Glueballs on the three-sphere

We study the non-perturbative effects of the global features of the configuration space for SU(2) gauge theory on the three-sphere. The strategy is to reduce the full problem to an effective theory for the dynamics of the low-energy modes. By explicitly integrating out the high-energy modes, the one-loop correction to the effective hamiltonian is obtained. Imposing the $\theta$ dependence through boundary conditions in configuration space incorporates the non-perturbative effects of the non-contractable loops in the full configuration space. After this we obtain the glueball spectrum of the effective theory with a variational method.Comment: 48 p LaTeX, 13 Postscript figures appende

Horocycle dynamics: new invariants and eigenform loci in the stratum H(1,1)

We study dynamics of the horocycle flow on strata of translation surfaces, introduce new invariants for ergodic measures, and analyze the interaction of the horocycle flow and real Rel surgeries. We use this analysis to complete and extend results of Calta and Wortman classifying horocycle-invariant measures in the eigenform loci. We classify the orbit-closures and prove that every orbit is equidistributed in its orbit-closure. We also prove equidistribution statements regarding limits of sequences of measures, some of which have applications to counting problems.Comment: 100 page

Euler characteristics of Teichm\"uller curves in genus two

We calculate the Euler characteristics of all of the Teichmuller curves in the moduli space of genus two Riemann surfaces which are generated by holomorphic one-forms with a single double zero. These curves can all be embedded in Hilbert modular surfaces and our main result is that the Euler characteristic of a Teichmuller curve is proportional to the Euler characteristic of the Hilbert modular surface on which it lies. The idea is to use techniques from algebraic geometry to calculate the fundamental classes of these Teichmuller curves in certain compactifications of the Hilbert modular surfaces. This is done by defining meromorphic sections of line bundles over Hilbert modular surfaces which vanish along these Teichmuller curves. We apply these results to calculate the Siegel-Veech constants for counting closed billiards paths in certain L-shaped polygons. We also calculate the Lyapunov exponents of the Kontsevich-Zorich cocycle for any ergodic, SL_2(R)-invariant measure on the moduli space of Abelian differentials in genus two (previously calculated in unpublished work of Kontsevich and Zorich).Comment: 178 pages, 14 figure

LieDetect: Detection of representation orbits of compact Lie groups from point clouds

We suggest a new algorithm to estimate representations of compact Lie groups from finite samples of their orbits. Different from other reported techniques, our method allows the retrieval of the precise representation type as a direct sum of irreducible representations. Moreover, the knowledge of the representation type permits the reconstruction of its orbit, which is useful to identify the Lie group that generates the action. Our algorithm is general for any compact Lie group, but only instantiations for SO(2), T^d, SU(2) and SO(3) are considered. Theoretical guarantees of robustness in terms of Hausdorff and Wasserstein distances are derived. Our tools are drawn from geometric measure theory, computational geometry, and optimization on matrix manifolds. The algorithm is tested for synthetic data up to dimension 16, as well as real-life applications in image analysis, harmonic analysis, and classical mechanics systems, achieving very accurate results.Comment: 84 pages, 16 figure

Signals on Networks: Random Asynchronous and Multirate Processing, and Uncertainty Principles

