Topology seminar - Fall 2010
September 10
Andrew Wilfong - The chi-y genus of quasitoric manifolds
Quasitoric manifolds can be viewed as a toplogical generalization of
non-singular projective toric varieties. In this talk, I will define
what a quasitoric manifold is, mainly focusing on its combinatorial
structure. I will then present a formula for the chi-y gensu of a
quasitoric manifold that utilizes this combinatorial stucture.
Finally, I will illustarte how this formula is
used with several basic examples.
November 5
Beth Kirby - Computing the elliptic genus
We will discuss work by Ochanine and Corbounov on the elliptic genus
of a complete intersection in a product of projective spaces. We will
consider an example for the level 2 elliptic genus and discuss how this
generalized to a level n elliptic genus. The computation is a result
of a residue theorem for several complex variables. We may also
discuss the challenges of computing the elliptic genus for a more
general toric variety.
November 19
Kate Ponto - Lefschetz fixed point theorem
Let f be an endomorphism of a finite simplicial complex X. Using the
trace from linear algebra and the homology of X, we can assign a
rational number to f. This invariant is called the Lefschetz number.
Surprisingly, this invariant has connections to the fixed points of f
(the points x in X where f(x)=x).
The Lefschetz fixed point theorem: If f has no fixed points the
Lefschetz number of f is zero.
We can also associate an integer, called the index, to each fixed point.
The index counts how essential the fixed point is. It is one way to
get a partial answer the question: if we make a small change to f can
we eliminate this fixed point?
The sum of the indices of all of the fixed points of f is called the
index of f. If f has no fixed points its index has to be zero. Then the
Lefschetz fixed point theorem is a consequence of the following:
Theorem: The Lefschetz number of f equals the index of f.
I will define the Lefschetz number and index and outline a standard
proof of the Lefschetz fixed point theorem. If I have time, I will
describe a nonstandard proof of the second theorem.
December 3 and 10
Kate Ponto - A proof of the Lefschetz fixed point theorem.
I will describe the proof of the Lefschetz fixed point theorem that I find most
useful. This proof uses category theory and has several advantages over the classical proof. One of
the most significant strengths is that it generalizes easily.