| Date | Speaker | Title | Notes |
| January 23 | Organizational meeting | ||
| January 30 | David Goldberg (CSU) | Braid Monodromy and Complex Line Arrangements: Take 2 | |
| February 6 | No talk | ||
| February 13 | No talk | ||
| February 20 | Holger Kley (CSU) | What is Enumerative Geometry? | Roughly speaking, enumerative geometry consists of those algebro-geometric problems in which one tries to count the number of of a certain geometric objects satisfying given conditions. For example, given 5 points in the plane, no three of which are colinear, how many conic sections pass through all of them? (Answer: 1.) In order for such a problem to have a chance of having a well-defined solution, certain conditions must be met. In this talk, I will explore the general scheme of enumerative geometry and the conditions needed to make a problem well-posed. The lecture will be liberally illustrated with examples. |
| February 27 | No talk | ||
| March 6 | Ken Driessel (CSU adjunct scholar) | On some algebraic questions concerning eigen-problems | Abstract: Recall that the QR algorithm preserves eigenvalues and also preserves tridiagonal structure. Can we find analogous algorithms for other classes of matrices? Here is an example. Let Z denote the n-by-n lower shift matrix, that is, the matrix with ones on the first sub-diagonal and zeros elsewhere. For define the operation D on n-by-n matrices by D.X := [ Z^T - Z , X ] where [U,V] := UV-VU is the usual matrix commutator or Lie bracket. (Note that D is a derivation for the algebra of square matrices.) I say that a matrix T is `D-structured' if the rank of D.T is small. Can we find an algorithm analogous to the QR algorithm which preserves eigenvalues and D-structure? In the seminar, I shall analyze this question by considering appropriate group actions. Why are the D-structured matrices important? Recall that a matrix T is `Toeplitz' if it has constant diagonals. (Also recall that Toeplitz matrices arise often in applied mathematics.) Note that if T is Toeplitz then D.T has small rank. In other words, Toeplitz matrices are D-structured. |
| March 13 | No talk (Spring Break) | ||
| March 20 | No talk | ||
| March 27 | Pamela Peters (CSU) | Groups formed by rational points on elliptic curves | |
| April 3 | Anamaria Dent (CSU) | A new approach to the Alexander-Hirschowitz theorem | Abstract: The general problem of computing the dimension of a space of polynomials satisfying certain multiplicity conditions at a set of general points can be formulated in any dimension. This problem, in its most general form, is still unsolved. The only statement known in higher dimension involves the multiplicity two case, which was solved in 1988 by J. Alexander and A. Hirschowitz. Their approach is from an alegbraic geometry point of view. In this talk I will discuss this problem and present an alternate proof, which I believe to be much more accessible than that given by Alexander and Hirschowitz. To prove this theorem I will use a slight variation of the methods developed by R.A. Lorentz and G.G. Lorenz, with which they have shown the dimension two case. (this is a preliminary result) |
| April 10 | Anamaria Dent (CSU) | A new approach to the Alexander-Hirschowitz theorem (contd.) | |
| April 17 | Alexander Hulpke (CSU) | Hilbert's Irreducibility Theorem | |
| April 24 | Talk postponed | On some algebraic questions concerning eigen-problems (Part ii) | |
| May 1 | Alexander Hulpke (CSU) | Hilbert's Irreducibility Theorem (continued) | |
| May 8th | Ken Driessel (CSU) adjunct scholar | On some algebraic questions concerning eigen-problems (Part ii) |