Spring 2003 Math Sciences colloquium

Math Sciences Colloquia - Spring 2003

All Mathematical Sciences Seminars take place Mondays at 4:15 p.m. in 655 Rhodes Hall with refreshments before the colloquium at 3:45 p.m in 657 Rhodes.

February 10 - Danny Abrams, Theoretical and Applied Mechanics, "Modelling Language Death"
Abstract: According to the Summer Institute of Linguistics' "Ethnologue," the world contains about 6,800 living languages, and linguists estimate that approximately 80% of those will disappear with the current generation of speakers. Many attempts have been made at active preservation of endangered languages, but these have thus far met with mixed results. By looking at a simplified model of the dynamics of language decline and death, it should be possible to anticipate language decline before the situation becomes critical, and perhaps implement more effective preservation mechanisms.
February 17 - NOTE -- Not regular Math Sciences Seminar--Neo Martinez, IGERT Visiting Professor of Nonlinear Systems, San Francisco State University, "Invertebrate-dominated food webs and the structure and nonlinear dynamics of complex ecological networks" 4:00pm in the Corson-Mudd Auditorium.
February 24 - Roland Roeder, Math, "Hamiltonian chaos, homoclinic tangles, and magnetic footprints in tokamaks"
Abstract: Hamiltonian systems and their associated chaos where originally studied in an effort to understand celestial mechanics and the stability of the solar system. However, more recently Hamiltonian dynamics has been used by fusion scientists to study the topology of the magnetic field in a tokamak. Associated with most Hamiltonian systems are complicated webs of stable and unstable manifolds known as homoclinic tangles. Historically, these tangles provided a mechanism for chaos in the dynamics of the solar system. However, more recently, accurate calculations of the "tangles" in the magnetic fields of tokamaks offer predictions of particle escape and the resulting patters of heat buildup on the tokamak wall. Amazingly enough these predictions are EASY to make upon seeing the geometry of the tangle and they offer ideas for experiments to verify these predictions. Although not horribly technical, this talk would be best received by an audience with some physics background.
March 3 - (room unavailable)
March 10 - Katharyn Boyle, CAM, "Hedging a Portfolio of Derivatives by Modeling Cost"
Abstract: We consider the problem of hedging the loss of a given portfolio of derivatives using a set of more liquid derivative instruments. We illustrate why the typical mathematical formulation for this hedging problem is ill-posed. We propose to determine a hedging portfolio by minimizing a proportional cost subject to an upper bound on the the hedge risk; this bound is typically slightly larger than the optimal hedge risk achievable without cost consideration. We illustrate that the optimal hedging portfolio obtained by the proposed method is attractive since it consists of fewer instruments with a comparable risk. Finally, we illustrate the importance of modeling volatility uncertainty in hedge risk minimization.
March 17 - ** SPRING BREAK **
March 24 - Evan Variano, Civil and Environmental Eng., "Evolution of modularity in dynamically-grown networks"
IGERT project, performed jointly with Jonathan McCoy, advised by Hod Lipson and Steve Strogatz
Abstract: We study large networks which are stable, both in the linear dynamics sense and in the sense or robustness to random mutations. Large linearly stable networks are extremely difficult to obtain, but we have done so by the use of a genetic algorithm. The resulting networks exhibit two types of modularity which are intricately related to their dynamical stability. We characterize this modularity, examine how it evolves, and connect it via matrix algebra to linear stability theory. Having such large stable matrices, we are able to further investigate some questions that Robert May posed about the stability of ecological networks. We also discuss possible implications of these results on understanding modularity and complexity in social and corporate networks.
March 31 - Alexei Egorov, Economics, "Forecasting the Joint Probability Density of Bond Yields: Can Affine Models Beat Random Walk?"
Abstract: The numerous empirical studies on affine term structure models have primarily focused on the in-sample fit of historical bond yields and ignored the out-of-sample forecast of future bond yields. Using a new econometric procedure for density forecast evaluation, we provide probably the first comprehensive empirical analysis of the out-of-sample performance of affine models in forecasting the joint conditional probability density of bond yields. We show that although it is extremely difficult to forecast the conditional mean of bond yields, some affine models have reasonably good forecasts of the joint conditional density and they significantly outperform the simple random walk models in density forecast. Our analysis demonstrates the great potential of affine models for financial risk management in fixed-income markets.
