Public Lecture: "Mathematical Modeling: How Powerful Is It?"

Wednesday, April 14, 2006, Albert C. Yates Hall, Room 104 at 7:00 pm

Abstract: Mathematical models have been widely used to understand, predict, or optimize many complex processes from a large variety of subject areas, from semiconductor or pharmaceutical design to global weather models to astrophysics and astronomy. In particular, use of mathematical models in aerospace engineering, effects of air and water pollution, production of hydrocarbons, protection of health, medical imaging, financial forecasting, and cryptography for security is extensive. Examples from several of these applications will be discussed.

There are five major stages to the modeling process. For example, for each process, a physical model must first be developed incorporating as much application-specific information as is deemed necessary to describe the essential phenomena. Second, a mathematical formulation of the physical model is obtained, often involving equations or coupled systems of non-linear equations. Third, the mathematical properties of the model must be sufficiently well understood. Fourth, a computer code capable of efficiently and accurately performing the necessary computations on a discrete version of the mathematical model must be developed. Finally, for complex solution sets, visualization techniques must be used to compare the discrete output with the original process to determine the effectiveness of the modeling process. The issues involved in each of the parts of this modeling process will be illustrated through a variety of applications.

Colloquium: "Mathematical Modeling in Energy and Environmental Applications"

Thursday, April 15, 2006, Engineering 120 at 4:10 pm

Abstract: Mathematical models are used extensively to understand the transport and fate of groundwater contaminants and to design effective in situ groundwater remediation strategies. Four basic problem areas must be addressed in the modeling and simulation of the flow of groundwater contamination. One must first obtain an effective mathematical model to describe the complex fluid/fluid interactions that control the transport of contaminants in groundwater. This includes the problem of obtaining accurate reservoir descriptions at various length scales to describe the underground reservoir in a statistical manner. One obtains coupled systems of nonlinear time-dependant equations. Next, one must develop accurate discretization techniques to discretize these continuous equations that retain the important physical properties of the continuous models. Then, one must develop efficient numerical solution methods that can solve the enormous resulting systems of linear equations. Finally, one must be able to visualize the results of the numerical models in order to ascertain the validity of the modeling process by comparing with data obtained from the physical process. Aspects of each of these steps will be presented.

Seminar: "Eulerian-Lagrangian Localized Adjoint Methods for Transport Problems"

Friday, April 16, 2006, Engineering 120 at 4:10 pm

Abstract: Convection-diffusion problems, which arise in the numerical simulation of groundwater contamination and remediation, often present serious numerical difficulties. Conventional Galerkin methods and classical viscosity methods usually exhibit some combination of nonphysical oscillation and excessive numerical dispersion. Many numerical methods have been developed to circumvent these difficulties.

Basically, there are two major classes of approximations. The first are the so-called optimal spatial methods, based upon the minimization of error in the approximation of spatial derivatives using optimal test functions that satisfy a localized adjoint condition. Optimal spatial methods yield time truncation errors that dominate the solution and potentially serious numerical dispersion. The second class, the so-called Eulerian Lagrangian Methods, accurately treat the advection along characteristics and show great potential. The methods described here combine the best aspects of both of these classes and have been applied succesfully to a wide variety of applications.

Wednesday, April 14, 2006, Albert C. Yates Hall, Room 104 at 7:00 pm

Abstract: Mathematical models have been widely used to understand, predict, or optimize many complex processes from a large variety of subject areas, from semiconductor or pharmaceutical design to global weather models to astrophysics and astronomy. In particular, use of mathematical models in aerospace engineering, effects of air and water pollution, production of hydrocarbons, protection of health, medical imaging, financial forecasting, and cryptography for security is extensive. Examples from several of these applications will be discussed.

There are five major stages to the modeling process. For example, for each process, a physical model must first be developed incorporating as much application-specific information as is deemed necessary to describe the essential phenomena. Second, a mathematical formulation of the physical model is obtained, often involving equations or coupled systems of non-linear equations. Third, the mathematical properties of the model must be sufficiently well understood. Fourth, a computer code capable of efficiently and accurately performing the necessary computations on a discrete version of the mathematical model must be developed. Finally, for complex solution sets, visualization techniques must be used to compare the discrete output with the original process to determine the effectiveness of the modeling process. The issues involved in each of the parts of this modeling process will be illustrated through a variety of applications.

