| 10:00 - 10:50
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Nathan Broaddus:
Finite rigid sets and homological non-triviality in the curve complex
Abstract: Aramayona and Leininger have provided a "finite rigid subset" $X(S)$ of
the curve complex $C(S)$ of a surface $S$, characterized by the fact that
any simplicial injection $X(S)\hookrightarrow C(S)$ is induced by a unique simplicial
automorphism $C(S)\cong C(S)$. We prove that, in the case of the sphere with
$n>4$ marked points, the reduced homology class of the finite rigid set
of Aramayona and Leininger is a $\mathrm{Mod}(S)$-module generator for the reduced
homology of the curve complex $C(S)$, answering in the affirmative a
question posed by Aramayona and Leininger. For the surface $S$ with genus
$g>2$ and $n=0$ or $n=1$ marked points we find that the finite rigid set
$X(S)$ of Aramayona and Leininger contains a proper subcomplex whose
reduced homology class is a $\mathrm{Mod}(S)$-module generator for the reduced
homology of $C(S)$ but which is not itself rigid. This is joint work with
J. Birman and W. Menasco.
|
| 10:50 - 11:10
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Coffee break
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| 11:10 - 12:00
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Chris Leininger:
Homology and dynamics of pseudo-Anosovs
Abstract: I'll explain a connection between a pseudo-Anosov homeomorphism's stretch factor, and its action on homology. This provides a kind of interpolation between a result of Penner and our prior work with Farb and Margalit, and answers a question of Ellenberg. This is joint work with Agol and Margalit.
|
| 12:00 - 14:00
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Lunch
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| 14:10 - 15:00
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Morwen Thistlethwaite:
Some interactions of computing with mathematics
Abstract: Within the last few years computers have become increasingly integrated with the pursuit of pure mathematics. Two examples of this phenomenon are given: first, an aid to investigation of a classical unsolved problem in number theory, and second an intensive application to the computation of representation varieties.
|
| 15:00 - 15:30
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Coffee break
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| 15:30 - 16:20
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Xingru Zhang:
Detection of knots in $S^3$
Abstract: We say that a given knot $J\subset S^3$ is detected by its knot Floer homology and $A$-polynomial if whenever a knot $K\subset S^3$ has the same knot Floer homology and the same $A$-polynomial as $J$, then $K=J$. We show that every torus knot $T(p,q)$ is detected by its knot Floer homology and $A$-polynomial. We also give a one-parameter family of infinitely many hyperbolic knots in $S^3$ each of which is detected by its knot Floer homology and $A$-polynomial. This is joint work with Yi Ni.
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