Kempner Colloquium Abstracts

Spring 2008

Let f be a continuous map from a "nice" compact topological space X to itself. Then f induces an endomorphism Hⁱ(f) of the cohomology groups Hⁱ(X,Q) of X for each i, and the classical Lefschetz trace formula asserts that the virtual trace ∑ _i (-1)ⁱ Tr(Hⁱ(f)) can be described in terms of the fixed points of f.

This result has various applications. For example, it gives a one line proof of the famous Brouwer's fixed point theorem.

In the 60s Grothendieck et al. showed that analogs of the Lefschetz trace formula also hold in algebraic geometry. This analog was used to prove the famous Weil conjectures on the number of points of algebraic varieties over finite fields.

In my lecture I will describe these results, give some of their applications and discuss recent developments. Namely I will discuss a recent (simpler) proof of a one of the most useful versions of the formula, conjectured by Deligne and proven first by Fujiwara. Our proof is based on an algebro-geometric notion of a "contracting morphism".

Later this year will be the one hundredth anniversary of the birth of a mathematician who solved a problem of Gauss.

I shall give an account of some parts of his interesting life, my personal impression of him and appreciate, as far as time will allow, another of his mathematical papers, i.e., not the one that solved the problem of Gauss.

This talk might be of interest to a wide audience.

The Reidemeister torsion is historically the first invariant of a manifold that is not preserved by homotopy. In 1971, Ray and Singer proposed an analytic analogue of the Reidemeister torsion, and conjectured that the two invariants are equal. This conjecture was proven 7 years later in the celebrated papers by Cheeger and Muller.

In the talk I will review the constructions of the Reidemeister and the Ray-Singer torsions and also will present a new invariant, the refined analytic torsion, recently introduced by T. Kappeler and myself. The refined analytic torsion is a holomorphic function on the space of representations of the fundamental group of a closed odd-dimensional manifold, whose absolute value is equal to the Ray-Singer torsion and whose phase is given by the Atiyah-Patody-Singer η-invariant. The fact that the Ray-Singer torsion and the η-invariant can be combined into one holomorphic function allows to use methods of complex analysis to study both invariants. I will present some applications of this method. In particular, I will prove a refinement of the Ray-Singer conjecture, establishing the relationship between the refined analytic torsion and the Turaev's refinement of the Reidemeister torsion.

The classical connection between quadratic forms and symmetric bilinear forms can be extended from degree n=2 to any n>2, but there are several surprises:

1) over infinite fields, n-forms need not be polynomial when n>3,

2) over finite fields, polynomial n-forms need not be homogeneous.

In order to understand these phenomena, we develop the notions of combinatorial polarization and combinatorial degree. The case n=3 is of interest in nonassociative algebra, where the associator A(x,y,z), defined by (xy)z = (x(yz))A(x,y,z), is often a trilinear symmetric form. This is joint research with A. Drapal (Charles University, Prague).

The famous Morse-Thue sequence has the "no BBb" property: it contains no block of the form

b₁b₂...b_n b₁b₂...b_n b₁.

In the early 1900s Axel Thue showed that the "friends" of the Morse-Thue sequence, i.e. the members of the closure of the orbit of the Morse-Thue sequence under the shift homeomorphism, are precisely the doubly infinite sequences on two symbols having the no BBb property.

What if the sequence is wearing a disguise? Here "wearing a disguise" means that the names of the symbols have been changed using some unknown local rule, "local" as in a cellular automaton. How do you determine whether or not the undisguised sequence is a member of the Morse Minimal Set?

I will tell you more than you want to know about the Morse-Thue sequence, answer the question above, and perhaps others about substitution minimal sets.

This is joint work with Mike Keane (Wesleyan) and Michelle LeMasurier (Hamilton College, 1988 UCB mathematics B.A.).