I have been working on my Columbia application as well. The Jan 10 deadline means that they will start reading applications on this date. In their chat forum, they say that applications received before this date will be collected in order, and then reviewed starting on this date in the order received. Applications received after Jan 10 will be added to the bottom of the pile and reviewed in order. They will then disclose decisions on a rolling basis 8-12 weeks later.
This differs from other schools that do not have rolling admissions. I believe other schools will collect all applications received before a given deadline, then (possibly) review them in random order and finally disclose all decisions at the same time at a given date.