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Wednesday, March 25, 2009 - 13:00 ,
Location: Skiles 255 ,
Junping Wang ,
NSF ,
Organizer: Haomin Zhou

This talk will first review domain decomposition methods for second order elliptic equations, which should be accessible to graduate students. The second part of the talk will deal with possible extensions to the Stokes equation when discretized by finite element methods. In particular, we shall point out the difficulties in such a generalization, and then discuss ways to overcome the difficulties.

Wednesday, March 25, 2009 - 11:00 ,
Location: Skiles 255 ,
Ruslan Rafikov ,
Medical College of Georgia ,
Organizer:

The stress condition calls for an adequate activity of key enzymatic systems of cellular defense. Massive protein destabilization and degradation is occurring in stressed cells. The rate of protein re-synthesis (DNA->RNA->protein) is inadequate to adapt to rapidly changing environment. Therefore, an alternative mechanism should exist maintaining sufficient activity of defense enzymes. Interestingly, more than 50% of enzymes consist of identical subunits which are forming multimeric interface. Stabilization of multimers is important for enzymatic activity. We found that it can be achieved by the formation of inter-subunit covalent bridges in response to stress conditions. It shows an elegance of the structure design that directs selective subunits linkage and increases enzyme's robustness and chances of cell survival during the stress. In contrast, modification of aminoacids involved in linkage leads to protein destabilization, unfolding and degradation. These results describe a new instantaneous mechanism of structural adaptation that controls enzymatic system under stress condition.

Series: Other Talks

There is much impressive observational evidence, mainly from the cosmic microwave background (CMB), for an enormously hot and dense early stage of the universe --- referred to as the Big Bang. Observations of the CMB are now very detailed, but this very detail presents new puzzles of various kinds, one of the most blatant being an apparent paradox in relation to the second law of thermodynamics. The hypothesis of inflationary cosmology has long been argued to explain away some of these puzzles, but it does not resolve some key issues, including that raised by the second law. In this talk, I describe a quite different proposal, which posits a succession of universe aeons prior to our own. The expansion of the universe never reverses in this scheme, but the space-time geometry is nevertheless made consistent through a novel geometrical conception. Some very recent analysis of the CMB data, obtained with the WMAP satellite, will be described, this having a profound but tantalizing bearing on these issues.

Series: PDE Seminar

We will give an overview of results on the global existence of solutions to the initial value problem for nonlinear elastic and viscoelastic materials in 3d without boundary. Materials will be assumed to be isotropic, but both compressible and incompressible cases will be discussed. In the compressible case, a key null condition must be imposed to control nonlinear interactions of pressure waves. This necessary assumption is consistent with the physical model. Initial conditions are small perturbations of a stress free reference state. Existence is proven using a fixed point argument which combines energy estimates and with some new dispersive estimates.

Series: ACO Seminar

Low-rank models are frequently used in machine learning and statistics. An important domain of application is provided by collaborative filtering, whereby a low-rank matrix describes the ratings that a large set of users attribute to a large set of products. The problem is in this case to predict future ratings from a sparse subset currently available. The dataset released for the Netflix challenge provides an ideal testbed for theory and algorithms for learning low-rank matrices. Given M, a random n x n matrix of rank r, we assume that a uniformly random subset E of its entries is observed. We describe an efficient procedure that reconstructs M from |E| = O(rn) observed entries with arbitrarily small root mean square error, whenever M is satisfies an appropriate incoherence condition. If r = O(1), the algorithm reconstructs M exactly from O(n log n) entries. This settles a recent open problem by Candes and Recht. In the process of proving these statements, we obtain a generalization of a celebrated result by Friedman-Kahn-Szemeredi and Feige-Ofek on the spectrum of sparse random matrices. [Based on joint work with R. H. Keshavan and S. Oh]

Series: CDSNS Colloquium

In this talk we will review results on local entropy theory for the past 15 years, introduce the current development and post some open questions for the further study.

Monday, March 23, 2009 - 13:00 ,
Location: Skiles 255 ,
Shigui Ruan ,
University of Miami ,
Organizer: Yingfei Yi

Understanding the seasonal/periodic reoccurrence of influenza will be very helpful in designing successful vaccine programs and introducing public health interventions. However, the reasons for seasonal/periodic influenza epidemics are still not clear even though various explanations have been proposed. In this talk, we present an age-structured type evolutionary epidemiological model of influenza A drift, in which the susceptible class is continually replenished because the pathogen changes genetically and immunologically from one epidemic to the next, causing previously immune hosts to become susceptible. Applying our recent established center manifold theory for semilinear equations with non-dense domain, we show that Hopf bifurcation occurs in the model. This demonstrates that the age-structured type evolutionary epidemiological model of influenza A drift has an intrinsic tendency to oscillate due to the evolutionary and/or immunological changes of the influenza viruses. (based on joint work with Pierre Magal).

Series: Graph Theory Seminar

Many results in asymptotic extremal combinatorics are obtained using just a handful of instruments, such as induction and Cauchy-Schwarz inequality. The ingenuity lies in combining these tools in just the right way. Recently,
Razborov developed a flag calculus which captures many of the available techniques in pure form, and allows one, in particular, to computerize the search for the right combination. In this talk we outline the general approach and describe its application to the conjecture that a digraph with minimum outdegree n/3 contains a directed triangle. This special case of the Caccetta-Haggkvist conjecture has been extensively investigated in the past. We show that a digraph with minimum outdegree a*n contains a directed triangle for a = 0.3465. The proof builds on arguments used to establish previously known bounds, due to Shen from 1998 (a = 0.3542) and Hamburger, Haxell and Kostochka from 2008 (a = 0.3531). It consists of a combination of ~80 computer generated inequalities. Based on joint work with Jan Hladky and Daniel Kral.

Series: PDE Seminar

We discuss the global regularity vs. finite time breakdown in Eulerian dynamics, driven by different models of nonlinear forcing. Finite time breakdown depends on whether the initial configuration crosses intrinsic, O(1) critical thresholds (CT). Our approach is based on spectral dynamics, tracing the eigenvalues of the velocity gradient which determine the boundaries of CT surfaces in configuration space. We demonstrate this critical threshold phenomena with several n-dimensional prototype models. For n=2 we show that when rotational forcing dominates the pressure, it prolongs the life-span of sub-critical 2D shallow-water solutions. We present a stability theory for vanishing viscosity solutions of the 2D nonlinear "pressureless" convection. We revisit the 3D restricted Euler and Euler-Poisson equations, and obtain a surprising global existence result for a large set of sub-critical initial data in the 4D case.

On the theory and applications of the longtime dynamics of 3-dimensional fluid flows on thin domains

Series: CDSNS Colloquium

The current theory of global attractors for the Navier-Stokes equations on thin 3D domains is motivated by the desire to better understand the theory of heat transfer in the oceans of the Earth. (In this context, the thinness refers to the aspect ratio - depth divided by expanse - of the oceans.) The issue of heat transfer is, of course, closely connected with many of the major questions concerning the climate. In order to exploit the tools of modern dynamical systems in this study, one needs to know that the global attractors are "good" in the sense that the nonlinearities are Frechet differentiable on these attractors. About 20 years ago, it was discovered that on certain thin 3D domains, the Navier-Stokes equations did possess good global attractors. This discovery, which was itself a major milestone in the study of the 3D Navier-Stokes equations, left open the matter of extending the theory to cover oceanic-like regions with the appropriate physical boundary behavior. In this lecture, we will review this theory, and the connections with climate modeling, while placing special emphasis on the recent developments for fluid flows with the Navier (or slip) boundary conditions