Part II, 2016

Algebraic Geometry

• Paper 1, Section II, H

  Algebraic Geometry | Part II, 2016

  Let $k$ be an algebraically closed field.

  (a) Let $X$ and $Y$ be affine varieties defined over $k$. Given a map $f: X \rightarrow Y$, define what it means for $f$ to be a morphism of affine varieties.

  (b) Let $f: \mathbb{A}^{1} \rightarrow \mathbb{A}^{3}$ be the map given by

  $$f(t)=\left(t, t^{2}, t^{3}\right)$$

  Show that $f$ is a morphism. Show that the image of $f$ is a closed subvariety of $\mathbb{A}^{3}$ and determine its ideal.

  (c) Let $g: \mathbb{P}^{1} \times \mathbb{P}^{1} \times \mathbb{P}^{1} \rightarrow \mathbb{P}^{7}$ be the map given by

  $g\left(\left(s_{1}, t_{1}\right),\left(s_{2}, t_{2}\right),\left(s_{3}, t_{3}\right)\right)=\left(s_{1} s_{2} s_{3}, s_{1} s_{2} t_{3}, s_{1} t_{2} s_{3}, s_{1} t_{2} t_{3}, t_{1} s_{2} s_{3}, t_{1} s_{2} t_{3}, t_{1} t_{2} s_{3}, t_{1} t_{2} t_{3}\right) .$

  Show that the image of $g$ is a closed subvariety of $\mathbb{P}^{7}$.

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• Paper 2, Section II, H

  Algebraic Geometry | Part II, 2016

  In this question we work over an algebraically closed field of characteristic zero. Let $X^{o}=Z\left(x^{6}+x y^{5}+y^{6}-1\right) \subset \mathbb{A}^{2}$ and let $X \subset \mathbb{P}^{2}$ be the closure of $X^{o}$ in $\mathbb{P}^{2} .$

  (a) Show that $X$ is a non-singular curve.

  (b) Show that $\omega=d x /\left(5 x y^{4}+6 y^{5}\right)$ is a regular differential on $X$.

  (c) Compute the divisor of $\omega$. What is the genus of $X$ ?

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• Paper 3, Section II, H

  Algebraic Geometry | Part II, 2016

  (a) Let $X$ be an affine variety. Define the tangent space of $X$ at a point $P$. Say what it means for the variety to be singular at $P$.

  Define the dimension of $X$ in terms of (i) the tangent spaces of $X$, and (ii) Krull dimension.

  (b) Consider the ideal $I$ generated by the set $\left\{y, y^{2}-x^{3}+x y^{3}\right\} \subseteq k[x, y]$. What is $Z(I) \subseteq \mathbb{A}^{2} ?$

  Using the generators of the ideal, calculate the tangent space of a point in $Z(I)$. What has gone wrong? [A complete argument is not necessary.]

  (c) Calculate the dimension of the tangent space at each point $p \in X$ for $X=$ $Z\left(x-y^{2}, x-z w\right) \subseteq \mathbb{A}^{4}$, and determine the location of the singularities of $X .$

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• Paper 4, Section II, H

  Algebraic Geometry | Part II, 2016

  (a) Let $C$ be a smooth projective curve, and let $D$ be an effective divisor on $C$. Explain how $D$ defines a morphism $\phi_{D}$ from $C$ to some projective space.

  State a necessary and sufficient condition on $D$ so that the pull-back of a hyperplane via $\phi_{D}$ is an element of the linear system $|D|$.

  State necessary and sufficient conditions for $\phi_{D}$ to be an isomorphism onto its image.

  (b) Let $C$ now have genus 2 , and let $K$ be an effective canonical divisor. Show that the morphism $\phi_{K}$ is a morphism of degree 2 from $C$ to $\mathbb{P}^{1}$.

  Consider the divisor $K+P_{1}+P_{2}$ for points $P_{i}$ with $P_{1}+P_{2} \nsim K$. Show that the linear system associated to this divisor induces a morphism $\phi$ from $C$ to a quartic curve in $\mathbb{P}^{2}$. Show furthermore that $\phi(P)=\phi(Q)$, with $P \neq Q$, if and only if $\{P, Q\}=\left\{P_{1}, P_{2}\right\}$.

  [You may assume the Riemann-Roch theorem.]

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Algebraic Topology

• Paper 1, Section II, G

  Algebraic Topology | Part II, 2016

  Let $T=S^{1} \times S^{1}$ be the 2-dimensional torus. Let $\alpha: S^{1} \rightarrow T$ be the inclusion of the coordinate circle $S^{1} \times\{1\}$, and let $X$ be the result of attaching a 2-cell along $\alpha$.

  (a) Write down a presentation for the fundamental group of $X$ (with respect to some basepoint), and identify it with a well-known group.

  (b) Compute the simplicial homology of any triangulation of $X$.

  (c) Show that $X$ is not homotopy equivalent to any compact surface.

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• Paper 2, Section II, G

  Algebraic Topology | Part II, 2016

  (a) Let $K, L$ be simplicial complexes, and $f:|K| \rightarrow|L|$ a continuous map. What does it mean to say that $g: K \rightarrow L$ is a simplicial approximation to $f ?$

  (b) Define the barycentric subdivision of a simplicial complex $K$, and state the Simplicial Approximation Theorem.

  (c) Show that if $g$ is a simplicial approximation to $f$ then $f \simeq|g|$.

  (d) Show that the natural inclusion $\left|K^{(1)}\right| \rightarrow|K|$ induces a surjective map on fundamental groups.

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• Paper 3, Section II, G

  Algebraic Topology | Part II, 2016

  Construct a space $X$ as follows. Let $Z_{1}, Z_{2}, Z_{3}$ each be homeomorphic to the standard 2-sphere $S^{2} \subseteq \mathbb{R}^{3}$. For each $i$, let $x_{i} \in Z_{i}$ be the North pole $(1,0,0)$ and let $y_{i} \in Z_{i}$ be the South pole $(-1,0,0)$. Then

  $$X=\left(Z_{1} \sqcup Z_{2} \sqcup Z_{3}\right) / \sim$$

  where $x_{i+1} \sim y_{i}$ for each $i$ (and indices are taken modulo 3 ).

  (a) Describe the universal cover of $X$.

  (b) Compute the fundamental group of $X$ (giving your answer as a well-known group).

  (c) Show that $X$ is not homotopy equivalent to the circle $S^{1}$.

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• Paper 4, Section II, G

  Algebraic Topology | Part II, 2016

  Let $T=S^{1} \times S^{1}$ be the 2-dimensional torus, and let $X$ be constructed from $T$ by removing a small open disc.

  (a) Show that $X$ is homotopy equivalent to $S^{1} \vee S^{1}$.

  (b) Show that the universal cover of $X$ is homotopy equivalent to a tree.

