$,

  (ii) $\left\{a^{m^{2}} b^{m^{2}} \mid m \geqslant 0\right\}$.

  [You may assume standard results without proof if clearly stated.]

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• Paper 3, Section II, F

  Automata and Formal Languages | Part II, 2020

  Give the definition of a deterministic finite state automaton and of a regular language.

  State and prove the pumping lemma for regular languages.

  Let $S=\left\{2^{n} \mid n=0,1,2, \ldots\right\}$ be the subset of $\mathbb{N}$ consisting of the powers of 2 .

  If we write the elements of $S$ in base 2 (with no preceding zeros), is $S$ a regular language over $\{0,1\}$ ?

  Now suppose we write the elements of $S$ in base 10 (again with no preceding zeros). Show that $S$ is not a regular language over $\{0,1,2,3,4,5,6,7,8,9\}$. [Hint: Give a proof by contradiction; use the above lemma to obtain a sequence $a_{1}, a_{2}, \ldots$ of powers of 2, then consider $a_{i+1}-10^{d} a_{i}$ for $i=1,2,3, \ldots$ and a suitable fixed d.]

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• Paper 4, Section I, $4 F$

  Automata and Formal Languages | Part II, 2020

  Define what it means for a context-free grammar (CFG) to be in Chomsky normal form $(\mathrm{CNF})$.

  Describe without proof each stage in the process of converting a CFG $G=$ $(N, \Sigma, P, S)$ into an equivalent CFG $\bar{G}$ which is in CNF. For each of these stages, when are the nonterminals $N$ left unchanged? What about the terminals $\Sigma$ and the generated language $\mathcal{L}(G)$ ?

  Give an example of a CFG $G$ whose generated language $\mathcal{L}(G)$ is infinite and equal to $\mathcal{L}(\bar{G})$.

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Classical Dynamics

• Paper 1, Section I, B

  Classical Dynamics | Part II, 2020

  A linear molecule is modelled as four equal masses connected by three equal springs. Using the Cartesian coordinates $x_{1}, x_{2}, x_{3}, x_{4}$ of the centres of the four masses, and neglecting any forces other than those due to the springs, write down the Lagrangian of the system describing longitudinal motions of the molecule.

  Rewrite and simplify the Lagrangian in terms of the generalized coordinates

  $$q_{1}=\frac{x_{1}+x_{4}}{2}, \quad q_{2}=\frac{x_{2}+x_{3}}{2}, \quad q_{3}=\frac{x_{1}-x_{4}}{2}, \quad q_{4}=\frac{x_{2}-x_{3}}{2}$$

  Deduce Lagrange's equations for $q_{1}, q_{2}, q_{3}, q_{4}$. Hence find the normal modes of the system and their angular frequencies, treating separately the symmetric and antisymmetric modes of oscillation.

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• Paper 2, Section I, B

  Classical Dynamics | Part II, 2020

  A particle of mass $m$ has position vector $\mathbf{r}(t)$ in a frame of reference that rotates with angular velocity $\boldsymbol{\omega}(t)$. The particle moves under the gravitational influence of masses that are fixed in the rotating frame. Explain why the Lagrangian of the particle is of the form

  $$L=\frac{1}{2} m(\dot{\mathbf{r}}+\boldsymbol{\omega} \times \mathbf{r})^{2}-V(\mathbf{r}) .$$

  Show that Lagrange's equations of motion are equivalent to

  $$m(\ddot{\mathbf{r}}+2 \boldsymbol{\omega} \times \dot{\mathbf{r}}+\dot{\boldsymbol{\omega}} \times \mathbf{r}+\boldsymbol{\omega} \times(\boldsymbol{\omega} \times \mathbf{r}))=-\boldsymbol{\nabla} V$$

  Identify the canonical momentum $\mathbf{p}$ conjugate to $\mathbf{r}$. Obtain the Hamiltonian $H(\mathbf{r}, \mathbf{p})$ and Hamilton's equations for this system.

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• Paper 2, Section II, B

  Classical Dynamics | Part II, 2020

  A symmetric top of mass $M$ rotates about a fixed point that is a distance $l$ from the centre of mass along the axis of symmetry; its principal moments of inertia about the fixed point are $I_{1}=I_{2}$ and $I_{3}$. The Lagrangian of the top is

  $$L=\frac{1}{2} I_{1}\left(\dot{\theta}^{2}+\dot{\phi}^{2} \sin ^{2} \theta\right)+\frac{1}{2} I_{3}(\dot{\psi}+\dot{\phi} \cos \theta)^{2}-M g l \cos \theta$$

  (i) Draw a diagram explaining the meaning of the Euler angles $\theta, \phi$ and $\psi$.

