Polynomial dynamical systems over finite fields or rings provide a useful tool for studying network dynamics, such as those of gene regulatory networks. In this talk, I will discuss linear dynamical systems over finite commutative rings. The limit cycles of a linear dynamical system over a finite field can be described by using the elementary divisors of the corresponding matrix. The extension of the study to a general finite commutative ring is natural and has applications. To address the difficulties in the commutative ring setting, we developed a computational approach. In an earlier work, we gave an efficient algorithm to determine whether such a system over a finite commutative ring is a fixed-point system or not. In a more recent work, we further analyzed the structure of such a system and provided a method to determine its limit cycles.

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## Upcoming Seminars

### Mon Nov 27, 2017

This talk will be a discussion of some interesting and novel subgroups of the mapping class group that arise via algebro-geometric constructions. Our talk will focus on the special case of how the theory of plane algebraic curves (essentially just polynomials in two variables!) interacts with the mapping class group in subtle ways. The motivating question can be formulated simply as, ``which mapping classes (of a surface of genus g) arise as one-parameter families of polynomials in two variables?’’ Perhaps surprisingly, the answer turns out to be ``either none at all, or else virtually all of them”. No familiarity with algebraic geometry will be assumed.

In this talk, I’ll sketch a way of unifying a wide variety of set theoretic approaches for generating new models from old models. The underlying methodology will draw from techniques in Sheaf Theory and the theory of Boolean Ultrapowers.

### Tue Nov 28, 2017

We introduce a new class of non-compact symplectic manifolds called

Liouville sectors and show they have well-behaved, covariantly functorial

Fukaya categories. Stein manifolds frequently admit coverings by Liouville

sectors, which can be used to understand the Fukaya category of the total

space (we will study this geometry in examples). Our first main result in

this setup is a local-to-global criterion for generating Fukaya categories.

Our eventual goal is to obtain a combinatorial presentation of the Fukaya

category of any Stein manifold. This is joint work (in progress) with John

Pardon and Vivek Shende.

We introduce a new class of non-compact symplectic manifolds called

Liouville sectors and show they have well-behaved, covariantly functorial

Fukaya categories. Stein manifolds frequently admit coverings by Liouville

sectors, which can be used to understand the Fukaya category of the total

space (we will study this geometry in examples). Our first main result in

this setup is a local-to-global criterion for generating Fukaya categories.

Our eventual goal is to obtain a combinatorial presentation of the Fukaya

category of any Stein manifold. This is joint work (in progress) with John

Pardon and Vivek Shende.

### Fri Dec 1, 2017

This is a report of some recent progress and challenges we have made and encountered in modelling and numerical simulation of materially nonlinear beam structures with applications in micro-electrical-mechanical systems. For simplicity, the fully nonlinear DE’s and the associated initial/boundary value problems arising from modelling Hollomon’s power-law material structures are presented as our representative mathematical models. While for linear elastic materials, the principal operator in the equations appears to be the Laplacian or the bi-Laplacian operator, for the Hollomon’s materials, the principal operator is the p-Laplacian or the bi-p-Laplacian. The main results presented are centered around approximations of solutions to the nonlinear wave equation by lumped-parameter models and numerical methods. Similar, but more challenging models are also introduced for further investigations.

This will be a continuation of the presentation from two weeks ago. The abstract is below.

Last time, we briefly discussed the notion of localization and some of its consequences and during this meeting we will finish with the rank one perturbation material and the promised proof.

**Abstract:**

The goal of this talk will be to discuss various issues related to the Anderson model as presented in Del Rio et. al "Operators with Singular Continuous Spectrum, IV." Firstly, we will explain the type of localization that allows one to make dynamical statements (i.e. given simple spectrum, we have 'SULE' iff 'SUDL'). We then present various facts relating to rank one perturbations of self adjoint operators. Finally, we connect the above two discussions to give the authors' proof that the singular continuous spectral measures produced by rank one perturbations of the Anderson model are supported on a set of Hausdorff dimension zero.

### Sat Dec 2, 2017

Uniform laws of large numbers provide theoretical foundations for statistical learning theory. This lecture will focus on quantitative uniform laws of large numbers for random matrices. A range of illustrations will be given in high dimensional geometry and data science.

*In this talk, we present some results on the existence of weak-solutions of the Navier-Stokes equation perturbed by transport-type rough path noise with periodic boundary conditions in dimensions two and three. The noise is smooth and divergence free in space, but rough in time. We will also discuss the problem of uniqueness in two dimensions. The proof of these results makes use of the theory of unbounded rough drivers developed by M. Gubinelli et al.*

*As a consequence of our results, we obtain a pathwise interpretation of the stochastic Navier-Stokes equation with Brownian and fractional Brownian transport-type noise. A Wong-Zakai theorem and support theorem follow as an immediate corollary. This is joint work with Martina Hofmanov\'a and Torstein Nilssen.*

We consider an i.i.d. balanced environment $\omega(x,e)=\omega(x,-e)$, genuinely d dimensional on the lattice and show that there exist a positive constant $C$ and a random radius $R(\omega)$ with streched exponential tail such that every non negative

$\omega$ harmonic function $u$ on the ball $B_{2r}$ of radius $2r>R(\omega)$,

we have $\max_{B_r} u <= C \min_{B_r} u$.

Our proof relies on a quantitative quenched invariance principle

for the corresponding random walk in balanced random environment and

a careful analysis of the directed percolation cluster.

This result extends Martins Barlow's Harnack's inequality for i.i.d.

bond percolation to the directed case.

This is joint work with N.Berger M. Cohen and X. Guo.

The random order of service (ROS) is a natural scheduling policy for systems where no ordering of customers can or should be established. Queueing models under ROS have been used to study molecular interactions of intracellular components in biology. However, these models often assume exponential distributions for processing and patience times, which is not realistic especially when operations such as binding, folding, transcription and translation are involved. We study a multi-class queueing model operating under ROS with reneging and generally distributed processing and patience times. We use measure-valued processes to describe the dynamic evolution of the network, and establish a fluid approximation for this representation. Obtaining a fluid limit for this network requires a multi-scale analysis of its fast and slow components, and to establish an averaging principle in the context of measure-valued process. In addition, under slightly more restrictive assumptions on the patience time distribution, we introduce a reduced, function-valued fluid model that is described by a system of non-linear Partial Differential Equations (PDEs). These PDEs, however, are non-standard and the analysis of their existence, uniqueness and stability properties requires new techniques.