  (c) Exhibit (finite) cell complexes $X, Y$, such that $X$ and $Y$ are not homotopy equivalent but their universal covers $\widetilde{X}, \widetilde{Y}$are.

  [State carefully any results from the course that you use.]

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Applications of Quantum Mechanics

• Paper 1, Section II, A

  Applications of Quantum Mechanics | Part II, 2016

  A particle in one dimension of mass $m$ and energy $E=\hbar^{2} k^{2} / 2 m(k>0)$ is incident from $x=-\infty$ on a potential $V(x)$ with $V(x) \rightarrow 0$ as $x \rightarrow-\infty$ and $V(x)=\infty$ for $x>0$. The relevant solution of the time-independent Schrödinger equation has the asymptotic form

  $$\psi(x) \sim \exp (i k x)+r(k) \exp (-i k x), \quad x \rightarrow-\infty$$

  Explain briefly why a pole in the reflection amplitude $r(k)$ at $k=i \kappa$ with $\kappa>0$ corresponds to the existence of a stable bound state in this potential. Indicate why a pole in $r(k)$ just below the real $k$-axis, at $k=k_{0}-i \rho$ with $k_{0} \gg \rho>0$, corresponds to a quasi-stable bound state. Find an approximate expression for the lifetime $\tau$ of such a quasi-stable state.

  Now suppose that

  $$V(x)= \begin{cases}\left(\hbar^{2} U / 2 m\right) \delta(x+a) & \text { for } x<0 \\ \infty & \text { for } x>0\end{cases}$$

  where $U>0$ and $a>0$ are constants. Compute the reflection amplitude $r(k)$ in this case and deduce that there are quasi-stable bound states if $U$ is large. Give expressions for the wavefunctions and energies of these states and compute their lifetimes, working to leading non-vanishing order in $1 / U$ for each expression.

  [ You may assume $\psi=0$ for $x \geqslant 0$ and $\lim _{\epsilon \rightarrow 0+}\left\{\psi^{\prime}(-a+\epsilon)-\psi^{\prime}(-a-\epsilon)\right\}=U \psi(-a)$.]

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• Paper 2, Section II, A

  Applications of Quantum Mechanics | Part II, 2016

  A particle of mass $m$ moves in three dimensions subject to a potential $V(\mathbf{r})$ localised near the origin. The wavefunction for a scattering process with incident particle of wavevector $\mathbf{k}$ is denoted $\psi(\mathbf{k}, \mathbf{r})$. With reference to the asymptotic form of $\psi$, define the scattering amplitude $f\left(\mathbf{k}, \mathbf{k}^{\prime}\right)$, where $\mathbf{k}^{\prime}$ is the wavevector of the outgoing particle with $\left|\mathbf{k}^{\prime}\right|=|\mathbf{k}|=k$.

  By recasting the Schrödinger equation for $\psi(\mathbf{k}, \mathbf{r})$ as an integral equation, show that

  $$f\left(\mathbf{k}, \mathbf{k}^{\prime}\right)=-\frac{m}{2 \pi \hbar^{2}} \int d^{3} \mathbf{r}^{\prime} \exp \left(-i \mathbf{k}^{\prime} \cdot \mathbf{r}^{\prime}\right) V\left(\mathbf{r}^{\prime}\right) \psi\left(\mathbf{k}, \mathbf{r}^{\prime}\right)$$

  [You may assume that

  $$\mathcal{G}(k ; \mathbf{r})=-\frac{1}{4 \pi|\mathbf{r}|} \exp (i k|\mathbf{r}|)$$

  is the Green's function for $\nabla^{2}+k^{2}$ which obeys the appropriate boundary conditions for a scattering solution.]

  Now suppose $V(\mathbf{r})=\lambda U(\mathbf{r})$, where $\lambda \ll 1$ is a dimensionless constant. Determine the first two non-zero terms in the expansion of $f\left(\mathbf{k}, \mathbf{k}^{\prime}\right)$ in powers of $\lambda$, giving each term explicitly as an integral over one or more position variables $\mathbf{r}, \mathbf{r}^{\prime}, \ldots$

  Evaluate the contribution to $f\left(\mathbf{k}, \mathbf{k}^{\prime}\right)$ of order $\lambda$ in the case $U(\mathbf{r})=\delta(|\mathbf{r}|-a)$, expressing the answer as a function of $a, k$ and the scattering angle $\theta$ (defined so that $\left.\mathbf{k} \cdot \mathbf{k}^{\prime}=k^{2} \cos \theta\right)$.

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• Paper 3, Section II, A

  Applications of Quantum Mechanics | Part II, 2016

  (a) A spinless charged particle moves in an electromagnetic field defined by vector and scalar potentials $\mathbf{A}(\mathbf{x}, t)$ and $\phi(\mathbf{x}, t)$. The wavefunction $\psi(\mathbf{x}, t)$ for the particle satisfies the time-dependent Schrödinger equation with Hamiltonian

  $$\hat{H}_{0}=\frac{1}{2 m}(-i \hbar \boldsymbol{\nabla}+e \mathbf{A}) \cdot(-i \hbar \boldsymbol{\nabla}+e \mathbf{A})-e \phi .$$

  Consider a gauge transformation

  $$\mathbf{A} \rightarrow \tilde{\mathbf{A}}=\mathbf{A}+\nabla f, \quad \phi \rightarrow \tilde{\phi}=\phi-\frac{\partial f}{\partial t}, \quad \psi \rightarrow \tilde{\psi}=\exp \left(-\frac{i e f}{\hbar}\right) \psi$$

  for some function $f(\mathbf{x}, t)$. Define covariant derivatives with respect to space and time, and show that $\tilde{\psi}$ satisfies the Schrödinger equation with potentials $\tilde{\mathbf{A}}$ and $\tilde{\phi}$.

  (b) Suppose that in part (a) the magnetic field has the form $\mathbf{B}=\boldsymbol{\nabla} \times \mathbf{A}=(0,0, B)$, where $B$ is a constant, and that $\phi=0$. Find a suitable $\mathbf{A}$ with $A_{y}=A_{z}=0$ and determine the energy levels of the Hamiltonian $\hat{H}_{0}$ when the $z$-component of the momentum of the particle is zero. Suppose in addition that the particle is constrained to lie in a rectangular region of area $\mathcal{A}$ in the $(x, y)$-plane. By imposing periodic boundary conditions in the $x$-direction, estimate the degeneracy of each energy level. [You may use without proof results for a quantum harmonic oscillator, provided they are clearly stated.]

  (c) An electron is a charged particle of spin $\frac{1}{2}$ with a two-component wavefunction $\psi(\mathbf{x}, t)$ governed by the Hamiltonian

  $$\hat{H}=\hat{H}_{0} \mathbb{I}_{2}+\frac{e \hbar}{2 m} \mathbf{B} \cdot \boldsymbol{\sigma}$$

  where $\mathbb{I}_{2}$ is the $2 \times 2$ unit matrix and $\sigma=\left(\sigma_{1}, \sigma_{2}, \sigma_{3}\right)$ denotes the Pauli matrices. Find the energy levels for an electron in the constant magnetic field defined in part (b), assuming as before that the $z$-component of the momentum of the particle is zero.

