CMSA Colloquium: Predicting non-continuous functions


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September 25, 2023 4:30 pm - 5:30 pm
CMSA, 20 Garden St, G10
Address: 20 Garden Street, Cambridge, MA 02138

Sean Cox - Virginia Commonwealth University

One of the strangest consequences of the Axiom of Choice is the following Hardin-Taylor 2008 result:  there is a "predictor" such that for every function $f$ from the reals to the reals---even nowhere continuous $f$---the predictor applied to $f \restriction (-\infty,t)$ correctly predicts $f(t)$ for *almost every* $t \in R$.  They asked how robust such a predictor could be, with respect to distortions in the time (input) axis; more precisely, for which subgroups $H$ of Homeo^+(R) do there exist $H$-invariant predictors?  Bajpai-Velleman proved an affirmative answer when H=Affine^+(R), and a negative answer when H is (the subgroup generated by) C^\infty(R).  They asked about the intermediate region; in particular, do there exist analytic-invariant predictors?  We have partially answered that question:  assuming the Continuum Hypothesis (CH), the answer is "no". Regarding other subgroups of Homeo^+(R), we have affirmative answers that rely solely on topological group-theoretic properties of the subgroup.  But these properties are very restrictive; e.g., all known positive examples are metabelian.  So there remain many open questions. This is joint work with Aldi, Buffkin, Cline, Cody, Elpers, and Lee.