What are the big topics in mobile user interface design discussions in 2013?
Web vs. native apps. Still. While the majority of companies are pretty firmly in the "native app" space, there are still debates and discussions taking place around HTML 5 and it's capabilities. The problem is, HTML 5 can't do everything a native app can, and it also (if not tweaked and optimized) doesn't provide the same type of fluid experience that users expect. I've seen many attempts at combining the two by using a thin "wrapper" around HTML5 content (most famously from LinkedIn and Facebook), but the "killer app" using such an approach has still not come out. People will still try, though.
Responsive Web Design. I heart Responsive Web Design. My personal blog is responsive, and I can see many many reasons for companies to make sure their app is responsive. However, the sad fact remains: existing companies already HAVE web sites, and changing these sites to make them responsive is a big investment of time and money, and the benefits aren't ones that align very well with most "20th Century" business drivers. It's going to be an uphill climb (I've fought for RWD on projects, and lost, because "it wouldn't have any noticeable ROI or impact on customer satisfaction").
What platform to design for? Do you design for Android and iPhone, or focus on only one? To again belabor the obvious, companies have limited budgets, and they look at where the users are. Right now, there's a LOT of Android users out there, but if you dig deeper into the stats you find that they aren't buying apps and don't actually use the device nearly as much as iPhone users do. And what about Windows Phone? What apps to create and for what devices is a big topic of discussion at companies big and small around the world.