  Consider $N$ such electrons confined to the rectangular region defined in part (b). Ignoring interactions between the electrons, show that the ground state energy of this system vanishes for $N$ less than some integer $N_{\max }$ which you should determine. Find the ground state energy for $N=(2 p+1) N_{\max }$, where $p$ is a positive integer.

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• Paper 4, Section II, A

  Applications of Quantum Mechanics | Part II, 2016

  Let $\Lambda \subset \mathbb{R}^{2}$ be a Bravais lattice. Define the dual lattice $\Lambda^{*}$ and show that

  $$V(\mathbf{x})=\sum_{\mathbf{q} \in \Lambda^{*}} V_{\mathbf{q}} \exp (i \mathbf{q} \cdot \mathbf{x})$$

  obeys $V(\mathbf{x}+l)=V(\mathbf{x})$ for all $l \in \Lambda$, where $V_{\mathbf{q}}$ are constants. Suppose $V(\mathbf{x})$ is the potential for a particle of mass $m$ moving in a two-dimensional crystal that contains a very large number of lattice sites of $\Lambda$ and occupies an area $\mathcal{A}$. Adopting periodic boundary conditions, plane-wave states $|\mathbf{k}\rangle$ can be chosen such that

  $$\langle\mathbf{x} \mid \mathbf{k}\rangle=\frac{1}{\mathcal{A}^{1 / 2}} \exp (i \mathbf{k} \cdot \mathbf{x}) \quad \text { and } \quad\left\langle\mathbf{k} \mid \mathbf{k}^{\prime}\right\rangle=\delta_{\mathbf{k} \mathbf{k}^{\prime}}$$

  The allowed wavevectors $\mathbf{k}$ are closely spaced and include all vectors in $\Lambda^{*}$. Find an expression for the matrix element $\left\langle\mathbf{k}|V(\mathbf{x})| \mathbf{k}^{\prime}\right\rangle$ in terms of the coefficients $V_{\mathbf{q}}$. [You need not discuss additional details of the boundary conditions.]

  Now suppose that $V(\mathbf{x})=\lambda U(\mathbf{x})$, where $\lambda \ll 1$ is a dimensionless constant. Find the energy $E(\mathbf{k})$ for a particle with wavevector $\mathbf{k}$ to order $\lambda^{2}$ in non-degenerate perturbation theory. Show that this expansion in $\lambda$ breaks down on the Bragg lines in k-space defined by the condition

  $$\mathbf{k} \cdot \mathbf{q}=\frac{1}{2}|\mathbf{q}|^{2} \quad \text { for } \quad \mathbf{q} \in \Lambda^{*}$$

  and explain briefly, without additional calculations, the significance of this for energy levels in the crystal.

  Consider the particular case in which $\Lambda$ has primitive vectors

  $$\mathbf{a}_{1}=2 \pi\left(\mathbf{i}+\frac{1}{\sqrt{3}} \mathbf{j}\right), \quad \mathbf{a}_{2}=2 \pi \frac{2}{\sqrt{3}} \mathbf{j}$$

  where $\mathbf{i}$ and $\mathbf{j}$ are orthogonal unit vectors. Determine the polygonal region in $\mathbf{k}$-space corresponding to the lowest allowed energy band.

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Applied Probability

• Paper 1, Section II, J

  Applied Probability | Part II, 2016

  (a) Define a continuous-time Markov chain $X$ with infinitesimal generator $Q$ and jump chain $Y$.

  (b) Prove that if a state $x$ is transient for $Y$, then it is transient for $X$.

  (c) Prove or provide a counterexample to the following: if $x$ is positive recurrent for $X$, then it is positive recurrent for $Y$.

  (d) Consider the continuous-time Markov chain $\left(X_{t}\right)_{t \geqslant 0}$ on $\mathbb{Z}$ with non-zero transition rates given by

  $$q(i, i+1)=2 \cdot 3^{|i|}, \quad q(i, i)=-3^{|i|+1} \quad \text { and } \quad q(i, i-1)=3^{|i|}$$

  Determine whether $X$ is transient or recurrent. Let $T_{0}=\inf \left\{t \geqslant J_{1}: X_{t}=0\right\}$, where $J_{1}$ is the first jump time. Does $X$ have an invariant distribution? Justify your answer. Calculate $\mathbb{E}_{0}\left[T_{0}\right]$.

  (e) Let $X$ be a continuous-time random walk on $\mathbb{Z}^{d}$ with $q(x)=\|x\|^{\alpha} \wedge 1$ and $q(x, y)=q(x) /(2 d)$ for all $y \in \mathbb{Z}^{d}$ with $\|y-x\|=1$. Determine for which values of $\alpha$ the walk is transient and for which it is recurrent. In the recurrent case, determine the range of $\alpha$ for which it is also positive recurrent. [Here $\|x\|$ denotes the Euclidean norm of $x$.]

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• Paper 2, Section II, J

  Applied Probability | Part II, 2016

  (a) Define an $M / M / \infty$ queue and write without proof its stationary distribution. State Burke's theorem for an $M / M / \infty$ queue.

  (b) Let $X$ be an $M / M / \infty$ queue with arrival rate $\lambda$ and service rate $\mu$ started from the stationary distribution. For each $t$, denote by $A_{1}(t)$ the last time before $t$ that a customer departed the queue and $A_{2}(t)$ the first time after $t$ that a customer departed the queue. If there is no arrival before time $t$, then we set $A_{1}(t)=0$. What is the limit as $t \rightarrow \infty$ of $\mathbb{E}\left[A_{2}(t)-A_{1}(t)\right]$ ? Explain.

  (c) Consider a system of $N$ queues serving a finite number $K$ of customers in the following way: at station $1 \leqslant i \leqslant N$, customers are served immediately and the service times are independent exponentially distributed with parameter $\mu_{i}$; after service, each customer goes to station $j$ with probability $p_{i j}>0$. We assume here that the system is closed, i.e., $\sum_{j} p_{i j}=1$ for all $1 \leqslant i \leqslant N$.

  Let $S=\left\{\left(n_{1}, \ldots, n_{N}\right): n_{i} \in \mathbb{N}, \sum_{i=1}^{N} n_{i}=K\right\}$ be the state space of the Markov chain. Write down its $Q$-matrix. Also write down the $Q$-matrix $R$ corresponding to the position in the network of one customer (that is, when $K=1$ ). Show that there is a unique distribution $\left(\lambda_{i}\right)_{1 \leqslant i \leqslant N}$ such that $\lambda R=0$. Show that

  $$\pi(n)=C_{N} \prod_{i=1}^{N} \frac{\lambda_{i}^{n_{i}}}{n_{i} !}, \quad n=\left(n_{1}, \ldots, n_{N}\right) \in S$$

  defines an invariant measure for the chain. Are the queue lengths independent at equilibrium?

