**Magnus Lectures Fall 2017 (October 23-24, 2017): M. Maria Chudovsky**, **MacArthur Fellow - 2012, Princeton University,
Department of Mathematics, PACM: The Program in Applied and Computational Mathematics
**

*Public Lecture*: Parties, doughnuts and coloring

Abstract: This is a gentle introduction to graph theory, connecting familiar concepts and classical problems with most recent cutting edge research in the field.

*Research Seminar*: Graphs with Forbidden Induced Subgraphs

Abstract:The problem of testing if a graph can be colored with a given number k of colors is NP-complete for every k>2. But what if we have more information about the input graph, namely that some fixed graph H is not present in it as an induced sub-graph? It is known that the problem remains NP-complete even for k=3, unless H is the disjoint union of paths. We consider the following two questions:

1. For which graphs H is there a polynomial time algorithm to 3-color (or in general k-color) an H-free graph?

2. For which graphs H are there finitely many 4-critical H-free graphs?

This talk will survey recent progress on these questions, and in particular give a complete answer to the second one.

*Colloquium*: Perfect and Beyond

Abstract: About 10 years ago one of the central open problems in graph theory at the time, the Strong Perfect Graph Conjecture, was solved. The proof used structural graph theory methods, and spanned 155 journal pages. The speaker was part of the team of authors of this mathematical beast. In this talk we will explain the problem, describe some of the ideas of the proof (that has since been shortened somewhat), and discuss related problems that have been a subject of more recent research.

**Magnus Lectures Spring 2016 (April 4 - 6, 2016): M. Gregory Forest**, **Grant Dahlstrom Distinguished Professor,
Departments of Mathematics & Biomedical Engineering,
Director, Carolina Center for Interdisciplinary Applied Mathematics, University of North Carolina at Chapel Hill**

*Public Lecture*: The Virtual Lung Project at UNC

Abstract: In the late 1990s, the Virtual Lung Project (VLP) at the University of North Carolina (UNC) at Chapel Hill began as an interdisciplinary response to the mucus transport problem for cystic fibrosis patients, integrating basic and medical science toward an understanding of disease and the potential for science-based, engineering solutions. Lung biologists Ric Boucher, John Sheehan, and Bill Davis of the UNC Cystic Fibrosis Center, now the Marsico Lung Institute, teamed with physicists Rich Superfine and Michael Rubinstein, applied mathematicians Greg Forest, Roberto Camassa, Tim Elston, Rich McLaughlin, and Sorin Mitran, and computer scientist Russ Taylor. All saw the opportunity to formulate open questions from mathematics, microscopy, stochastic processes, fluid mechanics, rheology, materials science, and computational science, in one remarkable biological system – lung transport of mucus. This lecture will survey the biology, the pathology, experiments and data, selected highlights of progress, with an emphasis on the role of mathematics and computation interwoven with medical science and biology. Contributions from many former PhD students, postdocs, and faculty will be acknowledged during the lecture.

*Colloquium*: Dynamic organization of DNA in living yeast

Abstract: DNA molecules are packaged in the cell nucleus in a sequence of compression steps by different molecular species, e.g., histones, condensin, and cohesin. While the genome has been sequenced from yeast to human, a current focus in biology is on the post-genome questions: how does the genome organize and interact throughout the cell cycle; what features of this dynamic organization can be explained simply from entropy of molecular confinement; and, what features require "intelligent design", guided by specific molecular species. My group, led by Paula Vasquez, together with David Adalsteinsson and several joint math-biology graduate students, has worked with Kerry Bloom's lab to weave experiments and modeling of chromosomal DNA in living yeast. This lecture will survey progress of that collaboration.

*Research Seminar*: Transient anomalous diffusion in mucus gels and other biological fluids

Abstract: Measurements of Brownian probes in viscous fluids have remarkably robust time series, e.g., mean-squared displacement (MSD) scales linearly with time, and one can use such data to determine the fluid viscosity. Passive microrheology generalizes this classical protocol to determine dynamic viscous and elastic moduli of complex fluids, i.e., the loss and storage modulus across a broad frequency range. One finds that in mucus gels, micron-scale particle paths are sub-diffusive (MSD scales sub-linearly with time) and transient (the local MSD power law exponent changes over time). Our group uses microbead time series data both as a viscoelasticity protocol and for inference of diffusive transport across mucus barriers. For the latter purpose, one must model the underlying stochastic process, since little is known about passage time distributions for transient anomalous diffusion. This lecture will cover the more technical aspects of data analysis, model inference, and direct simulations, based on joint work with a host of collaborators to be acknowledged.

