arXiv subjects

Felix Günther

Publications and source records attributed to Felix Günther.

13 recordsLinked to original sources

arXiv2025

Real Riemann Surfaces: Smooth and Discrete

This paper develops a discrete theory of real Riemann surfaces based on quadrilateral cellular decompositions (quad-graphs) and a linear discretization of the Cauchy-Riemann equations. We construct a discrete analogue of an antiholomorphic involution and classify the topological types of discrete real Riemann surfaces, recovering the classical results on the number of real ovals and the separation of the surface. Central to our approach is the construction of a symplectic homology basis adapted to the discrete involution. Using this basis, we prove that the discrete period matrix admits the same canonical decomposition $Π= \frac{1}{2} H + i T$ as in the smooth setting, where $H$ encodes the topological type and $T$ is purely imaginary. This structural result bridges the gap between combinatorial models and the classical theory of real algebraic curves.

math.CV↗

arXiv2023

The convergence of discrete period matrices

We study compact polyhedral surfaces as Riemann surfaces and their discrete counterparts obtained through quadrilateral cellular decompositions and a linear discretization of the Cauchy-Riemann equation. By ensuring uniformly bounded interior and intersection angles of diagonals, we establish the convergence of discrete Dirichlet energies of discrete harmonic differentials with equal black and white periods to the Dirichlet energy of the corresponding continuous harmonic differential with the same periods. This convergence also extends to the discrete period matrix, with a description of the blocks of the complete discrete period matrix in the limit. Moreover, when the quadrilaterals have orthogonal diagonals, we observe convergence of discrete Abelian integrals of the first kind. Adapting the quadrangulations around conical singularities allows us to improve the convergence rate to a linear function of the maximum edge length.

math.CV↗

arXiv2022

Nowcasting with leading indicators applied to COVID-19 fatalities in Sweden

The real-time analysis of infectious disease surveillance data, e.g., in the form of a time-series of reported cases or fatalities, is essential in obtaining situational awareness about the current dynamics of an adverse health event such as the COVID-19 pandemic. This real-time analysis is complicated by reporting delays that lead to underreporting of the number of events for the most recent time points (e.g., days or weeks). This can lead to misconceptions by the interpreter, e.g., the media or the public, as was the case with the time-series of reported fatalities during the COVID-19 pandemic in Sweden. Nowcasting methods provide real-time estimates of the complete number of events using the incomplete time-series of currently reported events by using information about the reporting delays from the past. Here, we consider nowcasting the number of COVID-19-related fatalities in Sweden. We propose a flexible Bayesian approach, extending existing nowcasting methods by incorporating regression components to accommodate additional information provided by leading indicators such as time-series of the number of reported cases and ICU admissions. By a retrospective evaluation, we show that the inclusion of ICU admissions as a leading signal improved the nowcasting performance of case fatalities for COVID-19 in Sweden compared to existing methods.

stat.ME↗

arXiv2019

The Gauss map of polyhedral vertex stars

In discrete differential geometry, it is widely believed that the discrete Gaussian curvature of a polyhedral vertex star equals the algebraic area of its Gauss image. However, no complete proof has yet been described. We present an elementary proof in which we compare, for a particular normal vector, its winding numbers around the Gauss image and its antipode with its critical point index. This index is closely related to the normal degree of the Gauss map. We deduce how the number of inflection faces is related to the numbers of positive and negative components in the Gauss image. The resulting formula significantly limits the possible shapes of the Gauss image of a polyhedral vertex star. For example, if a triangulated vertex star is in general position and its Gauss image has no self-intersections, then it is either a convex spherical polygon if the curvature is positive, or a spherical pseudo-quadrilateral if the curvature is negative.

math.DG↗

arXiv2017

The Status of Quantum-Based Long-Term Secure Communication over the Internet

Sensitive digital data, such as health information or governmental archives, are often stored for decades or centuries. The processing of such data calls for long-term security. Secure channels on the Internet require robust key establishment methods. Currently used key distribution protocols are either vulnerable to future attacks based on Shor's algorithm, or vulnerable in principle due to their reliance on computational problems. Quantum-based key distribution protocols are information-theoretically secure and offer long-term security. However, significant obstacles to their real-world use remain. This paper, which results from a multidisciplinary project involving computer scientists and physicists, systematizes knowledge about obstacles to and strategies for the realization of long-term secure Internet communication from quantum-based key distribution. We discuss performance and security particulars, consider the specific challenges arising from multi-user network settings, and identify key challenges for actual deployment.

cs.CR↗

arXiv2017

Smooth polyhedral surfaces

Polyhedral surfaces are fundamental objects in architectural geometry and industrial design. Whereas closeness of a given mesh to a smooth reference surface and its suitability for numerical simulations were already studied extensively, the aim of our work is to find and to discuss suitable assessments of smoothness of polyhedral surfaces that only take the geometry of the polyhedral surface itself into account. Motivated by analogies to classical differential geometry, we propose a theory of smoothness of polyhedral surfaces including suitable notions of normal vectors, tangent planes, asymptotic directions, and parabolic curves that are invariant under projective transformations. It is remarkable that seemingly mild conditions significantly limit the shapes of faces of a smooth polyhedral surface. Besides being of theoretical interest, we believe that smoothness of polyhedral surfaces is of interest in the architectural context, where vertices and edges of polyhedral surfaces are highly visible.

