advice.qmd

---
title: "Advice"
---

Here is some advice for PhD students in high-energy physics.

$$
d s^2 = - f(r) dt^2 + \frac{dr^2}{f(r)} + r^2 d\Omega_2^2
$$ {#eq-sch-metric}

Black-Scholes (@eq-black-scholes) is a mathematical model that seeks to explain the behavior of financial derivatives, most commonly options:

$$
\frac{\partial \mathrm C}{ \partial \mathrm t } + \frac{1}{2}\sigma^{2} \mathrm S^{2}
\frac{\partial^{2} \mathrm C}{\partial \mathrm C^2}
  + \mathrm r \mathrm S \frac{\partial \mathrm C}{\partial \mathrm S}\ =
  \mathrm r \mathrm C 
$$ {#eq-black-scholes}

```{=latex}
\begin{align}
d s^2 &= - f(r) dt^2 + \frac{dr^2}{f(r)} + r^2 d\Omega_2^2 \\
f(r) &= 1- \frac{r_+}{r} \label{eq-frblackhole}
\end{align}
```
We would like to compute the thermal partition function of this system. Since we know the spectrum, we can write down the answer

```{=tex}
\begin{align}
Z(\beta) = \sum_n e^{-\beta E_n} = \sum_{n\in \mathbb{Z}} \exp \left( 
- \beta \, \frac{n^2}{2R^2}
\right) \, .
\label{zbetaparticlering}
\end{align}
```