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Colloquium

Mathematics Colloquium

Date:
Time:
4:00 pm – 4:50 pm
Avery Hall Room: 115
1144 T St
Lincoln NE 68508
Additional Info: AVH
Contact:
Steve Cohn, (402) 472-7223, scohn1@math.unl.edu
Speaker: Paul Feehan
Affiliation: Rutgers
Local Host: Susan Hermiller
Title: Partial differential equations, physics, and low-dimensional manifolds

Additional Public Info:
Recent advances in cosmology (March 2014), in particular the
discovery of gravitational waves, have provided further support for
the physicists’ theory of the origin of the universe and the
unification of the long-range gravitational force and the three
short-range forces (electromagnetic, weak, and strong) at the big bang
space-time singularity. While a fully consistent (if not
mathematically rigorous) unified field theory appears to be at best a
distant hope at present, we do see examples of the power of physics in
helping us understand three and four-dimensional `closed universes’,
that is, manifolds. In this lecture, we shall begin by reviewing the
known classification of closed surfaces (two-dimensional manifolds)
using concepts of Morse theory. We shall then briefly sketch some
ideas behind the classification of smooth manifolds of dimension five
and higher due to Stephen Smale (Fields medal 1966), again using
concepts of Morse theory. Dimensions three and four, though of most
physical relevance to us, have long been known to be the most
challenging. Using ideas from nonlinear partial differential equations
developed by Richard Hamilton and having their origins in Einstein’s
general relativity theory (the `Hamilton-Ricci flow’), Grigori
Perelman (Fields medal 2006, declined) succeeded in completing the
essential ingredients in our understanding of the shape of
three-dimensional manifolds by proving the one hundred year old
Poincare conjecture. Dimension four remains the most wild and
mysterious. Intriguingly, however, ideas drawn from the physicists’
theory of short-range forces (the Yang-Mills equations) have provided
a powerful tool to help us understand the large-scale structure of
four-dimensional manifolds. We shall close with sketch of the
underlying ideas there due to Michael Freedman and Simon Donaldson
(Fields medals, 1996).

Refreshments will be served in 348 Avery, 3:30 - 4:00. The talk is free
and open to the public.

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