**Magnus Lectures Spring 20****15 (March 23, 2015): ****Andrea L. Bertozzi ****, ****Betsy Wood Knapp Chair for Innovation and Creativity, Director of Applied Mathematics, University of California Los Angeles**

Public Lecture: Math of Crime

Abstract: There is an extensive applied mathematics literature developed for problems in the biological and physical sciences. Our understanding of social science problems from a mathematical standpoint is less developed, but also presents some very interesting problems, especially for young researchers. This lecture uses crime as a case study for using applied mathematical techniques in a social science application and covers a variety of mathematical methods that are applicable to such problems. We will review recent work on agent based models, methods in linear and nonlinear partial differential equations, variational methods for inverse problems and statistical point process models. From an application standpoint we will look at problems in residential burglaries and gang crimes. Examples will consider both "bottom up'' and "top down'' approaches to understanding the mathematics of crime, and how the two approaches could converge to a unifying theory.

*Colloquium: *Swarming by Nature and by Design

Abstract: The cohesive movement of a biological population is a commonly observed natural phenomenon. With the advent of platforms of unmanned vehicles, such phenomena have attracted a renewed interest from the engineering community.

This talk will cover a survey of the speaker's research and related work in this area ranging from aggregation models in nonlinear partial differential equations to control algorithms and robotic testbed experiments. We will show how pair wise potential models are used to study biological movement and how to develop a systematic theory of such models. We also discuss how to use "designer potentials" to orchestrate cooperative movement in specific patterns, many of which may not be observed in nature but could be desirable for artificial swarms. Finally we conclude with some recent related work on emotional contagion in crowds.

*Research Seminar: **Geometric graph-based methods for high dimensional data *

Abstract: We present new methods for segmentation of large datasets with graph based structure. The method combines ideas from classical nonlinear PDE-based image segmentation with fast and accessible linear algebra methods for computing information about the spectrum of the graph Laplacian. The goal of the algorithms is to solve semi-supervised and unsupervised graph cut optimization problems. We discuss results for image processing applications such as image labeling and hyperspectral video segmentation, and results from machine learning and community detection in social networks, including modularity optimization posed as a graph total variation minimization problem.

**Magnus Lectures Spring 20****14 (February 3, 2014): ****Alex Lubotzky****, ****Maurice and Clara Weil Chair of Mathematics, Einstein Institute of Mathematics, Hebrew University of Jerusalem**

Public Lecture: Real applications of non real numbers

Abstract: Number theoretic considerations led mathematicians over a century ago to introduce the "field of p-adic numbers", which is just like the "field of real numbers" a completion of the familiar "field of rational numbers". This abstract system of numbers has found in the last 3 decades some unexpected applications in computer science and engineering. We will explain the basic ideas and some of these applications.

*Colloquium: **Ramanujan graphs and error correcting codes*

Abstract: While many of the classical error correcting codes are cyclic, a long standing conjecture asserts that there are no 'good' cyclic codes. In recent years the interest in symmetric codes has been promoted by Kaufman, Sudan, Wigderson and others (where symmetric means that the acting group can be any group). Answering their main question (and in contrary to the common expectation), we show that there DO exist symmetric good codes. In fact, our codes satisfy all the "golden standards" of coding theory. Our construction is based on the Ramanujan graphs constructed by Lubotzky-Samuels-Vishne as a special case of Ramanujan complexes. The crucial point is that these graphs are edge transitive and not just vertex transitive as in previous constructions of Ramanujan graphs. All notions will be explained. This is joint work with Tali Kaufman.

*Research Seminar: **From Ramanujan graphs to Ramanujan complexes*

Abstract: Ramanujan graphs are optimal expanders (from a spectral point of view). Explicit constructions of such graphs were given in the 80's as quotients of the Bruhat-Tits tree associated with GL(2) over a local field F, by suitable congruence subgroups. The spectral bounds were proved using works of Hecke, Deligne and Drinfeld on the "Ramanujan conjecture" in the theory of automorphic forms. The work of Lafforgue, extending Drinfeld from GL(2) to GL(n), opened the door for the construction of Ramanujan complexes as quotients of the Bruhat-Tits buildings. This gives finite simplical complexes, which on one hand are "random like", and at the same time have strong symmetries. Recently various applications have been found in combinatorics, coding theory and in relation to Gromov's overlapping properties. We will describe these developments and give some details on recent applications.

