Series: ACO Student Seminar
In the first part of the talk, I am going to give an introduction and overview of linear and semidefinite programming hierarchies. I will mostly review known integrality gaps for such programs and try to give an intuition of why we currently lack strong techniques for designing rounding algorithms. In the second part of the talk I will focus on the stronger SDP Lasserre hierarchy. In contrast with the previous LP and SDP hierarchies, very few examples of integrality gap instances are known to date. I will present a recent technique for designing such instances and discuss open problems in the area.
Series: School of Mathematics Colloquium
In this talk I will outline recent results of G-Q Chen, Dehua Wang, and me on the problem of isometric embedding a two dimensional Riemannian manifold with negative Gauss curvature into three dimensional Euclidean space. Remarkably there is very pretty duality between this problem and the equations of steady 2-D gas dynamics. Compensated compactness (L.Tartar and F.Murat) yields proof of existence of solutions to an initial value problem when the prescribed metric is the one associated with the catenoid.
Series: ACO Colloquium
One of the most interesting aspects of the Linear Complementarity Problem (LCP) is its range from relatively easy problems such as linear and convex quadratic programming problems to NP-hard problems. A major effort in LCP theory had been the study of the Lemke algorithm, a simplex-like algorithm which is guaranteed to terminate in finite number of iterations but not necessarily with a solution (or a certificate that no solution exists). Over the years, many subclasses of LCP were proven to be workable by the Lemke algorithm. Those subclasses were often characterized as ‘nice’ even when no polynomial upper bound for the algorithm was known to exist. In fact, for most of these classes, instances with exponential number of steps had been discovered. In this talk, I’ll discuss the close connection between these classes and the PPAD (Polynomial-time Parity Argument Directed) complexity class. The discovery that computing Nash equilibrium (which is an LCP) is PPAD complete is particularly significant in analyzing the complexity of LCP. I’ll also discuss the LCP reduction-via-perturbation technique and its relation to the PPAD class and the Lemke Algorithm. This talk is based on a joint work with Sushil Verma.
Mathematical and experimental considerations of density and physiological state effects on antimicrobial susceptibilityWednesday, April 1, 2009 - 11:00 , Location: Skiles 255 , Klas Udekwu , Emory University , Organizer:
The treatment of bacterial infections with antibiotics is universally accepted as one of (if not THE) most significant contributions of medical intervention to reducing mortality and morbidity during last century. Despite their widespread use over this extended period however, basic knowledge about how antibiotics kill or prevent the growth of bacteria is only just beginning to emerge. The dose and term of antibiotic treatment has long been determined empirically and intuitively by clinicians. Only recently have antibiotic treatment protocols come under scrutiny with the aim to theoretically and experimentally rationalize treatment protocols. The aim of such scrutiny is to design protocols which maximize antibiotics’ efficacy in clearing bacterial infections and simultaneously prevent the emergence of resistance in treated patients. Central to these endeavors are the pharmacodynamics, PD (relationship between bug and drug), and the pharmacokinetics, PK (the change antibiotic concentration with time) of each bacteria : drug : host combination. The estimation of PD and PK parameters is well established and standardized worldwide and although different PK parameters are commonly employed for most of these considerations, a single PD parameter is usually used, the minimum inhibitory concentration (MIC). MICs, also utilized as the criteria for resistance are determined under conditions that are optimal to the action of the antibiotic; low densities of bacteria growing exponentially. The method for estimating MICs which is the only one officially sanctioned by the regulatory authority (Clinical and Laboratory Standards Institute) defines conditions that rarely obtain outside of the laboratory and virtually never in the bacteria infecting mammalian hosts. Real infections with clinical symptoms commonly involve very high densities of bacteria, most of which are not replicating. These populations are rarely planktonic but rather reside as colonies or within matrices called biofilms which sometimes include other species of bacteria. In the first part of my talk, I will present newly published data that describes the pharmacodynamic relationship between the sometimes pathogenic bacterium Staphylococcus aureus and antibiotics of six classes and the effects of cell density on MICs. By including density dependent MIC in a standard mathematical model of antibiotic treatment (from our lab), I show that this density-dependence may explain why antibiotic treatment fails in the absence of inherited resistance. In the second part of my talk I will consider the effects of the physiological state of clinical isolates of S. aureus on their susceptibility to different antibiotics. I present preliminary data which suggests that the duration of an infection may contribute adversely to an antibiotics chance of clearing the infection. I conclude with a brief discussion of the implications of the theoretical and experimental results for the development of antibiotic treatment protocols. As a special treat, I will outline problems of antibiotic treatment that could well be addressed with some classy mathematics.
