**November 18**

**Brent Cody**,
Virginia Commonwealth University

**Sparse analytic systems**

Given a set $S$, an $S$-predictor $\mathcal{P}$ is a function that takes as inputs functions of the form $f:(-\infty,t)\rightarrow S$, where $t\in\mathbb{R}$, and outputs a guess $\mathcal{P}(f)$ for what $f(t)$ 'should be.' An $S$-predictor is good if for all total functions $F:\mathbb{R}\to S$ the set of $t\in\mathbb{R}$ for which the guess $\mathcal{P}(F\upharpoonright(-\infty,t))$ is not equal to $F(t)$ has measure zero. Hardin and Taylor proved that every set $S$ has a good $S$-predictor and they raised various questions asking about the extent to which the prediction $\mathcal{P}(f)$ made by a good predictor might be invariant after precomposing $f$ with various well-behaved functions - this leads to the notion of 'anonymity' of good predictors under various classes of functions. Bajpai and Velleman answered several of Hardin and Taylor's questions and asked: Does there exist, for every set $S$, a good $S$-predictor that is anonymous with respect to the strictly increasing analytic homeomorphisms of $\mathbb{R}$? We provide a consistently negative answer to this question by strengthening a result of Erdős, which states that the Continuum Hypothesis is equivalent to the existence of an uncountable family $F$ of (real or complex) analytic functions, such that $\{f(x):f\in F\}$ is countable for every $x$. We strengthen Erdős' result by proving that CH is equivalent to the existence of what we call *sparse analytic systems* of functions. This is joint work with Sean Cox and Kayla Lee.