The processing of signals defined on graphs has been of interest for many years, and finds applications in a diverse set of fields such as sensor networks, social and economic networks, and biological networks. In graph signal processing applications, signals are not defined as functions on a uniform time-domain grid but they are defined as vectors indexed by the vertices of a graph, where the underlying graph is assumed to model the irregular signal domain. Although analysis of such networked models is not new (it can be traced back to the consensus problem studied more than four decades ago), such models are studied recently from the view-point of signal processing, in which the analysis is based on the "graph operator" whose eigenvectors serve as a Fourier basis for the graph of interest. With the help of graph Fourier basis, a number of topics from classical signal processing (such as sampling, reconstruction, filtering, etc.) are extended to the case of graphs. The main contribution of this thesis is to provide new directions in the field of graph signal processing and provide further extensions of topics in classical signal processing. The first part of this thesis focuses on a random and asynchronous variant of "graph shift," i.e., localized communication between neighboring nodes. Since the dynamical behavior of randomized asynchronous updates is very different from standard graph shift (i.e., state-space models), this part of the thesis focuses on the convergence and stability behavior of such random asynchronous recursions. Although non-random variants of asynchronous state recursions (possibly with non-linear updates) are well-studied problems with early results dating back to the late 60's, this thesis considers the convergence (and stability) in the statistical mean-squared sense and presents the precise conditions for the stability by drawing parallels with switching systems. It is also shown that systems exhibit unexpected behavior under randomized asynchronicity: an unstable system (in the synchronous world) may be stabilized simply by the use of randomized asynchronicity. Moreover, randomized asynchronicity may result in a lower total computational complexity in certain parameter settings. The thesis presents applications of the random asynchronous model in the context of graph signal processing including an autonomous clustering of network of agents, and a node-asynchronous communication protocol that implements a given rational filter on the graph. The second part of the thesis focuses on extensions of the following topics in classical signal processing to the case of graph: multirate processing and filter banks, discrete uncertainty principles, and energy compaction filters for optimal filter design. The thesis also considers an application to the heat diffusion over networks. Multirate systems and filter banks find many applications in signal processing theory and implementations. Despite the possibility of extending 2-channel filter banks to bipartite graphs, this thesis shows that this relation cannot be generalized to M-channel systems on M-partite graphs. As a result, the extension of classical multirate theory to graphs is nontrivial, and such extensions cannot be obtained without certain mathematical restrictions on the graph. The thesis provides the necessary conditions on the graph such that fundamental building blocks of multirate processing remain valid in the graph domain. In particular, it is shown that when the underlying graph satisfies a condition called M-block cyclic property, classical multirate theory can be extended to the graphs. The uncertainty principle is an essential mathematical concept in science and engineering, and uncertainty principles generally state that a signal cannot have an arbitrarily "short" description in the original basis and in the Fourier basis simultaneously. Based on the fact that graph signal processing proposes two different bases (i.e., vertex and the graph Fourier domains) to represent graph signals, this thesis shows that the total number of nonzero elements of a graph signal and its representation in the graph Fourier domain is lower bounded by a quantity depending on the underlying graph. The thesis also presents the necessary and sufficient condition for the existence of 2-sparse and 3-sparse eigenvectors of a connected graph. When such eigenvectors exist, the uncertainty bound is very low, tight, and independent of the global structure of the graph. The thesis also considers the classical spectral concentration problem. In the context of polynomial graph filters, the problem reduces to the polynomial concentration problem studied more generally by Slepian in the 70's. The thesis studies the asymptotic behavior of the optimal solution in the case of narrow bandwidth. Different examples of graphs are also compared in order to show that the maximum energy compaction and the optimal filter depends heavily on the graph spectrum. In the last part, the thesis considers the estimation of the starting time of a heat diffusion process from its noisy measurements when there is a single point source located on a known vertex of a graph with unknown starting time. In particular, the Cramér-Rao lower bound for the estimation problem is derived, and it is shown that for graphs with higher connectivity the problem has a larger lower bound making the estimation problem more difficult.</p

Counting nodal components of boundary-adapted arithmetic random waves

The ‘nodal sets’ (zero sets) of Dirichlet Laplace eigenfunctions for the two dimensional unit square have historically raised many questions, and continue to do so today. Prominent amongst them is the question of the number of ‘nodal components’ (connected components of the zero set) of a typical eigenfunction. In this thesis, we attribute Gaussian random coefficients to a standard basis of eigenfunctions for each eigenspace, to form the ensemble of ‘boundary-adapted arithmetic random waves’. We then study the number of nodal components -- now a random variable -- of this ensemble as the eigenvalue grows to infinity, and establish the existence of a limiting mean nodal intensity which is non-universal, in the sense that it depends (indeed relies) upon restriction to subsequences of eigenvalues with specific arithmetic properties. We further show that the number of nodal components concentrates exponentially in probability about this limiting mean intensity