In this talk, I will present a strong (or pathwise) approximation of standard Brownian motion by a class of orthogonal polynomials. Most notably, the coefficients obtained from this expansion are independent Gaussian random variables. This will enable us to generate approximate Brownian paths by matching certain polynomial moments. To conclude the talk, I will discuss related works and applications to numerical methods for SDEs.
 

We investigate a nonlinear version of coevolving voter models, in which both node states and network structure update as a coupled stochastic dynamical process. Most prior work on coevolving voter models has focused on linear update rules with fixed rewiring and adopting probabilities. By contrast, in our nonlinear version, the probability that a node rewires or adopts is a function of how well it "fits in" within its neighborhood. To explore this idea, we incorporate a parameter σ that represents the fraction of neighbors of an updating node that share its opinion state. In an update, with probability σq (for some nonlinearity parameter q), the updating node rewires; with complementary probability 1−σq, the updating node adopts a new opinion state. We study this mechanism using three rewiring schemes: after an updating node deletes a discordant edge, it then either (1) "rewires-to-random" by choosing a new neighbor in a random process; (2) "rewires-to-same" by choosing a new neighbor in a random process from nodes that share its state; or (3) "rewires-to-none" by not rewiring at all (akin to "unfriending" on social media). We compare our nonlinear coevolving model to several existing linear models, and we find in our model that initial network topology can play a larger role in the dynamics, whereas the choice of rewiring mechanism plays a smaller role. A particularly interesting feature of our model is that, under certain conditions, the opinion state that is initially held by a minority of nodes can effectively spread to almost every node in a network if the minority nodes views themselves as the majority. In light of this observation, we relate our results to recent work on the majority illusion in social networks.

Reference: 

Kureh, Yacoub H., and Mason A. Porter. "Fitting In and Breaking Up: A Nonlinear Version of Coevolving Voter Models." arXiv preprint arXiv:1907.11608 (2019).

Thin film flows of nematic liquid crystal will be considered, using the Leslie-Ericksen formulation for nematics. Our model can account for variations in substrate anchoring, which may exert a strong influence on patterns that arise in the flow. A number of simulations will be presented using an "in house" code, developed to run on a GPU. Current modeling directions involving flow over interlaced electrodes, so-called "dielectrowetting", will be discussed.

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https://sites.google.com/view/geometryandstrings2019/home

We consider Schroedinger operators on the whole Euclidean space or on the half-space, subject to real Robin boundary conditions. I will present the construction of a non-real potential that decays at infinity so that the corresponding Schroedinger operator has infinitely many non-real eigenvalues accumulating at every point of the essential spectrum. This proves that the Lieb-Thirring inequalities, crucial in quantum mechanics for the proof of stability of matter, do no longer hold in the non-selfadjoint case.