Guest lectures and seminars - Page 30

The variety of sums of powers, VSP(F, r) of a homogeneous form F of rank r is the closure in the Hilbert scheme of apolar schemes of length r. A bad limit is a scheme in the closure that is not apolar to F. I will discuss examples of bad limits, including examples for quadrics found by Joachim Jelisiejew that contradicts earlier results on polar simplicies. This is report on work in progress with Jelisiejew and Schreyer and with Grzegorz and Michal Kapustka.

Counterexamples to the integral Hodge conjecture can arise either from torsion cohomology classes (as in Atiyah's and Hirzebruch's original counterexample from 1961) or from non-torsion classes (as first seen in Kollár's counterexample from 1991). After Voisin proved the IHC for uniruled threefolds, Schreieder found a unirational fourfold where the IHC fails. His construction of a non-algebraic Hodge class relies on abstract arguments with unramified cohomology. It was an open question whether this class is of torsion type. In this talk, I want to explain a new method that gives an explicit geometric description of the unramified cohomology class appearing in his argument. In particular, this approach allows to prove that Schreieder's unirational counterexample is of torsion type.

In this talk, I will go through my past research before joining UiO, particularly at The University of Texas at Austin. This will include a brief introduction to the development of stable and adaptive finite element methods for challenging problems in engineering science. Second, I will focus on modeling efforts in coastal ocean hydrodynamics, including a review of the underlying physics and assumption and a review of the current state-of-the-art. I will also introduce several related to my focus of storm surge modeling and how the models are used by stakeholders beyond academia.