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• Paper 3, Section II, J

  Applied Probability | Part II, 2016

  (a) State the thinning and superposition properties of a Poisson process on $\mathbb{R}_{+}$. Prove the superposition property.

  (b) A bi-infinite Poisson process $\left(N_{t}: t \in \mathbb{R}\right)$ with $N_{0}=0$ is a process with independent and stationary increments over $\mathbb{R}$. Moreover, for all $-\infty<s \leqslant t<\infty$, the increment $N_{t}-N_{s}$ has the Poisson distribution with parameter $\lambda(t-s)$. Prove that such a process exists.

  (c) Let $N$ be a bi-infinite Poisson process on $\mathbb{R}$ of intensity $\lambda$. We identify it with the set of points $\left(S_{n}\right)$ of discontinuity of $N$, i.e., $N[s, t]=\sum_{n} \mathbf{l}\left(S_{n} \in[s, t]\right)$. Show that if we shift all the points of $N$ by the same constant $c$, then the resulting process is also a Poisson process of intensity $\lambda$.

  Now suppose we shift every point of $N$ by $+1$ or $-1$ with equal probability. Show that the final collection of points is still a Poisson process of intensity $\lambda$. [You may assume the thinning and superposition properties for the bi-infinite Poisson process.]

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• Paper 4, Section II, J

  Applied Probability | Part II, 2016

  (a) Give the definition of a renewal process. Let $\left(N_{t}\right)_{t \geqslant 0}$ be a renewal process associated with $\left(\xi_{i}\right)$ with $\mathbb{E} \xi_{1}=1 / \lambda<\infty$. Show that almost surely

  $$\frac{N_{t}}{t} \rightarrow \lambda \quad \text { as } t \rightarrow \infty$$

  (b) Give the definition of Kingman's $n$-coalescent. Let $\tau$ be the first time that all blocks have coalesced. Find an expression for $\mathbb{E} e^{-q \tau}$. Let $L_{n}$ be the total length of the branches of the tree, i.e., if $\tau_{i}$ is the first time there are $i$ lineages, then $L_{n}=$ $\sum_{i=2}^{n} i\left(\tau_{i-1}-\tau_{i}\right)$. Show that $\mathbb{E} L_{n} \sim 2 \log n$ as $n \rightarrow \infty$. Show also that $\operatorname{Var}\left(L_{n}\right) \leqslant C$ for all $n$, where $C$ is a positive constant, and that in probability

  $$\frac{L_{n}}{\mathbb{E} L_{n}} \rightarrow 1 \quad \text { as } n \rightarrow \infty$$

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Asymptotic Methods

• Paper 2, Section II, C

  Asymptotic Methods | Part II, 2016

  What is meant by the asymptotic relation

  $$f(z) \sim g(z) \quad \text { as } \quad z \rightarrow z_{0}, \operatorname{Arg}\left(z-z_{0}\right) \in\left(\theta_{0}, \theta_{1}\right) ?$$

  Show that

  $$\sinh \left(z^{-1}\right) \sim \frac{1}{2} \exp \left(z^{-1}\right) \quad \text { as } \quad z \rightarrow 0, \operatorname{Arg} z \in(-\pi / 2, \pi / 2),$$

  and find the corresponding result in the sector $\operatorname{Arg} z \in(\pi / 2,3 \pi / 2)$.

  What is meant by the asymptotic expansion

  $$f(z) \sim \sum_{j=0}^{\infty} c_{j}\left(z-z_{0}\right)^{j} \quad \text { as } \quad z \rightarrow z_{0}, \operatorname{Arg}\left(z-z_{0}\right) \in\left(\theta_{0}, \theta_{1}\right) ?$$

  Show that the coefficients $\left\{c_{j}\right\}_{j=0}^{\infty}$ are determined uniquely by $f$. Show that if $f$ is analytic at $z_{0}$, then its Taylor series is an asymptotic expansion for $f$ as $z \rightarrow z_{0}\left(\right.$ for any $\left.\operatorname{Arg}\left(z-z_{0}\right)\right)$.

  Show that

  $$u(x, t)=\int_{-\infty}^{\infty} \exp \left(-i k^{2} t+i k x\right) f(k) d k$$

  defines a solution of the equation $i \partial_{t} u+\partial_{x}^{2} u=0$ for any smooth and rapidly decreasing function $f$. Use the method of stationary phase to calculate the leading-order behaviour of $u(\lambda t, t)$ as $t \rightarrow+\infty$, for fixed $\lambda$.

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• Paper 3, Section II, C

  Asymptotic Methods | Part II, 2016

  Consider the integral

  $$I(x)=\int_{0}^{1} \frac{1}{\sqrt{t(1-t)}} \exp [i x f(t)] d t$$

  for real $x>0$, where $f(t)=t^{2}+t$. Find and sketch, in the complex $t$-plane, the paths of steepest descent through the endpoints $t=0$ and $t=1$ and through any saddle point(s). Obtain the leading order term in the asymptotic expansion of $I(x)$ for large positive $x$. What is the order of the next term in the expansion? Justify your answer.

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• Paper 4, Section II, C

  Asymptotic Methods | Part II, 2016

  Consider the equation

  $$\epsilon^{2} \frac{d^{2} y}{d x^{2}}=Q(x) y$$

  where $\epsilon>0$ is a small parameter and $Q(x)$ is smooth. Search for solutions of the form

  $$y(x)=\exp \left[\frac{1}{\epsilon}\left(S_{0}(x)+\epsilon S_{1}(x)+\epsilon^{2} S_{2}(x)+\cdots\right)\right],$$

  and, by equating powers of $\epsilon$, obtain a collection of equations for the $\left\{S_{j}(x)\right\}_{j=0}^{\infty}$ which is formally equivalent to (1). By solving explicitly for $S_{0}$ and $S_{1}$ derive the Liouville- Green approximate solutions $y^{L G}(x)$ to (1).

  For the case $Q(x)=-V(x)$, where $V(x) \geqslant V_{0}$ and $V_{0}$ is a positive constant, consider the eigenvalue problem

  $$\frac{d^{2} y}{d x^{2}}+E V(x) y=0, \quad y(0)=y(\pi)=0$$

  Show that any eigenvalue $E$ is necessarily positive. Solve the eigenvalue problem exactly when $V(x)=V_{0}$.

  Obtain Liouville-Green approximate eigenfunctions $y_{n}^{L G}(x)$ for (2) with $E \gg 1$, and give the corresponding Liouville Green approximation to the eigenvalues $E_{n}^{L G}$. Compare your results to the exact eigenvalues and eigenfunctions in the case $V(x)=V_{0}$, and comment on this.

