What topics do you want to cover this semester? What activities do we want to plan? This meeting will mainly be a meet and greet with fellow club memebers so we can discuss out plans for the rest of the semester.
What topics do you want to cover this semester? What activities do we want to plan? This meeting will mainly be a meet and greet with fellow club memebers so we can discuss out plans for the rest of the semester.
In 2011, Dan Shechtman was awarded the Nobel Prize in Chemistry for his discovery of quasicrystals, novel materials with properties somewhere between the regularity of crystals and the disorder of random structures. In parallel with this scientific breakthrough, mathematicians have developed tools for understanding aperiodic order, such as Fibonacci substitutions and Penrose tilings. We will survey these mathematical models of quasicrystals, relying on linear algebra and graph theory. Eigenvalues play a central role, giving insight into how these exotic materials could behave. These problems can be subtle and surprising, opening opportunities for a wide range of mathematical contributions. We will describe our collaborative approach, which integrates numerical computation as a key tool in mathematical discovery, providing a bridge between pure and applied mathematics.
The field of Number theory is very diverse, as it has been studied for thousands of years. One of its newer subfields is known as Iwasawa/Tate Theory, first established in the 1950s by Kenkichi Iwasawa. This field was developed thanks to the discovery of p-adic numbers as well as Class Field Theory in the earlier parts of the century. In this talk we will give a brief introduction to Iwasawa Theory, and its relationship to traditional Number Theory. As well, we will discuss the history of the field and how it has changed since its inception.
The Riemann zeta function has enchanted mathematicians for over 150 years, and is the focus of one of the Clay Millenium Prize problems. In this talk, we will discuss some surprising geometry that arises from certain values of the zeta function, the genesis of which can be understood from solving some closely related linear ODEs. Emphasis will be placed on concrete computations that can be understood by students who have completed Calc II and ODEs.
Off-line states are periods during which the internal dynamics of the brain are relatively independent of external stimuli. The oscillatory dynamics that occur during these states are thought to be critical for learning and memory and are often disrupted by disease. An understanding of these oscillatory dynamics offers the possibility of both enhancing the cognitive capacities of healthy individuals and providing pharmacological and stimulation interventions for disease. In the first half of my talk, I will present a mechanistic model of alpha (8-13 Hz) oscillations during general anesthesia. In the induction of general anesthesia, behaviorally defined loss of consciousness coincides with anteriorization, the spatial shift of alpha power from posterior to anterior regions. We show that anteriorization can be explained by the differential effect of anesthetic drugs on thalamic nuclei with disparate spatial projections. In particular, we show that anesthetic drugs can disrupt the alpha activity generated at depolarized membrane potentials in posteriorly projecting thalamic nuclei while engaging a new, hyperpolarized alpha in frontally projecting thalamic nuclei. In the second half of my talk, I will present work examining oscillations during REM sleep. REM sleep, the period of sleep during which vivid dreams occur, is important for the processing of emotional memories. REM sleep is important, for example, in reducing the emotional charge of fear memories. Rhythmic interactions, especially in the theta band (4-8 Hz) between the medial prefrontal cortex (mPFC) and limbic structures, are known to play a role in reducing emotional charge, but the processing that occurs is largely unknown at the mechanistic and circuit levels. Using mathematical models, we show that theta inputs, but not other frequency inputs, from the mPFC are effective in producing synaptic changes that ultimately suppress the activity of fear expression cells in the amygdala associated with a given memory. We show how aberrant dynamics in this circuit may lead to the symptoms of Post-Traumatic Stress Disorder (PTSD). Our work also suggests potential neuromodulatory therapies for ameliorating PTSD symptoms.