April 7 - Siddharth Alexander, CAM, "Towards a Better Portfolio of Derivatives Based on CVaR"
Abstract: Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR) are the most frequently used risk measures in current risk management practice. As an alternative to VaR, CVaR is attractive since it is a coherent risk measure and a convex function of the portfolio instrument holdings. We analyze the problem of computing the optimal VaR and CVaR portfolios. In particular, we illustrate that the VaR and CVaR minimization problems for a portfolio of derivatives are typically ill-posed. For example, the VaR and CVaR minimizations based on delta-gamma approximations of the derivative values typically have an infinite number of solutions. We propose to include cost as an additional preference criterion for the CVaR optimization problem. We show that, with the addition of a proportional cost, it is possible to compute an optimal CVaR portfolio of derivatives with significantly fewer instruments and comparable CVaR and VaR. To handle large-scale portfolio selection problems, an efficient computational method based on a smoothing technique is proposed. Comparison is made with the standard interior point method and simplex method for solving the simulation based CVaR optimization problem.
The only background that could be assumed for this talk is some knowledge about financial derivatives such as options, and their dependency on uncertain risk factors such as stock prices and interest rates.
April 14 - Wen Xu, Electrical and Computer Eng, "Delay-Optimized Robust Transmission of Images over Multiple Channels"
Abstract: Transmission of delay-sensitive images over wireless channels can be hampered by insufficient bandwidth and unreliable channels. These challenges can be overcome by aggregation of multiple channels into a single, higher bandwidth logical channel and by employing strategic error correction. This talk addresses the problem of partitioning image data over multiple channels with unequal characteristics to minimize delay, and the robustness to channel losses and failures is provided by product codes. Data is unequally error protected based on both decoding priorities and channel reliabilities. The proposed product codes outperform a conventional product code due to the adaptivity of the partitions to the channel conditions. Two schemes for product coding are considered and their relative merits are analyzed and compared.
April 21 - Richard Casey, CAM, "Periodic orbits in neural models and algorithms for parameter optimization"
Abstract: Models based on the Hodgkin-Huxley equations tend to have a large number of parameters, many of which are often poorly known. One task frequently faced in modelling is to find parameter regimes that produce certain desired behaviour in the model neuron, typically to match firing properties observed experimentally. The difficulties inherent in this task motivate the development of better computational tools, and in this context I'll talk about algorithms we've been working on to better fit neural models to experimental data. We use automatic differentiation to compute periodic orbits in the model vector field, and combine this with a quasi-Newton algorithm to seek optimal fits between the model and the data. One of the key ideas is to define the distance between periodic orbits computed in the model and a reference orbit based on the data, and minimize this distance function. We give some applications to model refinement as well as data fitting.
April 28 - Mercedes Franco, CAM, "Numerical solutions of some semilinear elliptic problems"
Abstract: We studied the problem (1): -Delta u = lambda*u + u|u|^{p-1}$ in B, u = 0 on the Boundary, where B is the unit ball in R^3, p=5 (p+1 is the Sobolev critical exponent) and lambda is a real constant. Despite its simple form, (1) offers a very rich structure and is a source to many open problems and new ideas. Also it is significantly more difficult than the problem in higher dimensions (n > 3) or the problem when p is subcritical (p < (n+2)/(n-2)), both exhaustively studied. One open question for (1) is about existence of radially symmetric solutions that change sign. Such solutions satisfy (2): - u'' - {2\over r} u' = u^5 + \lambda u if 0 < r < 1, u'(0) = 0, u(1) =0, where u := u(|x|) = u(r). To avoid the singularity present in problem (2), we posed the problem in an ``approximated domain'': for 0 < 1 fixed we considered (3): - u'' - {2\over r} u' = u^5 + \lambda u, if r_0 < r < 1, u'(r_0) = 0, u(1) = 0. We showed that solutions to (2) (\lambda fixed) are limits of the solutions to (3) as r_0 goes to 0. We also studied (2) and (3) from the point of view of bifurcations, followed qualitative behavior along branches and established there the connection between both problems.