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Automata and Formal Languages

• Paper 1, Section I, $4 \mathrm{~F}$

  Automata and Formal Languages | Part II, 2016

  State the pumping lemma for context-free languages (CFLs). Which of the following are CFLs? Justify your answers.

  (i) $\left\{a^{2 n} b^{3 n} \mid n \geqslant 3\right\}$.

  (ii) $\left\{a^{2 n} b^{3 n} c^{5 n} \mid n \geqslant 0\right\}$.

  (iii) $\left\{a^{p} \mid p\right.$ is a prime $\}$.

  Let $L, M$ be CFLs. Show that $L \cup M$ is also a CFL.

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• Paper 1, Section II, F

  Automata and Formal Languages | Part II, 2016

  (a) Define a recursive set and a recursively enumerable (r.e.) set. Prove that $E \subseteq \mathbb{N}$ is recursive if and only if both $E$ and $\mathbb{N} \backslash E$ are r.e.

  (b) Define the halting set $\mathbb{K}$. Prove that $\mathbb{K}$ is r.e. but not recursive.

  (c) Let $E_{1}, E_{2}, \ldots, E_{n}$ be r.e. sets. Prove that $\bigcup_{i=1}^{n} E_{i}$ and $\bigcap_{i=1}^{n} E_{i}$ are r.e. Show by an example that the union of infinitely many r.e. sets need not be r.e.

  (d) Let $E$ be a recursive set and $f: \mathbb{N} \rightarrow \mathbb{N}$ a (total) recursive function. Prove that the set $\{f(n) \mid n \in E\}$ is r.e. Is it necessarily recursive? Justify your answer.

  [Any use of Church's thesis in your answer should be explicitly stated.]

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• Paper 2, Section $\mathbf{I}$, $4 F$

  Automata and Formal Languages | Part II, 2016

  (a) Which of the following are regular languages? Justify your answers.

  (i) $\left\{w \in\{a, b\}^{*} \mid w\right.$ is a nonempty string of alternating $a$ 's and $b$ 's $\}$.

  (ii) $\left\{w a b w \mid w \in\{a, b\}^{*}\right\}$.

  (b) Write down a nondeterministic finite-state automaton with $\epsilon$-transitions which accepts the language given by the regular expression $(\mathbf{a}+\mathbf{b})^{*}(\mathbf{b} \mathbf{b}+\mathbf{a}) \mathbf{b}$. Describe in words what this language is.

  (c) Is the following language regular? Justify your answer.

  $$\left\{w \in\{a, b\}^{*} \mid w \text { does not end in } a b \text { or } b b b\right\}$$

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• Paper 3, Section I, $4 F$

  Automata and Formal Languages | Part II, 2016

  (a) Define what it means for a context-free grammar (CFG) to be in Chomsky normal form (CNF). Can a CFG in CNF ever define a language containing $\epsilon$ ? If $G_{\text {Chom }}$ denotes the result of converting an arbitrary CFG $G$ into one in CNF, state the relationship between $\mathcal{L}(G)$ and $\mathcal{L}\left(G_{\text {Chom }}\right)$.

  (b) Let $G$ be a CFG in CNF, and let $w \in \mathcal{L}(G)$ be a word of length $|w|=n>0$. Show that every derivation of $w$ in $G$ requires precisely $2 n-1$ steps. Using this, give an algorithm that, on input of any word $v$ on the terminals of $G$, decides if $v \in \mathcal{L}(G)$ or not.

  (c) Convert the following CFG $G$ into a grammar in CNF:

  $$S \rightarrow a S b|S S| \epsilon$$

  Does $\mathcal{L}(G)=\mathcal{L}\left(G_{\text {Chom }}\right)$ in this case? Justify your answer.

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• Paper 3, Section II, F

  Automata and Formal Languages | Part II, 2016

  (a) Let $D=\left(Q, \Sigma, \delta, q_{0}, F\right)$ be a deterministic finite-state automaton. Define the extended transition function $\hat{\delta}: Q \times \Sigma^{*} \rightarrow Q$, and the language accepted by $D$, denoted $\mathcal{L}(D)$. Let $u, v \in \Sigma^{*}$, and $p \in Q$. Prove that $\hat{\delta}(p, u v)=\hat{\delta}(\hat{\delta}(p, u), v)$.

  (b) Let $\sigma_{1}, \sigma_{2}, \ldots, \sigma_{k} \in \Sigma$ where $k \geqslant|Q|$, and let $p \in Q$.

  (i) Show that there exist $0 \leqslant i<j \leqslant k$ such that $\hat{\delta}\left(p, \sigma_{1} \cdots \sigma_{i}\right)=\hat{\delta}\left(p, \sigma_{1} \cdots \sigma_{j}\right)$, where we interpret $\sigma_{1} \cdots \sigma_{i}$ as $\epsilon$ if $i=0$.

  (ii) Show that $\hat{\delta}\left(p, \sigma_{1} \cdots \sigma_{i} \sigma_{j+1} \cdots \sigma_{k}\right)=\hat{\delta}\left(p, \sigma_{1} \cdots \sigma_{k}\right)$.

  (iii) Show that $\hat{\delta}\left(p, \sigma_{1} \cdots \sigma_{i}\left(\sigma_{i+1} \cdots \sigma_{j}\right)^{t} \sigma_{j+1} \cdots \sigma_{k}\right)=\hat{\delta}\left(p, \sigma_{1} \cdots \sigma_{k}\right)$ for all $t \geqslant 1$.

  (c) Prove the following version of the pumping lemma. Suppose $w \in \mathcal{L}(D)$, with $|w| \geqslant|Q|$. Then $w$ can be broken up into three words $w=x y z$ such that $y \neq \epsilon,|x y| \leqslant|Q|$, and for all $t \geqslant 0$, the word $x y^{t} z$ is also in $\mathcal{L}(D)$.

  (d) Hence show that

  (i) if $\mathcal{L}(D)$ contains a word of length at least $|Q|$, then it contains infinitely many words;

  (ii) if $\mathcal{L}(D) \neq \emptyset$, then it contains a word of length less than $|Q|$;

  (iii) if $\mathcal{L}(D)$ contains all words in $\Sigma^{*}$ of length less than $|Q|$, then $\mathcal{L}(D)=\Sigma^{*}$.

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• Paper 4, Section I, $4 \mathrm{~F}$

  Automata and Formal Languages | Part II, 2016

  (a) Construct a register machine to compute the function $f(m, n):=m+n$. State the relationship between partial recursive functions and partial computable functions. Show that the function $g(m, n):=m n$ is partial recursive.

  (b) State Rice's theorem. Show that the set $A:=\left\{n \in \mathbb{N}|| W_{n} \mid>7\right\}$ is recursively enumerable but not recursive.

