Seminars and Colloquia by Series

An efficient numerical method for vesicle simulations

Series
Applied and Computational Mathematics Seminar
Time
Monday, October 27, 2008 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
George Biros – CSE, Georgia Tech
Fluid membranes are area-preserving interfaces that resist bending. They are models of cell membranes, intracellular organelles, and viral particles. We are interested in developing simulation tools for dilute suspensions of deformable vesicles. These tools should be computationally efficient, that is, they should scale well as the number of vesicles increases. For very low Reynolds numbers, as it is often the case in mesoscopic length scales, the Stokes approximation can be used for the background fluid. We use a boundary integral formulation for the fluid that results in a set of nonlinear integro-differential equations for the vesicle dynamics. The motion of the vesicles is determined by balancing the nonlocal hydrodynamic forces with the elastic forces due to bending and tension. Numerical simulations of such vesicle motions are quite challenging. On one hand, explicit time-stepping schemes suffer from a severe stability constraint due to the stiffness related to high-order spatial derivatives and a milder constraint due to a transport-like stability condition. On the other hand, an implicit scheme can be expensive because it requires the solution of a set of nonlinear equations at each time step. We present two semi-implicit schemes that circumvent the severe stability constraints on the time step and whose computational cost per time step is comparable to that of an explicit scheme. We discretize the equations by using a spectral method in space, and a multistep third-order accurate scheme in time. We use the fast multipole method to efficiently compute vesicle-vesicle interaction forces in a suspension with a large number of vesicles. We report results from numerical experiments that demonstrate the convergence and algorithmic complexity properties of our scheme. Joint work with: Shravan K. Veerapaneni, Denis Gueyffier, and Denis Zorin.

The triangle-free process

Series
Combinatorics Seminar
Time
Friday, October 24, 2008 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Tom Bohman – CMU
Consider the following random graph process. We begin with the empty graph on n vertices and add edges chosen at random one at a time. Each edge is chosen uniformly at random from the collection of pairs of vertices that do not form triangles when added as edges to the existing graph. In this talk I discuss an analysis of the triangle-free process using the so-called differential equations method for random graph processes. It turns out that with high probability the triangle-free process produces a Ramsey R(3,t) graph, a triangle-free graph whose independence number is within a multiplicative constant factor of the smallest possible.

A new topological bound for energy of fluid flows

Series
Geometry Topology Seminar
Time
Friday, October 24, 2008 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Rafal Komendarczyk – University of Pennsylvania
In many physical situations we are interested in topological lower bounds for L^2-energy of volume preserving vector fields. Such situations include for instance evolution of a magnetic field in ideal magnetohydrodynamics. Classical energy bounds involve topological invariants like helicity which measure linkage of orbits in the flow. In this talk I will present a new lower bound in terms of the third order helicity, which is an invariant measuring a third order linkage of orbits. I will also discuss how the third order helicity can be derived from the Milnor's \mu-bar invariant for 3-component links.

The differential equations method for random graph processes

Series
Graph Theory Seminar
Time
Thursday, October 23, 2008 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Tom Bohman – CMU
In this lecture I will introduce the method and sketch some recent applications. The main idea is to exploit a natural connection between the evolution of discrete random processes and continuous functions on the real numbers. Roughly speaking, the method is as follows: Given a discrete random process, we calculate the expected change in the random variable (or variables) of interest in one step of the process, write a differential equation suggested by the expected change, and show that the evolution of the random variable over the course of the process is sharply concentrated around the trajectory given by the solution of the differential equation. This allows us to translate simple facts (often combinatorial in nature) about expected changes in one step of the process into strong statements about sharp concentration of the random variable over the entire course of the process.

Quotient Correlation - A New Light of Measuring Variable Associations and Testing Hypotheses of Independence and Tail Independence

Series
Mathematical Finance/Financial Engineering Seminar
Time
Wednesday, October 22, 2008 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
ZhengJun Zhang – University of Wisconsin
Various correlation measures have been introduced in statistical inferences and applications. Each of them may be used in measuring association strength of the relationship, or testing independence, between two random variables. The quotient correlation is defined here as an alternative to Pearson's correlation that is more intuitive and flexible in cases where the tail behavior of data is important. It measures nonlinear dependence where the regular correlation coefficient is generally not applicable. One of its most useful features is a test statistic that has high power when testing nonlinear dependence in cases where the Fisher's Z-transformation test may fail to reach a right conclusion. Unlike most asymptotic test statistics, which are either normal or \chi 2, this test statistic has a limiting gamma distribution (henceforth the gamma test statistic). More than the common usages of correlation, the quotient correlation can easily and intuitively be adjusted to values at tails. This adjustment generates two new concepts -- the tail quotient correlation and the tail independence test statistics, which are also gamma statistics. Due to the fact that there is no analogue of the correlation coefficient in extreme value theory, and there does not exist an efficient tail independence test statistic, these two new concepts may open up a new field of study. In addition, an alternative to Spearman's rank correlation: a rank based quotient correlation is also defined. The advantages of using these new concepts are illustrated with simulated data, and real data analysis of internet traffic, tobacco markets, financial markets...

