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Math Colloquia: Fall 2006

Friday September 08

No talk today

Friday September 15

Title An elastic rod model with singular nonlocal self-interaction
Speaker Thomas I. Seidman
Department of Mathematics and Statistics
UMBC
http://www.math.umbc.edu/~seidman/

Abstract:
The constitutive law for stress/strain in elasticity models the local self-interaction of the rod in a now-standard way. On the other hand, if the rod curves around in space we will have a nonlocal self-interaction as it tries to touch or pass through itself. We will model this as involving a repulsive (electrostatic?) force which then blows up as the parts of the rod get very close. Our problem here is to model the situation so as to be able to prove existence of some energy-minimizing stationary configuration, necessarily without any self-contact.

This is joint work with Kathleen Hoffman

Friday September 22

Title Taming Infinity
Speaker Manil Suri
Department of Mathematics and Statistics
UMBC
http://www.math.umbc.edu/~suri

Abstract:
We present a proposal for a show entitled, “Taming Infinity” which is being developed as an exhibition for the UMBC Library gallery for Spring, 2009. This is in conjunction with collaborators from the Visual Arts Department. The show falls into the category of “Informal Science Education,” and is aimed primarily at non-mathematicians (the university community at large, as well as middle and high school students). The idea is to explore infinity in artistic and historical contexts before moving on to link it to the foundations of Calculus. The talk will focus on some of the issues that can arise in such outreach projects. A discussion on what mathematical content would be most suitable for the intended audience will conclude the presentation.

Friday September 29

11:00am–noon

Title Diagnosing Climate Sensitivity with help from the Fluctuation Dissipation Theorem
Speaker Daniel Kirk-Davidoff
Department of Atmospheric and Oceanic Science
University of Maryland College Park
http://www.atmos.umd.edu/~dankd

Abstract:
The availability of high spatial and spectral resolution infrared radiances from instruments such as AIRS places renewed emphasis on the development of analytic tools for the comparison of model output and climate data. It has previously been shown that lag-covariance based statistical measures, suggested by the Fluctuation Dissipation Theorem (FDT), may allow estimation of climate sensitivity in a climate model. Here we use a simple climate model to test the utility of this approach. We analyze the behavior of the model with adjustable heat capacity in two surface layers, subject to various stochastic forcings and for various climate sensitivities, modulated by albedo and water vapor feedbacks. We compare the equilibrium model sensitivity to these forcings with the sensitivity derived using the lag-covariance based measures, in order to demonstrate the precision and accuracy of these methods as a function of model parameter settings and time-series length. We show that these measures can produce accurate estimates of climate sensitivity, but that it usually requires at least as long as the model adjustment time to make an accurate prediction. For shorter time series, the lag-covariance-based estimators are influenced by both climate sensitivity and heat capacity.

Although such estimators do not allow accurate diagnosis of climate sensitivity, they make an extremely attractive basis for model-data comparison, since a useful model of climate must successfully reproduce both climate sensitivity and heat capacity. Using reanalysis data from the European Center for Medium Range Weather Forecasting, and model data from two independent modeling groups, we show that the lag-covariance based measures reveal inconsistencies between model output and data that are not revealed by simpler statistics.

Friday October 06

Title Pyroclastic Flows and Granular Avalanches: Modelling, Computing and Uncertainty
Speaker E. Bruce Pitman
Department of Mathematics
State University of New York, Buffalo
http://www.math.buffalo.edu/~pitman/

Abstract:
High-fidelity computational simulation of geophysical mass flows can be an invaluable tool in planning strategies for hazard risk mitigation. Accuracy and reliability of the model predictions are critical factors for success. In this talk we present a new simulation tool, TITAN2D, for solving model equations describing dry granular avalanches. Highlights of this methodology include the use of a depth- averaged model of the basic conservation laws, and an adaptive grid Godunov solver to solve the resulting PDEs. The software is designed to run on distributed memory supercomputers, but can also be run on desktop machines. A special feature of TITAN is its use of digital elevation data. TITAN dynamically refines the grid and input data to finer resolutions to better capture features as the flow evolves. Our simulations are validated using quantitative and qualitative comparisons to tabletop experiments and data from field observations.

