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Spring 2019

  • February 14
    Speaker: Steve Ferry (Rutgers and Binghamton)
    Title: How big is epsilon?

    Abstract (PDF): A 1979 theorem of Chapman-Ferry says that if $M$ is a compact connected topological $n$-manifold* without boundary with topological metric $d$, then there is an $\epsilon > 0$ so that if $f:M \to N$ is a map from $M$ to a connected manifold of the same dimension such that $diam(f^{-1}(x))<\epsilon$ for every $x \in N$, then $f$ is homotopic to a homeomorphism.

    This theorem and its descendants play a continuing role in the work of Farrell-Jones, Bartels-Lück, and others on the Novikov, Borel, and Farrell-Jones Conjectures, the general strategy being to apply ideas from dynamics to “squeeze” a given homotopy equivalence to an appropriately “controlled” equivalence to which some version of the theorem quoted above applies.

    We will show that the behavior of $\epsilon$ in our old theorem depends on results from algebraic topology on the vanishing of the $K$-homology of Eilenberg-MacLane spaces of torsion groups. An application to computational topology is suggested.

    This is joint work with Alexander Dranishnikov and Shmuel Weinberger.

    *Chapman-Ferry did the cases $n\geq5$. The case $n=4$ is due to Freedman-Quinn and $n=3$ follows from work of Perelman.

  • February 21
    Speaker: Ben Dozier (Stony Brook)
    Title: Equidistribution of billiard trajectories and translation surfaces

    Abstract: Consider a billiard ball bouncing around on a polygonal table. This dynamical system is surprisingly complex. When the angles of the table are rational, the billiard table can be “unfolded” to get a closed surface with a natural flat geometry. This is an example of a translation surface. Billiard trajectories on the original table unfold to straight lines on the translation surface. Translation surfaces form a moduli space (which is a bundle over $M_g$, the moduli space of genus g Riemann surfaces), and this space comes equipped with a natural action by $SL_2(\mathbb R)$. Through a technique of “renormalization”, questions about the dynamics on a fixed surface can be translated into questions about the dynamics associated with this $SL_2(\mathbb R)$ action. The $SL_2(\mathbb R)$ action is very rich, and analogies with homogeneous dynamics can be leveraged.

  • February 28
    Speaker: Cary Malkiewich (Binghamton)
    Title: Parametrized spectra and fixed-point theory

    Abstract: In 1980, Dold and Puppe presented a revisionist proof of the Lefschetz fixed point theorem. The main idea is that the Lefschetz number L(f) is secretly more than a number, it's actually a map of spectra. Their ideas can be generalized to the Reidemeister trace R(f), or to families of fixed-point problems, but these generalizations require us to work with parametrized spectra, in other words spectra that vary over a fixed base B. I'll talk about what these words mean, and some cool things we can prove once we have them in our toolbox.

seminars/topsem/topsem_spring2019.txt · Last modified: 2019/03/01 14:59 by jwilliams