Academic Year 2007-2008
| November 28-30, 2007 | |
| Speaker | Prof. James Keener (University of Utah) |
| Titles | Arrhythmias by Dimension; How Cells Make Measurements; Modelling Stochastic Calcium Oscillations |
| Abstracts | Arrhythmias by Dimension: Abnormalities of function of the cardiac conduction system are the cause of death of hundreds of people every day. For that reason, the study of cardiac arrhythmias is of great interest from a medical and scientific perspective. However cardiac arrhythmias are also interesting for mathematical reasons because the cardiac conduction system can be viewed as a dynamical system and the variety of its behaviors can be studied from the viewpoint of dynamical systems theory. In this talk, I will give a classification of cardiac arrhythmias that is based on spatial dimension, and is therefore useful for mathematicians, but probably not (as much) for physicians. I will describe examples of zero dimensional arrhythmias (abnormalities of single cells), one dimensional arrhythmias (Wolff-Parkinson White Syndrome), two dimensional arrhythmias (atrial flutter), and three dimensional arrhythmias (ventricular tachicardia and fibrillation). |
| How Cells Make Measurements: A fundamental problem of cell biology is to understand how cells make measurements and then make behavioral decisions in response to these measurements. The full answer to this question is not known but there are some underlying principles that are coming to light. The short answer is that the rate of molecular diffusion contains quantifiable information that can be transduced by biochemical feedback to give control over physical structures. In this talk, this principle will be illustrated by two specific examples of how rates of molecular diffusion contain information that is used to make a measurement and a behavioral decision. Example 1: Bacterial populations of P. aeruginosa are known to make a decision to secrete polymer gel on the basis of the size of the colony in which they live. This process is called quorum sensing and only recently has the mechanism for this been sorted out. It is now known that P. aeruginosa produces a chemical whose rate of diffusion out of the cell provides information about the size of the colony which when coupled with positive feedback gives rise to a hysteretic biochemical switch. Example 2: Salmonella employ a mechanism that combines molecular diffusion with a negative feedback chemical network to "know" how long its flagella are. As a result, if a flagellum is cut off, it will be regrown at the same rate at which it grew initially. |
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| Modelling Stochastic Calcium Oscillations: Calcium oscillations are an important means of signalling in many cell types. These oscillations have been studied extensively using ordinary differential equation (ode) models, some with a high degree of detail. These models generally show that the onset of oscillations is via a subcritical Hopf bifurcation. The experimental data, however, show that calcium release is not nearly as regular as the ode models suggest, but is highly stochastic in nature. In fact, these data show that oscillations are initially very irregular with large average interspike interval and large variance, eventually settling into a regular oscillation as the bifurcation parameter (in this case [IP_3]) is increased. The reason for the discrepancy between data and models is that calcium release is via events that are fundamentally stochastic in time and discrete in space. Furthermore, the assumptions that permit a whole cell ode model, namely uniformly distributed calcium and the law of large numbers, do not apply under many physiological conditions. In this talk, I will describe our recent attempts to develop and analyze mathematical models of the events underlying stochastic calcium oscillations, including a model for spark generation, a model for the spark to wave transition leading to a model of stochastic whole cell calcium release. The result of these is a description of the onset of calcium oscillations that shows improved agreement with experimental observations. |
| February 25-28, 2008 | |
| Speaker | Prof. Bernard Shiffman (Johns Hopkins University) |
| Titles | Random Polynomials and Waves; Zeros of random polynomials on Cn; Off-Diagonal Asymptotics of the Szego Kernel on Complex Manifolds and Applications to Random Zeros and Geometry |
| Abstracts | Random Polynomials and Waves: We discuss the values of random real and complex functions, focusing on harmonic functions and their complex analogues: complex polynomials and holomorphic functions. We regard zeros of complex polynomials as the nodes of "complex waves". We shall discuss the Kac-Rice and Edelman-Kostlan-Sodin formulas for the average number of zeros of random functions. We conclude with a survey of some recent results on the distributions of zeros and critical points of random holomorphic functions. |