Colloquium: "Mathematical Modeling in Energy and Environmental Applications"

Thursday, April 15, 2006, Engineering 120 at 4:10 pm

Abstract: Mathematical models are used extensively to understand the transport and fate of groundwater contaminants and to design effective in situ groundwater remediation strategies. Four basic problem areas must be addressed in the modeling and simulation of the flow of groundwater contamination. One must first obtain an effective mathematical model to describe the complex fluid/fluid interactions that control the transport of contaminants in groundwater. This includes the problem of obtaining accurate reservoir descriptions at various length scales to describe the underground reservoir in a statistical manner. One obtains coupled systems of nonlinear time-dependant equations. Next, one must develop accurate discretization techniques to discretize these continuous equations that retain the important physical properties of the continuous models. Then, one must develop efficient numerical solution methods that can solve the enormous resulting systems of linear equations. Finally, one must be able to visualize the results of the numerical models in order to ascertain the validity of the modeling process by comparing with data obtained from the physical process. Aspects of each of these steps will be presented.

Seminar: "Eulerian-Lagrangian Localized Adjoint Methods for Transport Problems"

Friday, April 16, 2006, Engineering 120 at 4:10 pm

Abstract: Convection-diffusion problems, which arise in the numerical simulation of groundwater contamination and remediation, often present serious numerical difficulties. Conventional Galerkin methods and classical viscosity methods usually exhibit some combination of nonphysical oscillation and excessive numerical dispersion. Many numerical methods have been developed to circumvent these difficulties.

Basically, there are two major classes of approximations. The first are the so-called optimal spatial methods, based upon the minimization of error in the approximation of spatial derivatives using optimal test functions that satisfy a localized adjoint condition. Optimal spatial methods yield time truncation errors that dominate the solution and potentially serious numerical dispersion. The second class, the so-called Eulerian Lagrangian Methods, accurately treat the advection along characteristics and show great potential. The methods described here combine the best aspects of both of these classes and have been applied succesfully to a wide variety of applications.

Extensions of these methods to spline-based test and trial functions
are extremely effective for pure transport problems in one and two
spatial dimensions. They also achieve accurate approximations under
minimal regularity assumptions on the solution. Finally, an
algorithm for the approximation of characteristics, a property required
by all of these methods, is developed in higher spatial dimensions.

This is joint work with Hong Wang (University of South Carolina) and James Liu (Colorado State University.)

Public Lecture: "Algebraic Statistics for Computational Biology"

Monday, March 21, 2005, Albert C. Yates Hall, Room 104 at 7:00 pm

Abstract: We discuss recent interactions between algebra and statistics and their emerging applications to computational biology. Statistical models of independence and alignments for DNA sequences will be illustrated by means of a fictional character, DiaNA, who rolls tetrahedral dice with face labels “A,” “C,” “G” and “T.” Reference.

Colloquium: "Tropical Geometry"

Tuesday, March 22, 2005, Clark A207 at 4:10 pm

Abstract: Tropical geometry is the geometry of the tropical semiring (min-plus-algebra.) Its objects are polyhedral cell complexes which behave like complex algebraic varieties. We offer an introduction to this theory, with an emphasis on plane curves and linear spaces, and we discuss applications to phylogenetics. This talk will be suitable for undegraduates.

Seminar: "Solving the Likelihood Equations"

Wednesday, March 23, 2005, Engineering 120 at 4:10 pm

Abstract: Given a model in algebraic statistics and some data, the likelihood function is a rational function on a projective variety. We discuss algebraic methods for computing all critical points of this function, with the aim of identifying the local maxima in the probability simplex. This is joint work with Serkan Hosten and Amit Khetan (math.ST/0408270.)

Monday, March 21, 2005, Albert C. Yates Hall, Room 104 at 7:00 pm

Abstract: We discuss recent interactions between algebra and statistics and their emerging applications to computational biology. Statistical models of independence and alignments for DNA sequences will be illustrated by means of a fictional character, DiaNA, who rolls tetrahedral dice with face labels “A,” “C,” “G” and “T.” Reference.

Colloquium: "Tropical Geometry"

Tuesday, March 22, 2005, Clark A207 at 4:10 pm

Abstract: Tropical geometry is the geometry of the tropical semiring (min-plus-algebra.) Its objects are polyhedral cell complexes which behave like complex algebraic varieties. We offer an introduction to this theory, with an emphasis on plane curves and linear spaces, and we discuss applications to phylogenetics. This talk will be suitable for undegraduates.

