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Algebraische Zahlentheorie

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.

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Mini-Workshop: Einstein Metrics, Ricci Solitons and Ricci Flow under Symmetry Assumptions

2014, Lauret, Jorge, Wang, McKenzie

Symmetry reduction methods play an important role in the study of Einstein metrics, Ricci solitons and Ricci flow. The general aim of this mini workshop was to gather researchers who have expertise in the construction of geometric examples and to survey and discuss the singularity properties of homogeneous Ricci flows and the existence question for Ricci solitons, in light of the known rigidity results and general properties. Particular topics focused on were the Alekseevskii conjecture for noncompact homogeneous Einstein spaces, the homogeneous Ricci flow and shrinking solitons.

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Algebraic Geometry

2015, Huybrechts, Daniel, Kawamata, Yujiro, Siebert, Bernd

The workshop covered a broad variety of areas in algebraic geometry and was the occasion to report on recent advances and works in progress. Special emphasis was put on the role of derived categories and various stability concepts for sheaves, varieties, complexes, etc. The mix of people working in areas like classification theory, mirror symmetry, derived categories, moduli spaces, $p$-adic geometry, characteristic $p$ methods, singularity theory led to stimulating discussions.

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Mini-Workshop: Coideal Subalgebras of Quantum Groups

2015, Kolb, Stefan, Stokman, Jasper V.

Coideal subalgebras of quantized enveloping algebras appear naturally if one considers quantum group analogs of Lie subalgebras. Examples appear in the theory of quantum integrable systems with boundary and in harmonic analysis on quantum group analogs of Riemannian symmetric spaces. Recently, much progress has been made to develop a deeper representation theoretic understanding of these examples. On the other hand, coideal subalgebras play a fundamental role in the theory of Nichols algebras. The workshop aimed to discuss these theories in view of the recent developments.

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Mini-Workshop: Singularities in G2-geometry

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.

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Mini-Workshop: Singular Curves on K3 Surfaces and Hyperkähler Manifolds

2015, Knutsen, Andreas Leopold, Sarti, Alessandra

The workshop focused on Severi varieties on $K3$ surfaces, hyperkähler manifolds and their automorphisms. The main aim was to bring researchers in deformation theory of curves and singularities together with researchers studying hyperkähler manifolds for mutual learning and interaction, and to discuss recent developments and open problems.

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Mini-Workshop: Friezes

2015, Jorgensen, Peter, Morier-Genoud, Sophie

Frieze patterns were introduced in the early 1970s by Coxeter. They are infinite arrays of numbers in which every four neighbouring entries always satisfy the same arithmetic relation. Amazingly, friezes appear in many situations from various areas of mathematics: projective geometry, number theory, algebraic combinatorics, difference equations, integrable systems, representation theory, cluster algebras… The mini-workshop aimed to gather researchers with diverse fields of expertise to present recent developments and to discuss new directions of investigation and open problems around friezes.

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Mini-Workshop: Deformation Quantization: between formal to strict

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.

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Algebraic Structures in Low-Dimensional Topology

2014, Manturov, Vassily Olegovich, Orr, Kent E., Schneiderman, Robert

The workshop concentrated on important and interrelated invariants in low dimensional topology. This work involved virtual knot theory, knot theory, three and four dimensional manifolds and their properties.

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Mini-Workshop: Discrete p-Laplacians: Spectral Theory and Variational Methods in Mathematics and Computer Science

2015, Lenz, Daniel, Mugnolo, Delio

The p-Laplacian operators have a rich analytical theory and in the last few years they have also offered efficient tools to tackle several tasks in machine learning. During the workshop mathematicians and theoretical computer scientists working on models based on p-Laplacians on graphs and manifolds have presented the latest theoretical developments and have shared their knowledge.