Step 1

1. Step 1 • The entire class will complete related readings due the day of your presentation

2. Step 2 • Your PowerPoint presentations on influences, barriers, support, diversity issues and mathematical style.

3. Carolyn Gordon1950-PresentGeometerHearing the Shape of a Drum

4. Influences, Support, Barriers and Diversity Issues • Older sister • Family expectations • Teased by a group of boys • Never met another woman mathematician until AWM gathering at a conference during her 3rd year in graduate school • Married to a mathematician, David Webb, and has a daughter

5. Mathematical Style • Collaborates with many mathematicians including Dr. Sarah • Describes herself as being “terrible at visualization and as having a bad memory.” • Good with numbers but says that does not help her in her geometry research • Paper models • Step away from a problem and let her subconscious work. • Doing research is like “being in a maze.”

6. Step 3 • Dr. Sarah will go over mathematics related to your mathematicians.

7. Indirect Observations • X-rays • CAT scans • Indirect ways of observing the world • Spectroscopy = chemical composition of stars • Natural vibration frequencies = pitches at which a molecule naturally rings are used to identify molecules in space. • Given a set of frequencies, what can we say about the structure?

8. Hearing the Shape of a Drum • Mathematical Drum = any shape that has an interior and a boundary. • Interior vibrates with each strike while the boundary frame remains rigid. • Mark Kac (1966) If two objects have the same vibration frequencies, must they have the same shape?

9. Earlier Work • 1911 Hermann Weyl proved that one can always hear the area of a drum. • Minakshisundaram and Pleijel proved that one can always hear the perimeter of a drum.

10. Hearing the Shape of a Drum • 1991, Carolyn Gordon, her husband David Webb, and Scott Wolpert came up with the answer: No! • One can sometimes, but not always hear the shape of a drum. • Two mathematical drums with different shapes, but the same vibration frequencies, therefore making the same exact sound.

11. Carolyn Gordon and husband David Webb • with their drums in 1991. • Must have the same area and perimeter, or they would sound different.

12. Their Proof • Used mathematics to prove they sounded the same without actually testing them in real life. • Mathematicians would not have been satisfied with experimental testing as a proof because a small frequency difference could escape experimental error. • Later physicists tested the drums and found they sounded the same, within the error attributed to their experimental procedures.

13. Conclusions • Carolyn Gordon's research on hearing the shape of a drum shows us there is not always just one conclusion that can be reached from a complete set of measurements. • The problem that spectroscopy hopes to solve is whether one can recover the chemical composition of stellar atmospheres by using vibration “fingerprints” in order to identify them. Carolyn’s research warns us that we must be careful with this method.