Sophie Germain (1776–1831)

• born in Paris, daughter of a rich merchant

• her home was meeting place for political discussions

• In and after 1789, she started reading her father’s math books, self-studies in Latin in Greek, read Newton and Euler.

• Germain could not enter university but obtained lecture notes from the École Polytech. in 1794 and sent her solutions to Lagrange under a false, male name (Antoine Le Blanc) • After a meeting with Lagrange, he was supportive and visited her at her home

• Read Gauss’s Disquisitiones Arithmeticae in 1801 and started a correspondence with him (again as M. Le Blanc)

• In 1806, the French occupied Gauss’s then hometown Braunschweig (Brunswick) and Germain was afraid Gauss would suffer Archimedes’s fate. Germain played her connections to ensure Gauss’s safety. In the course of this, Gauss learned about her true identity. • Gauss replied: How can I describe my astonishment and admiration on seeing my esteemed correspondent M le Blanc metamorphosed into this celebrated person. . . when a woman, because of her sex, our customs and prejudices, encounters inﬁnitely more obstacles than men in familiarizing herself with [number theory's] knotty problems, yet overcomes these fetters and penetrates that which is most hidden, she doubtless has the most noble courage, extraordinary talent, and superior genius.

• Germain then worked in number theory, in particular on FLT.

• She also did celebrated work on elastic surfaces, sparked by an Academy Prize competition of the Institut de France. She won the prize after several incomplete attempts.

• Her work was ignored by many contemporaries, such as Poisson and Laplace. Germain’s work on FLT

Theorem (Germain). Let n be an odd prime and p another prime such that

1. xn+yn+zn = 0 (mod p) implies p | xyz, and

2. xn = n (mod p) has no solution.

Then for any solution of xn+yn=zn, n must divide xyz.

Although this theorem looks incredibly technical, it is at the heart of the proof of FLT for n ≤ 100. Emmy Noether (1882–1935)

• Born in Erlangen (South Germany) • Father Max was math professor • Mother Ida Amalia daughter of a rich merchant • went to “Höhere Töchterschule”, graduated as girls’ school teacher in French and English 1900 • went to lectures in math, romance studies, history at the University of Erlangen • graduated from Gymnasium 1903, distance • studied math in Göttingen and Erlangen, PhD 1907 • helped her dad with teaching • moved to Göttingen 1916 • Habilitation 1919 (ﬁrst woman) • ﬂed 1933, taught at Bryn Mawr College (a women’s college in USA) Some comments on Emmy Noether

• Emmy Noether is the founder of modern algebra and one of the most inﬂuential mathematicians of the 20th century

• As a woman (moreover, Jew), it was incredibly difﬁcult for her to do mathematics professionally.

• She was forced to self-educate herself in math or take private lessons. Math education at schools was bad, but she was lucky in having a good teacher in the sciences.

• Sources tell about a rich social life (dance, sled riding…) Letter from Noether to Ernst Fischer 1915

“Your Cremona interpretation expresses exactly what I had in mind. My […] for dihedral groups looks as follows in this form:

Generating substitution:

x0 = x0; x10 = ⌃x1 xn0 1 = ⌃xn 1 ··· x0 = x0; x10 = xn 1; ...xn0 1 = x1 xi0 = xn i Under Cremona transformation, this becomes

yn0 = ✏yn; y10 = y1; ...yn0 1 = yn 1; y =1/y . Thus ( n0 n N = (y1,...,yn 1; yn) { n 2 n K = (y1,...,yn 1;(1 yn) /yn)= (y1,...,yn 1; Z) { where Z denotes Klein’s parameter (icosahedral fraction). On Monday, I shall travel with my father to Mannheim and Karlsruhe for 10 days. My mother is back, but unfortunately again under […] treatment. Hilbert writes that he knows how to elementarily prove the ﬁniteness theorem for ﬁnite groups, doesn’t know if such a proof is published. He probably found the proof himself. He has ordered Toeplitz to look for a printed proof. Oh, and he has looked at the Zermelo business, since he writes that he found looking at it interesting. By the way, what did Mertens have to say about your arguments? E.N. The mathematical content

• apparently taken from a longer conversation with her colleague E. Fischer, 1915

• Deals with invariant theory

• goes too far to investigate this in detail here; advanced mathematics.

Ernst Fischer, 1875–1954 Professor in Erlangen mathematical analysis The mathematical content

“Hilbert writes that he knows how to elementarily prove the ﬁniteness theorem for ﬁnite groups, doesn’t know if such a proof is published.” Math Ann. 77 (1915), 89–92 • The discussion resulted in him having a proof, which was non-constructive, though. Noether published her proof in 1915.

• “He probably found the proof himself. He has ordered Toeplitz to look for a printed proof.” – so Toeplitz had to do assistant jobs for Hilbert.

• “Oh, and he has looked at the Zermelo business, since he writes that he found looking at it interesting. By the way, what did Mertens have to say about your arguments?” — I don’t know what the “Zermelo business” is, or what the conversation between Fischer and Franz Mertens might have been about. • In the middle of the letter she writes: “On Monday, I shall travel with my father to Mannheim and Karlsruhe for 10 days. My mother is back, but unfortunately again under […] treatment.”

• Her mother died the same year.

• Noether is very occupied with her mathematics, but has family problems in the back of her mind. What do you think the letter implies about Noether’s motivation to do mathematics?

Which speciﬁc difﬁculties did both Germain and Noether have to overcome as women?

Where were there differences between them?