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Classical Dynamics

• Paper 1, Section I, E

  Classical Dynamics | Part II, 2016

  Consider a one-parameter family of transformations $q_{i}(t) \mapsto Q_{i}(s, t)$ such that $Q_{i}(0, t)=q_{i}(t)$ for all time $t$, and

  $$\frac{\partial}{\partial s} L\left(Q_{i}, \dot{Q}_{i}, t\right)=0$$

  where $L$ is a Lagrangian and a dot denotes differentiation with respect to $t$. State and prove Noether's theorem.

  Consider the Lagrangian

  $$L=\frac{1}{2}\left(\dot{x}^{2}+\dot{y}^{2}+\dot{z}^{2}\right)-V(x+y, y+z),$$

  where the potential $V$ is a function of two variables. Find a continuous symmetry of this Lagrangian and construct the corresponding conserved quantity. Use the Euler-Lagrange equations to explicitly verify that the function you have constructed is independent of $t$.

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• Paper 2, Section I, E

  Classical Dynamics | Part II, 2016

  Consider the Lagrangian

  $$L=A\left(\dot{\theta}^{2}+\dot{\phi}^{2} \sin ^{2} \theta\right)+B(\dot{\psi}+\dot{\phi} \cos \theta)^{2}-C(\cos \theta)^{k}$$

  where $A, B, C$ are positive constants and $k$ is a positive integer. Find three conserved quantities and show that $u=\cos \theta$ satisfies

  $$\dot{u}^{2}=f(u)$$

  where $f(u)$ is a polynomial of degree $k+2$ which should be determined.

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• Paper 2, Section II, E

  Classical Dynamics | Part II, 2016

  Define what it means for the transformation $\mathbb{R}^{2 n} \rightarrow \mathbb{R}^{2 n}$ given by

  $$\left(q_{i}, p_{i}\right) \mapsto\left(Q_{i}\left(q_{j}, p_{j}\right), P_{i}\left(q_{j}, p_{j}\right)\right), \quad i, j=1, \ldots, n$$

  to be canonical. Show that a transformation is canonical if and only if

  $$\left\{Q_{i}, Q_{j}\right\}=0, \quad\left\{P_{i}, P_{j}\right\}=0, \quad\left\{Q_{i}, P_{j}\right\}=\delta_{i j}$$

  Show that the transformation $\mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ given by

  $$Q=q \cos \epsilon-p \sin \epsilon, \quad P=q \sin \epsilon+p \cos \epsilon$$

  is canonical for any real constant $\epsilon$. Find the corresponding generating function.

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• Paper 3, Section I, E

  Classical Dynamics | Part II, 2016

  Consider a six-dimensional phase space with coordinates $\left(q_{i}, p_{i}\right)$ for $i=1,2,3$. Compute the Poisson brackets $\left\{L_{i}, L_{j}\right\}$, where $L_{i}=\epsilon_{i j k} q_{j} p_{k}$.

  Consider the Hamiltonian

  $$H=\frac{1}{2}|\mathbf{p}|^{2}+V(|\mathbf{q}|)$$

  and show that the resulting Hamiltonian system admits three Poisson-commuting independent first integrals.

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• Paper 4, Section I, E

  Classical Dynamics | Part II, 2016

  Using conservation of angular momentum $\mathbf{L}=L_{a} \mathbf{e}_{a}$ in the body frame, derive the Euler equations for a rigid body:

  $$I_{1} \dot{\omega}_{1}+\left(I_{3}-I_{2}\right) \omega_{2} \omega_{3}=0, \quad I_{2} \dot{\omega}_{2}+\left(I_{1}-I_{3}\right) \omega_{3} \omega_{1}=0, \quad I_{3} \dot{\omega}_{3}+\left(I_{2}-I_{1}\right) \omega_{1} \omega_{2}=0$$

  [You may use the formula $\dot{\mathbf{e}}_{a}=\boldsymbol{\omega} \wedge \mathbf{e}_{a}$ without proof.]

  Assume that the principal moments of inertia satisfy $I_{1}<I_{2}<I_{3}$. Determine whether a rotation about the principal 3-axis leads to stable or unstable perturbations.

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• Paper 4, Section II, $14 \mathrm{E}$

  Classical Dynamics | Part II, 2016

  A particle of unit mass is attached to one end of a light, stiff rod of length $\ell$. The other end of the rod is held at a fixed position, such that the rod is free to swing in any direction. Write down the Lagrangian for the system giving a clear definition of any angular variables you introduce. [You should assume the acceleration $g$ is constant.]

  Find two independent constants of the motion.

  The particle is projected horizontally with speed $v$ from a point where the rod lies at an angle $\alpha$ to the downward vertical, with $0<\alpha<\pi / 2$. In terms of $\ell, g$ and $\alpha$, find the critical speed $v_{c}$ such that the particle always remains at its initial height.

  The particle is now projected horizontally with speed $v_{c}$ but from a point at angle $\alpha+\delta \alpha$ to the vertical, where $\delta \alpha / \alpha \ll 1$. Show that the height of the particle oscillates, and find the period of oscillation in terms of $\ell, g$ and $\alpha$.

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Coding and Cryptography

• Paper 1, Section I, G

  Coding and Cryptography | Part II, 2016

  Find the average length of an optimum decipherable binary code for a source that emits five words with probabilities

  $$0.25,0.15,0.15,0.2,0.25$$

  Show that, if a source emits $N$ words (with $N \geqslant 2$ ), and if $l_{1}, \ldots, l_{N}$ are the lengths of the codewords in an optimum encoding over the binary alphabet, then

  $$l_{1}+\cdots+l_{N} \leqslant \frac{1}{2}\left(N^{2}+N-2\right) .$$

  [You may assume that an optimum encoding can be given by a Huffman encoding.]

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• Paper 1, Section II, G

  Coding and Cryptography | Part II, 2016

  What does it mean to say a binary code $C$ has length $n$, size $m$ and minimum distance d?

  Let $A(n, d)$ be the largest value of $m$ for which there exists an $[n, m, d]$-code. Prove that

  $$\frac{2^{n}}{V(n, d-1)} \leqslant A(n, d) \leqslant \frac{2^{n}}{V(n,\lfloor(d-1) / 2\rfloor)}$$

  where

  $$V(n, r)=\sum_{j=0}^{r}\left(\begin{array}{l} n \\ j \end{array}\right)$$

  Another bound for $A(n, d)$ is the Singleton bound given by

  $$A(n, d) \leqslant 2^{n-d+1}$$

  Prove the Singleton bound and give an example of a linear code of length $n>1$ that satisfies $A(n, d)=2^{n-d+1}$.

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• Paper 2, Section I, G

  Coding and Cryptography | Part II, 2016

  Show that the binary channel with channel matrix

  $$\left(\begin{array}{ll} 1 & 0 \\ \frac{1}{2} & \frac{1}{2} \end{array}\right)$$

  has capacity $\log 5-2$.