| Zeros of random polynomials on Cn: We discuss the distribution of the simultaneous zeros of random polynomials of several complex variables. The basic theme is how the distribution of zeros of typical polynomial systems depends on the probability measures. We shall consider two distinct types of Gaussian probability measures on spaces of polynomials. The first are obtained by orthonormalizing on a compact set K, and we show that in this case the zeros tend to be distributed uniformly with respect to the pluri-complex Green's function on the complement of K, as the degree tends to infinity. Secondly, we consider Gaussian random polynomials with coefficients in a given Newton polytope P, and we obtain a localized version of the Bernstein-Kouchnirenko theorem for the total number of simultaneous zeros of such polynomials. | |
| Off-diagonal asymptotics of the Szego kernel on complex manifolds and applications to random zeros and geometry: The Szego kernels for a positive Hermitian line bundle L over a compact Kaehler manifold M are the kernels for the orthogonal projections onto the spaces of holomorphic sections of the tensor powers L^N of the line bundle. We shall discuss the off-diagonal asymptotics of these Szego kernels and describe applications of these asymptotics to the distribution of random zeros and to complex geometry. For example, we give an asymptotic formula for the variance of the number of simultaneous zeros of m (=dim M) random holomorphic sections of L^N in a fixed domain U, showing that the number of zeros in U is highly unlikely to deviate much from the average. | |
| March 25-27, 2008 | |
| Speaker | Professor David Jerison (MIT) |
| Title | Elliptic Variational Problems |
| Abstract | Elliptic Variational Problems: In these lectures, I will discuss properties of level surfaces of solutions to semilinear elliptic equations of the form Δ u=f(u) for various functions f(u). Many features of level surfaces are studied by rescaling to obtain, in the limit, global solutions defined on all of Euclidean space. There is a deep and fruitful analogy between these level surfaces and minimal surfaces as suggested by E. De Giorgi. The approach to this equation using the calculus of variations is to study critical points of the functional
It is not hard to see that the critical point (Euler-Lagrange) equation for J is a generalized or weak form of the one above with f(u) = 2F(u). The first lecture will focus on one example, the case of the Heaviside function I will describe joint work with L. A. Caffarelli and C. E. Kenig and with D. De Silva. While the deep connection between free boundaries and minimal surfaces has been mined for some time, the smoothness of the free boundaries in dimension 3 was proved only recently, using the second variation of the functional J. The study of the second variation also leads to singular energy mimimizers, which demonstrate that there is a critical dimension for regularity theory for free boundaries like the one for minimal hypersurfaces. The key to further progress is to understand more about global minimizing solutions. I will discuss, in particular, the analogue of the classical Bernstein problem for minimal surfaces. In later lectures I will consider two-phase problems. In the two-phase case (when v is allowed to change sign) the solution u can represent the pressure of a fluid flow, and the free boundary a streamline of singular vorticity. The basic starting point for the existence and regularity theory is a beautiful monotonicity formula due to Alt, Caffarelli and Friedman. I will explain this formula and more recent, related inequalities and other aspects of one and two-phase regularity theory. |
(so that f is twice the delta function). The seminal work in this case is the 1980 paper by Alt and Caffarelli. The correct interpretation of the Euler-Lagrange equation is as a so-called free boundary problem. The minimization problem for J in the one-phase case v ≥ 0 corresponds to the question of determining the shape of a layer of insulation around an oven that insulates best. Let the domain 
, the complement of the oven K. The minimizing function u represents the temperature in equilibrium around the oven, normalized to be positive inside the insulating material and zero at the outer boundary of the insulation, the “free boundary” that we want to find. In addition to satisfying the equilibrium temperature condition Δ u=f(u) in the “positive phase” set (where u > 0), the minimizer u has slope 1 on the boundary where the insulation meets the air.
