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Tuesday, April 11, 2006

**Abstract:** Hardy fields (at 0+) are ordered differential fields of germs at 0+ of real-valued C^1 functions defined on intervals of the form (0,a) for some real number a. Every Hardy field comes equipped with a valuation and thus we can also consider Hardy fields as valued fields. It is well know that valued fields can be equipped with a valuation topology and that every valued field (K,v) possesses a completion with respect to this valuation topology. I will consider the question of whether this completion, which is a priori just a valued field, can be realized as a Hardy field. I will answer this question affirmatively for the simple case of the Hardy field R(t) and outline the proof. The proof requires two theorems from analysis and in order to answer the question affirmatively for more complicated Hardy fields, one would need generalizations of them to broader contexts. I will present a generalization of one of the theorems, Borel’s theorem on Taylor series, as half of the work necessary to complete the Hardy field R(t^1/d : d=1,2,…). I will define all of the notions involved and the talk should be accessible to anyone with knowledge of undergraduate real analysis.