Colloquium Archive

May 01, 2008

Big numbers with small prime factors

Speaker: Tonghai Yang, University of Wisconsin, Madison

April 30, 2008

Countable dense homogeneous spaces

Speaker: Jan van Mill, Vrije Universiteit, Amsterdam
Abstract: A topological space $X$ is countable dense homogeneous (abbreviated CDH) provided that for all countable dense subsets $D$ and $E$ of $X$ there is a homeomorphism $f$ of $X$ such that $f(D)=E$. By the now standard back-and-forth technique, Brouwer proved around 1910 that the real line is CDH. Most of the well-known homogeneous spaces are CDH: the Hilbert cube, the universal Menger continua, the Cantor set and manifolds without boundaries. Ungar claimed in 1978 that an open dense subset of a locally compact CDH-space is CDH. The proof of this fact is incomplete, and it is unclear whether it is true. Fitzpatrick and Zhou asked in 1988 whether there is a connected CDH-space with a dense open connected subspace that is not CDH. By using techniques from infinite-dimensional topology, we will present an example of a convex subset $S$ of Hilbert space $\ell^2$ with a dense convex open subspace $T$ such that (1) $S$ is CDH, $T$ is rigid, (3) $S\times S \approx T\times T\approx \ell^2$. Here a space is called rigid if the identity is its only homeomorphism, and $X\approx Y$ means that $X$ and $Y$ are homeomprhic topological spaces.

April 17, 2008

TBA

Speaker: Peter Shalen, University of Illinois at Chicago

April 03, 2008

On Floer homology and knots admitting lens space surgeries

Speaker: Matt Hedden, MIT
Abstract: There is a simple procedure called (Dehn) surgery which alters three dimensional manifolds using knots. The simplest three-manifolds are called lens spaces, and there is a conjecture regarding the knots on which one can perform surgery to obtain these manifolds. In this talk, I'll review these notions and discuss this conjecture, known as the Berge conjecture. I'll then discuss a strategy, developed jointly with Ken Baker and Eli Grigsby by which the knot Floer homology invariants of Ozsvath, Szabo, and Rasmussen could be used to prove this conjecture.

March 28, 2008

An introduction to exterior differential systems

Speaker: Jeanne Clelland, University of Colorado
Abstract: Exterior differential systems were first introduced by E. Cartan as a geometric approach to the study of partial differential equations. This talk will start with some definitions, followed by several examples which demonstrate this approach and illustrate some of its advantages.

This talk should be accessible to anyone with a reasonably good understanding of multivariable calculus; students are welcome!

March 07, 2008

Some extensions of holomorphic motions

Speaker: Professor S. Mitra, CUNY
Abstract: In this talk, we will discuss extensions of holomorphic motions over more general parameter spaces. It is well-known that Slodkowski's theorem cannot be generalized to higher dimensional parameter spaces. We will give a remarkably simple example of a holomorphic motion of a finite set over a simply connected space that cannot be extended to the whole sphere.

We will also discuss quasiconformal motions, an idea that was introduced by Sullivan and Thurston. We will show that a holomorphic motion of a closed setE (in the Riemann sphere) over a simply connected complex Banach manifold always extends to a quasiconformal motion of the sphere; furthermore, this extended motion has some interesting properties. The Teichm\"uller space of the closed set E will play an important role throughout our discussion.

All basic definitions will be given.

March 03, 2008

Hirzebruch-type invariants of complex algebraic varieties

Speaker: Laurentiu Maxim, CUNY-Lehman College
Abstract: I will report on recent progress on the study of genera and characteristic classes of algebraic varieties. I will describe Hodge-theoretic analogues of the Atiyah-Meyer signature formula, and discuss possible extensions of these results to the singular setting. This is joint work with S. Cappell, A. Libgober and J. Shaneson.

November 29, 2007

The length spectrum of a flat metric

Speaker: Moon Duchin, UC Davis/MSRI
Title: The length spectrum of a flat metric
Abstract: Trying to hear the shape of a drum via its spectrum of eigenvalues or lengths has by now been studied in many contexts. For a hyperbolic surface with a constant-curvature metric, knowing how long all curves are (and which curves correspond to which lengths) is enough to determine the metric. It turns out that 6g-5 curves suffice on the surface of genus g>1. It is natural to ask the same for a different class of metrics on hyperbolic surfaces, arising from the study of Teichmuller geometry: Metrics that are everywhere flat except at finitely many cone points, such as the genus-two surface obtained by a Euclidean octagon with opposite sides glued by translation. We show that the (marked) length spectrum does determine the flat metric, but no finite set of curves suffices. In fact, we give conditions for the set of curves to be spectrally rigid. This is joint work with Chris Leininger and Kasra Rafi.

November 26, 2007

Generalizing Pfaffian closure of an o-minimal structure

Speaker: Sergio Fratarcangeli, College of New Rochelle
Title: Generalizing Pfaffian closure of an o-minimal structure

November 08, 2007

Martin Bridgeman, Hausdorff dimension under bending deformations and the Weil-Petersson metric

Date: Thursday, November 8, 2007
Speaker: Martin Bridgeman, Boston College
Title: Hausdorff dimension under bending deformations and the Weil-Petersson metric
Time & Place: 4:15 P.M., Room 638SC

October 18, 2007

On the Enormity of the Knot Concordance Group

Speaker: Shelly Harvey, Rice University

Abstract: We say that a knot in S^3 (the boundary of B^4, the 4-dimensional ball) is slice if it is the boundary of 2-disk in B^4. Moreover, two knots K and J are concordant if their connected sum is slice. In 1966, Fox and Milnor defined the knot concordance group as the set of knots up to concordance where the addition is given by the connected sum. In 1997, Cochran, Orr, and Teichner defined a filtration of the knot concordance group, {F_n}, called the (n)-solvable filtration. This filtration is geometrically significant because it measures the successive failure of the Whitney trick for 2-disks in 4-manifolds. It was shown by Cochran and Teichner that each of the abelian groups F_n/F_n+1 has rank at least 1. We show for each n, that F_n/F_n+1 has infinite rank. This was only previously known to be true when n+0,1,2. We also resolve a long standing question as to whether certain natural families of knots considered by Casson, Gordon, Gilmer and others contain slice knots. This is joint work with Tim Cochran (Rice University) and Constance Leidy (Wesleyan University).

October 12, 2007

Virtual Knots and Assembly Graphs (joint work with Angela Angeleska and Masahico Saito)

Natasha Jonoska, University of Southern Florida
Abstract: We propose molecular models for homologous DNA recombination events that are guided by either double-stranded RNA or single-stranded RNA templates. This recombination can be seen as topological braiding of the DNA, with the template-guided alignment proceeding through DNA branch migration. We show that a virtual knot diagram can provide a physical representation of the DNA at the time of recombination. Schematically, the braiding process can be represented as a crossing in the virtual knot diagram. The homologous recombination corresponds to removal of the crossings in the knot diagram (called smoothing). We associate operations on words and graphs with such rearrangements and investigate their properties which lead to proper order of the DNA sequence.