Series: PDE Seminar
We study the problem of constructing systems of hyperbolic conservation laws with prescribed eigencurves, i.e. the eigenvector fields of the Jacobian of the flux are given. We formulate this as a (typically overdetermined) system of equations for the eigenvalues-to-be. Equivalent formulations in terms of differential and algebraic-differential equations are considered. The resulting equations are then analyzed with techniques from exterior differential systems (Cartan-Kahler theory). The cases of 2x2- and 3x3-systems can be treated in detail, and explicit examples show that already the 3x3-case is fairly complex. We also analyze general rich systems. We also characterize conservative systems with the same eigencurves as compressible gas dynamics. This is joint work with Irina Kogan (North Carolina State University).
Series: CDSNS Colloquium
Allocation of service capacity ('staffing') at stations in queueing networks is both of fundamental and practical interest. Unfortunately, the problem is mathematically intractable in general and one therefore typically resorts to approximations or computer simulation. This talk describes work in progress with M. Squillante and S. Ghosh (IBM Research) on an algorithm that serves as an approximation for the 'best' capacity allocation rule. The algorithm can be interpreted as a discrete-time dynamical system, and we are interested in uniqueness of a fixed point and in convergence properties. No prior knowledge on queueing networks will be assumed.
Series: Analysis Seminar
The contracted asymptotics for orthogonal polynomials whose recurrence coefficients tend to infinity will be discussed. The connection between the equilibrium measure for potential problems with external fields will be exhibited. Applications will be presented which include the Wilson polynomials.
Series: Geometry Topology Seminar
Let M be a hyperbolic 3-manifold, that is, a 3-manifold admitting a complete, finite volume Riemannian metric of constant section curvature -1. Let S be a Heegaard surface in M, that is, M cut open along S consists of two handlebodies. Our goal is to prove that is the volume of M (denoted Vol(M)) if small than S is simple. To that end we define two complexities for Heegaard surfaces. The first is the genus of the surface (denoted g(S)) and the second is the distance of the surface, as defined by Hempel (denoted d(S)). We prove that there exists a constant K>0 so that for a generic manifold M, if g(S) \geq 76KVol(M) + 26, then d(S) \leq 2. Thus we see that for a generic manifold of small volume, either the genus of S is small or its distance is at most two. The term generic will be explained in the talk.
Monday, March 30, 2009 - 13:00 , Location: Skiles 255 , Richardo March , Istituto per le Applicazioni del Calcolo "Mauro Picone" of C.N.R. , Organizer: Haomin Zhou
We consider ordered sequences of digital images. At a given pixel a time course is observed which is related to the time courses at neighbour pixels. Useful information can be extracted from a set of such observations by classifying pixels in groups, according to some features of interest. We assume to observe a noisy version of a positive function depending on space and time, which is parameterized by a vector of unknown functions (depending on space) with discontinuities which separate regions with different features in the image domain. We propose a variational method which allows to estimate the parameter functions, to segment the image domain in regions, and to assign to each region a label according to the values that the parameters assume on the region. Approximation by \Gamma-convergence is used to design a numerical scheme. Numerical results are reported for a dynamic Magnetic Resonance imaging problem.
Series: Probability Working Seminar
This talk is based in the article titled "On the convergence to equilibrium of Kac’s random walk on matrices" by Roberto Oliveira (IMPA, Brazil). We show how a strategy related to the path coupling method allows us to establish tight bounds for the L-2 transportation-cost mixing time of the Kac's random walk on SO(n).