Lectures are given on Thursdays in 103 McAllister.

Thursday, September 12 Andrei Zelevinsky (Northeastern University) 3:30 p.m. Laurent phenomenon ABSTRACT: Preparing for this lecture, I have browsed through the webpage for Penn State's group in algebra and number theory and stumbled upon a link (http://www-groups.dcs.st-and.ac.uk./~john/Zagier/Problems.html) to a beautiful set of problems posed by Don Zagier at the St Andrews Colloquium, 1996. Here is one of them (usually attributed to Michael Somos): Define a sequence t1, t2, t3, … by the recursion tn+5 = (tn+4 tn+1 + tn+3 tn+2)/tn with initial values (1,1,1,1,1). Prove that all of the tn are integers. In his outline of the solution, Zagier says: the proof comes from the theory of elliptic curves, and can be expressed either in terms of the denominators of the coordinates of the multiples of a particular point on a particular elliptic curve, or in terms of special values of certain Jacobi theta functions. I will discuss a new approach to the problem (found in a joint work with Sergey Fomin) using only elementary algebra. This approach allows us to prove integrality of several other sequences and number arrays conjectured by D.Gale-R.Robinson, J.Propp, N.Elkies, and M.Kleber. References 1. D.Gale, The strange and surprising saga of the Somos sequences, Math. Intelligencer, 13 (1991), no. 1, 40--43. 2. S.Fomin, A.Zelevinsky, The Laurent phenomenon, Adv. Applied Math., 28 (2002), no. 2, 119--144.
Thursday, September 19 Robin Forman (Rice University) 3:30 p.m. Topological problems in combinatorics ABSTRACT: At first glance the subjects of combinatorics—the study of finite sets, and the study of topology—the study of shapes, have little to do with each other. However, the notion of a simplicial complex lives comfortably in both worlds, and allows us to usefully restate a number of combinatorial problems in terms of topological notions. We will demonstrate this with a variety of examples. We will show how some problems in combinatorics can be successfully analyzed using basic notions from topology, while others give rise to new intriguing topological questions. (This talk represents just the tip of a fascinating iceberg. There are many other ways in which topology enters into combinatorics.)
Thursday, September 26 Serge Lvovski (Independent University of Moscow) On projective duality ABSTRACT: It is well known that there is a duality between a curve in the projective plane and the curve in the dual projective plane whose points are tangent lines to the curve in question. Unfortunately, it is less well known that this duality can be extended to curves in projective space of arbitrary dimension. To wit: assign to a curve C in a projective space the curve C* in the dual projective space whose points correspond to the so-called osculating hyperplanes of C. This curve is called the dual curve of C. It turns out that the dual of the dual coincides with the curve with which we started, exactly as in the plane case! Moreover, one can assign a linear differential operator of order n+1 to any curve in n-dimensional projective space; this operator determines the curve uniquely, and it turns out that the above-mentioned duality between curves can be extended to duality between operators; the operator corresponding to the dual curve C* is (up to a sign) the adjoint of the operator corresponding to the curve C. The aim of this colloquium talk is to explain this classical result and to speculate on its possible generalizations.
Thursday, October 3 Joseph Landsberg (Georgia Institute of Technology) 2:30 p.m. Asymptotic lines and the Griffiths-Harris rigidity of compact Hermitian symmetric spaces ABSTRACT: To study the differential geometry of surfaces, one takes derivatives. When studying the Euclidean geometry of surfaces in 3-space, that is the properties of surfaces invariant under translations and rotations, two derivatives are often enough to determine if two surfaces are congruent. What happens when one enlarges the group of motions to include all linear maps? In this talk I will describe (and answer!) questions like: how many derivatives does one need to take to recognize a familiar space like the variety of rank one 3×3 matrices (the Segre variety) in the set of all 3×3 matrices? If time remains I'll explain relations with open questions in algebraic geometry such as Hartshorne's conjecture on complete intersections and Salamon's conjecture on contact Fano varieties.