The work of a number of authors will be surveyed. Our works in these directions are in collaboration with various subsets of {S. Evra, K. Golubev, T. Kaufman, D. Kazhdan, R. Meshulam, S. Mozes, B. Samuels, U. Vishne}.

**Magnus Lectures Spring 20****13 (March 13-15): ****Pascal Chossat****, Director of Research, Department of Mathematics, University of Nice and French National Center for Scientific Research**

Public Lecture: Bifurcation and symmetry, a mathematical view on pattern formation in nature

Abstract: Patterns in Nature are not of so many types. The coat of a zebra is stripped while the coat of a leopard is spotted (and a cougar has a uniformly colored fur). Honey bees build incredibly regular hexagonal cells. Many plants or sea organisms present a high degree of symmetry, like the icosahedral shell of certain radiolarians. These quite simple patterns are extremely common, not only with living creatures but also in inanimate matter, think of the regular patterns in crystals like the cubic symmetry of salt for example, or the spiral patterns which can form on the heart muscle and provoke a heart attack. The common denominator of these examples is the underlying mathematics, which model the formation of regular patterns. Although more complex patterns have recently been observed, like quasi-crystals, the mathematical theory of pattern formation, which was initiated by the celebrated mathematician Alan Turing, is an example of the “unreasonable effectiveness of mathematics in natural science” as Nobel Prize winner Eugene Wigner used to say.

Colloquium: Pattern formation on compact Riemann surfaces and applications

Abstract: Pattern formation on the sphere and torus has been widely studied in relation to the occurrence of periodic patterns in classical hydrodynamical systems and in biochemical models of reaction-diffusion equations. Recently a model for images texture perception by the visual cortex was introduced, which involves neural field equations posed on the hyperbolic plane. Looking for pattern formation in this non euclidean geometric context comes back to analyzing the bifurcation of patterns on compact Riemann surfaces of genus > 1. This leads to new and sometimes unexpected results, which open the door to a classification of patterns on Riemann surfaces.

*Research Seminar: Pattern formation and the bifurcation of heteroclinic cycles*

Abstract: Robust heteroclinic cycles (RHC) are flow-invariant bounded sets that naturally occurs in certain types of dynamical systems (typically in systems with symmetry). The presence of a RHC can explain intermittent switching between steady-states or periodic orbits, which are sometimes observed in physical experiments. Heteroclinic cycles in pattern formation systems can exist but are usually associated with codimension 2 (or higher) bifurcations. I shall show an example of a generic, codimension 1 bifurcation of robust heteroclinic cycles when the domain is the hyperbolic plane.

**Magnus Lectures Spring 2012****: Pascal Chossat (canceled) **

**Magnus Lectures Spring 2011 (April 5-7)****: ****Ridgway Scott****, Louis Block Professor, University of Chicago, Computer Science and Mathematics**

*Public Lecture: **Mathematics in drug design*

Abstract: We show how mathematics can help in the complex process of drug discovery. We give an example of modification of a common cancer drug that reduces unwanted side effects. The mathematical model used to do this relates to the hydrophobic effect, something not yet fully understood. The hydrophobic effect modulates the dielectric behavior of water, and this has dramatic effects on how we process drugs. Future mathematical advances in this area hold the process of making drug discovery more rational, and thus more rapid and predictable, and less costly.

*Colloquium: **Two tales about Newton's method *

Abstract: We talk about Newton's method for solving nonlinear (systems of) equations, a common topic in Calculus. We describe two new areas of research that are related to Newton's method. We show that the "endgame" for Newton's method (that is, the behavior of the iterates viewed as a dynamical system) in multiple dimensions can be extremely complex, leading to tensor eigenvalue problems. We also show how Newton's method for solving nonlinear ODE's can provide a productive approach to creating parallelism in what would seem to be essentially sequential computations.

