ULL Topology Seminar, Spring 2009

webpage last modified: 5/1/09
  • January 21st: "Introduction to span," Thelma West (MDD 214, 3 p.m.).
  • January 28th: "Span, pt. II: a consideration of 'an atriodic tree-like continuum with positive span' (after Ingram)," Thelma West (MDD 309, 4 p.m.).
  • February 6th: "Sheaves of abelian groups on topological spaces," Daniel Davis (MDD 214, 1 p.m.).
  • February 13th: "Regular Sequences in unstable algebras over the Steenrod algebra," Mara Neusel (Texas Tech) (MDD 214, 1 p.m.). Abstract: An unstable algebra over the Steenrod algebra is a graded connected commutative algebra over a finite field equipped with an action of the Steenrod algebra. They appear for example as invariant rings over finite fields. Now, a basic question is: Which of the classical/nonclassical results in commutative algebra can be proven in the category of unstable algebras? Important results are for example the classification of injective modules in that category, the Lasker-Noether theorem and the prime filtration theorem. In my talk I will discuss an old result by Hartshorne, which reads for graded objects as follows: If H is a Noetherian (graded, commutative, connected) algebra over a field F, of Krull dimension n and depth d, then H is free as a module over F[h_1, ..., h_d], where h_1, ..., h_d \in H form a regular sequence of maximal length. Nick Kuhn asked: If H is an unstable algebra over the Steenrod algebra can we choose a regular sequence such that F[h_1, ..., h_d] is also an object in the category of unstable algebras? I am studying this question with Lars Christensen (TTU).
  • February 20th: "Connections between graph theory and knot theory," Maciej Niebrzydowski (MDD 214, 1 p.m.). Note: the 2009 Spring TGTC (at UH) is the 20th to the 22nd.
  • February 27th: "Defining sheaves with equalizer diagrams and Grothendieck sites," Daniel Davis (MDD 214, 1 p.m.). Note: Knots in Washington (at GWU) is the 27th to (March) the 1st.
  • March 6th: "Fibered products, sieves, and pretopologies on categories," Daniel (MDD 214, 1 p.m.).
  • March 13th: "Connections between graph theory and knot theory, Part II," Maciej Niebrzydowski (MDD 214, 1 p.m.).
  • March 20th: "Obtaining Grothendieck sites via bases," Daniel (MDD 214, 1 p.m.).
  • March 27th: "Topological quandles," Maciej (MDD 214, 1 p.m.).
  • April 3rd: "Introduction to hyperspaces," Thelma West (MDD 214, 1 p.m.). Note: the 2009 Rowlee Lecture and the conference "Homotopy theory and applications," at the University of Nebraska (Lincoln), is from the 3rd to the 5th.
  • April 10th: no seminar (Easter holiday & Spring break).
  • April 17th: no seminar (Easter holiday & Spring break).
  • April 24th: "An Application of the snake and horseshoe lemmas to the functors Hom( - , G) and Ext( - , G), in the category Ab," Chris Ryan (graduate student) (MDD 214, 1 p.m.). Abstract: The horseshoe and snake lemmas are essential tools in homological algebra. Given an extension in Mod_R of A by B and projective resolutions of A and B, a natural question is: "Is there a way to construct a projective resolution of the extension that makes the induced diagram commutative?" The horseshoe lemma answers this in the affirmative, and, additionally, it tells us how to construct the resolution in question. This result and the snake lemma can be used to prove that the functors Hom( - , G) and Ext( - , G) together form a contravariant homological \delta-functor in Ab.
  • May 1st: "Size levels of arcs, continued," Thelma West (MDD 214, 1 p.m.). Note: this was the last talk for this semester.
    organized by Daniel (Davis).