TGTS Tetrahedral Geometry/Topology Seminar

Sponsored by Elizabethtown College, Franklin and Marshall College,
Lebanon Valley College, Millersville University, and West Chester University

Speakers Spring 2025

  7 Feb 2025: Vince Coll, Lehigh University
  7 Mar 2025: David Johnson, Lehigh University
  4 Apr 2025: Subhajit Mishra, McMaster University

TGTS is a regional mathematics seminar/colloquium. We meet at 4:30
p.m. on the first Friday of each month during the academic year (with
some exceptions, as noted in the schedule above).  The public is
cordially invited to attend.

Next Talk

Date/Time: 7 February 2025, 4:30pm ET
 Location: Stager 215, Franklin & Marshall College
     Zoom: (zoom link to talk)
  Speaker: Vince Coll, Lehigh University
    Title: The Riemann Mapping Theorem and Curve Rounding Flows

 Abstract: Given a smooth Jordan curve $\alpha$, the Riemann
Mapping Theorem provides a biholomorphism $f(z)$ from the open unit disk
$D$ to the open planar region $\Omega$ bounded by $\alpha$.  The
\textit{smooth} Riemann mapping Theorem asserts that $f(z)$ extends
smoothly to the boundary, mapping $\overline{D}$ to $\overline{\Omega}$.
The pullback of the Euclidean metric $g_0$ has the form
$f^*(g_0)=e^{2\varphi}g_0$, where $\varphi$ is a harmonic function. We
call these \textit{harmonic metrics}.

In this presentation, we introduce and solve a flow equation for
harmonic metrics on $\overline{D}$ arising from a Riemann map. A
solution is a \textit{harmonic metric flow} that can be visualized by
isometrically embedding $\overline{D}$ in the Euclidean plane for each
harmonic metric in the flow.  A further enhancement to visualization is
achieved by focusing on the embeddings of the $S^1$ boundary of
$\overline{D}$. This yields a flow of smooth Jordan curves, starting
from the original Jordan curve $\alpha$.  For any harmonic metric flow
of curves, we compute arclenth, enclosed area, and curvature of the
intermediate curves in the flow.

These curve flows are best described as ``curve-rounding" flows since
the flow solutions always converge to a circle, never becoming
singular. Importantly, these flows are curve-shortening and
area-reducing, and once an intermediate curve becomes convex, all
subsequent curves are convex.

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