I'm not sure how recent "contemporary" mathematics means so I'll mention a few things, some somewhat classical but all connected to the fact that $\phi=\frac{1+\sqrt{5}}{2}$ is closely connected to the Fibonacci sequence. I will endeavor to order these roughly in oldest to most recent.

First, in a certain sense, $\phi$ is the hardest number to approximate with rational numbers. What do we mean by that? if you have an irrational number $\alpha$, and you want to approximate $\alpha$ with rational numbers of the form $\frac{n}{d}$, then you can get approximations as good as you want by making $d$ larger. However, suppose you are interested in getting as close as you can and wanting to know how bad a price you pay in terms of increasing $d$, it turns out that by multiple ways of making this precise, $\phi$ is the worst possible. This is closely related to the fact that it has continued fraction $[1,1,1,1...]$ and so the best possible approximates actually have numerator and denominator Fibonacci numbers.

Understanding the Fibonacci numbers better turns out to be closely connected to  the [Binet formula][1], which says that 
$$F_n = \frac{\left(\frac{1+\sqrt{5}}{2}\right)^n- \left(\frac{1-\sqrt{5}}{2} \right)^n}{\sqrt{5}}.$$

So for example, there's an old result that the Fibonacci sequence distributes over gcd, that is $$F_{\mathrm{gcd}(a,b)}= \mathrm{gcd}(F_a,F_b).$$ It turns out that one of the more enlightening ways of proving this is by using the Binet formula and then looking at how the Fibonacci numbers behave in the ring $\mathbb{Z}[\frac{1+\sqrt{5}}{2}]$.

This leads us to the more recent work. Define the order of apparition of  $n$, denoted by $z(n)$, as the least $k$ such that $n∣F_k$. The last few years have seen extensive work on trying to understand this function, and the closely related function of the [Pisano period][2], which says how long it takes $F_n$ to repeat mod $m$ for some $m$. In the early 2010s a whole bunch of papers on this topic were written by Diego Marques which are of note in this regard. One of the techniques here involves trying to understand  $\mathbb{Z}[\frac{1+\sqrt{5}}{2}]$ really closely. 

In a related note, there's a recent paper in the Journal of Number Theory by Roswitha Hofer which generalizes the golden ration to fields of formal power series. Hofer's paper can probably be used as a jumping off point for generalizing some of the results mentioned above to other contexts. 

  [1]: https://en.wikipedia.org/wiki/Fibonacci_number#Closed-form_expression
  [2]: https://en.wikipedia.org/wiki/Pisano_period