Vorträge in der Woche 08.01.2024 bis 14.01.2024

Dienstag, 09.01.2024: TBA

Dr. Christopher Cedzich (Uni Düsseldorf)

Mittwoch, 10.01.2024: Naive A^1-knot theory

Mario Kummer (Dresden)

We present an A^1-count of secant lines to a space curve from a joint work with Daniele Agostini and explain how it can be thought of as an arithmetic knot invariant. Building on this we make some first steps towards a naive A^1-knot theory and conclude with some open questions. This is one talk in a series of two, which is why we begin at 10 s.t.

Mittwoch, 10.01.2024: Tropical A^1-Enumerative Geometry

Andres Jaramillo Puentes (Essen)

Mikhalkin's correspondence theorem establishes a correspondence between algebraic curves on a toric surface and tropical curves. This translates the question of counting the number of algebraic curves through a given number of points to the question of counting tropical curves, i.e. certain graphs, with a given notion of multiplicity through a given number of points which can be solved combinatorially. In this talk we will present a version of Mikhalkin's correspondence theorem for an arbitrary base field for k-rational points in a joint work with Sabrina Pauli and we will discuss the combinatorial properties of the tropical counts in a joint work with Hannah Markwig, Sabrina Pauli and Felix Röhrle.

Donnerstag, 11.01.2024: Double-Commutator Formula for Interacting Lattice-Fermions

Marius Wesle (Tübingen)

The integer quantum Hall effect is a physical phenomenon that occurs at very low temperatures in various two-dimensional electron systems. When applying an electric potential across such a system, one can observe that the Hall conductance, which is the conductance associated with the current perpendicular to the electric field, takes on particular quantised values that are integer multiples of the so-called conductance quantum. The double-commutator formula for the Hall conductance, first derived by Avron, Seiler, and Simon, establishes a connection between the Hall conductance and the Chern number of a certain vector bundle. In this talk we will discuss a recent result for periodic non-interacting systems by Marcelli and Monaco, showing that the linear response coefficient of the Hall current is given by the double-commutator formula and that there are no power-law corrections. We will also see how one can use an operator-algebraic framework to extend the result to interacting lattice-fermions.

Donnerstag, 11.01.2024: Random currents and the Toric code

Sebastian Stengele (TU München)

The toric code is a key example of a topological quantum error-correcting code and a potential candidate for a scalable quantum memory. The random currents expansion is a powerful technique to analyze Ising model correlations. I will focus on introducing random currents, argue why this technique can also be applied to the toric code and present some preliminary results for the decay of correlations of the toric code.