All talks will be held in 203 TMCB
10:10 am Welcome!
10:30 am Sanghoon Kwak (University of Utah)
Title - Nonunique Ergodicity on the Boundary of Outer Space
Abstract - The Culler--Vogtmann's Outer space CVn is a space of marked metric graphs, and it compactifies to a set of Fn-trees. Each Fn-tree on the boundary of Outer space is equipped with a length measure, and varying length measures on a topological Fn-tree gives a simplex in the boundary. The extremal points of the simplex correspond to ergodic length measures. By the results of Gabai and Lenzhen--Masur, the maximal simplex of transverse measures on a fixed filling geodesic lamination on a complete hyperbolic surface of genus g has dimension 3g-4. In this talk, we give the maximal simplex of length measures on an arational Fn-tree has dimension in the interval [2n-7, 2n-2]. This is a joint work with Mladen Bestvina, and Elizabeth Field.
11:30 am Skyler Simmons (Utah Valey University)
Title - Periodic Collision-based Orbits in the Newtonian n-Body Problem
Abstract - The Newtonian n-body problem seeks to model the motion of n point masses acting under Newton’s inverse-square law for gravity. Singularities of the equations of motion occur when two or more point masses occupy the same point in space simultaneously. I will outline a few periodic orbits in two dimensions which feature collisions, highlighting the regularization techniques used to continue the orbits past these collision points. Then I will present some recent work done in three-dimensional orbits. Results relating to the stability of each orbit will also be presented.
12:30 pm Lunch
2:00 pm Izzy Zahalak (University of Utah)
Title - Exploring a Natural Extension of the Doubling Map on the Circle
Abstract - The doubling map on $S^1$ is a map whose straightforward definition belies its non-invertible structure. In order to overcome this non- invertiblity and understand the trajectories of points on $S^1$, we consider a natural extension of $S^1$ which ends up generating the 2-adic solenoid $\mathbb{S}_2$. Using the algebraic and topological properties of the circle and the solenoid, we show that the doubling action on the circle is topologically conjugate to the dynamic map associated to the Smale-Williams attractor.
2:30 pm Chase Ford (Brigham Young University)
Title - Improvement of Adam Optimizer via Introduction of Stochastic Time Delays
Abstract - The Adam optimizer is an optimization algorithm that has seen extensive use in deep learning applications in computer vision and natural language processing. It uses exponentially decaying averages of both gradients and the second moments of the gradients to produce its results. We propose a time-delayed adaptation of the Adam optimizer as a method for improving optimization on certain high-dimensional loss surfaces which simply wraps a given optimizer, in this case the Adam optimizer, allowing us to add a variety of time-delays. We find that adding stochastic time-delays, among others, significantly improves the Adam optimizer’s performance on a number of benchmark loss surfaces including the Rastrigen, Ackley, and Zacharov functions. This improvement is especially noticeable in high dimensions. We also show, under mild conditions, that the local and global minima of an arbitrary loss surface remain attracting fixed points of the time-delayed Adam optimizer by extending the theory of intrinsic stability to multistable systems.
3:00 pm Kurt Vinhage (University of Utah)
Title - The traps and trappings of a large centralizer
Abstract - Rigidity properties of hyperbolic smooth abelian group actions have been well studied from three main perspectives: measure rigidity, smooth rigidity and cocycle rigidity. We will discuss the rigidity features from each perspective, noting the origins of each prong of the rigidity phenomena, as well as recent developments in our understanding.