Kempner Colloquium Abstracts

Spring 2006

The integer quantum Hall effect (IQHE) entails a very precise quantization of the Hall conductance in a 2D sample at very low temperatures. Depending on whether the currents in the sample are ascribed to the bulk or the edge, two apparently different conductances σ_B and σ_E have been used to explain this effect. In this talk, the definition of the two conductances and a proof that σ_B = σ_E will be discussed along with related results involving the propagation of waves in 2D disordered magnetic systems.

It is well-known that there is a formula for the roots of a general quartic equation that is analogous to the quadratic formula. What is less well-known is that this equation may be visualized and understood as a pretty geometric object called a pencil. What is even more suprising is that this geometry is related to mathematical mechanics. This talk will describe this facinating geometry and the solution to the quartic.

In my talk I will describe R. Stanley's conjectures on the positivity of the h- and g-numbers of convex polytopes and their proof based on the speaker's joint work with V.Lunts and the recent proof of the combinatorial version of the "Hard Lefschetz Theorem" by Karu.

From the combinatorial point of view a polytope is reduced to the set of its faces partially ordered by inclusion and graded by the dimension. Examples of combinatorial invariants of polytopes are the f-numbers: these simply count the number of faces of the given dimension of the polytope. More complicated combinatorial invariants are the h- and the g-numbers. These are defined simultaneously by a recursion based on the structure of the partially ordered set of faces. For simplicial polytopes (but not in general) the h-numbers are related to the f-numbers in a simple way. A longstanding problem in combinatorics of convex polytopes has been to find the necessary and sufficient conditions for a string of integers to be the collection of the h-numbers (the so-called h-vector) of a convex polytope. Stanley's conjectures summarize the expected properties of the h-numbers. Namely, they say that the collection of h-numbers of a convex polytope have all the properties of the Betti numbers of a smooth projective variety or, more generally, of the intersection homology Betti numbers of a (possibly singular) projective variety. In fact, the conjectures are motivated by the intimate connection between rational convex polytopes and projective toric varieties. In fact, the conjectures were proven for rational polytopes by Stanley using this connection and a deep and far-reaching results on the topology of projective varieties (like the Hard Lefschetz theorem for the intersection homology).

Jointly with V. Lunts we developed an "elementary" theory to replace the application of algebraic geometry to combinatorics which, in particular, does not depend on the rationality hypothesis. The analog of the Hard Lefschetz Theorem in our context was proven recently by K.Karu.

In this talk, I will describe two common operations on symplectic (Kaehler) manifolds equipped with a group action: quantization and symplectic reduction. The quantization of a symplectic manifold is a Hilbert space, and the symplectic quotient of a symplectic manifold by a group action is (in nice cases) another symplectic manifold. It is a classic theorem of Guillemin and Sternberg that the quantization of a symplectic quotient has the same dimension as the group invariant subspace of the quantization of the original manifold. This result is known widely as ``quantization commutes with reduction''. To prove this result, Guillemin and Sternberg explicitly constructed a natural bijective map between the two spaces.

But the quantization of a symplectic manifold is a Hilbert space, that is, a vector space equipped with an inner product. This inner product is essential to both mathematical and physical interpretations. In general (even in simple examples), the Guillemin--Sternberg natural bijective map does not preserve the two inner products. On the other hand, if the metaplectic correction is made then the Guillemin--Sternberg map does become unitary in the semiclassical limit (which I will briefly discuss if time permits).

Calibrations were introduced into geometry and topology in 1982 with the ground-breaking work of Harvey and Lawson. They provide a means of studying area-minimizing submanifolds of general Riemannian spaces and have proved to be important in diverse applications to complex analysis, string theory, and topology. In this lecture, I will describe the basics of the theory and the fundamental motivating examples and illustrate these ideas with applications to several different areas of geometry and topology.

For a rich class of nonsmooth functions, it is possible to define a generalized (set-valued and densely defined) derivative which possesses a surprisingly complete calculus. This theory subsumes much of convex analysis. There are natural applications in control theory and optimization, where nonsmoothness is intrinsic.

We will see how hyperbolic geometry has become a powerful tool in knot theory and in 3-manifold theory, with lots of pictures. No previous background assumed.

Suppose R is the ring of algebraic integers in an algebraic number field k. The Kubota symbol is a homomorphism from a subgroup of finite index in SL_n(R) to the group of roots of unity in R. This map is related to both the reciprocity law for k and the congruence subgroup problem for SL_n(R). I'll explain some of the background to this and describe a new way of defining the Kubota symbol, which makes its properties more transparent.