School of Mathematics
I will introduce a new paradigm for comparing relative trace formulas, in order to prove instances of (relative) functoriality and relations between periods of automorphic forms.
We study the complexity of constructing a hitting set for the class of polynomials that can be infinitesimally approximated by polynomials that are computed by polynomial sized algebraic circuits, over the real or complex numbers. Specifically, we show that there is a PSPACE algorithm that given nsr in unary outputs a set of inputs from of size poly(nsr), with poly(nsr) bit complexity, that hits all $n$-variate polynomials of degree $r$ that are the limit of size $s$ algebraic circuits.
\[ x_1^2 + x_2^2 + \cdots + x_n^2 = ax_1x_2 \cdots x_n + k.\]