Discrete Differential Geometry

This is the collection of extended abstracts for the 26 lectures and the open problem session at the fourth Oberwolfach workshop on Discrete Differential Geometry

A design methodology for portable software on parallel computers

This final report for research that was supported by grant number NAG-1-995 documents our progress in addressing two difficulties in parallel programming. The first difficulty is developing software that will execute quickly on a parallel computer. The second difficulty is transporting software between dissimilar parallel computers. In general, we expect that more hardware-specific information will be included in software designs for parallel computers than in designs for sequential computers. This inclusion is an instance of portability being sacrificed for high performance. New parallel computers are being introduced frequently. Trying to keep one's software on the current high performance hardware, a software developer almost continually faces yet another expensive software transportation. The problem of the proposed research is to create a design methodology that helps designers to more precisely control both portability and hardware-specific programming details. The proposed research emphasizes programming for scientific applications. We completed our study of the parallelizability of a subsystem of the NASA Earth Radiation Budget Experiment (ERBE) data processing system. This work is summarized in section two. A more detailed description is provided in Appendix A ('Programming Practices to Support Eventual Parallelism'). Mr. Chrisman, a graduate student, wrote and successfully defended a Ph.D. dissertation proposal which describes our research associated with the issues of software portability and high performance. The list of research tasks are specified in the proposal. The proposal 'A Design Methodology for Portable Software on Parallel Computers' is summarized in section three and is provided in its entirety in Appendix B. We are currently studying a proposed subsystem of the NASA Clouds and the Earth's Radiant Energy System (CERES) data processing system. This software is the proof-of-concept for the Ph.D. dissertation. We have implemented and measured the performance of a portion of this subsystem on the Intel iPSC/2 parallel computer. These results are provided in section four. Our future work is summarized in section five, our acknowledgements are stated in section six, and references for published papers associated with NAG-1-995 are provided in section seven

Ahlfors circle maps and total reality: from Riemann to Rohlin

This is a prejudiced survey on the Ahlfors (extremal) function and the weaker {\it circle maps} (Garabedian-Schiffer's translation of "Kreisabbildung"), i.e. those (branched) maps effecting the conformal representation upon the disc of a {\it compact bordered Riemann surface}. The theory in question has some well-known intersection with real algebraic geometry, especially Klein's ortho-symmetric curves via the paradigm of {\it total reality}. This leads to a gallery of pictures quite pleasant to visit of which we have attempted to trace the simplest representatives. This drifted us toward some electrodynamic motions along real circuits of dividing curves perhaps reminiscent of Kepler's planetary motions along ellipses. The ultimate origin of circle maps is of course to be traced back to Riemann's Thesis 1851 as well as his 1857 Nachlass. Apart from an abrupt claim by Teichm\"uller 1941 that everything is to be found in Klein (what we failed to assess on printed evidence), the pivotal contribution belongs to Ahlfors 1950 supplying an existence-proof of circle maps, as well as an analysis of an allied function-theoretic extremal problem. Works by Yamada 1978--2001, Gouma 1998 and Coppens 2011 suggest sharper degree controls than available in Ahlfors' era. Accordingly, our partisan belief is that much remains to be clarified regarding the foundation and optimal control of Ahlfors circle maps. The game of sharp estimation may look narrow-minded "Absch\"atzungsmathematik" alike, yet the philosophical outcome is as usual to contemplate how conformal and algebraic geometry are fighting together for the soul of Riemann surfaces. A second part explores the connection with Hilbert's 16th as envisioned by Rohlin 1978.Comment: 675 pages, 199 figures; extended version of the former text (v.1) by including now Rohlin's theory (v.2

Natural Communication

In Natural Communication, the author criticizes the current paradigm of specific goal orientation in the complexity sciences. His model of "natural communication" encapsulates modern theoretical concepts from mathematics and physics, in particular category theory and quantum theory. The author is convinced that only by looking to the past is it possible to establish continuity and coherence in the complexity science

The classification of punctured-torus groups

Thurston's ending lamination conjecture proposes that a finitely generated Kleinian group is uniquely determined (up to isometry) by the topology of its quotient and a list of invariants that describe the asymptotic geometry of its ends. We present a proof of this conjecture for punctured-torus groups. These are free two-generator Kleinian groups with parabolic commutator, which should be thought of as representations of the fundamental group of a punctured torus. As a consequence we verify the conjectural topological description of the deformation space of punctured-torus groups (including Bers' conjecture that the quasi-Fuchsian groups are dense in this space) and prove a rigidity theorem: two punctured-torus groups are quasi-conformally conjugate if and only if they are topologically conjugate.Comment: 67 pages, published versio

