The Schroedinger Bridge problem as generative modelling tool has been recently studied. It is known that the solution to the problem can be cast as the optimal control (Foellmer drift) which steers a diffusion with fixed marginals at times t= 0 to t= 1 such that the law of the controlled diffusion has minimal KL-divergence with respect to the Wiener measure. In those note, we review this derivation and then use the deep stochastic optimal control- a hybrid method combining deep learning and trajectory optimization- to estimate the Foellmer drift directly.

In the first part of my talk I will recall Rost nilpotence and why this is an important property of varieties. Then I will discuss the following question: Given two smooth projective varieties $X$ and $Y$ satisfying Rost nilpotence, does this property holds for their disjoint union, i.e. the direct sum of their Chow motives? Beside its implications for blow-ups of varieties it turns out that this question has connections to Koethe's conjecture (in ring theory) and the theory of PI-algebras.

The talk discusses a general model using specific periodic special functions, that is degenerate elliptic Weierstrass P functions composed with the LambertW function, whose presence in the governing equations through the forcing terms simplify the periodic Navier Stokes equations (PNS). The properties of the LambertW function is discussed in context with numerical solutions of the Navier Stokes equations. Results in the literature can provide a systematic approach to study both large space and time behaviour for singular solutions to the Navier Stokes equations. The main part of the discussion will be to show that singular forcing can be used with the Duhamel principle to show numerical results occur and the existence of dipoles and vortices occur. How can we use singularities to show that numerical results with no singularities occur by shifting processes?