booklet - CUMC - Canadian Mathematical Society

the work that has been done in the area and the work that still needs to be completed,as well as extensions of the definition to other structures. Additionally, I prove two elementaryresults that clarify the definition and illustrate a nice property of these graphs,a property that suggests why the term "homogeneous" is used to describe them.VC DENSITY AND p-ADIC OPTIMISATIONNIGEL PYNN-COATESVC density is a measure of the combinatorial complexity of a family of sets firstdeveloped in computational learning theory but has been closely connected to a notionin model theory. This is the perspective I take. The p-adic numbers are a completion ofthe rationals different from the reals with many interesting properties.VC density in the p-adics is bounded but it is not known if the bound is optimal.We are using a version of the optimisation technique simulated annealing, that we haveadapted to work in the p-adics, to examine families of sets to test the bound.Required Background: Analysis and some elementary logic would be handy.CONTINUOUS LOGIC AND AN ISOMORPHISM THEOREMNIGEL SEQUEIRAWe often want to know how many different mathematical objects there of a particulartype up to isomorphism. In model theory, given a countable language L and anycountable L-structure M, Scott’s isomorphism theorem guarantees the existence of asentence φ of L ω1 ω, the language obtained from closing L under countable conjunctionand disjunction, such that for any countable model N where φ is true we have N ∼ = M.I will consider the analogous question for continuous logic: whether for every separablemetric language L we can find, for each separable L-structure M, a sentence ofL (or an extension thereof) classifying M up to isomorphism.We will see that such a sentence exists if we allow, along with countable conjunctionand disjunction, a ‘distance-to-zeroset’ operator ρ.Required Background: Analysis, linear algebra, and basic logic.40