*Research Seminar: Optimal algorithms using optimal meshes*

Abstract: We discuss two problems involving adaptive meshes. The first relates to non-nested multi-grid in two and three dimensions. We review what is known theoretically and describe some recent work related to optimal implementation. The second involves meshes in arbitrary dimensions. We show that there are meshes in which the number of nodes grows linearly in the dimension, and give some evidence via a quantum mechanics example that an h-P strategy can be effective to obtain good convergence behavior on these meshes. If time permits, we will describe some on-going work developing new formulations for nonlinear and nonlocal dielectric models for proteins.

**Magnus Lectures Spring 2010 (April 14-15)****: ****Günter Uhlmann****, Walker Family Endowment Professor of Mathematics, University of Washington, Department of Mathematics**

*Public Lecture: **Cloaking, Invisibility and Inverse Problems*

Abstract: We describe recent theoretical and experimental progress on making objects invisible to detection by electromagnetic waves, acoustic waves and quantum waves. For the case of electromagnetic waves, Maxwell's equations have transformation laws that allow for design of electromagnetic materials that steer light around a hidden region, returning it to its original path on the far side. Not only would observers be unaware of the contents of the hidden region, they would not even be aware that something was being hidden. The object, which would have no shadow, is said to be cloaked. We recount the recent history of the subject and discuss some of the mathematical issues involved.

*Colloquium: **30 years of Calderón's inverse problem*

Abstract: Calderón's problem consists in finding the electrical conductivity of a medium by making voltages and current measurements at the boundary. In mathematical terms one tries to determine the coefficient of a partial differential equation by measuring the corresponding Dirichlet-to-Neumann map. This problem arises in geophysical prospection and it has been proposed as a diagnostic tool in medical imaging, particular early breast cancer detection. We will also describe the progress that has been made on this problem since Calderón's seminal paper in 1980.

*Research Seminar: Travel Time Tomography and Boundary Rigidity*

Abstract: In this lecture we will describe a surprising connection between Calderón's inverse problem and travel time tomography. This latter problem consists in determining the index of refraction (sound speed) of a medium by measuring the travel times of sound waves going though the medium. In mathematical terms the question is to determine the Riemannian metric of a Riemannian manifold with boundary by measuring the distance function between points on the boundary. In differential geometry this is known as the boundary rigidity problem. This inverse problem arises in geophysics in determining the inner structure of the Earth by measuring the travel times of seismic waves as well as in ultrasound imaging.

**Magnus Lectures Spring 2009 (April 22-24): Roland Glowinski, Cullen Professor of Mathematics and Mechanical Engineering, University of Houston**

*Public Lecture: **Adventures in Computing*

Abstract: The main goal of this lecture is to present real life situations where Applied & Computational Mathematics can significantly contribute, via numerical simulation inparticular, to progress beneficial to Society. This presentation will be illustrated by examples related to real life applications, in Cardio-Vascular Medicine in particular, an area where methods developed by the speaker have found applications.

*General Seminar: **Particle clustering in rotating cylinders*

Abstract: In this lecture, we return, in some sense, to one of the *Magnus Lectures* given some years ago by *Tom Mullin*, and offer a computational science perspective to this former speaker exciting presentation. Indeed, in this lecture, we investigate computationally the clustering of rigid solid particles in rotating cylinders containing an incompressible viscous fluid, for particle populations ranging from 10 to more than 100. We study in particular the influence of the angular velocity on these clustering phenomena. The presentation will be illustrated by animations visualizing these truly three-dimensional phenomena, which to the best of our knowledge are not fully understood, as of today. The presentation will include a description of the numerical methodology retained for the solution of the differential system coupling the Navier-Stokes equations modeling the flow, with the Newton-Euler equations describing the particle motion.

Research Seminar: A Least-Squares/Fictitious Domain Method for Linear Elliptic Boundary Value Problems with Neumann or Robin Boundary Conditions: A Virtual Control Approach

Abstract: Motivated by the numerical simulation of particulate flow with slip-boundary conditions at the interface fluid-particles, we are going to address in this lecture the solution of the following elliptic problem

by a fictitious domain method (new to the best of our knowledge); in (1), (2), Ω denotes a bounded domain of Rd and ω a sub-domain of Ω. Our approach relies essentially on the transformation of (1), (2) in a (virtual) control problem (in the sense of J.L. Lions), involving an extension of (1) (completed by u =g0 on ∂Ω) on the whole Ω, the restriction of the extended solution to being the solution of (1), (2). From an algorithmic point of view, one solves the control problem by a least-squares/conjugate gradient algorithm whose finite element implementation is rather easy, even if the mesh associated with Ω does not match the geometry of ω. Numerical experiments, including the generalization to the solution of parabolic equations with moving ω, suggest optimal orders of convergence.

