[Binghamton Geometry Topology Seminar] Fwd: GT seminar: William Menasco

[Matthew Haulmark]

Due to technical difficulties, I am forwarding this announcement

MH
---------- Forwarded message ---------
From: Jose Roman Aranda Cuevas <jaranda at binghamton.edu>
Date: Mon, Sep 5, 2022 at 9:38 AM
Subject: GT seminar: William Menasco
To: <topsem at math.binghamton.edu>
Cc: William Menasco <menasco at buffalo.edu>, Cary Malkiewich <
malkiewich at math.binghamton.edu>, Jose Roman Aranda Cuevas <
jose at math.binghamton.edu>


Hi everyone,

This week we are pleased to have *William Menasco* from the University at
Buffalo speaking about projections of surfaces in 3-space, title and
abstract below. As usual, the talk will be on *Thursday at 2:50 pm in WH
100E.*

We will also have a lunch social, and meet at 12pm outside WH100E. See you
there!

Best,
Roman and Cary

%%%%%%%%%%%%%%%%%%%%%%%

Title: *Surface Embeddings in $\mathbb{R}^2 \times \mathbb{R}$ *
Abstract: // In this joint work with Margaret Nichols, we consider
$\mathbb{R}^3$ as having the product structure $\mathbb{R}^2 \times
\mathbb{R}$ and let $\pi : \mathbb{R}^2 \times \mathbb{R} \rightarrow
\mathbb{R}^2$ be the natural projection map onto the Euclidean plane.  Let
$ \epsilon : S_g \rightarrow \mathbb{R}^2 \times \mathbb{R}$ be a smooth
embedding of a closed oriented genus $g$ surface such that the set of
critical points for the map $\pi \circ \epsilon$ is a piece-wise smooth
(possibly multi-component) $1$-manifold, $\mathcal{C} \subset S_g$.  We say
$\mathcal{C}$ is the {\em crease set of $\epsilon$} and two embeddings are
in the same {\em isotopy class} if there exists an isotopy between them
that has $\mathcal{C}$ being an invariant set.  The case where $\pi \circ
\epsilon |_\mathcal{C}$ restricts to an immersion is readily accessible,
since the turning number function of a smooth curve in $\mathbb{R}^2$
supplies us with a natural map of components of $\mathcal{C}$ into
$\mathbb{Z}$.  The Gauss-Bonnet Theorem beautifully governs the behavior of
$\pi \circ \epsilon (\mathcal{C})$, as it implies $\chi(S_g) = 2
\sum_{\gamma \in \mathcal{C}} t( \pi \circ \epsilon (\gamma))$, where $t$
is the turning number function.  Focusing on when $S_g \cong S^2$, we give
a necessary and sufficient condition for when a disjoint collection of
curves $\mathcal{C} \subset S^2$ can be realized as the crease set of an
embedding $\epsilon: S^2 \rightarrow \mathbb{R}^2 \times \mathbb{R}$.
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