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: TBA

Dr. Philip Schwartz (Universität Hannover)

TBA

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 \[ \int\limits_{0} ^1 \log (1-f(x)^2) \frac{dx}{\sqrt{1-x^2}} > - \infty \, . \]