Seminar: "Solving the Likelihood Equations"

Wednesday, March 23, 2005, Engineering 120 at 4:10 pm

Abstract: Given a model in algebraic statistics and some data, the likelihood function is a rational function on a projective variety. We discuss algebraic methods for computing all critical points of this function, with the aim of identifying the local maxima in the probability simplex. This is joint work with Serkan Hosten and Amit Khetan (math.ST/0408270.)

**Magnus Lectures Spring 2004**: John G McWhirter FRS FREng, Senior Fellow, QinetiQ Ltd, Malvern Technology Centre-

*Graduate Student Lecture: "The Mathematics of Independent Component Analysis"*

Monday, March 22, 2004, Louis R. Weber Building, Room 202 at 10:00am

**Abstract:**

Independent component analysis (ICA) is a powerful new technique for signal and data processing. It extends the scope and capability of principal component analysis (PCA) by exploiting higher order statistics in circumstances where the statistics of the data or signal samples are non-Gaussian. The development of effective techniques for ICA leads to some interesting and challenging mathematical problems.

For example, the use of fourth order statistics to separate independent signals which have been mixed in an instantaneous manner involves approximate diagonalisation of a fourth order (i.e. four index) tensor. Whereas the problem of matrix diagonalisation is well understood, the diagonalisation of tensors of order greater than two poses some very challenging problems.

The use of independent component analysis to separate signals which have been mixed in a convolutive manner poses a particularly interesting challenge. The problem may be formulated in terms of polynomial matrices (i.e. matrices with polynomial elements) and involves identifying the elements of a paraunitary unmixing matrix.

In this seminar, I will introduce the basic concept of ICA, explain how some of these interesting mathematical problems arise and outline some of the progress which has already been made. I will then present some results obtained using ICA in practical applications such as HF communications and foetal heartbeat analysis.*Public Lecture: "MATHEMATICS: WHO NEEDS IT ANYWAY?"*

Monday, March 22, 2004 Albert C. Yates Hall, Room 104 at 7:00pm

**Abstract:**

Mathematics is not just a very interesting and beautiful subject in its own right. It is also the language of science and engineering and at the heart of many developments which we take for granted in this modern technological age. And yet there is a tendency in many of the worlds leading economic regions to assume that the teaching and learning of mathematics is less important than it used to be.

Many people are entirely unaware of the role that mathematics plays in their day to day lives. It is assumed that the need for mathematics has been diminished by the widespread availability of high performance computers. In this talk I will attempt to illustrate some of the areas where mathematics has made a vital contribution to our lives and argue that the need for mathematical skills has been increased rather than diminished by recent developments in computer technology.

Refreshments will be served immediately following the presentation.*Mathematics Lecture: "A NOVEL TECHNIQUE FOR BROADBAND SINGULAR VALUE DECOMPOSITION"*

Tuesday, March 23, 2003, Louis R. Weber Building, Room 202 at 4:00pm

**Abstract:**

The singular value decomposition (SVD) is a very important tool for narrowband adaptive sensor array processing. The SVD decorrelates the signals received from an array of sensors by applying a unitary matrix of complex scalars which serve to modify the signals in phase and amplitude. In broadband applications, or a situation where narrowband signals have been convolutively mixed, the received signals cannot be represented in terms of phase and amplitude. Instantaneous decorrelation using a unitary matrix is no longer sufficient to separate them. It is necessary to decorrelate the signals over a suitably chosen range of relative time delays. This process, referred to as strong decorrelation, requires a matrix of suitably chosen filters. Representing each filter (assumed to have finite impulse response) in terms of its z-transform, this takes the form of a polynomial matrix. The SVD may be generalized to broadband adaptive sensor arrays by requiring the polynomial matrix to be paraunitary so that it preserves the total energy at every frequency. In this talk, I will describe a novel technique for computing the required paraunitary matrix and show how the resulting broadband SVD algorithm can be used to identify the signal subspace for broadband adaptive beamforming. **Magnus Lectures Spring 2003:**Tom Mullin, Manchester Center for Nonlinear Dynamics-
*Public Lecture: "Patterns in the Sand: The Physics of Granular Flow"*, Wednesday, April 30 2003

**Abstract:**

Have you ever wondered why the sand dries around your foot when you walk along a wet beach? Or has it puzzled you that the fruit and nuts in your müsli are usually at the top of the packet? These fundamental physics questions will be discussed and videos of other spectacular effects in granular flows will be presented. Refreshments will be served before and after the presentation.

