The honest answer to the question is that I actually do not know. I will therefore rather talk about several famous examples that are widely called 'h-principle results' and try to explain some of the ideas behind the ones I am most familiar with.
Tag - Differential equations
This talk will be in two parts. The first part will be introductory, and will address the question: Given an ordinary differential equation (ODE) with certain physical/geometric properties (for example a preserved measure, first and/or second integrals), how can one preserve these properties under discretization? The second part of the talk will cover some more recent work, and address the question: How can one deduce hard to find properties of an ODE from its discretization?
In many situations where stochastic modeling is used, one desires to choose the coefficients of a stochastic differential equation which represents the reality as simply as possible. For example one desires to approximate a diffusion model with high complexity coefficients by a model within a class of simple diffusion models. To achieve this goal, we introduce a new Wasserstein type distance on the set of laws of solutions to d-dimensional stochastic differential equations.
This new distance W̃2 is defined similarly to the classical Wasserstein distance W̃2 but the set of couplings is restricted to the set of laws of solutions of 2d-dimensional stochastic differential equations. We prove that this new distance W̃2 metrizes the weak topology. Furthermore this distance W̃2 is characterized in terms of a stochastic control problem. In the case d = 1 we can construct an explicit solution. The multi-dimensional case, is more tricky and classical results do not apply to solve the HJB equation because of the degeneracy of the differential operator. Nevertheless, we prove that this HJB equation admits a regular solution.
The standard WDVV equations are PDEs in the potential function that generates Gromov-Witten invariants. These equations imply relations on the invariants, and sometimes allow computations thereof, as demonstrated by Kontsevich-Manin (1994). We prove analogous equations for open Gromov-Witten invariants that we defined in a previous work. For (ℂPn,ℝPn), the resulting relations allow the computation of all invariants. The formulation of the open WDVV requires a lift of the big quantum product to relative cohomology. Surprisingly, this brings us to use moduli spaces of disks with geodesic conditions. No prior knowledge of the subjects mentioned above will be assumed.
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