Andrea Bertozzi, UCLA
Self-organization in human, biological, and articial systems.

I will discuss recent work on self-organization in complex systems with a focus on human and biological models as well as articial systems. Specific case studies include (a) the formation of crime hotspots in urban settings and the mechanisms that lead to such behavior (b) collective motion of swarms, flocks, and schools in animal populations and (c) cooperative control of robotic vehicles using models motivated by biological examples. Such research problems have led to interesting work and open problems for the mathematics community, bringing together different research areas including dynamical systems, stochastic processes, statistical sampling, bifurcation theory, graph theory, and differential equations.

Helen Byrne, University of Oxford
Discrete and continuous approaches to modelling vascular tumour growth

By the time that they are clinically detectable, most solid tumours meet the abnormally high metabolic demands of their proliferating cells by stimulating the formation of a new blood supply from their host. This network of blood vessels typically forms over a short time period and is usually abnormal, comprising vessels that are highly tortuous, leaky and more compliant than their normal counterparts. To exacerbate matters further, the distribution of blood vessels within vascular tumours is often extremely heterogeneous with respect to both space and time. For example, cells in regions that are, at a given time, well-perfused will be nutrient-rich and proliferate rapidly. The associated increase in cell number increases the mechanical load on the compliant vessels which will collapse, halting the blood supply to the region and starving the cells of vital nutrients unless they can stimulate the ingrowth of new blood vessels before they die. The complexity of vascular tumours creates significant therapeutic challenges, including deliver cytotoxic drugs to the tumour cells.
When faced with such complexity, it is difficult to know what mathematical approach should be used to model vascular tumour growth. Continuum models, based on mixture theory, provide a coarse-grained description of the tumour's size and composition that is consistent with imaging modalities such as MRI. By contrast, multiscale models permit investigation of the interplay between subcellular, cellular and tissue scale process. In this talk I will introduce each approach, explain the complementary insight that they can provide and discuss techniques for investigating their inter-relationships. I will also explain the approaches we are using to validate the models against experimental data.

Theodore Kim, Media Arts and Technology Program,
University of California, Santa Barbara
Subspace Simulation of Non-linear Materials

Everyday materials such as biological tissue exhibit geometric and constitutive non-linearities which are crucial in applications such as surgical simulation and realistic human animation. However, these same non-linearities also make efficient finite element simulation difficult, as computing non-linear forces and their Jacobians can be computationally intensive. Reduced order methods, which limit simulations to a low-dimensional subspace, have the potential to accelerate these simulations by orders of magnitude. In this talk, I will describe a subspace method we have developed that efficiently computes all the necessary non-linear quantities by evaluating them at a discrete set of "cubature" points, an online integration method that can accelerate simulations even when the subspace is not known a priori, and a domain decomposition method that efficiently adds deformations to discretely partitioned, articulated simulations. Using our methods, we have observed speedups of one to four orders of magnitude.

Dilip Madan, University of Maryland

Jointly modeling three option surfaces, for an ADR, its local underlying stock and the exchange rate.
We apply the quadratic skew normal model with an underlying Bhattacharyya Density to jointly model the local stock and the exchange rate for the US dollar quoted in the local currency. Arbitrage free considerations are shown to imply a model for options on the stock's ADR. All three surfaces are simultaneously calibrated. Applications include the design of trades illustrated for VALE- BRL, UBS-CHF, BARCLAY-GBP, and SANTANDER-EUR.

Edriss S. Titi, University of California and Weizmann Institute
Recent Progress Regarding the Navier-Stokes, Euler and Related Geophysical
Equations

In this talk I will survey the status of, and the most recent advances concerning, the questions of global regularity of solutions to the three-dimensional Navier-Stokes and Euler equations of incompressible fluids. Furthermore, I will also present recent global regularity results concerning certain three-dimensional geophysical flows, including the three-dimensional viscous ”primitive equations” of oceanic and atmospheric dynamics.

back to top