booklet - CUMC - Canadian Mathematical Society
booklet - CUMC - Canadian Mathematical Society
booklet - CUMC - Canadian Mathematical Society
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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