### GEORGE MASON
UNIVERSITY

DEPARTMENT OF MATHEMATICAL
SCIENCES

COLLOQUIUM

NOVEMBER 1, 2019

**Speaker: **
Edwin O'Shea, James Madison University

**Title: *** The Parallel Postulate and Geometries without Rigidity
*

**Abstract:** The discovery of non-Euclidean geometry in the 19th century
elicited two distinct reactions from mathematicians wishing to preserve
the ethos of classical geometry: address the crises of foundations
(led by David Hilbert and others) and recast geometry vis-a-vis
transformation groups (led by Felix Klein and others).
This talk aims to build a bridge between these paradigms by exploring
axiomatic models of geometry without Hilbertian SAS congruence / Kleinian
rigidity. Using these models, we show that the classical equivalence of
Euclid’s parallel postulate and Playfair’s axiom collapses in the absence
of SAS and that a perfectly standard sum of angles property is equivalent
to SAS.

The first half or so of this talk will serve as an introduction to Euclid's
Elements and its progeny at a level suitable for undergraduates so an
in-depth knowledge of the axiomatic development of geometry will not be
assumed. This is joint work with my colleague, Elizabeth T. Brown and
our four REU students, Emily Castner, Stephen Davis, Edouard Seryozhenkov,
and AJ Vargas (J. Geom. (2019) 110: 42), and ongoing work with Brown.

**Time:** Friday, November 1, 2019, 3:30-4:20 p.m.

**Place:** Exploratory Hall, room 4106

**Refreshments** will be served at 3:00 p.m.

Department of Mathematical Sciences

George Mason University

4400 University Drive, MS 3F2

Fairfax, VA 22030-4444

http://math.gmu.edu/

Tel. 703-993-1460, Fax. 703-993-1491