  (ii) Derive expressions for the three integrals of motion $E, L_{3}$ and $L_{z}$.

  (iii) Show that the nutational motion is governed by the equation

  $$\frac{1}{2} I_{1} \dot{\theta}^{2}+V_{\text {eff }}(\theta)=E^{\prime}$$

  and derive expressions for the effective potential $V_{\mathrm{eff}}(\theta)$ and the modified energy $E^{\prime}$ in terms of $E, L_{3}$ and $L_{z}$.

  (iv) Suppose that

  $$L_{z}=L_{3}\left(1-\frac{\epsilon^{2}}{2}\right)$$

  where $\epsilon$ is a small positive number. By expanding $V_{\text {eff }}$ to second order in $\epsilon$ and $\theta$, show that there is a stable equilibrium solution with $\theta=O(\epsilon)$, provided that $L_{3}^{2}>4 M g l I_{1}$. Determine the equilibrium value of $\theta$ and the precession rate $\dot{\phi}$, to the same level of approximation.

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• Paper 3, Section I, B

  Classical Dynamics | Part II, 2020

  A particle of mass $m$ experiences a repulsive central force of magnitude $k / r^{2}$, where $r=|\mathbf{r}|$ is its distance from the origin. Write down the Hamiltonian of the system.

  The Laplace-Runge-Lenz vector for this system is defined by

  $$\mathbf{A}=\mathbf{p} \times \mathbf{L}+m k \hat{\mathbf{r}}$$

  where $\mathbf{L}=\mathbf{r} \times \mathbf{p}$ is the angular momentum and $\hat{\mathbf{r}}=\mathbf{r} / r$ is the radial unit vector. Show that

  $$\{\mathbf{L}, H\}=\{\mathbf{A}, H\}=\mathbf{0},$$

  where $\{\cdot, \cdot\}$ is the Poisson bracket. What are the integrals of motion of the system? Show that the polar equation of the orbit can be written as

  $$r=\frac{\lambda}{e \cos \theta-1},$$

  where $\lambda$ and $e$ are non-negative constants.

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• Paper 4, Section I, B

  Classical Dynamics | Part II, 2020

  Derive expressions for the angular momentum and kinetic energy of a rigid body in terms of its mass $M$, the position $\mathbf{X}(t)$ of its centre of mass, its inertia tensor $I$ (which should be defined) about its centre of mass, and its angular velocity $\boldsymbol{\omega}$.

  A spherical planet of mass $M$ and radius $R$ has density proportional to $r^{-1} \sin (\pi r / R)$. Given that $\int_{0}^{\pi} x \sin x d x=\pi$ and $\int_{0}^{\pi} x^{3} \sin x d x=\pi\left(\pi^{2}-6\right)$, evaluate the inertia tensor of the planet in terms of $M$ and $R$.

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• Paper 4, Section II, B

  Classical Dynamics | Part II, 2020

  (a) Explain how the Hamiltonian $H(\mathbf{q}, \mathbf{p}, t)$ of a system can be obtained from its Lagrangian $L(\mathbf{q}, \dot{\mathbf{q}}, t)$. Deduce that the action can be written as

  $$S=\int(\mathbf{p} \cdot d \mathbf{q}-H d t)$$

  Show that Hamilton's equations are obtained if the action, computed between fixed initial and final configurations $\mathbf{q}\left(t_{1}\right)$ and $\mathbf{q}\left(t_{2}\right)$, is minimized with respect to independent variations of $\mathbf{q}$ and $\mathbf{p}$.

  (b) Let $(\mathbf{Q}, \mathbf{P})$ be a new set of coordinates on the same phase space. If the old and new coordinates are related by a type-2 generating function $F_{2}(\mathbf{q}, \mathbf{P}, t)$ such that

  $$\mathbf{p}=\frac{\partial F_{2}}{\partial \mathbf{q}}, \quad \mathbf{Q}=\frac{\partial F_{2}}{\partial \mathbf{P}}$$

  deduce that the canonical form of Hamilton's equations applies in the new coordinates, but with a new Hamiltonian given by

  $$K=H+\frac{\partial F_{2}}{\partial t}$$

  (c) For each of the Hamiltonians (i) $H=H(p)$, (ii) $H=\frac{1}{2}\left(q^{2}+p^{2}\right)$,

  express the general solution $(q(t), p(t))$ at time $t$ in terms of the initial values given by $(Q, P)=(q(0), p(0))$ at time $t=0$. In each case, show that the transformation from $(q, p)$ to $(Q, P)$ is canonical for all values of $t$, and find the corresponding generating function $F_{2}(q, P, t)$ explicitly.

