Harold J. Gay Lecture Series

PDEs & Fractals

Geometry with its applications has been at the heart of the development of partial differential equations and boundary value problems since the very beginning. In physics, biology, economics, and other applied fields, a variety of new problems are now emerging that display unusual geometrical, analytical and scaling features, possibly of fractal type. The objective of these lectures is to acquire the view of outstanding mathematicians on the subject of differential equations and fractals, and their developments and applications, in a broad perspective encompassing both classical highlights and contemporary trends.

Bartlett Presentation Room
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Friday, December 9, 2011, 11:00 a.m.
Some New Uses of Functions with Finite Total Variation

1) Estimate the length of a continuous curve using drafting dividers.
2) Enhance an image by sharpening edges and smoothing roughness.
3) Find a spanning surface of least area in a very singular space X.
These 3 problems can all be attacked using the notion of the Total Variation (TV) of a function.
For 1) the usual formula for the length of a curve f :[a, b] -+ RN is the definition of TV( f).
For 2) one considers a two-variable function g :[a, b] x [c, d] -+ [0, 1] giving the grayscale intensity of an image in the rectangle [a, b] x [c, d]. Assuming TV(g), suitably defined, is finite, edges and roughness are described using different parts of the derivative of g. The models for image enhancement that we will discuss involve interesting POE's and many open questions.
For 3) we consider functions h whose values are finite sums of point masses in X. Assuming X has a distance function, we find a geometrically reasonable notion of the distance between 2 such sums. Then functions with TV(h) < oo essentially determine the surfaces and give a surprising amount of regularity. Analysis in singular spaces has had wide applications from algebraic geometry to data analysis.

Higgins Labs, Room 218
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Friday, December 2, 2011, 3:00 p.m.
Cloaking: Where Science Fiction Meets Science

Cloaking involves making an object partly or completely invisible to incoming waves such as sound waves, sea waves or seismic waves, but usually electromagnetic waves such as visible light, microwaves, infrared light, or radio waves. Camouflage and stealth technology achieve partial invisibility, but can one achieve true invisibility from such waves? This lecture will survey some of the wide variety of ideas on cloaking: these include cloaking by plasmonic covers, transformation based cloaking, non Euclidean cloaking, cloaking due to anomalous resonance, cloaking by complementary media, active interior cloaking and active exterior cloaking. Beautiful mathematics is involved.

Olin Hall, Room 107
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Friday, March 25, 2011, 3:00 p.m.
Topological Quantum Information and the Jones Polynomial

We give a quantum statistical interpretation for the Jones polynomial in terms of the Kauffman bracket polynomial state sum. The Jones polynomial is a well-known topological invariant of knots in three-dimensional space that is closely related to structures in statistical mechanics and quantum field theory. We use this interpretation to give a new quantum algorithm for computing the Jones polynomial. This algorithm is useful for its conceptual simplicity, and it applies to all values of the polynomial variable that lie on the unit circle in the complex plane. Letting H(K) denote the Hilbert space for this model, there is a natural unitary transformation U from H(K) to itself such that <K> = Trace(U) where <K> is the bracket polynomial for the knot K. The quantum algorithm for <K> arises directly from this formula via the Hadamard Test. We also review how we have implemented quantum algorithms for the Jones polynomial in NMR experiments and we show how the framework of the present model is related to recent work in knot theory such as Khovanov homology. This talk does not assume any background in either quantum computing or in the theory of knots and their invariants.

Salisbury Labs, Room 104
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Friday, October 8, 2010, 3:00 p.m.
The spreading of invasive species and related topics

In 1937 R.A. Fisher created a model for the spread of a ﬁtter mutant into an established population of the same species. The model was a semilinear parabolic equation for the fraction of the advantaged population. Fisher conjectured that such an invasion spreads with a ﬁnite asymptotic speed, and that this speed is also the slowest speed of a nontrivial traveling wave. This conjecture was proved by Kolmogorov, Petrovsky, and Piscounov in the same year. Since then, such properties have been shown to be true of an extensive set of models in the physical and biological sciences. The models can take the form of partial diﬀerential equations, ﬁnite diﬀerence equations, discrete-time integrodiﬀerence equations, or of more general discrete-time recursions in one or more space dimensions. They can also involve interactions between two or more species,

This lecture will give an outline of old and new results in the study of spreading speeds and traveling waves for such models.

(Refreshments at 2:30 p.m. in Salisbury Labs, Room 104)

Bartlett Center
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Friday, March 19, 2010, 11:00 a.m.
Imaging with noise

It is somewhat surprising at first that it is possible to locate a network of sensors from cross correlations of noise signals that they record. This is assuming that the speed of propagation in the ambient environment is known and that the noise sources are sufficiently diverse. If the sensor locations are known and the propagation speed is not known then it can be estimated from cross correlation information. Although a basic understanding of these possibilities had been available for some time, it is the success of recent applications in seismology that has revealed the great potential of correlation methods, passive sensors and the constructive use of ambient noise in imaging. I will introduce these ideas in an interdisciplinary, mathematical way and show that a great deal can be done with them. Things become more complicated, and mathematically more interesting, when the ambient medium is also strongly scattering. I will end with a review of what is known so far in this case, and what might be expected.