**Magnus Lectures Spring 200****8 (January 22-24):***Dr. Desmond J. Higham, FRSE**, **Professor of Mathematics University of Strathclyde, Glasgow, Scotland, UK*

*Elected Fellow of the Royal Society of Edinburgh *

*Public Lecture: **Network Science: Joining the Dot*

Abstract: Connections are important. In studying nature, technology, commerce and the social sciences it often makes sense to focus on the pattern of interactions between individual components. I will give examples of large, complex networks that arise:

**
**• in the cell: connecting proteins

• in the brain: connecting neural regions

• in the World Wide Web: connecting web pages

• in the Internet Movie Database: connecting actors

• in supermarkets: connecting products

Improvements in computing power have allowed us to store and analyze these massive data sets, and a new discipline, network science, has emerged. I will focus on contributions that mathematicians and other scientists have made towards understanding how large networks evolve, discovering universal properties and developing tools to pick out interesting details. Along the way we will see how Google ranks your home page and why Kevin Bacon is the center of the universe.

Abstract: Advances in experimental biology are creating challenging modelling and data analysis problems for researchers in bioinformatics. In particular, protein- protein interaction data sets can be viewed as large unweighted, undirected graphs that, when analyzed appropriately, may reveal important biological information. Researchers have considered high-level questions, such as “can we describe these networks in terms of a few parameters?” and low-level questions such as “can we identify interesting groups of proteins?” I will show how contributions at both levels can be made from a matrix computa- tion viewpoint. Results for real biological data sets will be given.

Abstract: To incorporate abrupt and unpredictable changes to the dynamics of a system, models are now being derived that incorprate a switch. Given a collection of stochastic differential equations (SDEs), the switch, taking the form of an independent continuous time Markov chain, determines which SDE is currently active. Important examples arise in mathematical finance and systems biology. I will look at two topics:

1. stability analysis of numerical methods, and

2. modelling/simulation issues for gene regulation.

**Magnus Lectures Spring 200****7 (March 26-28): **** Efmi Zelmanov****, ****Fields Medalist, Professor of Mathematics at University of California, San Diego.**

*Public Lecture: Algebra in the 20th Century *

*Colloquium:** Profinite Groups *

*Seminar: ** Some open problems concerning Infinite-Dimensional Algebras*

**Magnus Lectures Spring 200****6 (April 14-16): ****Richard Ewing****, Distinguished Professor of Mathematics, Applied Mathematics and Engineering, and Vice-President of Research, Texas A&M University**

*Public Lecture: **Mathematical Modeling: How Powerful Is It*

*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*

*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*

*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.)

**Magnus Lectures Spring 200****5 (March 21-23): ****Bernd Sturmfels****, Professor of Mathematics and Computer Science, University of California, Berkeley **

*Public Lecture - **Algebraic Statistics for Computational Biology*

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*

*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*

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 – (March 22-23)****: ****John G McWhirter FRS FREng****, Senior Fellow, QinetiQ Ltd, Malvern Technology Centre **

*Graduate Student Lecture - The Mathematics of Independent Component Analysis*

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 fetal heartbeat analysis.

*Public Lecture - Mathematics, Who Needs It Anyway*

*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 world’s 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.

*Mathematics Lecture - A Novel Technique for Boardband Signular Value Decomposition*

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 beam forming.

**Magnus Lectures Spring 2003 - (April 30- May 2):**** ****Tom Mullin****, Manchester Center for Nonlinear Dynamics **

*Public Lecture - Patterns in the Sand: The Physics of Granular Flow*

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?*

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*

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 (April 5 and April 8):**** Heinz-Otto Peitgen, University of Bremen and Florida Atlantic University **

*General Lecture - Harnessing Chaos*

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*

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 (April 17-20):**** Robert Calderbank, AT&T Laboratories **

*General Lecture - 50 Years of Information and Coding*

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*

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*

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 (April 10-11):**** Gilbert Strang, MIT**

*General Lecture - Partly Random Graphs and Small World Networks*

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*

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*

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 **

*Public Talk - Szego Polynomials Applied to Signal Processing*