# Harold J. Gay Lecture Series

## Current Series

### Viorel Barbu

Alexandru Ioan Cuza University, Iasi, Romania

#### PDEs and Variational-based Models for Image Restoring

**Friday, April 25, 2014
3:10pm, Kaven Hall 116**

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One surveys here a few nonlinear diffusion models in image restoration and denoising with main emphasis on that described by nonlinear parabolic equations of gradient type. The well-posedness of the corresponding Cauchy problem as well as stability of the derived finite difference scheme is studied from perspectives of nonlinear semigroup theory. Most of denoising PDE procedures existing in literature, though apparently are efficient at experimental level, are however mathematically ill posed and our effort here is to put them on more rigorous mathematical basis.

## Past Series

### Sergiu Klainerman

Princeton University

**Are Black Holes Real?**

**Friday, February 14, 2014
3:10pm, Salisbury Labs 402**

Black Holes are precise mathematical solutions of the Einstein field equations of General Relativity. Some of the the most exciting astrophysical objects in the Universe have been identified as corresponding to these mathematical Black Holes, but since no signals can escape their extreme gravitational pull can we be sure that we have made the right identification?

I will show how the issue of reality of Black Holes can be addressed by nothing more than pen and paper.

I will discuss three fundamental mathematical problems intimately connected to the issue of Reality of Black Holes: Rigidity, Stability and Collapse. I will then survey some of the main results which have been obtained in the last thirty years.

### Silvio Micali

Ford Professor of Engineering, MIT

#### Proofs, Secrets, and Computation

**Friday, November 15, 2013
3:15pm, Salisbury Labs 411**

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We show how Theory of Computation has revolutionized our millenary notion of a proof, revealing its unexpected applications to our new digital world. In particular, we shall demonstrate how interaction can make proofs much easier to verify, dramatically limit the amount of knowledge released, and yield the most secure identification schemes to date.

### Luis Caffarelli

Department of Mathematics and the Institute for Computational Engineering and Sciences, University of Texas at Austin

#### The Mathematical Idea of Diffusion

**Monday, February 11, 2013
2PM, Bartlett Center **

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I will discuss and review the many mathematicalm phenomena that can be considered a "diffusion process" from the classical heat diffusion and Brownian motion, to geometric configurations, like movement by mean curvature, free boundaries and surface diffusion.

### Irene Fonseca

Mellon College of Science Professor of Mathematics

Director, Center for Nonlinear Analysis, Carnegie Mellon

#### Variational Methods in Materials Science and Image Processing

**Friday, February 1, 2013
4pm, Salisbury Labs 104**

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Several questions in applied analysis motivated by issues in computer vision, physics, materials sciences and other areas of engineering may be treated variationally leading to higher order problems and to models involving lower dimension density measures. Their study often requires state-of-the-art techniques, new ideas, and the introduction of innovative tools in partial differential equations, geometric measure theory, and the calculus of variations. In this talk it will be shown how some of these questions may be reduced to well understood first order problems, while in others the higher order terms play a fundamental role. Applications to phase transitions, to the equilibrium of foams under the action of surfactants, imaging, micromagnetics, thin films, and quantum dots will be addressed.

### Jüri Engelbrecht

CENS – Centre for Nonlinear Studies

Institute of Cybernetics at Tallinn University of Technology

#### Modeling of Deformation Waves in Microstructured Solids

**Friday, January 18, 2013
**

**2pm, Salisbury Labs 105**

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The response of many materials (metals, alloys, composites, etc.) to external loading may essentially be influenced by an existing or emerging internal structure of materials at smaller scales. Over the past five decades, a number of advanced generalized continuum theories have been introduced for modeling the structural internal inhomogeneities on the macroscopic behavior of materials. We have found that the concept of internal variables is sufficiently general for modeling waves in such microstructured solids.

The models are based on Mindlin-type (micromorphic) theory but the thermodynamical considerations are added. The formalism includes the material (canonical) balance equations for material momentum and energy, while the internal structure is described by internal variables with the governing equations derived from the dissipation inequality. In such a way, internal fields of microdeformation and/or microtemperature can be taken into account. An essential generalization from the theoretical side is that the resulting governing equations for microstructure are not limited to first-order diffusion equations but can be hyperbolic.

The general nonlinear 3D theory is presented but for explanations, the 1D models are analysed in detail. In this case the basic model is a system of two second-order equations which takes the coupling of macro- and microstructure into account. This system allows several modifications: the full fourth-order equation, the approximated hierarchical equation, and the one-wave evolution equation. The fourth-order equation is of the Boussinesq type which allows to bridge solid mechanics with the fluid dynamics. The further modifications allow to model multiscale problems (scale within a scale) and internal temperature fields.