Data-driven methods in protein engineering: new ways to utilize sequence and structures of proteins

Series
Mathematical Biology Seminar
Time
Wednesday, October 22, 2008 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Andy Bommarius – School of Chemistry & Biochemistry, Georgia Tech
After rational protein design and combinatorial protein engineering (directed evolution), data-driven protein engineering emerges as a third generation of techniques for improving protein properties. Data-driven protein engineering relies heavily on the use of mathematical algorithms. In the first example, we developed a method for predicting the positions in the amino acid sequence that are critical for the catalytic activity of a protein. With nucleotide sequences of both functional and non-functional variants and a Support Vector Machine (SVM) learning algorithm, we set out to narrow the interesting sequence space of proteins, i.e. find the truly relevant positions. Variants of TEM-1 β-lactamase were created in silico using simulations of both mutagenesis and recombination protocols. The algorithm was shown to be able to predict critical positions that can tolerate up to two amino acids. Pairs of amino acid residues are known to lead to inactive sequences, unless mutated jointly. In the second example, we combine SVM, Boolean learning (BL), and the combination of the two, BLSVM, to find such interactive residues. Results on interactive residues in two fluorescent proteins, Discosoma Red Fluorescent Protein (Ds-Red) and monomeric Red Fluorescent Protein (mRFP), will be presented.

k-planes for classification

Series
Applied and Computational Mathematics Seminar
Time
Wednesday, October 22, 2008 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Arthur Szlam – UCLA

Please Note: SPECIAL TIME AND LOCATION FOR THIS WEEK ONLY

The k-planes method is the generalization of k-means where the representatives of each cluster are affine linear sets. In this talk I will describe some possible modifications of this method for discriminative learning problems.

Quantum Physics and Algebraic Graph Theory

Series
Joint School of Mathematics and ACO Colloquium
Time
Tuesday, October 21, 2008 - 16:30 for 2 hours
Location
Skiles 255
Speaker
Chris Godsil – University of Waterloo

Please Note: Refreshments will be served at 4PM in Skiles 236.

The possibility of a quantum computer has lead to much new work in theoretical physics and, naturally enough, this work has raised many new mathematical problems. What is perhaps surprising is that it has lead to interesting problems in algebraic graph theory. For example, questions about the relative power of quantum computer and classical computers lead to questions about the chromatic number of certain graphs. In my talk I will discuss some of these problems, and the progress that has been made.

A PDE and a stochastic model in cell polarity

Series
PDE Seminar
Time
Tuesday, October 21, 2008 - 15:15 for 1.5 hours (actually 80 minutes)
Location
Skiles 255
Speaker
Sigurd Angenent – University of Wisconsin, Madison
I will discuss a few ways in which reaction diffusion models have been used to pattern formation. In particular in the setting of Cdc42 transport to and from the membrane in a yeast cell I will show a simple model which achieves polarization. The model and its analysis exhibits some striking differences between deterministic and probabilistic versions of the model.

Timing Closure in Chip Design

Series
ACO Seminar
Time
Tuesday, October 21, 2008 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Stephan Held – University of Bonn
A central characteristic of a computer chip is the speed at which it processes data, determined by the time it takes electrical signals to travel through the chip. A major challenge in the design of a chip is to achieve timing closure, that is to find a physical realization fulfilling the speed specifications. We give an overview over the major tasks for optimizing the performance of computer chips and present several new algorithms. For the topology generation of repeater trees, we introduce a variant of the Steiner tree problem and present fast algorithm that balances efficiently between the resource consumption and performance. Another indispensable task is gate sizing, a discrete optimization problem with nonlinear or PDE constraints, for which a fast heuristic is introduced. The effectiveness in practice is demonstrated by comparing with newly developed lower bounds for the achievable delay. We conclude with a variant of the time-cost tradeoff problem from project management. In contrast to the usual formulation cycles are allowed. We present a new method to compute the time-cost tradeoff curve in such instances using combinatorial algorithms. Several problems in chip design can be modeled as time-cost tradeoff problems, e.g. threshold voltage optimization of plane assignment.

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