# Seminars & Colloquia

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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.

Let $X$ be an abelian variety of dimension $g$ over a field $k$. In general, the group $textrm{Aut}_k(X)$ of automorphisms of $X$ over $k$ is not finite. But if we fix a polarization $mathcal{L}$ on $X$, then the group $textrm{Aut}_k(X,mathcal{L})$ of automorphisms of the polarized abelian variety $(X,mathcal{L})$ over $k$ is known to be finite. Then it is natural to ask which finite groups can be realized as the full automorphism group of a polarized abelian variety over $k.$

In this talk, we give a classification of such finite groups for the case when $k$ is a finite field and $g$ is a prime number. If $g=2,$ then we need a notion of maximality in a certain sense, and for $g geq 3,$ we achieve a rather complete list without conveying maximality.

(This is a reading seminar for graduate students.)

Algebraic K-theory originated with the Grothendieck-Riemann-Roch theorem, a generalization of Riemann-Roch theorem to higher dimensional varieties. For this, we shall discuss the definitions of $K_0$-theory of a variety, its connection with intersection theory, $\lambda$-operation, $\gamma$-filtration, Chern classes and Adams operations.

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.

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IBS, Room B232 (DIMAG)
Discrete Math
Andreas Holmsen (KAIST)
Large cliques in hypergraphs with forbidden substructures

A result due to Gyárfás, Hubenko, and Solymosi, answering a question of Erdős, asserts that if a graph G does not contain K2,2 as an induced subgraph yet has at least c(n2) edges, then G has a complete subgraph on at least c210n vertices. In this paper we suggest a “higher-dimensional” analogue of the notion of an induced K2,2, which allows us to extend their result to k-uniform hypergraphs. Our result also has interesting consequences in topological combinatorics and abstract convexity, where it can be used to answer questions by Bukh, Kalai, and several others.