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Coding and Cryptography

• Paper 1, Section I, I

  Coding and Cryptography | Part II, 2020

  (a) Briefly describe the methods of Shannon-Fano and of Huffman for the construction of prefix-free binary codes.

  (b) In this part you are given that $-\log _{2}(1 / 10) \approx 3.32,-\log _{2}(2 / 10) \approx 2.32$, $-\log _{2}(3 / 10) \approx 1.74$ and $-\log _{2}(4 / 10) \approx 1.32$.

  Let $\mathcal{A}=\{1,2,3,4\}$. For $k \in \mathcal{A}$, suppose that the probability of choosing $k$ is $k / 10$.

  (i) Find a Shannon-Fano code for this system and the expected word length.

  (ii) Find a Huffman code for this system and the expected word length.

  (iii) Verify that Shannon's noiseless coding theorem is satisfied in each case.

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• Paper 1, Section II, I

  Coding and Cryptography | Part II, 2020

  (a) What does it mean to say that a binary code has length $n$, size $M$ and minimum distance d?

  Let $A(n, d)$ be the largest value of $M$ for which there exists a binary $[n, M, d]$-code.

  (i) Show that $A(n, 1)=2^{n}$.

  (ii) Suppose that $n, d>1$. Show that if a binary $[n, M, d]$-code exists, then a binary $[n-1, M, d-1]$-code exists. Deduce that $A(n, d) \leqslant A(n-1, d-1)$.

  (iii) Suppose that $n, d \geqslant 1$. Show that $A(n, d) \leqslant 2^{n-d+1}$.

  (b) (i) For integers $M$ and $N$ with $0 \leqslant N \leqslant M$, show that

  $$N(M-N) \leqslant\left\{\begin{array}{cl} M^{2} / 4, & \text { if } M \text { is even }, \\ \left(M^{2}-1\right) / 4, & \text { if } M \text { is odd } \end{array}\right.$$

  For the remainder of this question, suppose that $C$ is a binary $[n, M, d]$-code. For codewords $x=\left(x_{1} \ldots x_{n}\right), y=\left(y_{1} \ldots y_{n}\right) \in C$ of length $n$, we define $x+y$ to be the word $\left(\left(x_{1}+y_{1}\right) \ldots\left(x_{n}+y_{n}\right)\right)$ with addition modulo $2 .$

  (ii) Explain why the Hamming distance $d(x, y)$ is the number of 1 s in $x+y$.

  (iii) Now we construct an $\left(\begin{array}{c}M \\ 2\end{array}\right) \times n$ array $A$ whose rows are all the words $x+y$ for pairs of distinct codewords $x, y$. Show that the number of $1 \mathrm{~s}$ in $A$ is at most

  $$\begin{cases}n M^{2} / 4, & \text { if } M \text { is even } \\ n\left(M^{2}-1\right) / 4, & \text { if } M \text { is odd }\end{cases}$$

  Show also that the number of $1 \mathrm{~s}$ in $A$ is at least $d\left(\begin{array}{c}M \\ 2\end{array}\right)$.

  (iv) Using the inequalities derived in part(b) (iii), deduce that if $d$ is even and $n<2 d$ then

  $$A(n, d) \leqslant 2\left\lfloor\frac{d}{2 d-n}\right\rfloor$$

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• Paper 2, Section I, I

  Coding and Cryptography | Part II, 2020

  (a) Define the information capacity of a discrete memoryless channel (DMC).

  (b) Consider a DMC where there are two input symbols, $A$ and $B$, and three output symbols, $A, B$ and $\star$. Suppose each input symbol is left intact with probability $1 / 2$, and transformed into a $\star$ with probability $1 / 2$.

  (i) Write down the channel matrix, and calculate the information capacity.

  (ii) Now suppose the output is further processed by someone who cannot distinguish between $A$ and $\star$, so that the channel matrix becomes

  $$\left(\begin{array}{cc} 1 & 0 \\ 1 / 2 & 1 / 2 \end{array}\right)$$

  Calculate the new information capacity.

