Mutual stationarity is a notion of infinite products of stationary sets introduced by Foreman and Magidor. The assertion that an infinite sequence of stationary sets is mutually stationary has a natural model theoretic interpretation and can be viewed as a strengthening of the Loewenheim-Skolem property.

Epithelial-mesenchymal transition (EMT) is an instance of cellular plasticity that plays critical roles in development, regeneration and cancer progression. Utilizing a systems biology approach integrating modeling and experiments, we observed that adding the mutual inhibition relationship between Ovol2 and EMT inducer Zeb1 generates a novel four-state system consisting of two distinct intermediate phenotypes that differ in differentiation propensities and are favored in different environmental conditions. We then used mathematical models to show that multiple intermediate phenotypes in the EMT system help to attenuate the overall fluctuations of the cell population in terms of phenotypic compositions, thereby stabilizing a heterogeneous cell population in the EMT spectrum. Lastly, we attempted to bridge the gap between discrete and continuum modeling of the EMT system by incorporating the EMT core regulatory network into our heterogeneous cell population dynamics model to create a multiscale EMT model. Our model can capture the larger-scale population growth dynamics while acknowledging the intracellular interactions between proteins within each individual cell. This talk is aimed at a general audience.

For more information about MGSC: https://www.math.uci.edu/~mgsc/index.php

Computing the Unit Group of a Number Field

Starting from Pell’s equation and units in real quadratic fields, we will discuss the problem of computing the unit group of a number field. This will lead to a discussion of the Hidden Subgroup Problem, which arises in many quantum algorithms.  We will discuss recent work of Eisentrager, Hallgren, Kitaev, and Song, and of Biasse and Song, giving a quantum algorithm for this problem that runs in polynomial time.