MASS Colloquia 2004

Thursday, September 16 Professor Mark Levi (Penn State) 2:30 p.m. Riemann Mapping Theorem via Steepest Descent ABSTRACT : Conformal mappings of the plane are the ones which map infinitesimal squares to infinitesimal squares. Riemann Mapping Theorem is a fundamental fact which states, roughly speaking, that any simply connected domain in the plane can be conformally mapped to a circle. We give (an apparently new) proof of the Riemann Mapping using the idea of steepest descent.

Thursday, September 23 Professor Robert Connelly (Cornell University) 2:30 p.m. Volume and area formulas. ABSTRACT : Heron's formula gives the area of a triangle in terms of the lengths of its edges, and there is a similar formula for the volume of a tetrahedron in terms of the lengths of its edges. in 1995 I. Sabitov showed that for any triangulated surface in three-space, there is a polynomial that is satisfied by the volume bounded by the surface, and its coefficients are themselves polynomials in the lengths of the edges of the triangulated surface. This is related to another polynomial satisfied by the area enclosed by a polygon whose vertices lie on a circle in the plane which was studied by David Robbins before he died. These polynomials are interesting, but they can be very complicated with a degree that is exponential in the number of vertices.

Thursday, September 30 Professor Andrew Belmonte (W.G. Pritchard Laboratories, The Pennsylvania State University) 2:30 p.m. The dynamics of think flexible things: shaking and breaking ABSTRACT : The dynamics of a deformable continuum—say a solid or a fluid—can be considerably simplified mathematically by considering situations in which one of the dimensions is much smaller than the others. However, this restriction can introduce its own complications. I will discuss some surprising experimental observations which we have made in the lab, which have led us from classical mechanics of strings and rods to fundamental questions involving knots and fragmentation.

Thursday, October 7 Professor Andre Toom (Universidade Federal de Pernambuco, Brazil) 2:30 p.m. Spontaneous symmetry breaking in a 1-D process with variable length. ABSTRACT : For a long time it was a common opinion among physicists that phase transitions are impossible in one-dimensional systems.For example, Section 152 of Landau and Lifshitz's Statistical Physics was called The impossibility of the existence of phases in one-dimensional systems and an argument of physical nature was presented in support of this impossibility. However, mathematical objects are very general and may violate physical intuition. This is one of several attempts to show possibility of phases in 1-D systems. We present a 1-D random particle process with uniform local interaction,which displays some form of spontaneous symmetry breaking, that is non-symmetric distribution under symmetric rules. Particles, enumerated by integer numbers, interact at every step of the discrete time only with their nearest neighbors. Every particle has two possible states, called minus and plus. At every time step two transformations occur. The first one turns every minus into plus with probability $\BE$ and every plus into minus with probability $\GA$ independently from what happens at other places, where $\BE+\GA$ ≤ 1. Under the action of the second one, whenever a plus is a left neighbor of a minus, both disappear with probability $\AL$ independently from fate of other places. If $\BE$ is small enough by comparison with $\AL$2 and we start with all minuses, the minuses remain a majority forever. If $\GA$ is small enough by comparison with $\AL$2 and we start with all pluses, the pluses remain a majority forever. Therefore, if $\BE=\GA$ are small enough by comparison with $\AL$2, we have spontaneous symmetry breaking. If, in addition, $\AL$ < 1/8, we have at least two different invariant measures.

Thursday, October 21 Professor Anatoly Vershik (St. Petersburg State University and Steklov Institute) 2:30 p.m. What does the limit shape mean in geometry and combinatorics? ABSTRACT : Consider a configuration in the plane or in the 2-D lattice that grows in time following certain rules. The problem is to describe the limit shape of the configuration after a very long time. Another question of this type: what is a typical shape of a convex lattice polygon?

Thursday, October 28. Professor Vladimir Retakh (Rutgers University) 10:10 a.m. How many roots does a matrix polynomial equation have? ABSTRACT : Everybody knows how many roots a quadratic equation x2 + px +q = 0 has over the field of complex numbers. Not everybody knows how many roots this equation may have over the ring of complex matrices. In fact, the number of roots may be equal to 0, 1, …, ∞. I am going to discuss this and other related results in my talk.

