Oberwolfach Publications (Mathematisches Forschungsinst. Oberwolfach)
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2063 research outputs found
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Polyhedra and commensurability
No full textThis snapshot introduces the notion of commensurability of polyhedra. At its bottom, this concept can be developed from constructions with paper, scissors, and glue. Starting with an elementary example, we formalize it subsequently. Finally, we discuss intriguing connections with other fields of mathematics
Regularity and energy conservation for the compressible Euler equations
No full textResearch in Pairs 2016We give sufficient conditions on the regularity of solutions to the inhomogeneous incompressible Euler and the compressible isentropic Euler systems in order for the energy to be conserved. Our strategy relies on commutator estimates similar to those employed by P. Constantin et al. for the homogeneous incompressible Euler equations
Legendrian Lens Space Surgeries
No full textResearch in Pairs 2016We show that every tight contact structure on any of the lens spaces with , can be obtained by a single Legendrian surgery along a suitable Legendrian realisation of the negative torus knot in the tight or an overtwisted contact structure on the 3-sphere
Measured Group Theory
No full textThe workshop aimed to study discrete and Lie groups and their actions using measure theoretic methods and their asymptotic invariants, such as -invariants, the rank gradient, cost, torsion growth, entropy-type invariants and invariants coming from random walks and percolation theory. The participants came from a wide range of mathematics: asymptotic group theory, geometric group theory, ergodic theory, -theory, graph convergence, representation theory, probability theory, descriptive set theory and algebraic topology
Multiscale Interactions in Geophysical Fluids
No full textThe dynamics of the atmosphere and ocean involves a broad range of spatial and temporal scales, many of which emerge through complex nonlinear mechanisms from forcings at very different scales. This poses major challenges for the numerical prediction of the weather, ocean state and climate: many processes have scales that are too small to be resolved yet they play an essential role in determining large-scale features. This workshop examined how modern mathematical methods – ranging from multiscale asymptotics to adaptive numerical methods and stochastic modelling – can be applied to represent the large-scale impact of these small-scale processes and improve both deterministic and probabilistic predictions
The adaptive finite element method
No full textComputer simulations of many physical phenomena rely on approximations by models with a finite number of unknowns. The number of these parameters determines the computational effort needed for the simulation. On the other hand, a larger number of unknowns can improve the precision of the simulation. The adaptive finite element method (AFEM) is an algorithm for optimizing the choice of parameters so accurate simulation results can be obtained with as little computational effort as possible
Symmetry and characters of finite groups
No full textOver the last two centuries mathematicians have developed an elegant abstract framework to study the natural idea of symmetry. The aim of this snapshot is to gently guide the interested reader through these ideas. In particular, we introduce finite groups and their representations and try to indicate their central role in understanding symmetry
Totally Acyclic Complexes
No full textResearch in Pairs 2016We prove first (Proposition 3) that, over any ring , an acyclic complex of projective modules is totally acyclic if and only if the cycles of every acyclic complex of Gorenstein projective modules are Gorenstein projective. The dual result for injective and Gorenstein injective modules also holds over any ring (Proposition 4). And, when is a GF-closed ring, the analogue result for flat/Gorenstein flat modules is also true (Proposition 5). Then we show (Theorem 2) that over a left noetherian ring , a third equivalent condition can be added to those in Proposition 4, more precisely, we prove that the following are equivalent: 1. Every acyclic complex of injective modules is totally acyclic. 2. The cycles of every acyclic complex of Gorenstein injective modules are Gorenstein injective. 3. Every complex of Gorenstein injective modules is dg-Gorenstein injective. Theorem 3 shows that the analogue result for complexes of flat and Gorenstein flat modules holds over any left coherent ring . We prove (Corollary 1) that, over a commutative noetherian ring , the equivalent statements in Theorem 3 hold if and only if the ring is Gorenstein. We also prove (Theorem 4) that when moreover is left coherent and right -perfect (that is, every flat right -module has finite projective dimension ) then statements 1, 2, 3 in Theorem 2 are also equivalent to the following: 4. Every acyclic complex of projective right -modules is totally acyclic. 5. Every acyclic complex of Gorenstein projective right -modules is in . 6. Every complex of Gorenstein projective right -modules is dg-Gorenstein projective.
Corollary 2 shows that when is commutative noetherian of finite Krull dimension, the equivalent conditions (1)-(6) from Theorem 4 are also equivalent to those in Theorem 3 and hold if and only if is an Iwanaga-Gorenstein ring. Thus we improve slightly on a result of Iyengar's and Krause's; in [22] they proved that for a commutative noetherian ring with a dualizing complex, the class of acyclic complexes of injectives coincides with that of totally acyclic complexes of injectives if and only if is Gorenstein. We are able to remove the dualizing complex hypothesis and add more equivalent conditions. In the second part of the paper we focus on two sided noetherian rings that satisfy the Auslander condition. We prove (Theorem 7) that for such a ring that also has finite finitistic flat dimension, every complex of injective (left and respectively right) -modules is totally acyclic if and only if is an Iwanaga-Gorenstein ring
Mini-Workshop: Surreal Numbers, Surreal Analysis, Hahn Fields and Derivations
No full textNew striking analogies between H. Hahn’s fields of generalised series with real coefficients, G. H. Hardy’s field of germs of real valued functions, and J. H. Conway’s field No of surreal numbers, have been lately discovered and exploited. The aim of the workshop was to bring quickly together experts and young researchers, to articulate and investigate current key questions and conjectures regarding these fields, and to explore emerging applications of this recent discovery
Alexander r-Tuples and Bier Complexes
No full textResearch in Pairs 2016We introduce and study Alexander -Tuples of simplicial complexes, as a common generalization of pairs of Alexander dual complexes (Alexander 2-tuples) and r-unavoidable complexes of [BFZ-1]. In the same vein, the Bier complexes, defined as the deleted joins of Alexander -tuples, include both standard Bier spheres and optimal multiple chessboard complexes (Section 2.2) as interesting, special cases. Our main results are Theorem 4.3 saying that (1) the -fold deleted join of Alexander -tuple is a pure complex homotopy equivalent to a wedge of spheres, and (2) the -fold deleted join of a collective unavoidable -tuple is -connected, and a classification theorem (Theorem 5.1 and Corollary 5.2) for Alexander -tuples and Bier complexes