MASS Colloquia 2007

Thursday, September 13	Tadashi Tokieda, University of Cambridge, N/A
2:30pm	Toy modeling
ABSTRACT	Would you like to see some toys? A 'toy' for us is an object from daily life which you can find around the house or make yourself under 10 minutes (it must be that simple) but which, when played with in an imaginative fashion, reveals behaviors strange enough to puzzle a good scientist for over 10 weeks (it must be that mysterious). We will experiment with a dozen toys and study them mathematically. What we hope to learn is how to turn natural phenomena into mathematics and back----in short, {\it modeling}.

Thursday, September 6	Alexadre Kirillov, University of Pennsylvania, N/A
2:30pm	Analysis, Geometry and Arithmetics of fractals

Thursday, September 20	Dmitry Burago, Penn State, N/A
2:30pm	Playing zero-sum games and convex geometry after John Nash
ABSTRACT	We will discuss a brilliant and at the same time remarkably elementary work of a Nobel Medalist John Nash. In particular, a proof of the main theorem will be presented.

Thursday, September 27	Igor Rivin, Temple University, N/A
2:30pm	Geometry of Polyhedra
ABSTRACT	I will describe a set of related questions about polyhedra in three-dimensional Euclidean and non-Euclidean space. Example questions are: what does the metric of a surface of a polyhedron look like? What can you say about the angles of the faces? Dihedral angles? How much information do you need to give to determine a polyhedron? The questions go back to (at least) Euclid, and the answers keep coming even at present.

Thursday, October 11	Mark Levi, Penn State, N/A
2:30pm	Physical proofs of mathematical theorems
ABSTRACT	Based on a forthcoming book, we shall discuss examples of mathematical results proved by physical arguments.

Thursday, October 18	Zvezdelina Stankova, Mills College, N/A
2:30pm	What does the future hold for restricted patterns?
ABSTRACT	Restricted patterns made their major debut into the math arena in the 80's, with the works of Simion, Schmidt, Wilf, Lovasz, Lothaire, Rotem, Richards and many others. In truth, they had already permeated mathematical research since the 60's via Robinson-Schensted's, Knuth's and Stanley's earlier results. In fact, any time you stumble upon the Catalan, Fibonacci or Stirling numbers, Dyck paths, Young diagrams, random matrices, generating trees or Chebychev or Kazhdan-Lusztig polynomials, restricted patterns are likely to appear in one incarnation or another. The more recent rebirth of the topic was initiated by West in 1992, and taken up by a number of researchers, including Bona, Albert, Arratia, Backelin, Babson, Mansour, Marcus and Tardos. Yet, the ever-tempting Wilf-classification of restricted patterns is still a wide open question. In this talk, we shall walk along several paths of pattern-exploration that originated at the Duluth REU in 1991-92, and discuss whether, among the array of generated ideas and methods, there is a "true'' way of approaching pattern-avoidance equivalence and ordering.

Thursday, October 25	Dmitry Khavinson, University of South Florida, N/A
2:30pm	From the Fundamental Theorem of Algebra to astrophysics: a "harmonious" journey
ABSTRACT	The Fundamental Theorem of Algebra first rigorously proved by Gauss states that each complex polynomial of degree n has precisely n complex roots. In recent years various extensions of this celebrated result have been considered. We shall discuss the extension of the FTA to harmonic polynomials of degree n. In particular, the 2003 theorem of D. Khavinson and G. Swiatek that shows that the harmonic polynomial \bar{z}-p(z), deg p=n>1 has at most 3n-2 zeros as was conjectured in the early 90's by T. Sheil-Small and A. Wilmshurst. More recently L. Geyer was able to show that the result is sharp for all n. In 2004 G. Neumann and D. Khavinson showed that the maximal number of zeros of rational harmonic functions \bar{z}-r(z), deg r=n>1 is 5n-5. It turned out that this result resolved the conjecture by an astrophysist S. H. Rhie dealing with the estimate on maximal number of images of a star if the light from it is deflected by n co-planar masses. The first non-trivial case of one mass was already investigated by A. Einstein around 1912. We shall also discuss the problem of gravitational lensing of a point source of light, e.g., a star, by an elliptic galaxy, more precisely the problem of the maximal number of images that one can observe. Under some more or less "natural'' assumptions on the mass distribution within the galaxy one can prove that the number of visible images can never be more than four. Interestingly, the latter situation can actually occur and has been observed by astronomers. Still there are much more open questions than there are answers.