*Mathematics Lecture: "Can Granular Segregation be considered as a Phase Transition?"*, Thursday, May 1 2003

**Abstract:**

Segregation of mixtures of granular materials is a topic of interest to a broad range of scientists from Physicists, to Geologists and Engineers. The process can be driven by either simple avalanching in binary mixtures when the angle of repose of the constituents are different or it can be promoted using an external drive or perturbation. We will discuss these issues and present the results of a new experimental study of particle segregation in a binary mixture that is subject to a periodic horizontal forcing. A surprising self-organization process is observed which shows critical behavior in its formation. Connections with concepts from equilibrium phase transitions will be discussed.

*Mathematics Lecture: "Balls in Syrup: A ‘Simple’ Dynamical System"*, Friday, May 2 2003

**Abstract:**

We present the results of an experimental investigation of a novel dynamical system in which one, two or three solid spheres are free to move in a horizontal rotating cylinder filled with highly viscous fluid. At low rotation rates steady motion is found where the balls adopt stable equilibrium positions rotating adjacent to the rising wall at a speed which is in surprisingly close agreement with available theory. At higher cylinder speeds, time-dependent motion sets in via Hopf bifurcations. When one or two balls are present the motion is strictly periodic. However, low dimensional chaos is found with three balls. **Magnus Lectures Spring 2002:**Heinz-Otto Peitgen, University of Bremen and Florida Atlantic University*General Lecture: "Harnessing Chaos"*, Friday, April 5

Refreshments and Reception immediately following in Weber 117

Abstract:

We will discuss how chaos theory has changed our view of nature and has an impact on how we do science. The lecture will include a historical treatment of how our current scientific view of the world has evolved and changed with chaos theory, its impact on the arts and culture, and finish with state of the art applications in information technology and medicine.

*Mathematics Lecture: "Mathematical Methods in Medical Imaging: Analysis of Vascular Structures for Liver Surgery Planning".*

Monday, April 8, 4 pm, Plant Sciences Room C146.

Abstract:

Mathematics and medicine do not have a long history of close and fruitful cooperation. The application of mathematical models in medical applications is becoming viable due to the increasing performance of computers and because more and more image data are acquired digitally. With mathematical methods these data may be quantitatively analyzed and visualized such that medical diagnosis and the assessment of therapeutic strategies become more and more reliable and reproducible. As an example we will demonstrate our work for the planning of oncological and transplantation liver surgery.**Magnus Lectures Spring 2001:**Robert Calderbank, AT&T Laboratories*General Lecture: "50 Years of Information and Coding"*

Tuesday, April 17, 4:10-5:00 pm, Lory Student Center 203-205

Abstract:

Over 50 years have passed since the appearance of Shannon's landmark paper "A Mathematical Theory of Communication". This talk is a personal perspective on what has been achieved in certain areas over the past 50 years and what the future challenges might be. The focus will be on connecting coding theory with coding practice in areas like digital data storage, deep space communication and wireless networks.*Mathematics Colloquium: "Combinatorics, Quantum Computing and Cellular Telephones"*

Thursday, April 19, 4:10-5:00 pm, Hammond Auditorium

Abstract:

This talk explores the connection between quantum error correction and wireless systems that employ multiple antennas at the base station and the mobile terminal. The two topics have a common mathematical foundation, involving orthogonal geometry - the combinatorics of binary quadratic forms. We explain these connections, and describe how the wireless industry is making use of a mathematical framework developed by Radon and Hurwitz about a hundred years ago.*Algebraic Combinatorics Seminar: "Tailbiting Representations of the Binary Golay Code"*

Friday, April 20, 4:10-5:00 pm, Weber 117

Abstract:

This talk introduces tailbiting representations of a most extraordinary binary code - the [24,12,8] Golay code. The problem is to represent Golay codewords as paths in a graph - the graph has a particular form - it is the concatenation of 24 identical sections - section per coordinate symbol - and the paths have to start and end at the "same" node. The objective is to minimize the number of nodes, and the analysis will involve Conway's Miracle Octad Generator.**Magnus Lectures Spring 2000:**Gilbert Strang, Massachusetts Institute of Technology*General Lecture: "Partly Random Graphs and Small World Networks"*

Tuesday, April 11, 4-5pm, Lory Student Center 220-224 (Refreshments at 3:30)

Abstract:

It is almost true that any two people in the US are connected by less than six steps from one friend to another. What are models for large graphs with such small diameters ? This is the "6 Degrees of Separation" that appeared in a movie title.

