While the properties of families of abelian varieties with respect to the Newton polygon stratification in characteristic p > 0 are generally well understood, relatively little is known about families of smooth curves of genus g as soon as g > 3. Supersingular curves are those with the most unusual Newton polygon. Their existence for g = 4 and any prime p > 0 was only recently established by Kudo-Harashita-Senda and independently by Pries. In this talk, we present some results on the dimensions of the loci of supersingular curves of genus g = 4, as well as on their automorphism groups. As an outcome, our results confirm Oort's conjecture about the generic automorphism group of the supersingular locus of principally polarized abelian varieties for g = 4 and p > 2.

Speaker: Nathanael Berestycki (U Vienna)

Title: On the spectral geometry of Liouville quantum gravity (3^rd part)

Abstract:

In these talks we will discuss the spectral geometry of the Laplace-Beltrami operator associated to Liouville quantum gravity. Over the course of two lectures, our goals will be to:

- Explain how eigenvalues and eigenfunctions for LQG are defined;

- Show that the eigenvalues a.s. obey a Weyl law (joint work with Mo Dick Wong). This is closely related to the short time asymptotics of the LQG heat kernel;

- Discuss the second term in the Weyl law and its relation to the KPZ (Knizhnik-Polyakov-Zamolodchikov) scaling relation;

- Finally we will talk about some conjectures which suggest a rather beautiful connection to a phenomenon called quantum chaos.

Time: 11-12.45

Afternoon

Speaker: Dirk Schuricht (UU)

Title: Majoranas, parafermions, and what one can do with them
Abstract:
In the first part we study the effect of interactions on Kitaev's toy model for Majorana wires [1]. To this end we map the model onto the axial next-nearest neighbour Ising chain and discuss the link between spinless fermions and Majoranas. We demonstrate that even though strong repulsive interaction eventually drive the system into a Mott insulating state the competition between the (trivial) band-insulator and the (trivial) Mott insulator leads to an interjacent topological insulating state for arbitrary strong interactions. We show that the exact ground states can be obtained analytically even in the presence of interactions when the chemical potential is tuned to a particular function of the other parameters [2]. Finally, we investigate the effect of disorder [3].

In the second part we generalise our analysis to parafermions and clock variables, with the Jordan-Wigner transformation being replaced by the so-called Fradkin-Kadanoff one. The resulting parafermion chain is shown to be equivalent to the non-chiral Z3 axial next-nearest neighbour Potts model. The phase diagram contains several gapped phases, including a topological phase where the system possesses three (nearly) degenerate ground states, and a gapless Luttinger-liquid phase [4]. We also extent Witten’s conjugation argument [5] to spin chains and use it to construct various frustration-free models [6]. If time permits, we may also briefly discuss Fock parafermions [7], which generalise spinless fermions to Z3 symmetry.  

References:  
[1] F Hassler and D Schuricht, New J. Phys. 14, 125018 (2012)
[2] H Katsura, D Schuricht and M Takahashi, Phys. Rev. B 92, 115137 (2015)
[3] N M Gergs, L Fritz and D Schuricht, Phys. Rev. B 93, 075129 (2016)
[4] J Wouters, F Hassler, H Katsura and D Schuricht, SciPost Phys. Core 5, 008 (2022)
[5] E. Witten, Nucl. Phys. B 202, 253 (1982)
[6] J Wouters, H Katsura and D Schuricht, SciPost Phys. Core 4, 027 (2021)
[7] E Cobanera and G Ortiz, Phys. Rev. A 89, 012328 (2014)

Time: 14.15-16.00

In the first lecture I will explain the black hole stability problem in classical general relativity and some of the recent results on it — these involve a fascinating combination of geometry and the analysis of partial differential equations. I will also give at least some indication of some of the tools that went into proving this. In the second lecture, I will discuss in more detail the analytic and geometric tools that lead to the understanding of black hole stability, especially with a positive cosmological constant (Kerr-de Sitter spacetimes), though also mentioning aspects of the vanishing cosmological constant case (Kerr). The third lecture will discuss the stability of the expanding (cosmological) region of Kerr-de Sitter spacetimes. I will also discuss the smoothness of the metric up to the future conformal boundary, with a Fefferman–Graham type asymptotic expansion, which is already of interest in de Sitter spaces, for which the global stability theorem of Friedrich in the 1980s was the first general stability result! This is based on joint works with Dietrich Häfner, Peter Hintz and Oliver Petersen.

The late-time behavior of solutions to the wave equation on Kerr spacetime is governed by inverse polynomial decay. However, at earlier time-scales, numerical simulations are found to be dominated by quasinormal modes (QNMs). These are exponentially damped oscillatory solutions with complex frequencies characteristic of the system. In this talk, I will present a rigorous characterization of QNMs for the scalar wave equation on Kerr. They are obtained as the discrete set of poles of the meromorphically continued cutoff resolvent. The construction combines the method of complex scaling near asymptotically flat infinity with microlocal methods near the black hole horizon. I will also discuss the distribution of QNMs in both the high and low energy regimes. In particular, I will present uniform low energy resolvent estimates, which exclude the accumulation of QNMs at zero energy.

Given the sharp logarithmic decay of linear waves on the Kerr-AdS black hole (Holzegel, Smulevici '13), it is expected that the Kerr-AdS spacetime is unstable as a solution of the Einstein vacuum equations. However, the scattering construction presented here for exponentially decaying nonlinear waves on a fixed Kerr-AdS background serves as a first step to confronting the scattering problem for the full Einstein system. In this context, one may hope to derive a class of perturbations of Kerr-AdS which remain 'close' and dissipate sufficiently fast.