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• Paper 2, Section II, G

  Coding and Cryptography | Part II, 2016

  Define a $B C H$ code of length $n$, where $n$ is odd, over the field of 2 elements with design distance $\delta$. Show that the minimum weight of such a code is at least $\delta$. [Results about the Vandermonde determinant may be quoted without proof, provided they are stated clearly.]

  Let $\omega \in \mathbb{F}_{16}$ be a root of $X^{4}+X+1$. Let $C$ be the $\mathrm{BCH}$ code of length 15 with defining set $\left\{\omega, \omega^{2}, \omega^{3}, \omega^{4}\right\}$. Find the generator polynomial of $C$ and the rank of $C$. Determine the error positions of the following received words:

  (i) $r(X)=1+X^{6}+X^{7}+X^{8}$,

  (ii) $r(X)=1+X+X^{4}+X^{5}+X^{6}+X^{9}$.

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• Paper 3, Section I, $3 G$

  Coding and Cryptography | Part II, 2016

  Describe in words the unicity distance of a cryptosystem.

  Denote the cryptosystem by $\langle M, K, C\rangle$, in the usual way, and let $m \in M$ and $k \in K$ be random variables and $c=e(m, k)$. The unicity distance $U$ is formally defined to be the least $n>0$ such that $H\left(k \mid c^{(n)}\right)=0$. Derive the formula

  $$U=\frac{\log |K|}{\log |\Sigma|-H}$$

  where $H=H(m)$, and $\Sigma$ is the alphabet of the ciphertext. Make clear any assumptions you make.

  The redundancy of a language is given by $R=1-\frac{H}{\log |\Sigma|}$. If a language has zero redundancy what is the unicity of any cryptosystem?

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• Paper 4, Section I, G

  Coding and Cryptography | Part II, 2016

  Describe the Rabin-Williams scheme for coding a message $x$ as $x^{2}$ modulo a certain $N$. Show that, if $N$ is chosen appropriately, breaking this code is equivalent to factorising the product of two primes.

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Cosmology

• Paper 1, Section I, C

  Cosmology | Part II, 2016

  The expansion scale factor, $a(t)$, for an isotropic and spatially homogeneous universe containing material with pressure $p$ and mass density $\rho$ obeys the equations

  $$\begin{gathered} \dot{\rho}+3(\rho+p) \frac{\dot{a}}{a}=0, \\ \left(\frac{\dot{a}}{a}\right)^{2}=\frac{8 \pi G \rho}{3}-\frac{k}{a^{2}}+\frac{\Lambda}{3}, \end{gathered}$$

  where the speed of light is set equal to unity, $G$ is Newton's constant, $k$ is a constant equal to 0 or $\pm 1$, and $\Lambda$ is the cosmological constant. Explain briefly the interpretation of these equations.

  Show that these equations imply

  $$\frac{\ddot{a}}{a}=-\frac{4 \pi G(\rho+3 p)}{3}+\frac{\Lambda}{3} .$$

  Hence show that a static solution with constant $a=a_{\mathrm{s}}$ exists when $p=0$ if

  $$\Lambda=4 \pi G \rho=\frac{k}{a_{\mathrm{s}}^{2}}$$

  What must the value of $k$ be, if the density $\rho$ is non-zero?

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• Paper 1, Section II, C

  Cosmology | Part II, 2016

  The distribution function $f(\mathbf{x}, \mathbf{p}, t)$ gives the number of particles in the universe with position in $(\mathbf{x}, \mathbf{x}+\delta \mathbf{x})$ and momentum in $(\mathbf{p}, \mathbf{p}+\delta \mathbf{p})$ at time $t$. It satisfies the boundary condition that $f \rightarrow 0$ as $|\mathbf{x}| \rightarrow \infty$ and as $|\mathbf{p}| \rightarrow \infty$. Its evolution obeys the Boltzmann equation

  $$\frac{\partial f}{\partial t}+\frac{\partial f}{\partial \mathbf{p}} \cdot \frac{d \mathbf{p}}{d t}+\frac{\partial f}{\partial \mathbf{x}} \cdot \frac{d \mathbf{x}}{d t}=\left[\frac{d f}{d t}\right]_{\mathrm{col}}$$

  where the collision term $\left[\frac{d f}{d t}\right]_{c o l}$ describes any particle production and annihilation that occurs.

  The universe expands isotropically and homogeneously with expansion scale factor $a(t)$, so the momenta evolve isotropically with magnitude $p \propto a^{-1}$. Show that the Boltzmann equation simplifies to

  $$\frac{\partial f}{\partial t}-\frac{\dot{a}}{a} \mathbf{p} \cdot \frac{\partial f}{\partial \mathbf{p}}=\left[\frac{d f}{d t}\right]_{\mathrm{col}}$$

  The number densities $n$ of particles and $\bar{n}$ of antiparticles are defined in terms of their distribution functions $f$ and $\bar{f}$, and momenta $p$ and $\bar{p}$, by

  $$n=\int_{0}^{\infty} f 4 \pi p^{2} d p \quad \text { and } \quad \bar{n}=\int_{0}^{\infty} \bar{f} 4 \pi \bar{p}^{2} d \bar{p}$$

  and the collision term may be assumed to be of the form

  $$\left[\frac{d f}{d t}\right]_{\mathrm{col}}=-\langle\sigma v\rangle \int_{0}^{\infty} \bar{f} f 4 \pi \bar{p}^{2} d \bar{p}+R$$

  where $\langle\sigma v\rangle$ determines the annihilation cross-section of particles by antiparticles and $R$ is the production rate of particles.

  By integrating equation $(*)$ with respect to the momentum $\mathbf{p}$ and assuming that $\langle\sigma v\rangle$ is a constant, show that

  $$\frac{d n}{d t}+3 \frac{\dot{a}}{a} n=-\langle\sigma v\rangle n \bar{n}+Q$$

  where $Q=\int_{0}^{\infty} R 4 \pi p^{2} d p$. Assuming the same production rate $R$ for antiparticles, write down the corresponding equation satisfied by $\bar{n}$ and show that

  $$(n-\bar{n}) a^{3}=\mathrm{constant}$$

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• Paper 2, Section I, C

  Cosmology | Part II, 2016

  A spherical cloud of mass $M$ has radius $r(t)$ and initial radius $r(0)=R$. It contains material with uniform mass density $\rho(t)$, and zero pressure. Ignoring the cosmological constant, show that if it is initially at rest at $t=0$ and the subsequent gravitational collapse is governed by Newton's law $\ddot{r}=-G M / r^{2}$, then

  $$\dot{r}^{2}=2 G M\left(\frac{1}{r}-\frac{1}{R}\right) .$$

  Suppose $r$ is given parametrically by

  $$r=R \cos ^{2} \theta$$

  where $\theta=0$ at $t=0$. Derive a relation between $\theta$ and $t$ and hence show that the cloud collapses to radius $r=0$ at

  $$t=\sqrt{\frac{3 \pi}{32 G \rho_{0}}},$$

  where $\rho_{0}$ is the initial mass density of the cloud.