Thursday, October 25

Dmitry Khavinson, University of South Florida, N/A

2:30pm

From the Fundamental Theorem of Algebra to astrophysics: a "harmonious" journey

ABSTRACT

The Fundamental Theorem of Algebra first rigorously proved by Gauss states that each complex polynomial of degree n has precisely n complex roots. In recent years various extensions of this celebrated result have been considered. We shall discuss the extension of the FTA to harmonic polynomials of degree n. In particular, the 2003 theorem of D. Khavinson and G. Swiatek that shows that the harmonic polynomial \bar{z}-p(z), deg p=n>1 has at most 3n-2 zeros as was conjectured in the early 90's by T. Sheil-Small and A. Wilmshurst. More recently L. Geyer was able to show that the result is sharp for all n.

In 2004 G. Neumann and D. Khavinson showed that the maximal number of zeros of rational harmonic functions \bar{z}-r(z), deg r=n>1 is 5n-5. It turned out that this result resolved the
conjecture by an astrophysist S. H. Rhie dealing with the estimate on maximal number of images of a star if the light from it is deflected by n co-planar masses. The first non-trivial case of one
mass was already investigated by A. Einstein around 1912.

We shall also discuss the problem of gravitational lensing of a point source of light, e.g., a star, by an elliptic galaxy, more precisely the problem of the maximal number of images that one can observe. Under some more or less "natural'' assumptions on the mass distribution within the galaxy one can prove that the number of visible images can never be more than four. Interestingly, the latter situation can actually occur and has been observed by astronomers. Still there are much more open questions than there are answers.

Thursday, November 1	Andrei Gudkov, Roswell Park Cancer Institute, Buffalo, NY, N/A
2:30pm	Can understanding cancer be translated into effective anticancer therapy?
ABSTRACT	Multicellular organisms, such as human, can be viewed as a social society of cells differing in their professional skills and functions with all major properties of social system in place (birth, growth, education, choice of profession, ageing, death, etc.). Rules of social behavior determining stability and functionality of the society involve strict multifold negative control over cell multiplication and place of residence by external (immune system, an equivalent of law enforcing agencies) and internal (mechanisms of self-evaluation, an equivalent of personal moral rules and cell ageing) mechanisms. In extreme cases of cell "misbehavior" these mechanisms may lead to enforced or suicidal cell death. Cancer development requires not only loss of normal response to negative growth regulation (equivalents of criminals) but also acquisition by tumor cells the ability to avoid regulated cell death. All the above-mentioned mechanisms are relatively well understood as a result of recent advancements of molecular and cellular biology. However, the majority of currently used cancer treatment approaches were developed long before these mechanisms become understood. Perspectives and principles of translation of new knowledge of mechanisms into effective cancer treatment strategies and new anticancer drugs will be presented and discussed.

Thursday, November 8	Alexandra Shlapentokh, East Carolina University, N/A
2:30pm	Hilbert's Tenth over Subrings of Q
ABSTRACT	Hilbert's Tenth Problem was a question concerning existence of an algorithm to solve arbitrary polynomial equations over Z. Yuri Matijasevich building on work of Julia Robinson, Martin Davis and Hilary Putnam showed that such an algorithm does not exist. However the analogous question is still open if we look for solutions in rational numbers. We discuss the status of this open problem today, as well as related issues of Diophantine definability.

Thursday, November 15	Steven Krantz, American Institute of Mathematics, N/A
2:30pm	A Glimpse of Several Complex Variables
ABSTRACT	We present two of the key results from the foundations of the function theory of several complex variables. These illustrate the fundamental differences between one complex variable and several complex variables. The first result is Hartogs's theorem about domains that are NOT domains of holomorphy. Contrast with one variable, where every domain is a domain of holomorphy. The second result is Poincare's theorem that the ball and the bidisc are biholomorphically inequivalent. Contrast with one variable, where the Riemann mapping theorem says that any two simply connected domains (except the plane) are conformally equivalent. Discussion will be provided to put these results in context, and to point to research areas of current interest.

Thursday, November 29	Alexander Razborov, Institute for Advanced Study, N/A
2:30pm	Inverse problems in Arithmetic Combinatorics
ABSTRACT	Arithmetic Combinatorics studies behavior of subsets of algebraic structures under their operations. In inverse problems we know something about this behaviour; a typical assumption would indicate that the subset A in question expands under those operations much less than expected. And we want to derive some conclusions about the internal structure of A. It turns out that even the most basic questions of this sort about the simplest structures like Z or F_p lead to either very deep and difficult theorems or to widely open important questions, and I hope to discuss some of them in my talk. It is worth noting that in recent years many of these ideas and results have found rather unexpected applications in many different areas, such as Harmonic Analysis or Theoretical Computer Science.