we show that certain singularities do not arise as blow-ups of mean curvature flow starting at a generic hypersurface. this is joint work with kyeongsu choi, christos mantoulidis, and felix schulze.

in 1998, p. biane and r. speicher developed a theory of stochastic integration against free additive self-adjoint brownian motion, the large-$n$ limit of $n \times n$ hermitian brownian motion. this development included a kind of it\={o} formula for certain functions of free it\={o} processes. in this talk, we discuss -- for motivation -- the finite-$n$ version of this it\={o} formula and use objects called \textit{double operator integrals} to give a new analytic interpretation of the quantities therein, which to date have been understood and interpreted mostly in combinatorial ways. this analytic interpretation yields an extension, which we shall also discuss, of biane and speicher's original formula.

langlands's functoriality conjectures predict the existence of
``liftings'' of automorphic representations along morphisms of l-groups. a
basic case of interest comes from the irreducible algebraic
representations of gl(2), thought of as the l-group of the reductive
group gl(2) over q. i will discuss the proof, joint with james newton,
of the existence of the corresponding functorial liftings for a broad
class of holomorphic modular forms, including ramanujan's delta function.