math.MG↗

arXiv2017

Discrete Riemann surfaces based on quadrilateral cellular decompositions

Our aim in this paper is to provide a theory of discrete Riemann surfaces based on quadrilateral cellular decompositions of Riemann surfaces together with their complex structure encoded by complex weights. Previous work, in particular of Mercat, mainly focused on real weights corresponding to quadrilateral cells having orthogonal diagonals. We discuss discrete coverings, discrete exterior calculus, and discrete Abelian integrals. Our presentation includes several new notions and results such as branched coverings of discrete Riemann surfaces, the discrete Riemann-Hurwitz Formula, double poles of discrete one-forms and double values of discrete meromorphic functions that enter the discrete Riemann-Roch Theorem, and a discrete Abel-Jacobi map.

math.CV↗

arXiv2015

Discrete complex analysis on planar quad-graphs

We develop a linear theory of discrete complex analysis on general quad-graphs, continuing and extending previous work of Duffin, Mercat, Kenyon, Chelkak and Smirnov on discrete complex analysis on rhombic quad-graphs. Our approach based on the medial graph yields more instructive proofs of discrete analogs of several classical theorems and even new results. We provide discrete counterparts of fundamental concepts in complex analysis such as holomorphic functions, derivatives, the Laplacian, and exterior calculus. Also, we discuss discrete versions of important basic theorems such as Green's identities and Cauchy's integral formulae. For the first time, we discretize Green's first identity and Cauchy's integral formula for the derivative of a holomorphic function. In this paper, we focus on planar quad-graphs, but we would like to mention that many notions and theorems can be adapted to discrete Riemann surfaces in a straightforward way. In the case of planar parallelogram-graphs with bounded interior angles and bounded ratio of side lengths, we construct a discrete Green's function and discrete Cauchy's kernels with asymptotics comparable to the smooth case. Further restricting to the integer lattice of a two-dimensional skew coordinate system yields appropriate discrete Cauchy's integral formulae for higher order derivatives.

math.CV↗

arXiv2015

On discrete integrable equations with convex variational principles

We investigate the variational structure of discrete Laplace-type equations that are motivated by discrete integrable quad-equations. In particular, we explain why the reality conditions we consider should be all that are reasonable, and we derive sufficient conditions (that are often necessary) on the labeling of the edges under which the corresponding generalized discrete action functional is convex. Convexity is an essential tool to discuss existence and uniqueness of solutions to Dirichlet boundary value problems. Furthermore, we study which combinatorial data allow convex action functionals of discrete Laplace-type equations that are actually induced by discrete integrable quad-equations, and we present how the equations and functionals corresponding to (Q3) are related to circle patterns.

nlin.SI↗

arXiv2015

The maximum number of intersections of two polygons

We investigate the maximum number of intersections between two polygons with p and q vertices, respectively, in the plane. The cases where p or q is even or the polygons do not have to be simple are quite easy and already known, but when p and q are both odd and both polygons are simple, the problem is more difficult. The conjectured maximum is (p-1)(q-1)+2 for all odd p and q.

math.CO↗

arXiv2015

The local geometry of compact homogeneous Lorentz spaces

In 1995, S. Adams and G. Stuck as well as A. Zeghib independently provided a classification of non-compact Lie groups which can act isometrically and locally effectively on compact Lorentzian manifolds. In the case that the corresponding Lie algebra contains a direct summand isomorphic to the two-dimensional special linear algebra or to a twisted Heisenberg algebra, Zeghib also described the geometric structure of the manifolds. Using these results, we investigate the local geometry of compact homogeneous Lorentz spaces whose isometry groups have non-compact connected components. It turns out that they all are reductive. We investigate the isotropy representation and curvatures. In particular, we obtain that any Ricci-flat compact homogeneous Lorentz space is flat or has compact isometry group.

math.DG↗

arXiv2013

On a game on graphs

We start with the well-known game below: Two players hold a sheet of paper to their forehead on which a positive integer is written. The numbers are consecutive and each player can only see the number of the other one. In each time step, they either say nothing or tell what number they have. Both of them will eventually figure out their number after a certain amount of time. The game is rather cooperative than competitive, and employs the notions of common knowledge and mutual knowledge. We generalize this game to arbitrary (directed and non-directed) simple graphs and try to establish for which graphs one or both of them will figure out the solution, and how long they do need to find it. We give a complete answer for the case of two players, even if they are both allowed to discuss before the start of the game.

math.CO↗

arXiv2011

Isometry groups of Lorentzian manifolds of finite volume and The local geometry of compact homogeneous Lorentz spaces

Based on the work of Adams and Stuck as well as on the work of Zeghib, we classify the Lie groups which can act isometrically and locally effectively on Lorentzian manifolds of finite volume. In the case that the corresponding Lie algebra contains a direct summand isomorphic to the two-dimensional special linear algebra or to a twisted Heisenberg algebra, we also describe the geometric structure of the manifolds if they are compact. Using these results, we investigate the local geometry of compact homogeneous Lorentz spaces whose isometry groups have non-compact connected components. It turns out that they all are reductive. We investigate the isotropy representation, curvatures and holonomy. Especially, we obtain that any Ricci-flat compact homogeneous Lorentz space is flat or has compact isometry group.

math.DG↗