Thursday, October 28 Professor Arek Goetz (San Francisco State University) 2:30 p.m. The dynamics and geometry of microscopic structures in piecewise rotations. ABSTRACT : Let f(x): [0,1] ->[0,1] be the fractional part of (x + t). Iteration of f(x) results in the dynamics we understand. The sequence x, f(x), f(f(x)),..., is finite if t is a rational number or it is infinite and uniformly distributed if t is irrational. The map f(x) is discontinuous at x=1-t, it exchanges two intervals, [0,1-t) and [1-t,1). This is an example of the simplest interval exchange transformation. Such maps when the number of exchanged intervals is greater than two have been extensively studied, partially due tons is their connection with rational billiards. Piecewise rotations are two dimensional generalizations of interval exchanges. In this multimedia talk, we will invite the audience to take a tour of fractal structures arising from the action of the piecewise rotations. These structures are produced on a computer using rigorous algorithms with roots in basic algebraic number theory. We propose open questions and make available a rigorous computer package for a later use in the exciting process of discovery. ( Examples of piecewise rotations include, exchanges of two triangles, or the pizza map. The pizza map T rearranges a finite number of cones (pizza slices) and then T acts as translation on all pieces. The resulting orbit behavior includes familiar behavior from dimension one as well as it features a rich and tantalizing structure of polygons, sets whose iteration never breaks into smaller pieces. A computer zoom on this structure unravels a new landscape of dynamical and geometric phenomena. Unlike in dimension one, here often we observe many periodic domains. The key to begin understanding the dynamics of piecewise rotations is to investigate return actions to smaller domains. However, unlike in one dimensional case where the number of pieces that come back to an interval is finite, in two dimensional dynamics, such a number may be infinite. If, for example, the return action looks like the original map, just smaller, then we are very lucky and can conclude that the map gives rise to a fractal. Often the return actions are very complicated and in order to keep track of details, we introduce using numbers in cyclotomic fields, that is sets of rational polynomial expressions in roots of unity. Using such a tool allows us to prove new rigorous results.

Thursday, November 4 Professor Richard Schwartz (University of Maryland) 2:30 p.m. Experiments with triangular billiards. ABSTRACT : A billiard path on a triangle describes the trajectory taken by a frictionless and infinitely small billiard ball as it rolls around on a billiard table shaped like the triangle. A periodic billiard path is one which endlessly repeats itself. Amazingly, it is not known if every triangle has a periodic billiard path. For acute triangles the affirmative result was known since the late 1770's; for right triangles the affirmative result was established in the 1990's independently by Holt and Galperin-Stepin-Vorobets. Not much is known about the obtuse case. In my talk I will demonstrate a computer program I wrote, which searches for periodic billiard paths in triangles. I will demonstrate, at least experimentally, how every triangle with angles less than 100 degrees has a periodic billiard path and I will discuss how one converts the numerical evidence from the plots into a rigorous mathematical proof. If you want to see the program in advance of the talk, check out the "billiards" link on my website:

Thursday, November 11 Professor Walter Neumann (Columbia University) 2:30 p.m. Polynomials and Knots. ABSTRACT : There are deep unsolved problems relating to polynomials, already for polynomials in two variables. The most famous such problem is the Jacobian Conjecture, giving a conjectural characterization of polynomial changes of coordinates. The talk will describe some of these questions, as well as connections with knot theory.

Thursday, November 18 Professor Mariusz Lemanczyk (Torun, Poland) 2:30 p.m. On the filtering problem of stationary processes. ABSTRACT : A signal (which is meant to be a stationary stochastic process X=(Xn), {n ∈ Z}) is sent through a communication channel. We assume that a noise (another stationary process Y=(Yn)) is present so, as an output, we obtain a process Z=(Zn) which is a function of the processes X and Y. Can we reconstruct the process X from Z? By that we mean an algorithm allowing us to get X from Z. Assuming joint stationarity of all the processes under consideration we will show how this problem is leading to some pure ergodic theory questions. To have a chance for a positive solution of the filtering problem we need to assume that the two processes X and Y are sufficiently different, they have to be at least independent. We will present some partial solutions (due to Furstenberg) of the filtering problem under some integrability assumptions on the processes in the case Zn=Xn+Yn. However even in this simple case the non-integrable case remains open. At the end of the lecture I will present a full (positive) solution of the filtering problem when the time takes value in the group Z2, i.e., in the case of random Z2-fields.