Hybrid seminar

Titel: The unitary almost-Mathieu operator and (possibly) beyond

Abstract: In this talk, we introduce the unitary almost-Mathieu operator (UAMO) and discuss its connections to several model systems in physics and mathematics. We draw parallels to the self-adjoint almost Mathieu operator and discuss how the UAMO originates in a two-dimensional quantum walk in a uniform magnetic field as well as its connection to one-dimensional split-step quantum walks and CMV matrices. We exhibit a version of Aubry–André duality for this model, which partitions the parameter space into three regions: a supercritical region and a subcritical region that are dual to one another, and a critical regime that is self-dual. In each parameter region, we characterize the cocycle dynamics of the transfer matrix cocycle generated by the associated generalized eigenvalue equation. As a practical application, we show how to obtain a lower bound on the Lyapunov exponent, which in turn allows us to exclude point spectrum in the subcritical regime. As a main result, we characterize the spectral type for each value of the coupling constant, almost every frequency, and almost every phase. If time allows, we go beyond that and discuss characteristics of complex extensions of the UAMO.

Titel: Dimensional Properties for the Spectra of Sturmian Hamiltonians in a Small Coupling Regime

Abstract: We prove that the Hausdorff dimension of the spectrum of a Sturmian Hamiltonian of bounded type tends to one as the coupling constant tends to zero. We give a sketch of the proof which relies on the methods of trace map formalism.

Titel: Cheeger type inequalities associated with isocapacitary constants on graphs

Abstract: In this talk, we introduce Cheeger type constants via isocapacitary constants introduced by Maz'ya to estimate first Dirichlet, Neumann and Steklov eigenvalues on a finite subgraph of a graph. Moreover, we estimate the bottom of the spectrum of the Laplace operator and the Dirichlet-to-Neumann operator for an infinite subgraph. Estimates for higher-order Steklov eigenvalues on a finite or infinite subgraph are also discussed.

Titel: Construction of signed distance functions with an elliptic equation

Abstract: Driven by recent advancements in structural optimization problems, this paper introduces a new method for constructing the distance function to the boundaries of specific sets of interest, thereby simplifying the optimization process. We build upon Varadhan's celebrated elliptic equation theory from 1967 by incorporating a source term into the equation, which allows us to encode information about the set. Additionally, we will establish the convergence rate within this new framework, which is proven to be sharp, at least in the one-dimensional case. This is a collaborative effort with T. Hasebe, T. Oka, K. Sakai, and T. Yamada.

Titel: Effects of time-reversal invariance violation in quantum graphs.

Titel: Anti-Faber-Krahn property for the heat content on metric graphs (or “Building better batteries”)

Abstract: We speak about the heat content at time t on metric graphs. This quantity measures how much mass of a constant initial configuration remains in the graph at time t under the action of the heat semigroup. It has been known that its integral over all times satisfies a Faber-Krahn property: it is maximized by path graphs - the analogon of balls. 

We prove that surprisingly, the heat content radically contradicts the Faber-Krahn property at some times, but not for others: Intervals are no longer minimizers for small times, but are asymptotically optimal for large times. Consequently, when designing an optimal topology for a battery-like network under leakage, one must take the timescale into account. 

Our proofs rely on probabilistic arguments: the Feynman-Kac formula and results from the theory of discrete random walks. This is joint work with Delio Mugnolo and Patrizio Bifulco (both Hagen).

Titel: A stability phase transition for cosmological fluids

Abstract: Fluids as modelled by the Euler equations are known to form shocks in finite time from regular initial data, those are singularities in the first derivatives of the fluid variables.

These shocks correspond to a breakdown of the solution but do have relevant physical interpretations in various contexts. We are interested under which circumstances shocks can form from arbitrarily small inhomogeneities. In other words: When are homogeneous fluid solutions unstable? This question is relevant in the context of cosmology where it asks how structures can form from homogeneous matter distributions. In the context of the (relativistic) Euler equations this is a well-posed problem. In the cosmological context, the fluid is considered in an expanding space corresponding to the cosmological model. The expansion has a dissipative effect on the fluid, which resembles a friction term. This dissipative effect can tame the fluid in case the expansion is sufficiently fast.  On the other side, the speed of sound of the fluid, which enters the equations via the equation of state, increases the tendency of the fluid to form shocks. These two effects compete with each other. We investigate the two dimensional parameter space of expansion-speed and speed of sound and find a non-trivial critical curve where a phase transition between stable and unstable behaviour occurs by using classical methods from hyperbolic PDEs. This curve has rigorous  implications for matter dynamics in the early universe. 

Titel: The normal injectivity radius of an embedded hypersurface