The dispersion analysis has revealed many explanations about the distortion mechanisms of wave profiles, the solutions of nonlinear Boussinesq-type equations and evolution equations (by the pseudospectral method) have demonstrated the emergence and interaction of solitary waves in microstructured solids and the distortion of their profiles which is due to the nonlinear properties of a microstructure. The direct computations (by the finite volume method) permit to compare the wave motion in regular (laminated) materials with microstructured solids modeled by continuum theories. Several numerical results will be shown to illustrate the various aspects of the modeling including the solving of inverse problems.

The talk gives an overview on research in CENS over the last years.

### Professor Luc Tartar, Dr.Sc.

University of Paris

Carnegie Mellon University

#### From ODE to "beyond PDE"

**Thursday, November 29, 2012
Salisbury Labs 104**

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In the late 1960s, Benoît Mandelbrot wrote that a piece of the coast of Brittany has Hausdorff dimension 1.6, but one difficulty with his statement is that there is no coast of Brittany! Indeed, there are tides and the boundary between the Atlantic Ocean and the land changes in a few hours, but the coast also erodes on a much longer time scale, and the rough nature of the coast which interested him took an extremely long time to appear. Using classical models in continuum mechanics leads to systems of PDE (partial differential equations), but modeling erosion involves some chemistry, and not so classical boundary conditions in some parts: even the movement of grains of sand resulting from the waves on a sandy beach is a much too hard mathematical question at the moment! In short, one has a complex evolution problem involving various length scales and time scales, and the mathematical tools of GTH (the general theory of homogenization) are not good enough yet for handling such questions, but it is already known that one should not expect an effective equation to be a system of PDE, since nonlocal effects appear in simpler problems: I coined the term "beyond PDE" for the new class of systems to consider, which is not clearly understood at the moment, in part because no good mathematical theory of nonlinear nonlocal effects exists.

In the mid-1980s, Alain LeMéhauté wrote that some electrodes develop a Hausdorff dimension between 2 and 3, adding that the Hausdorff dimension is different for old electrodes (he also pretended to invent fractional derivatives, which I had been taught by Laurent SCHWARTZ twenty years before); here, erosion involves electricity and chemistry coupled with continuum mechanics at various length scales and time scales, since electrons and ions moving near the electrode interact and create new chemical compounds, which must be evacuated for letting others approach the electrode.

Rough objects should not always be modeled by self-similar fractal sets, and something about the evolution processes which create the roughness should be studied in order to select better mathematical questions to consider. The creation of fjords in Norway or Greenland resulted from the erosion by glaciers (at a time when the level of the Atlantic ocean was much below its actual level), while the coast of Brittany resulted from the erosion by salted water, but the same word erosion corresponds to something more mechanical in the case of

a glacier, and more chemical in the case of salted water: the two questions then correspond to two quite different evolution equations, creating different kinds of roughness at different scales.

Using ODE (ordinary differential equations) in mechanics corresponds to the 18th century point of view of classical mechanics, while the 19th century point of view of continuum mechanics uses PDE, but the 20th century point of view requires going “beyond PDE”. In the absence of adapted mathematical tools, physicists sometimes guessed wrongly, and did not correct some obvious mistakes (like EINSTEIN creating a "fake Brownian motion" by confusing the jumps in position used by BACHELIER for modeling the stock market and the jumps in velocity observed by R. BROWN); correcting some less obvious mistakes made in the 19th and 20th centuries will require the work of 21st century mathematicians (with probably a good physical intuition) for improving the needed mathematical tools for describing the evolution of rough objects.

In the talk, I shall describe a few basic examples of the mathematical questions concerning effective equations for mixtures (i.e., roughness occurring in the bulk), and why nonlocal effects appear, without relying on probabilistic methods, which are not well adapted to the PDE of continuum mechanics or physics.

### Professor Alfio Quarteroni

École Polytechnique Fédérale, Lausanne, Switzerland;

Politecnico di Milano, Milan, Italy

#### Modeling and Complexity Reduction in PDEs for Multiphysics

**Friday, April 13, 2012, 11:00 a.m.
Atwater Kent Labs 219**

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The numerical solution of complex physical problems typically requires the setup of appropriate PDE models and of accurate numerical methods. Often, the numerical problem is so large that a reduction of its complexity becomes mandatory. This can be achieved by a manifold strategy with the attempt of simplifying the original mathematical model, devising novel numerical approximation methods, and developing efficient parallel algorithms that exploit the dimensional reduction paradigm. In different circumstances, especially in control and optimization problems for parametrized PDEs, reduced order models, such as the reduced basis method and the simplified shape parametrization method, can be used to alleviate the computational complexity. After introducing the proper mathematical setting, in this presentation a variety of representative applications to blood flow modeling, environmental modeling, and sports design will be illustrated.