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• Paper 2, Section II, I

  Coding and Cryptography | Part II, 2020

  Let $C$ be the Hamming $(n, n-d)$ code of weight 3 , where $n=2^{d}-1, d>1$. Let $H$ be the parity-check matrix of $C$. Let $\nu(j)$ be the number of codewords of weight $j$ in $C$.

  (i) Show that for any two columns $h_{1}$ and $h_{2}$ of $H$ there exists a unique third column $h_{3}$ such that $h_{3}=h_{2}+h_{1}$. Deduce that $\nu(3)=n(n-1) / 6$.

  (ii) Show that $C$ contains a codeword of weight $n$.

  (iii) Find formulae for $\nu(n-1), \nu(n-2)$ and $\nu(n-3)$. Justify your answer in each case.

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• Paper 3, Section I, I

  Coding and Cryptography | Part II, 2020

  Let $N$ and $p$ be very large positive integers with $p$ a prime and $p>N$. The Chair of the Committee is able to inscribe pairs of very large integers on discs. The Chair wishes to inscribe a collection of discs in such a way that any Committee member who acquires $r$ of the discs and knows the prime $p$ can deduce the integer $N$, but owning $r-1$ discs will give no information whatsoever. What strategy should the Chair follow?

  [You may use without proof standard properties of the determinant of the $r \times r$ Vandermonde matrix.]

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• Paper 4, Section I, I

  Coding and Cryptography | Part II, 2020

  (a) What does it mean to say that a cipher has perfect secrecy? Show that if a cipher has perfect secrecy then there must be at least as many possible keys as there are possible plaintext messages. What is a one-time pad? Show that a one-time pad has perfect secrecy.

  (b) I encrypt a binary sequence $a_{1}, a_{2}, \ldots, a_{N}$ using a one-time pad with key sequence $k_{1}, k_{2}, k_{3}, \ldots .$ I transmit $a_{1}+k_{1}, a_{2}+k_{2}, \ldots, a_{N}+k_{N}$ to you. Then, by mistake, I also transmit $a_{1}+k_{2}, a_{2}+k_{3}, \ldots, a_{N}+k_{N+1}$ to you. Assuming that you know I have made this error, and that my message makes sense, how would you go about finding my message? Can you now decipher other messages sent using the same part of the key sequence? Briefly justify your answer.

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Cosmology

• Paper 1, Section I, D

  Cosmology | Part II, 2020

  The Friedmann equation is

  $$H^{2}=\frac{8 \pi G}{3 c^{2}}\left(\rho-\frac{k c^{2}}{R^{2} a^{2}}\right)$$

  Briefly explain the meaning of $H, \rho, k$ and $R$.

  Derive the Raychaudhuri equation,

  $$\frac{\ddot{a}}{a}=-\frac{4 \pi G}{3 c^{2}}(\rho+3 P),$$

  where $P$ is the pressure, stating clearly any results that are required.

  Assume that the strong energy condition $\rho+3 P \geqslant 0$ holds. Show that there was necessarily a Big Bang singularity at time $t_{B B}$ such that

  $$t_{0}-t_{B B} \leqslant H_{0}^{-1}$$

  where $H_{0}=H\left(t_{0}\right)$ and $t_{0}$ is the time today.

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• Paper 1, Section II, D

  Cosmology | Part II, 2020

  A fluid with pressure $P$ sits in a volume $V$. The change in energy due to a change in volume is given by $d E=-P d V$. Use this in a cosmological context to derive the continuity equation,

  $$\dot{\rho}=-3 H(\rho+P),$$

  with $\rho$ the energy density, $H=\dot{a} / a$ the Hubble parameter, and $a$ the scale factor.

  In a flat universe, the Friedmann equation is given by

  $$H^{2}=\frac{8 \pi G}{3 c^{2}} \rho .$$

  Given a universe dominated by a fluid with equation of state $P=w \rho$, where $w$ is a constant, determine how the scale factor $a(t)$ evolves.

  Define conformal time $\tau$. Assume that the early universe consists of two fluids: radiation with $w=1 / 3$ and a network of cosmic strings with $w=-1 / 3$. Show that the Friedmann equation can be written as

  $$\left(\frac{d a}{d \tau}\right)^{2}=B \rho_{\mathrm{eq}}\left(a^{2}+a_{\mathrm{eq}}^{2}\right)$$

  where $\rho_{\mathrm{eq}}$ is the energy density in radiation, and $a_{\mathrm{eq}}$ is the scale factor, both evaluated at radiation-string equality. Here, $B$ is a constant that you should determine. Find the solution $a(\tau)$.