Watts and Strogatz observed (in Nature, June 1998) that a few random edges in a graph could quickly reduce its diameter (longest distance between two nodes). We report on an analysis by Newman and Watts (using mathematics of physicists) to estimate the average distance between nodes, starting with a circle of N friends and M random shortcuts, 1 << M << N.

We also study a related model, which adds N edges around a second (but now random) cycle. The average distance between pairs becomes nearly A log n + B. The eigenvalues of the adjacency matrix are surprisingly close to an arithmetic progression; for each cycle they would be cosines, the sum changes everything.

We will discuss some of the analysis (with Alan Edelman and Henrik Eriksson at MIT) and also some applications. We also report on the surprising eigenvalue distribution for trees (large and growing ) found by Li He and Xiangwei Liu. And a nice work by Jon Kleinberg discusses when the short paths can actually be located efficiently.

*Colloquium Lecture: "Cosine Transforms and Wavelet Transforms and Signal Processing"*

Monday, April 10, 10-11am, Shepardson 102

Abstract:

Each Discrete Cosine Transform uses N real basis vectors whose components are cosines. These basis vectors are orthogonal and the transform is much used in image processing (we will point out drawbacks). The cosine series is quickly computed by the FFT. But a direct proof of orthogonality, by calculating inner products, does not reveal how natural these cosine vectors are in applications.

We prove orthogonality in a different way. Each DCT comes from the eigenvectors of a symmetric "second-difference matrix". By varying the boundary conditions we get the established transforms DCT-1 through DCT-4 (and also four more orthogonal bases of cosines). The boundary condition determines the centering (at a meshpoint or a midpoint) and decides on the entries cos [j or j+0.5] [k or k+0.5] pi/N .

Then we discuss bases from filter banks and wavelets. The key is to create a banded *block Toeplitz* matrix whose inverse is also banded. The algebra shows how the approximation properties of the wavelet basis are determined by the polynomials that can be reproduced exactly by wavelets. In signal processing, so much depends on the choice of a good basis.

*Seminar Lecture: "Teaching Applied Mathematics"*

Monday, April 10, 4-5pm, EE104 (Refreshments at 3:30 in the Department of Mathematics Coffee Room)

Abstract:

We will discuss the possibilities (and the problems) of teaching applied mathematics and engineering mathematics. I have found this a very positive experience -- the students are interested and more motivated, I have new ideas to learn about, the mathematics is interesting and not simply formulas. It is pleasing to see how a few key ideas appear in many different genuine applications.**Magnus Lectures Spring 1999:**Cheryl Praeger, University of Western Australia- General Lecture: "Symmetries of Designs"

Colloquium Lecture: "Algorithms for computing with groups of matrices over finite fields"

Seminar Lecture: "Quasiprimitive permutation groups and their actions on graphs and linear spaces" **Magnus Lectures Spring 1998:**Raghu Varadhan, Courant Institute of Mathematics- General Lecture: "How Rare Is Rare?"

Colloquium Lecture: "Problems of Hydrodynamic Scaling"

Seminar Lecture: "Large Deviations for the Simple Exclusion Process" -
**Magnus Lectures Spring 1997:**Marty Golubitsky, University of Houston - General Lecture:"Symmetry and Chaos: Patterns on Average"

Colloquium Lecture: "Oscillations in Coupled Systems and Animal Gaits"

Seminar Lecture: "Spiral Waves and Other Planar Patterns" -
**Magnus Lectures Spring 1996:**Fabrizio Catanese, University of Pisa - General Lecture: "Moduli of surfaces and differentiable 4-manifolds"

Colloquium Lecture: "Enriques' rough classification of algebraic surfaces and the fine classification"

Seminar Lecture: "Homological algebra and algebraic surfaces" -
**Magnus Lectures Spring 1994:**Lawrence Sirovich, Brown University - General Lecture: "Image Analysis"

Colloquium Lecture: "Dynamics of Wall-Bounded Turbulence"

Seminar Lecture: "EOF Analysis of TOMS Ozone Image Data" **Magnus Lecture Spring 1993:**Bill Jones, University of Colorado- "Szego Polynomials Applied to Signal Processing"