강연영상 17 페이지 > 서울대학교 수리과학부

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.

Many problems in financial mathematics are closely related to the stochastic optimization problem because the optimal decision must be made under the uncertainty. In particular, optimal stopping, singular control, and optimal switching problems in the stochastic optimization problem arising from financial mathematics are formulated into the free boundary problem when the uncertainty follows the Markov process. The optimal strategies to each optimization problem is determined by the free boundary. In this talk, I introduce various free boundary problems in financial mathematics.

Coulomb Gases are point processes consisting of particles whose pair interaction is governed by the Coulomb potential. There is also an external potential which confines the particles to a region. Wigner introduced this toy model for the Gibbs states of electrons in a crystal, and in the 1950s, connections with random matrix theory were established. In this talk we will discuss edge statistics of one and two dimensional Coulomb gases.

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We obtain a quantitative and robust proof that incompressible fluid models are strongly ill-posed in critical Sobolev spaces, in the sense that norm inflation and even nonexistence occur for critical initial data. We then show how to use the result to obtain short-time enhanced dissipation for the Navier-Stokes equations.

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