Georgia Institute of Technology College of Sciences

Link: https://gatech.zoom.us/j/91390791493?pwd=QnpaWHNEOHZTVXlZSXFkYTJ0b0Q0UT09

Abstract: The timing of human sleep is strongly modulated by the 24 hour circadian rhythm, our internal biological clock, and the homeostatic sleep drive, one’s need for sleep which depends on prior awakening. The parameters dictating the evolution of the homeostatic sleep drive may vary with development and have been identified as important parameters for generating the transition from multiple sleeps to a single sleep episode per day. We employ piecewise-smooth ODE-based mathematical models to analyze developmentally-mediated transitions of sleep-wake patterns, including napping and non-napping behaviors. Our framework includes the construction of a circle map that captures the timing of sleep onsets on successive days. Analysis of the structure and bifurcations in the map reveals changes in the average number of sleep episodes per day in a period-adding-like structure. In two-state models of sleep-wake regulation, namely models that generate sleep and wake states, we observe saddle-node and border collision bifurcations in the maps. However, in our three-state model of sleep-wake regulation, which captures wake, rapid eye movement (REM) sleep, and non-REM sleep, these sequences are disrupted by period-doubling bifurcations and can exhibit bistability.

This week, we'll continue discussing the rational blowdown and use it to construct small exotic 4-manifolds. We will see how we can view the rational blowdown as a "monodromy substitution." Finally, if time allows, we will discuss knot surgery on 4-manifolds. 

We will finish the proof of the unique continuation theorem, starting with a brief discussion of the growth lemma discussed at our previous talk. After this, we will reduce unique continuation for untitled squares to unique continuation for tilted squares, and using the tilted square growth lemma prove such unique continuation result.

Understanding the route to thermalization of a physical system is a fundamental problem in statistical mechanics. When a system is initialized far from thermodynamical equilibrium, many interesting phenomena may arise. Among them, a lot of interest is attained by systems subjected to periodic driving (Floquet systems), which under certain circumstances can undergo a two-stage long dynamics referred to as "prethermalization", showing nontrivial physical features. In this talk, we present some prethermalization results for a class of lattice systems with quasi-periodic external driving in time. When the quasi-periodic driving frequency is large enough or the strength of the driving is small enough, we show that the system exhibits a prethermal state for exponentially long times in the perturbative parameter. Moreover, we focus on the case when the unperturbed Hamiltonian admits constants of motion and we prove the quasi-conservation of a dressed version of them. We discuss applications to perturbations of the Fermi-Hubbard model and the quantum Ising chain.

Join Zoom Meeting

https://gatech.zoom.us/j/96817326631