Bartlett Center
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Friday, January 22, 2010, 3:15 p.m.
Progress in Hyperbolic Conservation Laws

Hyperbolic Conservation Laws, namely nonlinear hyperbolic systems of first order partial differential equations in divergence form, have an illustrious pedigree and diverse applications to physics and beyond. Despite dramatic progress in recent years, this area is still replete with challenging open problems. The lecture will provide a glimpse to the history, the current state of affairs and the emerging trends in this field.

Salisbury Labs, Room 105
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Tuesday, April 14, 2009, 4:00 p.m.
Are Most Triangles Acute or Obtuse?

This talk has two separate parts, both about shapes. First, we ask how a change from circle to polygon affects the solution to a differential equation inside. Key examples are the eigenvalue problem for Laplace's equation, and Poisson's equation u_xx + u_yy = 1. The area between the circle and polygon becomes a crucial quantity and we ask how this leading term in the error might be removed--to improve the accuracy of the eigenvalues and the solution.

Part 2 is about an innocent question--Is a random triangle acute or obtuse? Everything depends on the meaning of "random." Are the angles random or the sides? Is the distribution uniform or normal? New answers keep coming, and some are surprising.

Bartlett Center
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Monday, March 16, 2009, 4:00 p.m.
Scaling Limits of Large Systems

We will discuss the longtime behavior of large systems of interacting particles that evolve in time. The number of particles is conserved. When we rescale space, the local density as a function of space will evolve slowly to its equilibrium value, which is a constant indicating uniform density. With suitable recaling of time, it will evolve, in the limit, according to some nonlinear PDE. We will examine several examples of this behavior.

Bartlett Center
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Thursday, November 20, 2008, 3:00 p.m.
Steady Rotational Water Waves

Precise study of water waves began with the derivation of the basic mathematical equations of fluids by the great Euler in 1752. In the two and a half centuries since then, the theory of fluids has played a central role in the development of mathematics. Water waves are fluids with a free surface. I will discuss waves that travel at a constant speed. Using local and global bifurcation theory, we now know how to prove that there exist very many such waves. They may have either small or large amplitudes. I will outline the existence proof and then exhibit some recent computations of the waves using numerical continuation. The computations illustrate certain relationships between the amplitude, energy and mass flux of the waves. If the vorticity is sufficiently large, the first stagnation point of the wave occurs either at the crest, on the bed directly below the crest, or in the interior of the fluid. This work is a perfect example of the synergy between theory and computation.

Higgins Labs 202
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Friday, April 18, 2008, 2:00 p.m.
Transport in Small Systems with a Look at Motor Proteins

Motion in small live systems has many challenges. Prominent environmental conditions are high viscosity and warmth. Not only is it difficult to move, but maintaining a course is rendered difficult by immersion in a highly fluctuating bath. This holds especially for the motor proteins responsible for much of eukaryotic cellular traffic. The situation falls under the rubric of diffusion mediated transport. We give some brief historical notes, including the original work of many distinguished scientists, and then turn to an approach based on the Monge transport problem (1787) and its modern version, Monge-Kantorovich Theory, which offers us a means of studying these systems with analysis. We arrive at a precipice: does this help? Can we say anything about the behavior of the cellular process? An exciting venue for math in the natural world!

Bartlett Center
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Friday, September 28, 2007, 3:30 p.m.
Scaling, Self-similarity, and the Renormalization Group in Partial Differential Equations

Scaling (self-similar) solutions to the partial differential equations entered the applied mathematics field 200 years ago. Until recently they were treated mostly as "exact special" solutions to some very specific problems--elegant, sometimes useful for qualitative investigations of the models but, in general, very limited in their significance elements of the general theories.

Gradually, it was recognized that the value of these solutions is much more significant: they are the intermediate asymptotics to the solutions to wider classes of problems when the influence of the details of the initial and/or boundary conditions already disappeared, but the solution is still far from its ultimate form. The appearance of computers did not reduce but increased the value of the scaling solutions.

In some cases (in fact, such cases are rather rare) the scaling solutions can be obtained using the dimensional analysis. However, as a rule this is not the case: scaling solutions appear due to the invariance of the problem to an additional group (note,--group, not semigroup), which we identify as the renormalization group.

A survey of these topics will be presented in this lecture; illustrative examples will be used.

