The Conference on 'Mathematical Technology of Networks-QGraphs 2013' was held in December 2013. Interdisciplinary in nature, the meeting was attended mostly by mathematicians and physicists, but also by engineers and neuroscientists. This blend of scientific backgrounds ensured vibrant discussions; a broad and dynamical research landscape emerged.
The meeting was devoted to the study of a certain class of structures, called 'graphs' or 'networks': They arise as idealisations of thin pipelines that appear in different fields (like interconnected neurons or road networks) or as abstract, purely mathematical schemes that encode binary relations of proximity between points (e.g., to describe societies or computer grids).
The mathematical analysis of networks is a highly topical subject in both pure and applied sciences. Networks and related discrete structures arise e.g. in neuroscience, genetics, ecology, medical mathematics, as well as in many social sciences. In most of these cases the usage of a network formalism is currently limited to a mere description of a system's statistical features-e.g., in order to determine emergence of hubs or to quantify connectivity of relevant sub-networks. More and more researchers, though, have turned in the last few decades to more sophisticated investigations that put further mathematical objects on the stage of networks. The Bielefeld conference was mainly devoted to the latter approach, which may be dubbed 'analytical network theory', by which the properties of such discrete systems are described by difference or differential equations.
After four days and roughly 50 talks, one can attempt to draw up a balance. Current research in the field of analytical network theory can be apparently clustered in four main directions:
All of the above fields have been consistently represented at this conference. The first two topics marked the birth of this theory, in the early 1980s (in mathematics) and later again, independently, in the late 1990s (in theoretical physics). Interest in them is still broad, as networks have proved in the last decades an efficient source of models: Unlike other mathematical structures "so-called manifolds" that still represent the most common setting of differential equations, networks are much easier to deal with but are still capable to capture many non-trivial behaviours: These can then be studied and analysed in a friendlier environment.
The study of periodicity is currently rather popular in view of its applications to solid state physics, and in particular to graphene technology, for which an accurate description of spectral properties of molecules is indispensable. Weaker, periodicity-like properties are enjoyed by many systems, in particular by nonlinear ones in the neuroscience, but the theory is much less mature.
More generally, the study of nonlinear equations on networks is still in an early stage of development (with the possible exception of road traffic analysis) but seems to be gaining momentum both in mathematical analysis and in theoretical physics.
Ricardo Adami (Turin, ITA), Martin Johannes Adler (Tübingen, GER), Mehrdad Agha Mohammad Ali Kermani (Teheran, IRI), Victor Aleruchi (Rumumodomaya, NGR), Felix Ali Mehmeti (Valencienne, FRA), Patricia Alonso Ruiz (Ulm, GER), Omid Askari Sichani (Teheran, IRI), Fatihcan Atay (Leipzig, GER), Ram Band (Haifa, ISR), Fatima Zahra Benzahra Belkacem (Oran, ALG), Jens Bolte (Egham, GBR), José Luis Casadiego Bastidas (Göttingen, GER), Gianfausto Dell Antonio (Triest, ITA), Semra Demirel-Frank (Pasadena, USA), Nikita Evseev (Novosibirsk, RUS), Pavel Exner (Rez, CZE), Christian Himpe (Münster, GER), Muhammad Imran (Islamabd, PAK), Mahdi Jalili (Teheran, IRI), Matthias Keller (Jena, GER), James Kennedy (Stuttgart, GER), Evgeny Korotyaev (St. Petersburg, RUS), Pavel Kurasov (Stockholm, SWE), Anton Kutsenko (Talence, FRA), Fereshteh Lagzi (Freiburg i.Br., GER), Jirí Lipovský (Hradec Králové, CZE), Annemarie Luger (Stockholm, SWE), Benedict J. Lünsmann (Göttingen, GER), Gabriela Malenová (Ulm, GER), Claudio Marchi (Padua, ITA), Diego Noja (Mailand, ITA), Svetlana Pastukhova (Moskau, RUS), Shariefuddin Pirzada (Srinagar, IND), Matteo Polettini (Luxemburg, LUX), Olaf Post (Durham, GBR), Stefan Rotter (Freiburg i.Br., GER), Ralf Rückriemen (Delft, NED), Natalia Saburova (Arkhangelsk, RUS), Khidir R. Sharaf (Zakho, IRQ), Uzy Smilansky (Rehovot, ISR), Rune Suhr (Stockholm, SWE), Dimitri Volchenkov (Bielefeld, GER), Joachim von Below (Calais, FRA), Hans-Arwed Weidenmüller (Heidelberg, GER), Wolfgang Woess (Graz, AUT), Wu Ye (Peking, CHN)