Extra comments:Talk can be livestreamed on zoom meeting ID 986 2892 7517 - send request to jonathan.pakianathan@rochester.edu at least 2 days before talk to guarantee this option. Video of the lecture will be made available after the talk also (not live).

Abstract: My talk will discuss the basics of higher ramification over local fields (i.e. fields that carry an integer-valued `valuation’, which is similar to an absolute value, that satisfies some nice properties). I will first discuss some preliminaries about discrete valuations, extensions of local fields, and ramification in general, and then define the higher ramification groups and give some of their properties. I will talk about the lower and upper numberings of the ramification groups, why we should care about them (which comes down to the fact that the lower numbering behaves nicely with subgroups of the galois group while the upper numbering behaves nicely with quotients), and how to translate in between them (in particular, I will give an example or two of how to use something called Tate’s lemma to do this easily in certain cases). I will finish by discussing in broad detail a result of Deligne, which tells how the galois groups of extensions that aren’t “too highly ramified” can be determined even only with knowledge of the base field up to (modulo) a certain power of its maximal ideal (and so hopefully will provide some motivation for how higher ramification can be useful in general for studying local fields).