2012, Carstensen, Carsten, Schröder, Jörg, Wriggers, Peter

The finite element method is the established simulation tool for the numerical solution of partial differential equations in many engineering problems with many mathematical developments such as mixed finite element methods (FEMs) and other nonstandard FEMs like least-squares, nonconforming, and discontinuous Galerkin (dG) FEMs. Various aspects on this plus related topics ranging from order-reduction methods to isogeometric analysis has been discussed amongst the pariticpants form mathematics and engineering for a large range of applications.

2013, Montgomery, Hugh L., Vaughan, Robert C., Wooley, Trevor D.

Analytic number theory has florished over the past few years, and this workshop brought together world leaders and young talent to discuss developments in various branches of the subject.

2012, Szymanski, Wojciech, Zacharias, Joachim

The main aim of the workshop was to explore recent progress in the study of endomorphisms of $C*$-algebras, semigroup crossed products, graph algebras, ring $C*$-algebras, purely infinite $C*$-algebras and related algebraic constructions, such as dilations or Leavitt path algebras, by bringing together experts from several different fields.

2012, Helminck, Aloysius

This workshop brought together experts from the areas of algebraic Lie theory, invariant theory, Kac–Moody theory and the theories of Tits buildings and of symmetric spaces. The main focus was on topics related to symmetric spaces in order to stimulate progress in current research projects or trigger new collaboration via comparison, analogy, transfer, generalization, and unification of methods. Specific topics that were covered include Kac–Moody symmetric spaces, double coset decompositions of (groups of rational points of) algebraic groups and Kac–Moody groups, and symmetric/Gelfand pairs in Lie algebras.

2013, Huber-Klawitter, Annette, Jannsen, Uwe, Levine, Marc

Algebraic K-theory and motivic cohomology are strongly related tools providing a systematic way of producing invariants for algebraic or geometric structures. The definition and methods are taken from algebraic topology, but there have been particularly fruitful applications to problems of algebraic geometry, number theory or quadratic forms. 19 one-hour talks presented a wide range of latest results on the theory and its applications.

2013, Jantzen, Jens Carsten, Reichstein, Zinovy

Linear algebraic groups is an active research area in contemporary mathematics. It has rich connections to algebraic geometry, representation theory, algebraic combinatorics, number theory, algebraic topology, and differential equations. The foundations of this theory were laid by A. Borel, C. Chevalley, T. A. Springer and J. Tits in the second half of the 20th century. The Oberwolfach workshops on algebraic groups, led by Springer and Tits, played an important role in this effort as a forum for researchers, meeting at approximately 3 year intervals since the 1960s. The present workshop continued this tradition, featuring a number of important recent developments in the subject.

2015, Esposito, Chiara, Nest, Ryszard, Waldmann, Stefan

The philosophy of deformation was proposed by Bayen, Flato, Fronsdal, Lichnerowicz, and Sternheimer in the seventies and since then, many developments occurred. Deformation quantization is based on such a philosophy in order to provide a mathematical procedure to pass from classical mechanics to quantum mechanics. Basically, it consists in deforming the pointwise product of functions to get a non-commutative one, which encodes the quantum mechanics behaviour. In formal deformation quantization, the non-commutative product (also said, star product) is given by a formal deformation of the pointwise product, i.e. by a formal power series in the deformation parameter which physically play the role of Planck’s constant $\hbar$. From a physical point of view this is clearly not sufficient to provide a reasonable quantum mechanical description and hence one needs to overcome the formal power series aspects in some way. One option is strict deformation quantization, which produces quantum algebras not just in the space of formal power series but in terms of $C$*-algebras, as suggested by Rieffel, with e.g. a continuous dependence on $\hbar$. There are several other options in between continuous and formal dependence on $\hbar$ like analytic or smooth deformations. The Oberwolfach workshop Deformation quantization: between formal to strict consolidated, continued, and extended these research activities with a focus on the study of the connection between formal and strict deformation quantization in their various flavours and their applications in particular those in quantum physics and non-commutative geometry. It brought together specialists in differential geometry, operator algebras, non-commutative geometry, and quantum field theory with research interests in the mentioned quantization procedures. The aim of the workshop was to develop a coherent viewpoint of the many recent diverse developments in the field and to initiate new lines of research.

2014, Kings, Guido, Sujatha, Ramdorai, Venjakob, Otmar

The workshop brought together leading experts in Algebraic Number Theory. The talks presented new methods and results that intertwine a multitude of topics ranging from classical diophantine themes to modern arithmetic geometry, modular forms and p-adic aspects in number theory.

2015, Haskins, Mark, Weiss, Hartmut

All currently known construction methods of smooth compact $\mathrm G_2$-manifolds have been tied to certain singular $\mathrm G_2$-spaces, which in Joyce’s original construction are $\mathrm G_2$-orbifolds and in Kovalev’s twisted connected sum construction are complete G2-manifolds with cylindrical ends. By a slight abuse of terminology we also refer to the latter as singular $\mathrm G_2$-spaces, and in fact both construction methods may be viewed as desingularization procedures. In turn, singular $\mathrm G_2$-spaces comprise a (conjecturally large) part of the boundary of the moduli space of smooth compact $\mathrm G_2$-manifolds, and so their deformation theory is of considerable interest. Furthermore, singular $\mathrm G_2$-spaces are also important in theoretical physics. Namely, in order to have realistic low-energy physics in M-theory, one needs compact singular $\mathrm G_2$-spaces with both codimension 4 and 7 singularities according to Acharya and Witten. However, the existence of such singular $\mathrm G_2$-spaces is unknown at present. The aim of this workshop was to bring reserachers from special holonomy geometry, geometric analysis and theoretical physics together to exchange ideas on these questions.

2012, Suciu, Alexander I.

The purpose of this workshop was to bring together researchers with a common interest in the objects mentioned in the title from, respectively, the points of view of toric and tropical geometry, arrangement theory, and geometric group theory.