Guest lectures and seminars - Page 23

QOMBINE seminar talk by Vebjørn Hallberg Bakkestuen (UiO)

Constructing fast solution schemes often involves deciding which errors are acceptable and which approximations can be made for the sake of computational efficiency. Herein, we consider a mixed formulation of Darcy flow in porous media and take the perspective that the physical law of mass conservation is significantly more important than the constitutive relationship, i.e. Darcy's law. From this point of view, we propose an inexact solution technique that nevertheless guarantees local mass conservation. The method is based on first solving the mass balance equation and then computing a solenoidal correction using the curl of a potential field. We extend the method to flows in fractured porous media and present numerical experiments that indicate the efficiency of the scheme.

Fano manifolds are complex projective manifolds having positive first Chern class. The positivity condition on the first Chern class has far reaching geometric and arithmetic implications. For instance, Fano manifolds are covered by rational curves, and families of Fano manifolds over one dimensional bases always admit holomorphic sections. In recent years, there has been some effort towards defining suitable higher analogues of the Fano condition. Higher Fano manifolds are expected to enjoy stronger versions of several of the nice properties of Fano manifolds.

In this talk, I will discuss higher Fano manifolds which are defined in terms of positivity of higher Chern characters. After a brief survey of what is currently known, I will present recent joint work with Carolina Araujo, Roya Beheshti, Kelly Jabbusch, Svetlana Makarova, Enrica Mazzon and Nivedita Viswanathan, regarding toric higher Fano manifolds. I will explain a strategy towards proving that projective spaces are the only higher Fano manifolds among smooth projective toric varieties.

For millennia, origami and kirigami artists have used folds and cuts to create beautiful shapes from a simple sheet of paper. I will describe our recent scientific attempts to catch up with these remarkably imaginative arts phrased as inverse problems in physical geometry that aim to control the shape and rigidity of a thin surface. Using discrete operations that vary the number, size, orientation and coordination of folds and cuts, I will show how to create piecewise isometric kirigami and origami tessellations and control their local and global morphology and mechanical response, mixing experimental, computational and theoretical approaches.