26. Theorietag Automaten und Formale Sprachen 23. Jahrestagung Logik in der Informatik: Tagungsband

Der Theorietag ist die Jahrestagung der Fachgruppe Automaten und Formale Sprachen der Gesellschaft für Informatik und fand erstmals 1991 in Magdeburg statt. Seit dem Jahr 1996 wird der Theorietag von einem eintägigen Workshop mit eingeladenen Vorträgen begleitet. Die Jahrestagung der Fachgruppe Logik in der Informatik der Gesellschaft für Informatik fand erstmals 1993 in Leipzig statt. Im Laufe beider Jahrestagungen finden auch die jährliche Fachgruppensitzungen statt. In diesem Jahr wird der Theorietag der Fachgruppe Automaten und Formale Sprachen erstmalig zusammen mit der Jahrestagung der Fachgruppe Logik in der Informatik abgehalten. Organisiert wurde die gemeinsame Veranstaltung von der Arbeitsgruppe Zuverlässige Systeme des Instituts für Informatik an der Christian-Albrechts-Universität Kiel vom 4. bis 7. Oktober im Tagungshotel Tannenfelde bei Neumünster. Während des Tre↵ens wird ein Workshop für alle Interessierten statt finden. In Tannenfelde werden • Christoph Löding (Aachen) • Tomás Masopust (Dresden) • Henning Schnoor (Kiel) • Nicole Schweikardt (Berlin) • Georg Zetzsche (Paris) eingeladene Vorträge zu ihrer aktuellen Arbeit halten. Darüber hinaus werden 26 Vorträge von Teilnehmern und Teilnehmerinnen gehalten, 17 auf dem Theorietag Automaten und formale Sprachen und neun auf der Jahrestagung Logik in der Informatik. Der vorliegende Band enthält Kurzfassungen aller Beiträge. Wir danken der Gesellschaft für Informatik, der Christian-Albrechts-Universität zu Kiel und dem Tagungshotel Tannenfelde für die Unterstützung dieses Theorietags. Ein besonderer Dank geht an das Organisationsteam: Maike Bradler, Philipp Sieweck, Joel Day. Kiel, Oktober 2016 Florin Manea, Dirk Nowotka und Thomas Wilk

Integrability and the AdS/CFT correspondence

The description of gauge theories at strong coupling is one of the long-standing problems in theoretical physics. The idea of a relation between strongly coupled gauge theories and string theory was pioneered by 't Hooft, Wilson and Polyakov. A decade ago, Maldacena made this relation explicit by conjecturing the exact equivalence of a conformally invariant theory in four dimensions, the maximally supersymmetric Yang-Mills theory, with string theory in the AdS5 x S5 background. Other examples of correspondence between a conformally invariant theory and string theory in an AdS background were discovered recently. The comparison of the two sides of the correspondence requires the use of non-perturbative methods. The discovery of integrable structures in gauge theory and string theory led to the conjecture that the two theories are integrable for any value of the coupling constant and that they share the same integrable structure defined non-perturbatively. The last eight years brought remarkable progress in identifying this solvable model and in explicitly solving the problem of computing the spectrum of conformal dimensions of the theory. The progress came from the identification of the dilatation operator with an integrable spin chain and from the study of the string sigma model. In this thesis, I present the evolution of the concept of integrability in the framework of the AdS/CFT correspondence and the the main results obtained using this approach.Comment: 106 pages, 9 figures, habilitation thesis; minor corrections, published versio

Hybrid analysis of memory references and its application to automatic parallelization

Executing sequential code in parallel on a multithreaded machine has been an elusive goal of the academic and industrial research communities for many years. It has recently become more important due to the widespread introduction of multicores in PCs. Automatic multithreading has not been achieved because classic, static compiler analysis was not powerful enough and program behavior was found to be, in many cases, input dependent. Speculative thread level parallelization was a welcome avenue for advancing parallelization coverage but its performance was not always optimal due to the sometimes unnecessary overhead of checking every dynamic memory reference. In this dissertation we introduce a novel analysis technique, Hybrid Analysis, which unifies static and dynamic memory reference techniques into a seamless compiler framework which extracts almost maximum available parallelism from scientific codes and incurs close to the minimum necessary run time overhead. We present how to extract maximum information from the quantities that could not be sufficiently analyzed through static compiler methods, and how to generate sufficient conditions which, when evaluated dynamically, can validate optimizations. Our techniques have been fully implemented in the Polaris compiler and resulted in whole program speedups on a large number of industry standard benchmark applications

Advances in Discrete Differential Geometry

Differential Geometr

Nonarchimedean Holographic Entropy from Networks of Perfect Tensors

We consider a class of holographic quantum error-correcting codes, built from perfect tensors in network configurations dual to Bruhat-Tits trees and their quotients by Schottky groups corresponding to BTZ black holes. The resulting holographic states can be constructed in the limit of infinite network size. We obtain a p-adic version of entropy which obeys a Ryu-Takayanagi like formula for bipartite entanglement of connected or disconnected regions, in both genus-zero and genus-one p-adic backgrounds, along with a Bekenstein-Hawking-type formula for black hole entropy. We prove entropy inequalities obeyed by such tensor networks, such as subadditivity, strong subadditivity, and monogamy of mutual information (which is always saturated). In addition, we construct infinite classes of perfect tensors directly from semiclassical states in phase spaces over finite fields, generalizing the CRSS algorithm, and give Hamiltonians exhibiting these as vacua