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• Paper 2, Section I, D

  Cosmology | Part II, 2020

  During inflation, the expansion of the universe is governed by the Friedmann equation,

  $$H^{2}=\frac{8 \pi G}{3 c^{2}}\left(\frac{1}{2} \dot{\phi}^{2}+V(\phi)\right)$$

  and the equation of motion for the inflaton field $\phi$,

  $$\ddot{\phi}+3 H \dot{\phi}+\frac{\partial V}{\partial \phi}=0 \text {. }$$

  The slow-roll conditions are $\dot{\phi}^{2} \ll V(\phi)$ and $\ddot{\phi} \ll H \dot{\phi}$. Under these assumptions, solve for $\phi(t)$ and $a(t)$ for the potentials:

  (i) $V(\phi)=\frac{1}{2} m^{2} \phi^{2}$ and

  (ii) $V(\phi)=\frac{1}{4} \lambda \phi^{4}, \quad(\lambda>0)$.

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• Paper 3, Section I, D

  Cosmology | Part II, 2020

  At temperature $T$, with $\beta=1 /\left(k_{B} T\right)$, the distribution of ultra-relativistic particles with momentum $\mathbf{p}$ is given by

  $$n(\mathbf{p})=\frac{1}{e^{\beta p c} \mp 1},$$

  where the minus sign is for bosons and the plus $\operatorname{sign}$ for fermions, and with $p=|\mathbf{p}|$.

  Show that the total number of fermions, $n_{\mathrm{f}}$, is related to the total number of bosons, $n_{\mathrm{b}}$, by $n_{\mathrm{f}}=\frac{3}{4} n_{\mathrm{b}}$.

  Show that the total energy density of fermions, $\rho_{\mathrm{f}}$, is related to the total energy density of bosons, $\rho_{\mathrm{b}}$, by $\rho_{\mathrm{f}}=\frac{7}{8} \rho_{\mathrm{b}}$.

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• Paper 3, Section II, D

  Cosmology | Part II, 2020

  In an expanding spacetime, the density contrast $\delta(\mathbf{x}, t)$ satisfies the linearised equation

  $$\ddot{\delta}+2 H \dot{\delta}-c_{s}^{2}\left(\frac{1}{a^{2}} \nabla^{2}+k_{J}^{2}\right) \delta=0,$$

  where $a$ is the scale factor, $H$ is the Hubble parameter, $c_{s}$ is a constant, and $k_{J}$ is the Jeans wavenumber, defined by

  $$c_{s}^{2} k_{J}^{2}=\frac{4 \pi G}{c^{2}} \bar{\rho}(t)$$

  with $\bar{\rho}(t)$ the background, homogeneous energy density.

  (i) Solve for $\delta(\mathbf{x}, t)$ in a static universe, with $a=1$ and $H=0$ and $\bar{\rho}$ constant. Identify two regimes: one in which sound waves propagate, and one in which there is an instability.

  (ii) In a matter-dominated universe with $\bar{\rho} \sim 1 / a^{3}$, use the Friedmann equation $H^{2}=8 \pi G \bar{\rho} / 3 c^{2}$ to find the growing and decaying long-wavelength modes of $\delta$ as a function of $a$.

  (iii) Assuming $c_{s}^{2} \approx c_{s}^{2} k_{J}^{2} \approx 0$ in equation $(*)$, find the growth of matter perturbations in a radiation-dominated universe and find the growth of matter perturbations in a curvature-dominated universe.

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• Paper 4 , Section I, D

  Cosmology | Part II, 2020

  At temperature $T$ and chemical potential $\mu$, the number density of a non-relativistic particle species with mass $m \gg k_{B} T / c^{2}$ is given by

  $$n=g\left(\frac{m k_{B} T}{2 \pi \hbar^{2}}\right)^{3 / 2} e^{-\left(m c^{2}-\mu\right) / k_{B} T},$$

  where $g$ is the number of degrees of freedom of this particle.

  At recombination, electrons and protons combine to form hydrogen. Use the result above to derive the Saha equation

  $$n_{H} \approx n_{e}^{2}\left(\frac{2 \pi \hbar^{2}}{m_{e} k_{B} T}\right)^{3 / 2} e^{E_{\mathrm{bind}} / k_{B} T}$$

  where $n_{H}$ is the number density of hydrogen atoms, $n_{e}$ the number density of electrons, $m_{e}$ the mass of the electron and $E_{\text {bind }}$ the binding energy of hydrogen. State any assumptions that you use in this derivation.

