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We explore the C2-equivariant spectra T mf1(3) and T M F1(3). In particular, we compute their C2-equivariant Picard groups and the C2-equivariant Anderson dual of T mf1(3). This implies corresponding results for the fixed point spectra T M F0(3) and T mf0(3). Furthermore, we prove a Real Landweber exact functor theorem.

Abstract. Given an algebraic stack X, one may compare the derived category of quasi-coherent sheaves on X with the category of dg-modules over the dg-ring of functions on X. We study the analogous question in stable homotopy theory, for derived stacks that arise via realizations of diagrams of Landweber-exact homology theories. We identify a condition (quasi-affineness of the map to the moduli stack of formal groups) under which the two categories are equivalent, and study applications to topological modular forms. In particular, we provide new examples of Galois extensions of ring spectra and vanishing results for Tate spectra.

Interval-based synchronization provides the nodes of a distributed system with guaranteed bounds on a common time. This is a crucial piece of infrastructure in many distributed sensing and actuating systems. In this paper, we propose a modification to a known interval-based synchronization algorithm; our new algorithm obtains substantially better results in sensor-network scenarios by taking advantage of the typical drift diversity of the nodes' clocks.We propose a model for synchronization in ad-hoc, sporadiccommunication scenarios. The model allows us to identify the worst and the best case in terms of achievable time uncertainty and to show the worst-case optimality of the discussed algorithms. Simulations show that in the average case, our modification significantly reduces the time uncertainty.

we study the C 2 -spectra BPRhni and ER.n/ that refine the usual truncated Brown-Peterson and the Johnson-Wilson spectra. In particular, we show that they satisfy Gorenstein duality with a representation grading shift and identify their Anderson duals. We also compute the associated local cohomology spectral sequence in the cases n D 1 and 2.

Clock synchronization is a crucial basic service in typical sensor networks, since the observations of distributed sensors more often than not need to be ordered ("a happened before b") or otherwise related ("a and b happened within a time window of size x") in time. Ad-hoc networks may exhibit characteristics which make the use of traditional clock-synchronization algorithms infeasible. Recently, algorithms suitable for ad-hoc networks have been presented.We first propose an improvement to an existing algorithm. While needing less computation and no more communication or memory than the original algorithm, our new algorithm always yields equal or better results and thus outperforms the original algorithm. We then examine how even better synchronization can be obtained, possibly at the cost of additional computation, communication, and memory. To this end, we introduce a model for internal synchronization. This model allows us to find an algorithm which makes use of all the data a node can obtain from the network for a given communication pattern and thus provides optimal synchronization in our model.

We study rings of integral modular forms for congruence subgroups as modules over the ring of integral modular forms for SL2Z. In many cases these modules are free or decompose at least into well-understood pieces. We apply this to characterize which rings of modular forms are Cohen-Macaulay and to prove finite generation results. These theorems are based on decomposition results about vector bundles on the compactified moduli stack of elliptic curves.

Abstract. We study vector bundles on the moduli stack of elliptic curves over a local ring R. If R is a field or a discrete valuation ring of (residue) characteristic not 2 or 3, all these vector bundles are sums of line bundles. For R = Z (3) , we construct higher rank indecomposable vector bundles and give a classification of vector bundles that are iterated extensions of line bundles. If R = Z (2) , we show that there are even indecomposable vector bundles of arbitrary high rank.