Mathematical Colloquium

Electricity is a very peculiar commodity - it cannot be stored and has to be delivered over given time slots. Network stability demands that production and consumption are matched at any given point of time. Due to that, prices react sharply to short-term over- or under-supply. To reduce the resulting risks, electricity producers and consumers trade large amounts of electricity beforehand. In order to quantify these transactions on this market one needs adequate models and techniques. In this talk, we will discuss electricity markets, their modelling and some observations around them.

Gromov’s non-squeezing theorem states that if one tries to put a ball into a cylinder symplectically, one shall not squeeze the ball. Consequently, the Gromov width of a symplectic manifold is given by the largest ball that could be embedded into it symplectically. It is a natural problem to determine the Gromov widths of symplectic manifolds. We are aiming to give an answer for the case of (generalized partial) flag varieties using representation theory of Lie algebras. The link is via toric degenerations, standard monomial theory and Newton-Okounkov bodies.

Zeta functions give an approach to the study of solution sets of polynomial equations. In the talk I will explain how to associate zeta functions to such solution sets and indicate in examples how Galois representations enter the picture.

Many algebraic properties of a complex algebraic variety can be read from topological properties of the space induced by the norm. In contrast, the naïve analogues of such results radically fail in $\mathbb{C}_p$ (the $p$-adic analogue of the complex numbers). In 1990, V. Berkovich associated a new topological space to an algebraic variety over $\mathbb{C}_p$, obtaining the right analogy between $\mathbb{C}$ and $\mathbb{C}_p$. Some of his results were later generalized and extended by E. Hrushovski and F. Loeser to arbitrarily algebraically closed valued fields. After informally reviewing the above context, I will report in this talk on a joint project with Mário Edmundo and Jinhe Ye in which, building on Hrushovski and Loeser's work, we extended results of V. Berkovich on the cohomology of algebraic varieties.

Lattices are an important ingredient in algebraic geometry, linking it to several other areas of mathematics. I will survey the theory, with emphasis on surfaces, and report on recent applications to finite automorphism groups of K3 surfaces.