Friday October 13

Title Multiple Equilibria in Biochemical Reaction Networks
Speaker Gheorghe Craciun
Department of Mathematics and Department of Biomolecular Chemistry
University of Wisconsin-Madison
http://www.math.wisc.edu/~craciun/

Abstract:
In nature there are millions of distinct networks of biochemical reactions that might present themselves for study at one time or another. Each reaction network gives rise to its own system of mass-action differential equations. These are usually high dimensional, nonlinear, and have many unknown parameters. Nevertheless, each reaction network induces its differential equations (up to parameter values) in a precise way. This raises the possibility that qualitative properties of the induced differential equations might be tied directly to reaction network structure. We show that reaction diagrams, similar to those that biochemists usually draw, carry subtle information about a reaction network’s capacity to exhibit multiple equilibria. We will point out connections with open problems in algebraic geometry and graph theory.

References:

  1. Gheorghe Craciun, Yangzhong Tang, Martin Feinberg (2006). Understanding bistability in complex enzyme-driven reaction networks, PNAS 103:23, 8697–8702.
  2. Gheorghe Craciun, Martin Feinberg (2005). Multiple Equilibria in Complex Chemical Reaction Networks: I. The Injectivity Property, SIAM Journal on Applied Mathematics 65:5, 1526–1546.
  3. Gheorghe Craciun, Martin Feinberg (2006). Multiple Equilibria in Complex Chemical Reaction Networks: II. The Species-Reactions Graph, SIAM Journal on Applied Mathematics 66:4, 1321–1338.

Friday October 20

No talk today due to conflict with Statistics Colloquium

Friday October 27

11:00am–noon

Title Nonholonomic Mechanics and Control
Speaker Anthony Bloch
Department of Mathematics
University of Michigan
http://www.math.lsa.umich.edu/~abloch

Abstract:
In this talk I will discuss the relationship between nonholonomic mechanics and nonlinear control theory. Nonholonomic mechanics concerns the mechanics of systems with velocity constraints and has important applications to robotics and vehicle control. Systems subject to nonholonomic constraints have natural links to nonlinear control systems as the constraints often induce good controllability properties. I will discuss the important distinction between kinematic and dynamic nonholonomic systems and will describe the different optimal control problems that arise for these two classes of systems. I will discuss stability and stabilization of nonholonomic systems and show how one can get asymptotic stability in certain classes of nonholonomic systems even in the absence of external dissipation. An interesting related issue is that momentum is not necessarily conserved in nonholonomic systems even in the absence of external forces or torques. I will illustrate these ideas with a number of simple examples.

Biosketch: Dr Anthony Bloch is a professor and department chair at the mathematics dept of University of Michigan. His expertise is in the area of geometry and dynamics of Hamiltonian systems, and dynamics and control of mechanical systems. He holds two masters degrees; one from Caltech in Physics, one from Cambridge University in control engineering and operations research; and a PhD from Harvard University in Applied Mathematics. He has won numerous awards and grants including the Guggenheim fellowship for mathematics of mechanical systems and the presidential young investigator award.

Friday November 03

Title Computational Geometric Mechanics and its Applications to Geometric Optimal Control Theory
Speaker Melvin Leok
Department of Mathematics
Purdue University
http://www.math.purdue.edu/~mleok/

Abstract:
The geometric approach to mechanics serves as the theoretical underpinning of innovative control methodologies in geometric control theory. These techniques allow the attitude of satellites to be controlled using changes in its shape, as opposed to chemical propulsion, and are the basis for understanding the ability of a falling cat to always land on its feet, even when released in an inverted orientation.

We will discuss the application of geometric structure-preserving numerical schemes to the control of the 3D pendulum system, and more generally, the applications of discrete mechanics and geometry to the discretization of optimal control problems. In particular, we consider Lie group variational integrators, which are based on a discretization of Hamilton’s principle that preserves the Lie group structure of the configuration space, without the use of local charts, reprojection, or constraints.

In addition, we will introduce a numerically robust shooting based optimization algorithm that relies on the conservation properties of geometric integrators to accurately compute sensitivity derivatives, thereby yielding an optimization algorithm for the control of mechanical systems that is exceptionally efficient. The role of geometric phases in these control algorithms will also be addressed.

