강연영상 16 페이지 > 서울대학교 수리과학부

The invariant subspace problem is one of the longstanding open problem in the field of functional analysis and operator theory. It is due to J. von Neumann (in 1932) and is stated as: Does every operator have a nontrivial invariant subspace? An invariant subspace for an operator means a subspace such that the image of the subspace under the operator returns to the subspace. This problem remains still open for infinite-dimensional separable Hilbert spaces. In this talk we consider the reason why so much attention to the invariant subspace problem and discuss the invariant subspaces of the shifts. In the context of this problem, we come across G.H. Hardy and A. Beurling.

The thirteen books "Elements" were written or collected by Euclid of Alexandria about 300 BCE. Many think that "Elements" is the most important example of deductive mathematics. In fact, the Common Notions and the Postulates of Elements are not just arbitrary axioms. They should be regarded as marvelous pearls in the ocean of geometry. Discovering and proving theorems are the Arts of Mathematics.

For each natural number k, the C^k diffeomorphisms of the circle form a group with function compositions. This definition even extends to real numbers k no less than one by Hölder continuity. We survey algebraic properties of this group, many of which can distinguish various values of k. (Joint work with Thomas Koberda)

We describe some of the history of the three-body problem and how it lead to symplectic geometry. We start by sketching Poincare’s prize-winning work, and discuss how it lead to the birth of the fields of dynamical systems and symplectic geometry. To make the talk accessible we begin with the basics and give an illustrated guide of some of the developments.

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.