Mathias Hudoba de Badyn

Emergent geometry of membranes

A bstractIn work [1], a surface embedded in flat ℝ3 is associated to any three hermitian matrices. We study this emergent surface when the matrices are large, by constructing coherent states corresponding to points in the emergent geometry. We find the original matrices determine not only shape of the emergent surface, but also a unique Poisson structure. We prove that commutators of matrix operators correspond to Poisson brackets. Through our construction, we can realize arbitrary noncommutative membranes: for example, we examine a round sphere with a non-spherically symmetric Poisson structure. We also give a natural construction for a noncommutative torus embedded in ℝ3. Finally, we make remarks about area and find matrix equations for minimal area surfaces.

Network entropy: A system-theoretic perspective

In this paper, we highlight the importance of two measures associated with networked dynamic systems, namely the loop entropy and the Kolmogorov-Sinai entropy, that quantify the notion of information content in the network. We then proceed to show connections between these measures and certain system theoretic properties that these networks exhibit for two classes of network dynamics. Throughout the paper, we also provide relevant bounds and insights into what these network measures quantify.

Controllability and stabilizability analysis of signed consensus networks

Signed networks have been a topic of recent interest in the network control community as they allow studying antagonistic interactions in multi-agent systems. Although dynamical characteristics of signed networks have been well-studied, notions such as controllability and stabilizability for signed networks for protocols such as consensus are missing in the literature. Classically, graph automorphisms with respect to the input nodes have been used to characterize uncontrol-lability of consensus networks. In this paper, we show that in addition to the graph symmetry, the topological property of structural balance facilitates the derivation of analogous sufficient conditions for uncontrollability for signed networks. In particular, we provide an analysis which shows that a gauge transformation induced by structural balance allows symmetry arguments to hold for signed consensus networks. Lastly, we use fractional automorphisms to extend our observations to output controllability and stabilizability of signed networks.

Growing controllable networks via whiskering and submodular optimization

The topology of a network directly influences the behaviour and controllability of dynamical processes on that network. Therefore, the design of network topologies is an important area of research when examining the control of distributed systems. We discuss a method for growing networks known as whiskering, as well as generalizations of this process, and prove that they preserve controllability. We then use techniques from submodular optimization to analyze optimization algorithms for adding new nodes to a network to optimize certain objectives, such as graph connectivity.

On the Hilbert-Pólya and Pair Correlation Conjectures

The Hilbert-Pólya Conjecture supposes that there exists an operator in a Hilbert space whose eigenvalues are the zeroes of the Riemann Zeta function ζ(s). This conjecture, if true, would very likely expedite the proof of the Riemann Hypothesis, namely that the non-trivial zeroes of ζ(s) have real part 12 . In this thesis we summarize work by Berry, Keating and others in constructing such an operator. Although the work so far has not yet yielded such an operator, some have been found that have properties very close to what is desired. We also summarize a (partially proven) conjecture by Montgomery that motivates the search for this operator. He conjectures that the pair correlation function for the spacing between the imaginary parts of the Riemann zeroes is the same as the correlation function for the spacing between eigenvalues of random Gaussian unitary matrices. Acknowledgements I would like to first thank my parents for the many things they have done that have helped contribute to my successes so far, one of which was their decision to give me a name which contains ‘math’ as a subsequence. I would also like to thank the laws of physics and mathematics which allow the Universe to exist the way it does and for allowing the atoms which comprise my body to have