This is joint work with Anthony Bloch (Math, UM), Mathieu Desbrun (CS, Caltech), Anil Hirani (CS, UIUC), Islam Hussein (Aero, UM), Taeyoung Lee (Aero, UM), Jerrold Marsden (CDS, Caltech), N. Harris McClamroch (Aero, UM), Amit Sanyal (MAE, ASU), Alan Weinstein (Math, Berkeley), and Dmitry Zenkov (Math, NCSU).

The research has been supported in part by NSF grant DMS-0504747.

Biosketch: Melvin Leok is a tenure-track assistant professor of mathematics at Purdue University, West Lafayette, where his research is supported in part by a National Science Foundation applied mathematics grant. Prior to joining Purdue, he was a T.H. Hildebrandt research assistant professor of mathematics at the University of Michigan, Ann Arbor, where he received a Horace H. Rackham Faculty Fellowship and Grant, and a Margaret and Herman Sokol Spring/Summer Research Grant.

He received his B.S. with honors and M.S. in Mathematics in 2000, and his Ph.D. in Control and Dynamical Systems with a minor in Applied and Computational Mathematics under the direction of Jerrold Marsden in 2004, all from the California Institute of Technology.

His primary research interests are in computational geometric mechanics, computational geometric control theory, discrete geometry, and structure-preserving numerical schemes, and particularly how these subjects relate to systems with symmetry and multiscale systems.

He was the recipient of the SIAM Student Paper Prize, and the Leslie Fox Prize (second prize) in Numerical Analysis, both in 2003, for his work on Foundations of Computational Geometric Mechanics. While a doctoral student at Caltech, he held a Poincaré Fellowship (2000–2004), a Josephine de Kármán Fellowship (2003–2004), an International Fellowship from the Agency for Science, Technology, and Research (2002–2004), a Tau Beta Pi Fellowship (2000–2001), and a Tan Kah Kee Foundation Postgraduate Scholarship (2000).

As a Caltech undergraduate, he received the Loke Cheng-Kim Foundation Scholarship (1996–2000), the Carnation Scholarship (1998–2000), the Herbert J. Ryser Scholarship (1999), the E.T. Bell Undergraduate Mathematics Research Prize (1999), and the Jack E. Froehlich Memorial Award (1999).

Friday November 10

11:00am–noon, PP 105

Title Increasing the Number of Minority Ph.D.s in the Sciences
Speaker David Manderscheid
Department of Mathematics
University of Iowa
http://www.math.uiowa.edu/~mander

Abstract:
Over twenty percent of the graduate students in Mathematics at the University of Iowa are US minorities from groups underrepresented in Mathematics. This year we expect that four US minority students will earn Ph.D.s in Mathematics at Iowa, over ten percent of the total nationally. In 2005 the Department received the Presidential Award for Excellence in Science, Mathematics and Engineering Mentoring for its work toward increasing the number of US minority Ph.D.s in Mathematics. In this talk I will discuss our efforts toward this end. These efforts include the building and fostering of community, intensive mentoring by both faculty and peers, and the identification and nurturing of previously untapped talent. I will also discuss our efforts to institutionalize the changes that we have made. Finally I will discuss the replicability of our program. Throughout I will emphasize the benefits that accrue to all students.

Friday November 17

Title 2-Dimensional Sloshing Flows for the Shallow Water Equations
Speaker James M. Greenberg
Department of Mathematical Sciences
Carnegie Mellon University
http://www.math.cmu.edu/people/fac/greenberg.html

Abstract:
In this talk I’ll discuss “sloshing” 2-Dimensional flows for the shallow-water equations. The model describes the motion of a finite volume of viscous fluid taking place in a container whose bottom is described by a parabolic like surface z = a(x,y), where a → ∞ as x2 + y2 → ∞. The model includes gravity, coriolis, and viscous forces.

Friday November 24

Thanksgiving

Friday December 01

Title Modelling with discontinuities
Speaker Thomas I. Seidman
Department of Mathematics and Statistics
UMBC
http://www.math.umbc.edu/~seidman/

Abstract:
A considerable variety of problems involve rapid transitions, approximately interpretable as discontinuities. While these are often properly modelled as multiscale problems, we may not know much about the fine-scale model and hope that is only minimally relevant, subsumed by some macroscopic `rule’.

The talk will be largely expository, discussing how one could approach such situations in the contrasting contexts of descriptive and prescriptive modelling.

Friday December 08

No talk today