Recently, the classification of isoparametric hypersurfaces in spheres has been completed. Therefrom, various new research projects in geometry have been initiated. The study of minimal lagrangian submanifolds via isoparametric hypersurfaces is one of the most active projects a la mode. In this talk, we have an introduction to the study of Isoparametric hypersurfaces and minimal Lagrangian submanfolds, and discuss the relationship between them.

Many modern applications such as machine learning require solving large-dimensional optimization problems. First-order methods are widely used to solve such problems, since their computational cost per iteration mildly depends on the problem dimension. However, they suffer from relatively slow convergence rates, and this talk will discuss recent progress on the acceleration of first-order methods, particularly using the close relationship between convex optimization methods and maximally monotone operators.

Finite element discretization of solutions with respect to simplicial/cubical meshes has been studied for decades, resulting in a clear understanding of both  the relevant mathematics and computational engineering challenges. Recently, there has been both a desire and need for an equivalent body of research regarding discretization with respect to generic polygonal/polytopal meshes. General meshes offer a very convenient framework for mesh generation, mesh deformations, fracture problems, composite materials, topology optimizations, mesh refinements and coarsening; for instance, to handle hanging nodes, different cell shapes within the same mesh and  non-matching interfaces. Such a flexibility represents a powerful tool towards the efficient solution of problems with complex inclusions as in geophysical applications or posed on very complicated or possibly deformable geometries as encountered in basin and reservoir simulations, in fluid-structure interaction, crack propagation or contact problems.

In this talk,  a new computational paradigm for discretizing PDEs is presented via staggered Galerkin approach on general meshes. First, a class of locally conservative, lowest order staggered discontinuous Galerkin method on general quadrilateral/polygonal meshes for elliptic problems are proposed.  The method can be flexibly applied to rough grids such as highly distorted  meshes. Next, adaptive mesh refinement is an attractive tool for general meshes due to their flexibility and simplicity in handling hanging nodes.

I report on work with M. Gubinelli and T. Oh on the
renormalized nonlinear wave equation
in 2d with monomial nonlinearity and in 3d with quadratic nonlinearity.
Martin Hairer has developed an efficient machinery to handle elliptic
and parabolic problems with additive white noise, and many local
existence questions are by now well understood. In contrast not much is
known for hyperbolic equations. We study the simplest nontrivial
examples and prove local existence and weak universality, i.e. the
nonlinear wave equations with additive white noise occur as scaling
limits of wave equations with more regular noise.