Talks are Tuesdays at 4:00 and will be held in O'Connor Hall room 204 unless otherwise noted. There will be refreshments before all talks in O'Connor 31 starting around 3:30pm.
Title: What is algebraic geometry?
My talk will begin with some of the fundamental ideas of algebraic geometry. I will explain why it is useful to work in complex projective space, and how this framework actually simplifies otherwise difficult problems relating to intersections of curves. Next, I will discuss surfaces and threefolds. The final part of the talk will deal with an application of algebraic geometry to elementary calculus in the spirit of Leibniz.
Title: Computational Geometry: An Introduction
I will overview some of the basic problems and solution techniques in the field of computational geometry. Computational geometry encompasses the study of inherently geometric problems for which efficient algorithmic solutions are needed. Problems in point location, rectangle intersection testing, convex hull computation, motion planning and and polygon guarding will be discussed with an emphasis on general techniques.
Title: Real-valued functions in H^infty + C
Let g be the function on the unit circle such that g is one on the upper half and zero on the lower half. Is g in H^infty + C? We answer this and a related question.
Title: Where does a power series in two variables converge?
Upon encountering a power series, one often investigates its region of convergence. In a calculus course one learns that a power series converges in an interval. In a complex analysis course one learns that a power series in one complex variable converges in a disk. For functions of more than one variable, the question of convergence of power series is more intriguing. In this talk we describe classical analytic and geometric characterizations of convergence regions of power series in two complex variables. Note: It is not necessary to have studied functions of several complex variables in order to understand this talk.
Title: Does mathematics need new axioms?
Yes and No: it depends on what you mean by "mathematics" and what you mean by "need". And prior to that, you might well ask whether mathematics needs axioms at all - and the answer to that depends on what you mean by "axioms". Let's start with the last: when the logician speaks of axioms he or she usually means those for the laws of valid reasoning and for some fundamental concepts like number, set and function. When the working mathematician (I mean, other than logicians) speaks of axioms (s)he usually means those for some particular part of mathematics, such as groups, rings, fields, topological spaces, metric spaces, Hilbert Spaces, etc., etc.; the value of those kinds of axioms is indisputable.
My talk is about the logical view of our question, and so the first thing we'll do is go over the case for having axioms at all of fundamental concepts. Then we'll look at some specific systems of axioms for number theory and set theory which that leads up to, and ask the question relative to those systems. On the Yes side: according to Goedel's incompleteness theorems, the systems in question (and, in fact, any systems like them) can't prove all truths in their respective subject matters, so it does seem that we need new axioms to get at those truths. Goedel himself was one of the leading proponents of the need for new axioms to settle both arithmetical and set-theoretical truths, and he suggested how this might be pursued via so-called axioms of higher infinity, or "large cardinal" axioms, examples of which will be given in the talk. On the No side, I will explain work going back to Weyl which shows that all (or practically all) of scientifically applicapable mathematics can be carried out in a system which is justified on the basis of the Peano Axioms for number theory. And practically all of everyday mathematics, pure and applied, can be formalized in relatively weak portions of set theory, and there is no evidence that any more than that will really be needed for those purposes. In the lecture I'll give some criticisms of both answers, and let you know where I stand on this question.
Title: Women at Cambridge
The Educational Times, a monthly periodical devoted to pedagogical interest, contained a section devoted to mathematical problems and their solutions. The Oxford mathematician, W.K. Clifford, claimed that the journal did more to encourage original mathematical research than any other European periodical in the late nineteenth century. Approximately three percent of the published mathematical problems and solutions were contributed by women. We discuss the accomplishments of several problem solvers including Christine Ladd-Franklin, who received her Ph.D. from Johns Hopkins, Hertha Ayrton, first women to have been nominated a Fellow in London's Royal Society; Charlotte Scott, first women to receive first class honors on the Cambridge Mathematical Tripos; and Philippa Fawcett who placed above the Senior Wrangler on the 1890 Tripos. Selected problems from The Educational Times and the 1880 and 1890 Mathematical Tripos will be discussed.
Title: On a problem of Diophantus
In a drill and practice section of the ``Arithmetica'', Diophantus shows us how to find ``A sum of three squares which equals a square, the first square being the side (ie: square root) of the second, and the second square being the side of the third'' (VI.17 in the ``Arabic'' books). In modern terminology, Diophantus finds a non-trivial rational solution to the equation x^2+x^4+x^8=y^2. Are there any other solutions? 17 centuries years later this is still a very difficult problem, at the very edge of what modern arithmetic theory and computational algebra can accomplish. After a brief discussion of the historical context of this problem, I will outline the techniques used to prove that Diophantus had, in fact, found all of the non-trivial rational solutions to this problem.
Title: Spaces of Wallpaper Functions
We take a new approach to wallpaper patterns using real- or complex-valued functions to indicate shadings or colors in repeat patterns. This is in contrast to traditional approaches which view wallpaper patterns as consisting of point sets, possibly colored by a finite set of colors. We review the 17 wallpaper groups and outline our approach, which involves the partial differential equations that model what might happen if your wallpaper were to vibrate like a drum. This talk should be of special interest to students who have taken Survey of Geometry, as well as to anyone with a strong interest in visual mathematics.
Anyone wishing further information about any of these talks should contact the coordinator of the Colloquium Series, Prof. Ed Schaefer, email@example.com