A random walk through subRiemannian geometry 

Tom Laetsch (U Conn)
A subRiemannian manifold $M$ is a connected smooth manifold such that the only smooth curves in $M$ which are admissible are those whose tangent vectors at any point are restricted to a subset $\mathcal{H} \subset TM$, called the horizontal distribution. Such spaces have several applications in physics and engineering, as well as in the study of hypoelliptic operators. In this talk we will construct a family of geometrically natural hypoelliptic Laplaciantype operators and discuss the trouble with defining one which is canonical. We will also attempt to construct a random walk on a subRiemannian manifold which converges weakly to a process whose infinitesimal generator is one of our hypoelliptic Laplaciantype operators.
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Invariant densities for dynamical systems with random switching 

Tobias Hurth (Georgia Tech)
Consider a finite family of smooth vector fields on a finitedimensional smooth manifold $M$. For a fixed starting point on $M$ and an initial vector field, we follow the solution trajectory of the corresponding initialvalue problem for an exponentially distributed random time. Then, a new vector field is selected at random from the given family, and we start following the induced trajectory for another exponentially distributed time. Iterating this construction, we obtain a stochastic process $X$ on $M$. To $X$, we adjoin a second process $A$ that records the driving vector field at any given time. The twocomponent process $(X,A)$ is Markov. In the talk, I will present sufficient conditions for uniqueness and absolute continuity of its invariant measure. These consist of a Hörmandertype hypoellipticity condition that holds at a point on M that can be approached from all other points on the manifold. If $M$ is the real line, one can show that the densities of the invariant measure are smooth away from critical points. For analytic vector fields, we can derive the asymptotically dominant term of the densities at critical points. This is joint work with Yuri Bakhtin. top of page

Random zero sets under repeated differentiation of an analytic function 

Sneha Subramanian (U Penn)
We study the result of repeatedly differentiating a random entire function whose zeros are points of a Poisson process of intensity 1 on $\mathbb{R}$. For the first part of the talk, we shall discuss a toymodel based on a random polynomial, whose zeros are i.i.d. Rademacher random variables. For the second, we shall use our results from the first part as a poorism to discuss the main result regarding Poisson process. Based on joint work with Robin Pemantle. top of page

An Unconventional Functional Central Limit Theorem for the Queue Length Process in a Shortest Remaining Processing Time Queue 

Amber Puha (CSU San Marcos)
In a shortest remaining processing time (SRPT) queue, the job that requires the least amount of processing time is preemptively served first. One effect of this is that the queue length is small in comparison to the total amount or work in the system (measured in units of processing time). In fact, it is minimized so well that the sequence of queue length processes associated with a sequence of SRPT queues rescaled with standard functional central limit theorem scaling and satisfying standard heavy traffic conditions converges in distribution to the process that is identically equal to zero. This happens despite the fact that under this same regime the rescaled workload processes converge to a nondegenerate reflected Brownian motion. In particular, the queue length process is of smaller order magnitude than the workload process. In the case of processing time distributions that satisfy a rapid variation condition, we implement an alternative, unconventional scaling that leads to a nontrivial limit for the queue length process. This result quantities this order of magnitude difference between queue length and workload processes. We illustrate this result for Weibull processing time distributions. top of page



