Elie Cartan's celebrated paper ˝Pfaffian systems in 5 variables˝ in 1910 studied the equivalence problem for general Pfaffian systems of rank 3 in 5 variables as the curved version of the Pfaffian system with G_2 symmetry. The G_2 case admits a beautiful geometric correspondence with Engel's PDE system. We give a historical overview of Cartan's paper and discuss recent works extending the correspondence to curved cases, which is based on ideas from geometric control theory and algebraic geometry.

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pw: 2021math

In this talk, we consider stochastic systems with several stable sets. Typical examples are low-temperature physical systems and stochastic optimization algorithms. The macroscopic description of such systems is usually carried out via a so-called model reduction. We explain a necessary and sufficient condition for model reduction in terms of solutions of certain form of partial differential equations.

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pw: 2021math

Generative adversarial networks (GAN) are a widely used class of deep generative models, but their minimax training dynamics are not understood very well. In this work, we show that GANs with a 2-layer infinite-width generator and a 2-layer finite-width discriminator trained with stochastic gradient ascent-descent have no spurious stationary points. We then show that when the width of the generator is finite but wide, there are no spurious stationary points within a ball whose radius becomes arbitrarily large (to cover the entire parameter space) as the width goes to infinity.

It is a gentle introduction to the mean curvature flow and its application to knot theory for undergraduate students. J.W.Alexander discovered a knotted sphere embedded in 3-dimensional Euclidean space in 1924. This example has provoked curiosity to find simple conditions under which embedded spheres are unknotted. In this talk we will sketch theorems and conjectures in the mean curvature flow for the knot theory, in analogy to the Ricci flow for the smooth 4-dimensional Poincare conjecture.

The compressible Euler system was first formulated by Euler in 1752, and was complemented by Laplace and Clausius in the 19th century, by introducing the energy conservation law and the concept of entropy based on thermodynamics. The most important feature of the Euler system is the finite-time breakdown of smooth solutions, especially, appearance of a shock wave as severe singularity to irreversibility(-in time) and discontinuity(-in space). Therefore, a fundamental question (since Riemann 1858) is on what happens after a shock occurs. This is the problem on well-posedness (that is, existence, uniqueness, stability) of weak solutions satisfying the 2nd law of thermodynamics, which is called entropy solution. This issue has been conjectured as follows: Well-posedness of entropy solutions for CE can be obtained in a class of vanishing viscosity limits of solutions to the Navier-Stokes system. This conjecture for the fundamental issue remains wide open even for the one-dimensional CE. This talk will give an overview of the conjecture, and recent progress on it.