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Differential Geometry

• Paper 1, Section II, I

  Differential Geometry | Part II, 2020

  (a) Let $X \subset \mathbb{R}^{N}$ be a manifold. Give the definition of the tangent space $T_{p} X$ of $X$ at a point $p \in X$.

  (b) Show that $X:=\left\{-x_{0}^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=-1\right\} \cap\left\{x_{0}>0\right\}$ defines a submanifold of $\mathbb{R}^{4}$ and identify explicitly its tangent space $T_{\mathbf{x}} X$ for any $\mathbf{x} \in X$.

  (c) Consider the matrix group $O(1,3) \subset \mathbb{R}^{4^{2}}$ consisting of all $4 \times 4$ matrices $A$ satisfying

  $$A^{t} M A=M$$

  where $M$ is the diagonal $4 \times 4$ matrix $M=\operatorname{diag}(-1,1,1,1)$.

  (i) Show that $O(1,3)$ forms a group under matrix multiplication, i.e. it is closed under multiplication and every element in $O(1,3)$ has an inverse in $O(1,3)$.

  (ii) Show that $O(1,3)$ defines a 6-dimensional manifold. Identify the tangent space $T_{A} O(1,3)$ for any $A \in O(1,3)$ as a set $\{A Y\}_{Y \in \mathfrak{S}}$ where $Y$ ranges over a linear subspace $\mathfrak{S} \subset \mathbb{R}^{4^{2}}$ which you should identify explicitly.

  (iii) Let $X$ be as defined in (b) above. Show that $O^{+}(1,3) \subset O(1,3)$ defined as the set of all $A \in O(1,3)$ such that $A \mathbf{x} \in X$ for all $\mathbf{x} \in X$ is both a subgroup and a submanifold of full dimension.

  [You may use without proof standard theorems from the course concerning regular values and transversality.]

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• Paper 2, Section II, I

  Differential Geometry | Part II, 2020

  (a) State the fundamental theorem for regular curves in $\mathbb{R}^{3}$.

  (b) Let $\alpha: \mathbb{R} \rightarrow \mathbb{R}^{3}$ be a regular curve, parameterised by arc length, such that its image $\alpha(\mathbb{R}) \subset \mathbb{R}^{3}$ is a one-dimensional submanifold. Suppose that the set $\alpha(\mathbb{R})$ is preserved by a nontrivial proper Euclidean motion $\phi: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$.

  Show that there exists $\sigma_{0} \in \mathbb{R}$ corresponding to $\phi$ such that $\phi(\alpha(s))=\alpha\left(\pm s+\sigma_{0}\right)$ for all $s \in \mathbb{R}$, where the choice of $\pm \operatorname{sign}$ is independent of $s$. Show also that the curvature $k(s)$ and torsion $\tau(s)$ of $\alpha$ satisfy

  $$\begin{gathered} k\left(\pm s+\sigma_{0}\right)=k(s) \text { and } \\ \tau\left(\pm s+\sigma_{0}\right)=\tau(s) \end{gathered}$$

  with equation (2) valid only for $s$ such that $k(s)>0$. In the case where the sign is $+$ and $\sigma_{0}=0$, show that $\alpha(\mathbb{R})$ is a straight line.

  (c) Give an explicit example of a curve $\alpha$ satisfying the requirements of (b) such that neither of $k(s)$ and $\tau(s)$ is a constant function, and such that the curve $\alpha$ is closed, i.e. such that $\alpha(s)=\alpha\left(s+s_{0}\right)$ for some $s_{0}>0$ and all $s$. [Here a drawing would suffice.]

  (d) Suppose now that $\alpha: \mathbb{R} \rightarrow \mathbb{R}^{3}$ is an embedded regular curve parameterised by arc length $s$. Suppose further that $k(s)>0$ for all $s$ and that $k(s)$ and $\tau(s)$ satisfy (1) and (2) for some $\sigma_{0}$, where the choice $\pm$ is independent of $s$, and where $\sigma_{0} \neq 0$ in the case of + sign. Show that there exists a nontrivial proper Euclidean motion $\phi$ such that the set $\alpha(\mathbb{R})$ is preserved by $\phi$. [You may use the theorem of part (a) without proof.]

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• Paper 3, Section II, I

  Differential Geometry | Part II, 2020

  (a) Show that for a compact regular surface $S \subset \mathbb{R}^{3}$