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

We are interested in determining the most likely control network(s) that govern the regulation of human colon crypt stem cell lineages, where lineages are comprised of stem cells, transit amplifying cells, and differentiated cells. We started with a theoretically known set of 32 smallest control networks compatible with tissue stability. We proposed and implemented an algorithm of tests where we compared the networks' simulated behavior with the measured observations, and we discovered only 3 candidate networks that are most compatible with the measurements.

See the MGSC website for more details: http://www.math.uci.edu/~mgsc/index.php

The mean curvature flow provides a way to generate paths in the space of hypersurfaces so can be used to prove connectedness theorems, but it has its limitations. In this talk we explain how an extension of the mean curvature flow, mean curvature flow with surgery, can be used to partially overcome these limitations and prove stronger connectedness results.

Sums of two Cantor sets arise naturally in homoclinic bifurcations, Markov and Lagrange dynamical spectra, and the spectrum of the square Fibonacci Hamiltonian. In the 1970s Palis conjectured that for generic pairs of regular Cantor sets either the sum has zero Lebesgue measure or else it contains an interval. This problem is known to be extremely difficult, and is still open for affine Cantor sets. In this talk, we will discuss the history of sums of two Cantor sets, and also introduce my recent results about sums of two homogeneous Cantor sets. 

Credit ratings have been an important variable in the measurement and management of credit risk. In this talk I will present a Markovian model of credit risk that takes into account an individual's migration between different credit ratings. I will also discuss the portfolio case and introduce a model for the correlation that takes place in a portfolio. I will present a way of measuring the associated Value at Risk and using it to set interest rates. Finally, I will present some results using data.