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• Paper 3, Section I, C

  Cosmology | Part II, 2016

  A universe contains baryonic matter with background density $\rho_{B}(t)$ and density inhomogeneity $\delta_{B}(\mathbf{x}, t)$, together with non-baryonic dark matter with background density $\rho_{D}(t)$ and density inhomogeneity $\delta_{D}(\mathbf{x}, t)$. After the epoch of radiation-matter density equality, $t_{\mathrm{eq}}$, the background dynamics are governed by

  $$H=\frac{2}{3 t} \quad \text { and } \quad \rho_{D}=\frac{1}{6 \pi G t^{2}}$$

  where $H$ is the Hubble parameter.

  The dark-matter density is much greater than the baryonic density $\left(\rho_{D} \gg \rho_{B}\right)$ and so the time-evolution of the coupled density perturbations, at any point $\mathbf{x}$, is described by the equations

  $$\begin{aligned} &\ddot{\delta}_{B}+2 H \dot{\delta}_{B}=4 \pi G \rho_{D} \delta_{D} \\ &\ddot{\delta}_{D}+2 H \dot{\delta}_{D}=4 \pi G \rho_{D} \delta_{D} \end{aligned}$$

  Show that

  $$\delta_{D}=\frac{\alpha}{t}+\beta t^{2 / 3}$$

  where $\alpha$ and $\beta$ are independent of time. Neglecting modes in $\delta_{D}$ and $\delta_{B}$ that decay with increasing time, show that the baryonic density inhomogeneity approaches

  $$\delta_{B}=\beta t^{2 / 3}+\gamma$$

  where $\gamma$ is independent of time.

  Briefly comment on the significance of your calculation for the growth of baryonic density inhomogeneities in the early universe.

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• Paper 3, Section II, C

  Cosmology | Part II, 2016

  The early universe is described by equations (with units such that $c=8 \pi G=\hbar=1$ )

  $$3 H^{2}=\rho, \quad \dot{\rho}+3 H(\rho+p)=0$$

  where $H=\dot{a} / a$. The universe contains only a self-interacting scalar field $\phi$ with interaction potential $V(\phi)$ so that the density and pressure are given by

  $$\begin{aligned} \rho &=\frac{1}{2} \dot{\phi}^{2}+V(\phi) \\ p &=\frac{1}{2} \dot{\phi}^{2}-V(\phi) . \end{aligned}$$

  Show that

  $$\ddot{\phi}+3 H \dot{\phi}+V^{\prime}(\phi)=0$$

  Explain the slow-roll approximation and apply it to equations (1) and (2) to show that it leads to

  $$\sqrt{3} \int \frac{\sqrt{V}}{V^{\prime}} d \phi=-t+\text { const. }$$

  If $V(\phi)=\frac{1}{4} \lambda \phi^{4}$ with $\lambda$ a positive constant and $\phi(0)=\phi_{0}$, show that

  $$\phi(t)=\phi_{0} \exp \left[-\sqrt{\frac{4 \lambda}{3}} t\right]$$

  and that, for small $t$, the scale factor $a(t)$ expands to leading order in $t$ as

  $$a(t) \propto \exp \left[\sqrt{\frac{\lambda}{12}} \phi_{0}^{2} t\right]$$

  Comment on the relevance of this result for inflationary cosmology.

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• Paper 4, Section I, C

  Cosmology | Part II, 2016

  The external gravitational potential $\Phi(r)$ due to a thin spherical shell of radius $a$ and mass per unit area $\sigma$, centred at $r=0$, will equal the gravitational potential due to a point mass $M$ at $r=0$, at any distance $r>a$, provided

  $$\frac{M r \Phi(r)}{2 \pi \sigma a}+K(a) r=\int_{r-a}^{r+a} R \Phi(R) d R$$

  where $K(a)$ depends on the radius of the shell. For which values of $q$ does this equation have solutions of the form $\Phi(r)=C r^{q}$, where $C$ is constant? Evaluate $K(a)$ in each case and find the relation between the mass of the shell and $M$.

  Hence show that the general gravitational force

  $$F(r)=\frac{A}{r^{2}}+B r$$

  has a potential satisfying $(*)$. What is the cosmological significance of the constant $B$ ?

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Differential Geometry

• Paper 1, Section II, G

  Differential Geometry | Part II, 2016

  Define what is meant by the regular values and critical values of a smooth map $f: X \rightarrow Y$ of manifolds. State the Preimage Theorem and Sard's Theorem.

  Suppose now that $\operatorname{dim} X=\operatorname{dim} Y$. If $X$ is compact, prove that the set of regular values is open in $Y$, but show that this may not be the case if $X$ is non-compact.

  Construct an example with $\operatorname{dim} X=\operatorname{dim} Y$ and $X$ compact for which the set of critical values is not a submanifold of $Y$.

  [Hint: You may find it helpful to consider the case $X=S^{1}$ and $Y=\mathbf{R}$. Properties of bump functions and the function $e^{-1 / x^{2}}$ may be assumed in this question.]

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• Paper 2, Section II, G

  Differential Geometry | Part II, 2016

  If an embedded surface $S \subset \mathbf{R}^{3}$ contains a line $L$, show that the Gaussian curvature is non-positive at each point of $L$. Give an example where the Gaussian curvature is zero at each point of $L$.

  Consider the helicoid $S$ given as the image of $\mathbf{R}^{2}$ in $\mathbf{R}^{3}$ under the map

  $$\phi(u, v)=(\sinh v \cos u, \sinh v \sin u, u) .$$

  What is the image of the corresponding Gauss map? Show that the Gaussian curvature at a point $\phi(u, v) \in S$ is given by $-1 / \cosh ^{4} v$, and hence is strictly negative everywhere. Show moreover that there is a line in $S$ passing through any point of $S$.

  [General results concerning the first and second fundamental forms on an oriented embedded surface $S \subset \mathbf{R}^{3}$ and the Gauss map may be used without proof in this question.]

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• Paper 3, Section II, G

  Differential Geometry | Part II, 2016

  Explain what it means for an embedded surface $S$ in $\mathbf{R}^{3}$ to be minimal. What is meant by an isothermal parametrization $\phi: U \rightarrow V \subset \mathbf{R}^{3}$ of an embedded surface $V \subset \mathbf{R}^{3}$ ? Prove that if $\phi$ is isothermal then $\phi(U)$ is minimal if and only if the components of $\phi$ are harmonic functions on $U$. [You may assume the formula for the mean curvature of a parametrized embedded surface,

  $$H=\frac{e G-2 f F+g E}{2\left(E G-F^{2}\right)}$$

  where $E, F, G$ (respectively $e, f, g$ ) are the coefficients of the first (respectively second) fundamental forms.]

  Let $S$