WPI Higgins Labs 116
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Friday, March 23, 2007, 2:00 p.m.
Quasistaticity

A problem for the evolution in time of some system is said to have a quasistatic approximation when the velocity and acceleration are neglected. These derivatives can usually be neglected if they have coefficients that are small parameters. In this case, formal asymptotic methods might exhibit the detailed effects of these parameters. Rigorous asymptotic justifications, which provide error estimates and are typically far harder to carry out, are used by those compulsive about mathematical hygiene, but seldom say more that the formal methods.

The purpose of this lecture is to give rigorous justifications of the quasistatic behavior of solutions of the differential equations governing a couple of conceptually simple problems from particle and continuum mechanics. The justification for these justifications is that the solutions of these simple problems exhibit strange and surprising behavior.

WPI Bartlett Center
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Monday, February 26, 2007, 4:00 p.m.
The geometrical basis of numerical stability

The accuracy of a numerical solution to a partial differential equation depends on the consistency and stability of the discretization method. While consistency is usually elementary to establish, stability of numerical methods can be subtle, and for some key PDE problems the development of stable methods is extremely challenging. After illustrating the situation through simple but surprising examples, we will describe a powerful new approach--the finite element exterior calculus---to the design and understanding of discretizations for a variety of elliptic PDE problems. This approach achieves stability by developing discretizations which are compatible with the geometrical and topological structures, such as de Rham cohomology and Hodge decompositions, which underlie well-posedness of the PDE problem being solved.

WPI Salisbury Labs 104
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Wednesday, December 6, 2006, 11:00 a.m.
From Collisionless Shocks to Integrable Systems

Collisionless shocks have been studied twice. First in the 1950s they were proposed as a mechanism for heating up a controlled nuclear fusion machine for creating energy. But their mathematical structure was an open question. Now such a shock has been observed by Voyager 2 in its travels through space. The lecture will first describe how collisionless shocks occur in the solar system .Then we will examine in a simple model what we mean mathematically by a collisionless shock and why its structure is a puzzle. Finally we look at how these investigations led to the study of completely integrable systems of partial differential equations.

WPI Bartlett Center
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Friday, December 1, 2006, 11:00 a.m.
Multidimensional Conservation Laws

The analysis of quasilinear hyperbolic partial differential equations presents a number of challenges. Although equations of this type are important in a number of applications, ranging from high-speed aerodynamics, through magnetohydrodynamics, to multiphase flows important in industrial technology, there is little theory against which even to check the reliability of numerical simulations.

Development of a theory for conservation laws in a single space variable has led to remarkable advances in analysis, including the theory of compensated compactness and the study of novel function spaces. Recently, a number of groups have begun to approach multidimensional systems via self-similar solutions.

In this talk, I will give some history of the development of conservation law theory, including an indication of why the applications are important. I will describe some of the recent results on self-similar solutions, and the interesting results in analysis that they involve. Finally, I will outline some of the paradoxical questions that remain.

WPI Olin Hall 107
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Friday, November 10, 2006, 3:00 pm
Fractal Roughness
Beautiful, damn hard, and surprisingly useful

Some fractals imitate mountains, clouds, stock markets, and many other aspects of nature and culture. Others yield wild and wonderful new patterns that a child can draw but great masters struggle or fail to understand. All are shapes that look the same from any distance, far away or close by. Since time immemorial, some have been used by great artists. A hundred years ago, mathematicians called them monsters and an excuse to split from physics. Now—especially since my 1975 term fractal—they help heal this split. Fractal geometry helps mathematics and the sciences to cross a long avoided boundary between the smooth and the rough. Partial differential equations must allow very rough solutions. Man's basic sensation of roughness can now be measured intrinsically by fractal dimension, first step to being mastered. An introduction to fractal geometry with updates on some current developments including finance.

WPI Stratton Hall 203
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Friday, March 31, 2006, 2:00 pm
Degenerate Symmetric Matrices

A symmetric matrix is called degenerate by physicists if it has a multiple eigenvalue.Wigner and von Neumann have shown long ago that the degenerate matrices form a variety of codimension two in the space of all symmetric matrices.This explains the phenomenon of "avoidance of crossing".On the other hand the degenerate matrices are characterised by the single equation discr(S)=0, where discr(S) is the discriminant of S.In this talk we investigate the nature of the discriminant, especially its representation as a sum of squares. In the second part it will be shown that some pencils of real symmetric matrices always contain a degenerate one.

WPI Stratton Hall 203
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Tuesday, October 4, 2005 11:00 am
Distance to the boundary and Hamilton-Jacobi equations

We study the set of points where the distance function to the boundary is not smooth. Its dimension is estimated. A similar result is derived for the singular set of solutions of some Hamilton-Jacobi equations.

Friday, October 7, 2005 11:00 am
Estimates for laminar materials

Some problems on laminar materials lead to elliptic systems, with coefficients that are smooth in subregions but may jump from region to region. It is of interest to get estimates on the solution and its derivatives, in each subregion, which are independent of the narrowness of the regions. Some such estimates are presented.

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Last modified: November 17, 2011 09:26:17