Vorträge in der Woche 04.11.2024 bis 10.11.2024

Montag, 04.11.2024: The field of p-adic numbers and some basics about its absolute Galois group

Elmar Große-Klönne

PRE_COLLOQUIUM: I want to briefly explain the construction of the field of p-adic numbers and explain some of its basic structural features as well as its absolute Galois group. I will then give some motivation for studying the representation theory of the latter.

Montag, 04.11.2024: Weights in Families: How Galois representations live together

Elmar Große-Klönne

Abstract: Let Q_p denote the field of p-adic numbers for a prime p and let G be its Galois group. A group homomorphisms r:G -> GL_n(k) for a field k is called a Galois representation. The famous Langlands program suggests that such representations r have ‘automorphic incarnations’ in the world of modular forms (n=2) and in this way give rise to a set W(r) of ‘weights’. There is a natural notion of ‘families’ of such Galois representations and the corresponding variations of the sets W(r) obey intricate combinatorial laws.

Dienstag, 05.11.2024: L2-Betti numbers of totally disconnected groups and their approximation by Betti numbers of lattices

Paul Vögele

Dienstag, 05.11.2024: L2-Betti numbers of totally disconnected groups and their approximation by Betti numbers of lattices

Paul Vögele

Mittwoch, 06.11.2024: Tropical trigonal curves

Angelina Zheng (Tübingen)

The moduli of algebraic curves admits a well known stratification by gonality. We would like to study the locus of d-gonal curves, but in the moduli space of tropical curves. We will first review the hyperelliptic case, studied by Melody Chan. In particular, for hyperelliptic curves, the divisorial 2-gonality of the metric graph defining the curve is precisely equivalent to its combinatorial analogue, the 2-gonality of its underlying graph. We will show that, if we allow tropical modifications, then such a relation can be generalized also to the trigonal case. This is a work in progress with Margarida Melo.

Donnerstag, 07.11.2024: Geometric descriptions of (post-)Newtonian modified gravity

Dr. Philip Schwartz (Universität Hannover)

Newton--Cartan gravity is a differential-geometric reformulation of Newtonian gravity, bringing the latter closer to general relativity (GR): as in GR, gravitational effects are encoded into spacetime geometry, which however is of non-Lorentzian nature. Newton—Cartan gravity can be shown to arise as Newtonian limit of GR in a rigorous sense, hence providing a coordinate-free geometric formulation of this limiting process. In this talk, we will discuss recent advances in analogous geometric descriptions of the Newtonian behaviour of modified theories of gravity. Concretely, after giving an introduction to standard Newton--Cartan gravity, we are going to discuss the construction of a `teleparallelised' version of Newton--Cartan gravity. We show how this theory arises as a formal large-speed-of-light limit of the so-called teleparallel equivalent of general relativity (TEGR). Thus, it provides a geometric formulation of the Newtonian limit of TEGR, analogous to the GR case. We end with an outlook on extensions to (a) post-Newtonian expansions, and (b) more general modified theories of gravity.

Donnerstag, 07.11.2024: How to represent a function in a quantum computer

Dr. Michel Alexis (Uni Bonn)

Quantum Signal Processing (QSP) is an algorithmic process by which one represents a signal $f: [0,1] \to (-1,1)$ as the upper left entry of a product of $SU(2)$ matrices parametrized by the input variable $x \in [0,1]$ and some ``phase factors'' $\{\psi_k\}_{k \geq 0}$ depending on $f$. We show that, after a change of variables, QSP is actually the SU(2)-valued nonlinear Fourier transform, and the phase factors $\{\psi_k\}_k$ correspond to the nonlinear Fourier coefficients. By exploiting a nonlinear Plancherel identity and using some basic spectral theory, we then show that a QSP representation exists for every $f$ satisfying the log integrability condition \[ \int\limits_{0} ^1 \log (1-f(x)^2) \frac{dx}{\sqrt{1-x^2}} > - \infty \, . \]