Home Program Location

All talks will be held in LCB222

12:20pm Welcome!
Please join us in the JWB lounge for some snacks and sodas before the talks.
12:40pm Arithmeticity for Smooth Maximal Rank Positive Entropy Actions of R^k
Alp Uzman (University of Utah)
We prove an arithmeticity theorem in the context of nonuniform measure rigidity. Adapting machinery developed by A. Katok and F. Rodriguez Hertz [J. Mod. Dyn. 10 (2016), 135–172; MR3503686] for Z^k systems to R^k systems, we show that any maximal rank positive entropy system on a manifold generated by k>=2 commuting vector fields of regularity C^r for r>1 is measure theoretically isomorphic to a constant time change of the suspension of some action of Z^k on the (k+1)-torus or the (k+1)-torus modulo {id,-id} by affine automorphisms with linear parts hyperbolic. Further, the constructed conjugacy has certain smoothness properties. This in particular answers a problem and a conjecture from a prequel paper of Katok and Rodriguez Hertz, joint with B. Kalinin [Ann. of Math. (2) 174 (2011), no. 1, 361–400; MR2811602].
1:40pm Minimal Specialization: Coevolution of Network Structure and Dynamics
Annika King (Brigham Young University)
In real-world systems, the changing topology of the network is driven by the need to optimize the network's function, which is often related to moving quantities efficiently through the network. Dynamical processes such as traffic flow, information transfer, etc., put pressure on the network's topology to evolve in specific ways. In order to model this behavior, we use the dynamics on the network, or the dynamical processes the network models, to influence the dynamics of the network structure, i.e., to determine where to modify the network structure. We model the dynamics on the network using Jackson network dynamics and the dynamics of the network structure using the newly proposed mechanism of network specialization called minimal specialization. The resulting mode coevolves both the structure and the dynamics of the network. This model produces networks with real-world properties, such as right-skewed degree distributions, sparsity, the small-world property, and non-trivial equitable partitions. Additionally, when compared to other growth models, our model creates networks with small diameter, minimizing distances across the network. Along with producing these structural features, this model also sequentially removes the network's largest bottlenecks. The result is networks that have both dynamic and structural features that allow quantities to efficiently move through the network.
2:40PM Break
3:00pm Slow Entropy of Sturmian Shifts and 3-IETs
Carlos Ospina (University of Utah)
Entropy is essential in studying dynamical systems for different reasons. Some of them are that it is an invariant of equivalent systems and because it classifies dynamical systems by describing the growth of relevant orbits. The classical definition of entropy puts zero-entropy systems into a vast class. In this talk, we will review the description of slow entropy, explain why it is helpful in studying the class of systems mentioned before, and explain how to calculate it for Sturmian systems and 3-interval exchange transformations. This talk is ongoing work with Minhua Chen, Kurt Vinhage, and Yibo Zhai.
4:00pm Intrinsic Stability: Stabilizing Against Time-Delays and Switching
Ben Webb (Brigham Young University)
In many natural and technological systems the rule that governs the system's dynamics changes over time. Additionally these systems experience time-delays due to the need to process and transport quantities over distances. Both time-delays and switching between distinct types of dynamics can be a significant source of instability and, more generally, poor performance. However, not all systems can be destabilized by time-delays and/or switching. In a series of papers it has been shown that if a system is intrinsically stable, which is a stronger form of global stability, then the system maintains stability even when time-delays or switching are introduced into the system. In this talk we describe two types of criteria. For the first, we show that intrinsically stable systems remain stable in the presence of any type of time-delay and prove that the asymptotic state of an intrinsically stable switched system is independent of both the system's initial conditions and the time-delays it experiences. We then extend these results to i.i.d. stochastically switched systems and give a simple criterium under which such systems are patiently stable, i.e. cannot be destabilized by time delays. These results side step the need to use Lyapunov, linear matrix inequalities, and semi-definite programming-type methods.