Thursday, October 17 Adrian Ocneanu (Penn State) 3:00 p.m. The exponential in mathematics and physics ABSTRACT: The exponential is the most important function in mathematics, since it transforms addition to multiplication. It appears in a beautiful way in the structure of continuous symmetry, the Lie groups such as the group of non-degenerate n by n matrices. The additive structure of diagonal matrices maps to the multiplicative structure of off diagonal matrices. In the physical world, what happens if we exponentiate 2 apples? (Answer: we get quantum mechanics apples). What happens if we exponentiate the result again? (Answer: we get a quantum field theory in which apples are created and annihilated).
Thursday, October 24 Roger Penrose (Oxford University and Penn State) 3:00 p.m. Impossible Crystals
Thursday, October 31 Mark Sapir (Vanderbilt University) 3:00 p.m. Some connections between combinatorics, group theory, topology and computer science ABSTRACT: We consider a connection between tilings of polygons with rubber tiles, isoperimetric functions of topological spaces, computational complexity of groups, and computational complexity of real numbers.
Thursday, November 7 Mikhail Shubin (Northeastern University) 2:30 p.m. The Nash Equilibrium ABSTRACT: In 1949-50 John Nash established his famous result about the existence of an equilibrium for any non-cooperative game. This result eventually brought a Nobel prize in economics to Nash in 1994. It became very influential due to a great success of the model which Nash proposed. Nash originally published his result in Proceedings of the National Academy of Sciences (USA), v. 36 (1950), 48-49. This remarkable 1.5 page paper contains a complete proof of his equilibrium theorem. The proof is quite elegant. It is based on a Kakutani fixed point theorem for multi-valued maps. I will explain the proof of the Nash theorem together with all necessary definitions and prerequisites. I will also discuss some related topics and applications.
Thursday, November 14 Joel Anderson (Penn State) 3:00 p.m. Iterated exponentials ABSTRACT: Fix a > 0 and consider the sequence {a, aa, aaa, …}, which I call the sequence of iterated exponentials. A natural problem is to try to determine the values of a for which this sequence converges. This question has a remarkable history. It was first considered (and solved) by Euler in the eighteenth century. Since then at least 5 other authors, (4 of whom were unaware of other work on the problem) have published solutions. The problem is also remarkable for its unexpected answer and the different threads that interweave in the deveopment of its solution. I will present a solution of the problem along with some historical notes. The arguments used will be elementary in the sense that they only require a knowledge of freshman calculus.
Thursday, November 21 Mira Bernstein (Wellesley) 3:00 p.m. Enumerative Geometry ABSTRACT: A typical problem in enumerative geometry asks for the number of geometric objects of a certain type that satisfy a given set of conditions. For example: * Easy: given 4 general points in the plane, how many parabolas go through all of them? (Answer: exercise.) * Medium: how many lines lie on a general cubic surface? (Famous answer: 27) * Hard: how many plane quartic curves (i.e. curves defined by a 4th degree equation in two variables) are tangent to 14 general lines in the plane? (Answer: 23,011,191,144. Computed by Zeuthen in 1873, proved by Vakil in 1998!) The history of enumerative geometry is a bit like the history of Fermat's Last Theorem. The questions have a similar sort of appeal: they are concise and easy to state—many mathematicians are drawn by this clear-cut challenge. Also, as with Fermat, the answers themselves are not so important: what really matters is the deep and beautiful mathematics that has been developed in the process. There are also mysteries in this story, more acute, in some ways, than that of Fermat. For while we do not know if Fermat's proof was correct, we do know that Zeuthen, e.g., got most of his numbers right—yet in many cases (including the one given above), we do not know exactly how he did it! In the 19th century, geometers developed a powerful calculus for solving enumerative problems. Their method had no rigorous theoretical foundation, but despite some flagrant errors, bitter arguments, and unexplained leaps of intuition, it worked remarkably well. Justifying their results was the subject of Problem #15 on Hilbert's famous list. In the 20th century, enumerative geometry has been reconceptualized and made rigorous in terms of intersection theory on parameter spaces. In the first half of my talk, I will explain roughly what this means, using some classical 19th century examples. The second half of the talk will be devoted to a key issue in the study of parameter spaces: compactification. Again, to keep things simple, I will focus on a 19th century construction: the space of complete conics. However, the problems that motivate this construction are much the same as those facing algebraic geometers today. In fact, the space of complete conics turns out to be a special case of the Kontsevich moduli space of stable maps—an extremely important object of study in recent years, about which I might say a few words at the end, if time allows.