Departmental Colloquium

The absolute Galois group of a number field is a mysterious object, that one can try to understand by means of its representations. It is known that holomorphic cuspidal modular newforms are, in a suitable sense, a source of many p-adic Galois representations. More generally, it is conjectured that also torsion classes in the coherent cohomology of Shimura varieties have attached Galois representations, with prescribed local properties. I will give an introduction to these themes, and present results obtained in collaboration with Matthew Emerton and Liang Xiao toward the conjecture.

Since the revolution of molecular biology in the early 1980s, mathematical modeling has been widely used to understand complex biological systems, which consist of non-linear and stochastic biochemical interactions. Typically, the process of applying mathematical models to biological systems includes mathematical representation of biological systems, model fitting to data, analysis and simulations, and experimental validation. In this talk, I will describe our efforts to develop and integrate mathematical tools across the different steps of the modeling process. I will also discuss the shortcomings of our present approach and how they point to the parts of current toolbox of mathematical biology that need further mathematical development.

In the theory of finite groups, there are basic but subtle differences in the behavior of ordinary representations and of projective representations. After introducing these elementary phenomena, I will discuss their ramifications in an `arithmetic' setting, where abstract groups are replaced by Galois groups of field extensions of the rational numbers. Questions then naturally arise in `classical' Galois theory that seem to be impossible to understand within this framework. But an extension of classical Galois theory envisioned by Grothendieck--motivic Galois theory, an arithmeto-geometric subject in which the classical theory amounts to the study of zero-dimensional algebraic varieties--provides exactly the right language for thinking about these questions. There are many reasons for embracing this motivic Galois formalism, but I believe the one to be sketched here is among the most concrete and accessible. I will end by describing the deep problems that arise from transposing our initial question about projective representations to this motivic Galois setting.

Understanding the interplay between network structure and dynamical properties is a key problem in systems theory. Although much progress has been made to understand systems, there is still no framework that focuses on the system itself instead of the representation of its components. In this talk, I will present frameworks to study the problem of predicting dynamical properties from network structure and the problem of inferring network structure from dynamical properties. These approaches focus on the way the components of the system interact instead of their representation; thus, this "systems mathematics" approach is a formalization of the systems biology paradigm. To infer dynamical properties of an ODE from its structure, we use the topological features of the wiring diagram to first infer dynamical properties of a "Boolean representation" of the original system. Then, based on the relationship between ODE and Boolean systems, we can use the predicted "Boolean dynamics" to infer dynamical features of the original ODE. To infer network structure from time series data, we first encode the data as an algebraic object. Then, using algebraic tools we can find the decomposition of this algebraic object, which gives precisely the best networks that are consistent with the time series data. Furthermore, given enough data, the method guarantees perfect recovery of the network structure. These approaches use tools from dynamical systems, algebraic geometry, and graph theory. February 10: SPECIAL COLLOQUIUM Room and Day: JWB 335, Tuesday

A new methodology is discussed for the fitting of non-stationary time series that exhibit non-linearity, asymmetry, local persistence and changes in location, scale and shape of the underlying distribution. To do this, model selection techniques are developed for the class of piecewise stationary quantile autoregressive processes. The best model is defined in terms of minimizing a minimum description length criterion derived from an asymmetric Laplace likelihood. Its practical minimization is achieved with the use of genetic algorithms. If the data generating process follows indeed a piecewise quantile autoregressive structure, it can be shown that the proposed method is consistent for estimating the number of pieces and the autoregressive parameters. Empirical work suggests that the proposed method performs well in finite samples. The talk is based on joint work with R. Cheung, T. Lee and M. Zhong.

A classical theorem due to H. Weyl states that if P is a real polynomial such that at least one of its coefficients (other than the constant term) is irrational, then the sequence P(n), n=1,2,... is uniformly distributed mod 1. After briefly reviewing various approaches to the proof of Weyl's theorem, we will discuss some recent extensions which involve "generalized polynomials", that is, functions which are obtained from the conventional polynomials by the use of the greatest integer function, addition and multiplication. We will explain the role of dynamical systems on nil-manifolds in obtaining these results and discuss the intrinsic connection between the generalized polynomials and the polynomial extensions of Szemeredi's theorem on arithmetic progressions. We will conclude with formulating and discussing some natural open problems and conjectures.

A mean curvature flow is an evolving submanifold M_t whose velocity is equal to its mean curvature. Mean curvature flow is in some respects the most natural evolution equation for a moving submanifold: it is the gradient flow of the area functional, as well as the analog of the heat equation for submanifolds. The lecture will survey mean curvature flow for a general audience.

Blood is a suspension of particles of various shapes, sizes and mechanical properties and the distribution of these particles during blood flow is important in many contexts. Red blood cells (RBCs) tend to migrate toward the center of a blood vessel, leaving a so-called cell-free layer at the vessel wall, while white blood cells (WBCs) and platelets are preferentially found near the walls, a phenomenon called margination that is critical for the physiological responses of inflammation and hemostasis. Potential beneficial effects on hemodynamics arise from addition of high molecular weight long-chain polymer molecules known as drag-reducing additives (DRAs) to blood; one effect of these additives is the reduction of the cell- free-layer thickness. Additionally, the segregation properties of WBCs, platelets, and RBCs can be employed for their separation or detection in microfluidic devices. Finally, drug delivery particles in the bloodstream will also undergo segregation phenomena – the influence of these phenomena on the efficacy of such particles is unknown. This talk describes efforts to gain a systematic understanding of flow- induced segregation phenomena in blood and other complex mixtures, using a combination of theory and direct simulations of flowing suspensions. Two specific issues are addressed here: (1) the origin of the margination phenomenon and its dependence on the relative properties of the different types of suspended particles in a mixture and (2) the effect of DRAs on the formation of the cell-free layer. A kinetic theory model based on pair collisions and wall-induced hydrodynamic migration can capture the key effects observed in direct simulations, including a “drainage transition” in which one component is completely depleted from the bulk of the flow. In the case of polymer additives, the experimentally observed thinning of the cell-free layer is reproduced in simulations and the mechanism underlying it is described. Having in hand an understanding of the mechanisms underlying these phenomena now allows more rational approaches to development of quantitative models of them and processes that exploit them. This knowledge will also lead to a better understanding of the consequences of these phenomena in physiology and medicine.

In the last 35 years renormalization became one of the main tools in the theory of dynamical systems. We shall discuss renormalization theory in the simplest setting of circle dynamics, and present the results in the cases of diffeomorphisms, critical circle maps, and maps with breaks. The relation between hyperbolicity of renormalizations and rigidity theory will be also discussed.

Most living or social systems consist of a large number of agents interacting through elementary rules involving only neighbouring agents. In spite of their simplicity, these interactions drive the system towards a self-organized coherent collective behavior. The emergence of collective dynamics poses many mathematical challenges which will be outlined in this talk. We will use the example of the Vicsek model (in which self-propelled particles interact through local alignment) to illustrate some of these challenges and examples of applications to real biological or social systems will be presented. Fall 2014