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Here is the second movement of my sonatina for trumpet an piano. Played by Michael Brest and Enae Han.

The world is shaped by events. Xenakis happened, Ligeti happened, Boulez happened, Grisey happened, Steve Reich happened, Lachenmann happened, Arvo Pärt happened, Eric Whitacre happened. Anyway, even if you hate what these composers have done and what they stand for, they have left a mark on the musical community and their ideas bear engaging with - even if that means doing the exact opposite from them. There is no unhearing spectralism - the idea that you can use timbre and the ideas related to timbre as musical material has directly challenged the concept of timbre and even the concept of "musical material" in the first place. Likewise, the complete dissolution of traditional instrumental roles by people like Lachenmann, Sciarrino and Berio has changed the way people think about writing for instruments. That doesn't mean that every piece from now on must only use extended techniques, but the playing field has been changed forever and in order to be relevant one must acknowledge these events. But this isn't really neo-romanticism by itself. If the intent is only to write music in a purebred Romantic style, then by all means - but not if you want to comment on the new, through the old (and vice versa). Then you need to synthesize old ideas and new ideas in a self-consistent manner and weave them into a homogenous tapestry, like what Berg did (unlike Schönberg). That gives you an interesting and probably non-obvious perspective on both which could change your perception. But if you use old ideas in an old context, there is nothing "neo" about it. EDIT: No reason to keep it purely classical either. Free jazz happened, rock 'n' roll happened, heavy metal happened (with all its variations), EDM happened. What does it mean to write music in a world where all of these things exist? I mean, take these things by themselves. Free jazz was a response to a jazz community oversaturated with the same ideas which became decreasingly relevant. Rock 'n' roll (and metal after that) is what happened when somebody figured out that by turning the amp volume way up you distort the amplified guitar signal. EDM, which is probably the most interesting thing that has ever happened to popular music, is the result of somebody discovering the crazy experiments of the early proponents of electronic instruments and crafting an entirely new sound out of it. The style of these genres is directly informed by the central ideas of the music, in other words the philosophy of the musicians making it.

Rapuns, particularly because you have limited time to put together a portfolio, I wouldn't suggest you try to learn everything about contemporary theory in the next few months. Try to think like the people who are reviewing your application. Why have they included this requirement? I suspect it's so that you can show your ability to write in more than one musical style, so they know you are in control of your compositions and thinking about theory, instead of just splatting some notes on a page and saying, "I'm a genius, if you don't understand my brilliance it's because you are stupid." (A surprisingly common way of thinking.) :P So, don't be overwhelmed by this assignment. Pick one specific aspect of contemporary theory that you find personally interesting. Write a composition that lets you try out the things you read about it in a few different ways. Don't try to mash 50 years of techniques by 100 different composers in there. Think of this as an exercise. Because, I expect that's what the people who asked for this are looking for. They want to know, if they let you into their program, and you are given an exercise to develop a piece using a technique that is new to you, can you do that in an organized manner. (Because you'll be doing that all the time as a composition student.) There are always lots of students popping up on these and other forums who complain that their professors asked them to (gasp!) learn about a musical style that was new to them instead of letting them write whatever they felt like in the style they were already comfortable with. That's the point of going to school to study. To learn new musical styles. To add them to your tool kit so you can pull them out confidently later in life, whenever you feel like it. So let your application show that you are open to exploring new ideas and can be organized about it when you do so.

This isn't really my arena so I have no recommendations. ... it's just funny to me that >30-year-old texts are ideal for studying "contemporary" music ... Why not look towards actual contemporary music...like, from this decade?