First, I introduce quantum graph theory. I also discuss a known lower bound on the independence numbers and derive from it an upper bound on the chromatic numbers of quantum graphs. Then, I construct a family of quantum graphs that can be described as tropical, cyclical, and commutative. I also define a step logarithm function and express with it the bounds on quantum graph invariants in closed form. Finally, I obtain an upper bound on the independence numbers and a lower bound on the chromatic numbers of the constructed quantum graphs that are similar in form to the existing bounds. I also show that the constructed family contains graphs of any valence with arbitrarily high chromatic numbers and conclude by it that quantum graph colorings are quite different from classical graph colorings.

This mini-course described the Thue-Siegel method, as used in the proof of Faltings' theorem on the Mordell conjecture. The exposition followed Bombieri's variant of this proof, which avoids the machinery of Arakelov theory.

The underlying mechanisms of alpha band (8-12 Hz) neural oscillations are of importance to the functioning of attention control systems as well as to neuropsychiatric conditions that are characterized by deficits of that system, such as attention deficit hyperactivity disorder (ADHD). The objectives of the present study were to test if visual encoding-related alpha event-related desynchronization (ERD) correlates with fronto-parieto-occipital connectivity, and whether this is disrupted in ADHD during spatial working memory (SWM) performance. We acquired EEG concurrently with fMRI in thirty boys (12-16 yrs. old, 15 with ADHD), during SWM encoding. Psychophysiological connectivity analyses indicated that alpha ERD during SWM encoding was associated with both occipital activation and fronto-parieto-occipital functional connectivity, a finding that expands on prior associations between alpha ERD and occipital activation. This finding provides novel support for the interpretation of alpha ERD (and the associated changes in occipital activation) as a phenomenon that involves, and perhaps arises as a result of, top-down network interactions. Alpha ERD was associated less strongly with occipital activity, but associated more strongly with fronto-parieto-occipital connectivity in ADHD, consistent with a compensatory attentional response. Additionally, we illustrate that degradation of EEG data quality by MRI-amplified motion artifacts is robust to existing cleaning algorithms and is significantly correlated with hyperactivity symptoms and the ADHD Combined Type diagnosis. We conclude that persistent motion-related MR artifacts in EEG data can increase variance and introduce bias in interpretation of group differences in populations characterized by hypermobility--a clear limitation of current-state EEG-fMRI methodology.