Vorträge in der Woche 16.01.2023 bis 22.01.2023

Dienstag, 17.01.2023: Sheaves and categories

Lou-Jean Cobigo

Mittwoch, 18.01.2023: Crossing the transcendental divide: from translation surfaces to algebraic curves

Yelena Mandelshtam (MPI Leipzig)

A translation surface is obtained by identifying edges of polygons in the plane to create a compact Riemann surface equipped with a nonzero holomorphic one-form. Every Riemann surface can be given as an algebraic curve via its Jacobian variety. We aim to construct explicitly the underlying algebraic curves from their translation surfaces, given as polygons in the plane. The key tools in our approach are discrete Riemann surfaces, which allow us to approximate the Riemann matrices, and then, via theta functions, the equations of the curves. In this talk, I will present our algorithm and numerical experiments. From the newly found Riemann matrices and equations of curves, we can then make several conjectures about the curves underlying the Jenkins-Strebel representatives, a family of examples that until now, lived squarely on the analytic side of the transcendental divide between Riemann surfaces and algebraic curves.

Donnerstag, 19.01.2023: Hawking mass monotonicity for initial data sets

Sven Hirsch (Duke University)

An interesting feature of General Relativity is the presence of singularities which can happen in even the simplest examples such as the Schwarzschild spacetime. However, in this case the singularity is cloaked behind the event horizon of the black hole which has been conjectured to be generically the case. To analyze this so-called Cosmic Censorship Conjecture Penrose proposed in 1973 a test which involves Hawking's area theorem, the final state conjecture and a geometric inequality on initial data sets (M,g,k). For k=0 this Penrose inequality has been proven by Huisken-Ilmanen and by Bray using different methods, but in general the question is wide open. Huisken-Ilmanen's proof relies on the Hawking mass monotonicity formula under inverse mean curvature flow (IMCF), and the purpose of this talk is to generalize the Hawking mass monotonicity formula to initial data sets. For this purpose, we start with recalling spacetime harmonic functions and their applications which have been introduced together with Demetre Kazaras and Marcus Khuri in the context of the spacetime positive mass theorem.

Donnerstag, 19.01.2023: Wave scattering from nontrivial boundary conditions

Jörg Nick (Universität Tübingen)

The talk presents the numerical approximation of time-dependent acoustic and electromagnetic wave scattering problems in the presence of non-standard boundary conditions. Of particular interest is the numerical treatment of generalized impedance boundary conditions, effective models that approximate the wave-material interaction of partially penetrable obstacles. Classical applications of such boundary conditions are the scattering of highly absorbing materials and perfectly reflecting obstacles with a thin coating. Finally, a class of nonlinear boundary conditions is covered in the context of electromagnetic scattering. Formulated on the time domain, these boundary conditions contain surface differential operators and temporal convolution operators. The resulting boundary value problems on exterior domains are reformulated to retarded boundary integral equations, which are themselves nonlocal in time and space, but fully formulated on the boundary. To derive numerical methods, the boundary integral equations are discretized in time and space. The temporal discretization is carried out using the Runge--Kutta convolution quadrature method. Fully discrete schemes are derived by combining the time discretization with appropriate boundary element methods in space. Error bounds with specific convergence rates are presented for all boundary conditions. Numerical experiments illustrate the theoretical results.

Donnerstag, 19.01.2023: Boundary superconductivity in the BCS model

Dr. Barbara Roos (IST Austria)

We consider the linear BCS equation, determining the BCS critical temperature, in the presence of a boundary, where Dirichlet boundary conditions are imposed. In the one-dimensional case with point interactions, we prove that the critical temperature is strictly larger than the bulk value, at least at weak coupling. We also show that the relative shift in critical temperature vanishes if the coupling constant either goes to zero or to infinity.

Donnerstag, 19.01.2023: Unitary designs from spectral gaps

Dr. Nick Hunter-Jones (Stanford University)

Random quantum circuits (RQCs) are a solvable model of strongly-interacting quantum dynamics and have been the central focus of recent demonstrations of quantum computational advantage. This stems in part from their ability to rapidly generate quantum pseudorandomness and converge to approximate designs. In this talk we'll compute the depth at which RQCs form approximate unitary designs (probability distributions which 'look like' Haar random unitaries) using a statistical-mechanical mapping to lattice model as well as by bounding the spectral gap of a local Hamiltonian. We'll also make use of some improved spectral gap bounds to study more general random circuit architectures.