**Date/Time**

Date(s) - 17/03/2023*1:30 pm - 2:30 pm*

**Speaker:** Patrick Speissegger, McMaster University

**Title:** *O-minimal Borel lemma*

**Abstract: ** A Lemma of Borel’s states that if a real function is increasing and greater than or equal to 1, and if r>1 is a real number, then the set of all x such that f(x + 1/f(x)) is greater than or equal to r f(x) has outer measure at most r/(r-1). In the o-minimal context, we can replace “outer measure” with “total length”, and in this case I will show that Borel’s Lemma is true in any o-minimal expansion of an ordered field. The proof is very elementary and only uses the o-minimality axiom (no derived results). This lemma is a bit of a proof in search of a theorem; the classical version is often used in Nevanlinna theory for entire holomorphic functions.

**Location:** Hamilton Hall, Room 312