<div dir="ltr"><div dir="auto"></div><div>Due to technical difficulties, I am forwarding this announcement</div><div><br></div><div>MH<br></div><div><div class="gmail_quote"><div dir="ltr"><div class="gmail_quote"><div dir="ltr" class="gmail_attr">---------- Forwarded message ---------<br>From: <b class="gmail_sendername" dir="auto">Jose Roman Aranda Cuevas</b> <span dir="auto"><<a href="mailto:jaranda@binghamton.edu" rel="noreferrer" target="_blank">jaranda@binghamton.edu</a>></span><br>Date: Mon, Sep 5, 2022 at 9:38 AM<br>Subject: GT seminar: William Menasco<br>To: <<a href="mailto:topsem@math.binghamton.edu" rel="noreferrer" target="_blank">topsem@math.binghamton.edu</a>><br>Cc: William Menasco <<a href="mailto:menasco@buffalo.edu" rel="noreferrer" target="_blank">menasco@buffalo.edu</a>>, Cary Malkiewich <<a href="mailto:malkiewich@math.binghamton.edu" rel="noreferrer" target="_blank">malkiewich@math.binghamton.edu</a>>, Jose Roman Aranda Cuevas <<a href="mailto:jose@math.binghamton.edu" rel="noreferrer" target="_blank">jose@math.binghamton.edu</a>><br></div><br><br><div dir="ltr">Hi everyone,<br><br>This week we are pleased to have <b>William Menasco</b> from the University at Buffalo speaking about projections of surfaces in 3-space, title and abstract below. As usual, the talk will be on <b>Thursday at 2:50 pm in WH 100E.</b><br><br>We will also have a lunch social, and meet at 12pm outside WH100E. See you there!<br><br>Best,<br>Roman and Cary<br><br>%%%%%%%%%%%%%%%%%%%%%%%<br><br>Title: <b>Surface Embeddings in $\mathbb{R}^2 \times \mathbb{R}$ </b><div>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 $\epsilon: S^2 \rightarrow \mathbb{R}^2 \times \mathbb{R}$. <br></div></div>
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