MASS Colloquia 1999

Thur, Sep 2 Greg Swiatek (Penn State) 3:00 pm The Meaning of Chaos

Thur, Sep 23 Dale Brownawell (Penn State) 3:00 pm When Do Polynomials Have Common Zeros? Abstract: The answer is easy when the polynomials involve only one variable or when they are linear. The elegant general answer is still being investigated more than 100 years after its discovery. Related questions are: How hard can this question be, anyway? How small can polynomial values be? These are active areas of investigation.

Thur, Oct 14 Arkady Vaintrob (University of New Mexico) 3:00 pm Knots and Knot Invariants Abstract: A systematic study of knots was initiated by physicists in the second half of the last century. In the beginning of this century mathematicians took over the field and introduced knot invariants - tools for comparing different knot types. Near the end of the century physicists struck back by inventing a host of new more powerful invariants whose geometric meaning, however, remained mysterious. Recently as a result of joint efforts of physicists and mathematicians, unexpected connections between old and new knot invariants have been discovered raising expectations for new exciting results to come. In the lecture, after a brief introduction to knot theory, we will discuss the current developments by comparing the classical Alexander invariant and the most famous invariant of the new era, the Jones polynomial.

Thur, Oct 21 Yulij Ilyashenko (Cornell) 3:00 pm Hilbert's 16th Problem Near Its Centenary Abstract: The second part of the 16th problem appeared to be one of the most difficult in the Hilbert's list. Smale included it in his list of the main problems for the forthcoming century. In the expiring century the problem had a long and dramatic history. In the last 20 years interest in the problem has grown substantially, and some progress was achieved. Several filial problems about "Hilbert type numbers" were stated and partially solved. The subject of the talk will be a sketch of this history and achievements.

Tues, Oct 26 Viorel Nitica (Notre Dame) 3:00 pm Replicating Tiles Abstract: A plane figure (or tile) is defined to be replicating of order k (or rep-k) if it can be dissected into k replicas, each congruent to the other and similar to the original. If k=4, an equivalent formulation is that four identical figures are to be assembled into a scale model, twice as long and twice as high. All triangles and parallelograms are rep-4 tiles. I shall start my lecture by showing a list of rep-k tiles for various values of k. Then I shall relate the study of rep-tiles to more general questions that are subject to current research, such as when can a finite region consisting of cells in a square lattice be perfectly tiled by tiles drawn from a finite set of shapes?

Thur, Nov 4 Yuri Latushkin (University of Missouri) 3:00 pm Nightmares and Dreams of Lyapunov: Stability and Semigroups Abstract: After a five-minute mini-course in functional analysis, we will discuss how semigroups of linear operators are related to the stability of linear differential equations in infinite dimensional spaces. A classical theorem of A. M. Lyapunov says that the equation is stable provided the spectrum of its (bounded) coefficient belongs to the open left half-plane. If the coefficient is an unbounded operator, then this theorem, generally, does not hold. We will discuss a replacement of the Lyapunov Theorem that uses so-called evolution semigroups.

Thur, Nov 11 Alexander Dranishnikov (Penn State) 3:00 pm On the Hilbert-Smith Conjecture Abstract: The Hilbert-Smith conjecture is a remaining branch of the fifth Hilbert problem. We will discuss how it leads to p-adic numbers and strange phenomena in dimension theory.

Thur, Nov 18 Serge Tabachnikov (University of Arkansas) 3:00 pm The DNA Geometric Inequality: An Open-Ended Story Abstract: Consider a closed plane curve, possibly self-intersecting, inside a closed convex plane curve; the former curve is referred to as DNA and the latter as Cell. The DNA geometric inequality asserts that the average absolute curvature of DNA is not less than that of Cell. I will discuss this simple result (surprisingly, proved only a few years ago) and give five different proofs of the particular case when Cell is a circle.

Thur, Dec 2 Misha Guysinsky (Tufts University) 3:00 pm The Banach-Tarski Paradox and Amenable Groups Abstract: Is it possible to cut up a pea into finitely many pieces that can be rearranged to form a ball the size of the sun? The answer to this question looks obvious, but it is not. We define a special class of groups called amenable groups and discuss its connection with questions like this.

Conrad Plaut (University of Tennessee) 3:00 pm Group Reconstruction Abstract: What happens when you reconstruct a group using only information from a small piece of the group? In 1928 O. Schreier discovered a simple way to do such reconstuction, which was rediscovered later by A. Mal'tsev, and later still by J. Tits. We begin with a little known and disastrous solution to Hilbert's Fifth Problem that was published in the Annals of Mathematics in 1957. We then go on to show how Schreier's fundamental construction is connected to covering group theory and finitely represented froups.