Project Group A

Project Group A "Rough Analysis, Signatures and SPDEs"

With a significant algebraic and geometric component, ‘rough analysis’ has been a revolution in our ability to analyze large classes of ‘sub-critical’ singular stochastic dynamics. A key example is given by the KPZ equation, a stochastic model for interface growth, first solved with rough paths, later with regularity structures and paracontrolled distributions; the ‘KPZ’ fixed point is a key question in this field. Central to group A is an outward looking investigation of rough and stochastic (partial, backward) differential equations  and their robustness properties (A03, A06, A07); pressing questions concerning controlled McKean-Vlasov dynamics, non-Markov by nature, will be addressed (A10). The algebraic-geometric notion of ‘(expected) signature’ will be studied through the lenses of  algebraic geometry (A04, A05), feeding further into algebraic statistics (→ B). Inspired  by models of mathematical biology, rough super-Brownian motion is studied (A09). We  seek to cross-fertilize ’optimal transport’ with problems from rough analysis, e.g. in the  context of certain singular Langevin dynamics (A02). The new notion of energy solution, a sort of Itô calculus for singular dynamics, gives hope to our long-term goal of  overcoming sub-criticality, first steps will be taken (A01, A08).