In this talk we will address the potential dependence of the density of states and related spectral functions for discrete Schr\"odinger operators on infinite graphs. Following ideas by J. Bourgain and A. Klein, we will consider the density of states {\em{outer}} measure (DOSoM), a {\em{deterministic}} quantity, which is well defined for {\em{all}} Schr\"odinger operators. We will explicitly quantify the potential dependence of the DOSoM in weak topology by proving a modulus of continuity with respect to the potential in $\ell{l}^\infinity$-norm. The resulting modulus of continuity reflects the geometry of the graph at infinity. For the special case of operators on $\mathbb{Z}^d$ our result implies Lipschitz continuity of the DOSoM, in the case of the Bethe lattice, we obtain that the DOSoM is $\frac{1}{2}$-log-H\"older continuous. Applications of this result to ergodic, and in further consequence, for random Schr\"odinger operators will be presented. This talk is based on joint work with Peter Hislop (University of Kentucky).

Semiclassical gravity is the theory in which gravity is treated classically and matter is treated in the framework of quantum field theory. The spacetime metric tensor, which encodes gravitational effects in its curvature, interacts with matter through the semiclassical Einstein equations: matter sources the dynamics of the spacetime metric via the expectation value of the stress-energy tensor of the quantum fields, while the quantum fields propagate in curved spacetime. It is currently unknown whether semiclassical gravity has a well-posed initial value formulation even for free fields. In this talk, I will discuss the situation in static spacetimes, where time-translation symmetry greatly reduces the difficulty of the problem, and where one can show well-posedness. I will also discuss the main ideas on how to generalise these results to non-static spacetimes under some special circumstances. Based on arXiv:2011.05947 and on some unpublished work in progress with S. Modak.

Quantum graphs surprised researchers by their extraordinary spectral properties and possibility to carry out explicit analysis. One such example is given by the trace formula connecting spectra of quantum graphs to the set of periodic orbits on the underlying metric graph (without any additional correction terms). It appears that the corresponding spectral measure provides an explicit example of so-called crystalline measures generalising classical Dirac comb and Poisson summation formula.

Crystalline measures are tempered distributions given by locally finite purely atomic measures whose Fourier transform is also a purely atomic measure. Such measures were studied by J.-P. Kahane, A.-P. Guinand, and S. Mandelbrojt in the fifties. It remained unclear whether Dirac combs provide the only type of examples of crystalline measures with uniformly discrete support. We are going to show how to construct a wide family of crystalline measures using quantum graphs (differential operators on metric graphs) and more generally via stable polynomials. The measure we obtain are:

• positive crystalline measures with uniformly discrete support;
• Fourier quasicrystals for which every arithmetic progression meets the support in a finite set;
• Fourier quasicrystals for which the support is a Delone set, but the support of the Fourier transform not.

Our results complement recent studies by Y. Meyer, N. Lev, A. Olevskii, and others. This is a joint work with Peter Sarnak.

We analyze the spectrum of the (scaled) adjacency matrix A of the Erdős- Rényi graph G(N, d/N ) in the critical regime d = b log N. We establish a one-to-one correspondence between vertices of degree at least 2d and nontrivial eigenvalues outside the asymptotic bulk [−2, 2]. This correspondence implies a transition at an explicit b*. For d>b* log N the spectrum is just the bulk [−2, 2] and the eigenvectors are completely delocalized. For d< b* log N another phase appears. The spectrum outside [−2, 2] is not empty and there the eigenvectors concentrate around the large degree vertices.(joint work with Antti Knowles and Johannes Alt)

This talk concerns a naive model of bicycle motion: a bicycle is a segment of fixed length that can move so that the velocity of the rear end is always aligned with the segment. Surprisingly, this simple model is quite rich and has connections with several areas of research, including completely integrable systems. Here is a sampler of problems that I hope to touch upon: 1) The trajectory of the front wheel and the initial position of the bicycle uniquely determine its motion and its terminal position; the monodromy map sending the initial position to the terminal one arises. This mapping is a Moebius transformation, a remarkable fact that has various geometrical and dynamical consequences. 2) The rear wheel track and a choice of the direction of motion uniquely determine the front wheel track; changing the direction to the opposite, yields another front track. These two front tracks are related by the bicycle (Backlund, Darboux) correspondence, which defines a discrete time dynamical system on the space of curves. This system is completely integrable and it is closely related with another, well studied, completely integrable dynamical system, the filament (a.k.a binormal, smoke ring, local induction) equation. 3) Given the rear and front tracks of a bicycle, can one tell which way the bicycle went? Usually, one can, but sometimes one cannot. The description of these ambiguous tire tracks is an open problem, intimately related with Ulam's problem in flotation theory (in dimension two): is the round ball the only body that floats in equilibrium in all positions? This problem is also related to the motion of a charge in a magnetic field of a special kind. It turns out that the known solutions are solitons of the planar version of the filament equation.