Poset pinball, GKM-compatible subspaces, and Hessenberg varieties

This paper has three main goals. First, we set up a general framework to address the problem of constructing module bases for the equivariant cohomology of certain subspaces of GKM spaces. To this end we introduce the notion of a GKM-compatible subspace of an ambient GKM space. We also discuss poset-upper-triangularity, a key combinatorial notion in both GKM theory and more generally in localization theory in equivariant cohomology. With a view toward other applications, we present parts of our setup in a general algebraic and combinatorial framework. Second, motivated by our central problem of building module bases, we introduce a combinatorial game which we dub poset pinball and illustrate with several examples. Finally, as first applications, we apply the perspective of GKM-compatible subspaces and poset pinball to construct explicit and computationally convenient module bases for the $S^1$-equivariant cohomology of all Peterson varieties of classical Lie type, and subregular Springer varieties of Lie type $A$. In addition, in the Springer case we use our module basis to lift the classical Springer representation on the ordinary cohomology of subregular Springer varieties to $S^1$-equivariant cohomology in Lie type $A$.Comment: 32 pages, 4 figure

Community Structure in Congressional Cosponsorship Networks

We study the United States Congress by constructing networks between Members of Congress based on the legislation that they cosponsor. Using the concept of modularity, we identify the community structure of Congressmen, as connected via sponsorship/cosponsorship of the same legislation, to investigate the collaborative communities of legislators in both chambers of Congress. This analysis yields an explicit and conceptually clear measure of political polarization, demonstrating a sharp increase in partisan polarization which preceded and then culminated in the 104th Congress (1995-1996), when Republicans took control of both chambers. Although polarization has since waned in the U.S. Senate, it remains at historically high levels in the House of Representatives.Comment: 8 pages, 4 figures (some with multiple parts), to appear in Physica A; additional background info and explanations added from last versio

QPACE 2 and Domain Decomposition on the Intel Xeon Phi

We give an overview of QPACE 2, which is a custom-designed supercomputer based on Intel Xeon Phi processors, developed in a collaboration of Regensburg University and Eurotech. We give some general recommendations for how to write high-performance code for the Xeon Phi and then discuss our implementation of a domain-decomposition-based solver and present a number of benchmarks.Comment: plenary talk at Lattice 2014, to appear in the conference proceedings PoS(LATTICE2014), 15 pages, 9 figure

A Data-Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition

The Koopman operator is a linear but infinite dimensional operator that governs the evolution of scalar observables defined on the state space of an autonomous dynamical system, and is a powerful tool for the analysis and decomposition of nonlinear dynamical systems. In this manuscript, we present a data driven method for approximating the leading eigenvalues, eigenfunctions, and modes of the Koopman operator. The method requires a data set of snapshot pairs and a dictionary of scalar observables, but does not require explicit governing equations or interaction with a "black box" integrator. We will show that this approach is, in effect, an extension of Dynamic Mode Decomposition (DMD), which has been used to approximate the Koopman eigenvalues and modes. Furthermore, if the data provided to the method are generated by a Markov process instead of a deterministic dynamical system, the algorithm approximates the eigenfunctions of the Kolmogorov backward equation, which could be considered as the "stochastic Koopman operator" [1]. Finally, four illustrative examples are presented: two that highlight the quantitative performance of the method when presented with either deterministic or stochastic data, and two that show potential applications of the Koopman eigenfunctions

Mining and Analyzing the Italian Parliament: Party Structure and Evolution

The roll calls of the Italian Parliament in the XVI legislature are studied by employing multidimensional scaling, hierarchical clustering, and network analysis. In order to detect changes in voting behavior, the roll calls have been divided in seven periods of six months each. All the methods employed pointed out an increasing fragmentation of the political parties endorsing the previous government that culminated in its downfall. By using the concept of modularity at different resolution levels, we identify the community structure of Parliament and its evolution in each of the considered time periods. The analysis performed revealed as a valuable tool in detecting trends and drifts of Parliamentarians. It showed its effectiveness at identifying political parties and at providing insights on the temporal evolution of groups and their cohesiveness, without having at disposal any knowledge about political membership of Representatives.Comment: 27 pages, 14 figure

Voting Behavior, Coalitions and Government Strength through a Complex Network Analysis

We analyze the network of relations between parliament members according to their voting behavior. In particular, we examine the emergent community structure with respect to political coalitions and government alliances. We rely on tools developed in the Complex Network literature to explore the core of these communities and use their topological features to develop new metrics for party polarization, internal coalition cohesiveness and government strength. As a case study, we focus on the Chamber of Deputies of the Italian Parliament, for which we are able to characterize the heterogeneity of the ruling coalition as well as parties specific contributions to the stability of the government over time. We find sharp contrast in the political debate which surprisingly does not imply a relevant structure based on establised parties. We take a closer look to changes in the community structure after parties split up and their effect on the position of single deputies within communities. Finally, we introduce a way to track the stability of the government coalition over time that is able to discern the contribution of each member along with the impact of its possible defection. While our case study relies on the Italian parliament, whose relevance has come into the international spotlight in the present economic downturn, the methods developed here are entirely general and can therefore be applied to a multitude of other scenarios.Comment: 6 pages, 4 figure