### Professor David Jerison

Professor of Mathematics

Massachusetts Institute of Technology

#### Internal Diffusion-Limited Aggregation

**Friday, March 23, 2012, 11:00 a.m.
Bartlett Presentation Room**

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In 1986, chemists Paul Meakin and John Deutch proposed a model for the process of removal of material such as occurs in electropolishing, erosion and etching. Their model, known as internal diffusion-limited aggregation or internal DLA, is a random model in which corrosive fluid is represented as a growing blob of lattice sites that eats away at its solid surroundings as it grows. The main question is, how smooth is the boundary of the blob? To answer this question we'll need partial differential equations, Fourier series, martingales, and methods used in analytic number theory to count the number of lattice points in a disk or ball.

### Professor Robert Hardt

W.L. Moody Professor of Mathematics

Rice University

#### Some New Uses of Functions with Finite Total Variation

**Friday, December 9, 2011, 11:00 a.m.
Bartlett Presentation Room**

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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.

### Professor Graeme Milton

Department of Mathematics

University of Utah

#### Cloaking: Where Science Fiction Meets Science

**Friday, December 2, 2011, 3:00 p.m.
Higgins Labs, Room 218**

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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.

### Professor Louis H. Kauffman

Department of Mathematics, Statistics, and Computer Science

University of Illinois at Chicago

#### Topological Quantum Information and the Jones Polynomial

**Friday, March 25, 2011, 3:00 p.m.
Olin Hall, Room 107**

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- Download the talk (pdf)

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

### Professor Hans Weinberger

Professor Emeritus, Mathematics

University of Minnesota

#### The spreading of invasive species and related topics

**Friday, October 8, 2010, 3:00 p.m.
Salisbury Labs, Room 104**

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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)

### Professor George C. Papanicolaou

Robert Grimmett Professor in Mathematics

Stanford University

#### Imaging with noise

**Friday, March 19, 2010, 11:00 a.m.
Bartlett Center**

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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.

### Professor Constantine M. Dafermos

Brown University

#### Progress in Hyperbolic Conservation Laws

**Friday, January 22, 2010, 3:15 p.m.
Bartlett Center**

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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.

### Professor Gilbert Strang

Massachusetts Institute of Technology

#### Are Most Triangles Acute or Obtuse?

**Tuesday, April 14, 2009, 4:00 p.m.
Salisbury Labs, Room 105**

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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.

### Professor Srinivasa Varadhan

Courant Institute of Mathematical Sciences

New York University

#### Scaling Limits of Large Systems

**Monday, March 16, 2009, 4:00 p.m.
Bartlett Center**

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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.

### Professor Walter Strauss

Department of Mathematics

Brown University

#### Steady Rotational Water Waves

**Thursday, November 20, 2008, 3:00 p.m.
Bartlett Center**

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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.

### Professor David Kinderlehrer

Center for Nonlinear Analysis and Department of Mathematical Sciences

Carnegie Mellon University

#### Transport in Small Systems with a Look at Motor Proteins

**Friday, April 18, 2008, 2:00 p.m.
Higgins Labs 202**

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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!

### Professor Grigory Isaakovich Barenblatt

Professor in Residence, Department of Mathematics

University of California at Berkeley

#### Scaling, Self-similarity, and the Renormalization Group in Partial Differential Equations

**Friday, September 28, 2007, 3:30 p.m.
Bartlett Center**

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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.

### Professor Stuart S. Antman

Institute for Physical Science and Technology

Institute for Systems Research

University of Maryland, College Park

#### Quasistaticity

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

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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.

### Professor Douglas N. Arnold

Institute for Mathematics and its Applications

University of Minnesota

#### The geometrical basis of numerical stability

**Monday, February 26, 2007, 4:00 p.m.
WPI Bartlett Center**

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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.

### Professor Cathleen Morawetz

Courant Institute, New York University

#### From Collisionless Shocks to Integrable Systems

**Wednesday, December 6, 2006, 11:00 a.m.
WPI Salisbury Labs 104**

- Download the flyer (pdf)

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.

### Professor Barbara Lee Keyfitz

Fields Institute and University of Houston

#### Multidimensional Conservation Laws

**Friday, December 1, 2006, 11:00 a.m.
WPI Bartlett Center**

- Download the flyer (pdf)

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.

### Professor Benoit Mandelbrot

Yale and Pacific Northwest National Laboratory

#### Fractal Roughness

Beautiful, damn hard, and surprisingly useful

**Friday, November 10, 2006, 3:00 pm
WPI Olin Hall 107**

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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.

### Professor Peter D. Lax

Courant Institute of Mathematical Sciences, NYU

#### Degenerate Symmetric Matrices

**Friday, March 31, 2006, 2:00 pm
WPI Stratton Hall 203**

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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.

### Professor Louis Nirenberg

Courant Institute of Mathematical Sciences, NYU

#### Distance to the boundary and Hamilton-Jacobi equations

**Tuesday, October 4, 2005 11:00 a.m.
WPI Stratton Hall 203**

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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.

#### Estimates for laminar materials

**Friday, October 7, 2005 11:00 a.m.
WPI Stratton Hall 203**

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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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