Group 8 Sophie Germain (April 1, 1776-June 27, 1831) ● “Attended” École Polytechnique under the pseudonym “M. LeBlanc” ● In correspondence with Lagrange, Legendre, Gauss and Poisson ● Her interest in Number Theory sparked with Legendre’s ​Essai sur le Théorie des Nombres (1798) and Gauss’ ​Disquisitiones Arithmeticae ​(1801) ● Her interest in Elastic Theory sparked with the Institut de France’s competition in 1811 ○ Germain was one of the earliest pioneers of Elastic Theory ● Important philosophical contributions, precursor of Comte’s positivism Contributions to Elastic Theory ● Previous work done by Newton, members of the Bernoulli family, Euler ● Chladni’s 1808 experiment: existence of modes of vibration on 2D surfaces ● 3 attempts made (1811, 1813, 1815) for the Paris Academy of Sciences Prize: finding a mathematical theory for elastic surfaces ● Self-taught, read previous works by Lagrange and Euler ● Final attempt postulate: the force of elasticity is proportional to the applied force (deformation of the surface) ○ Suggests a process using integrals to find the curvature of the surface Contributions to Number Theory ● Germain did not submit anything to the Fermat problem competition (established around 1816), but her correspondence with Legendre and Gauss show that she was working on it (Laubenbacher, 2010) ● Major contribution to FLT: ​Germain’s Theorem ○ Let p be an odd prime and N be an integer. If there exists an auxiliary prime θ = 2Np + 1 such that: (i) if x​p ​ + y​p ​ + z​p ​ ≡ 0 (mod θ) then θ divides x, y, or z; and (ii) p is not a pth power residue (mod θ). Then the first case of FLT holds true for p. ​(Del Centina, 2007, p. 372)). ○ Legendre refers to this theorem in a footnote of his 1823 memoir and it can be applied successfully for p < 100 (Laubenbacher, 2010) ○ Germain breaks down the possible solutions of FLT into 2 types: ■ Case 1: Solutions where x, y, and z are not divisible by p ■ Case 2: Solutions where one of x, y and z is divisible by p (Del Centina, 2008, p. 372) ○ Eliminates the existence of solutions from Case 1 whenever θ can be found that satisfies conditions 1 and 2. (Del Centina, 2008) ● Germain’s primes​ are the primes p such that 2p+1 is also a prime. If p > 3 and 2p +1 are primes, then the first case of FLT holds true for exponent p. (Del Centina, 2008, p. 372) Extra ● Germain continued writing under her pseudonym “M. LeBlanc” because she was afraid of being ignored as a woman ● There were around 18 female mathematicians in Germain’s time compared to the 243 female mathematicians in 2018 (Linke & Hunsicker, 2018) ● Germain was the first woman to win the Paris Academy of Sciences prize ○ The award she won is now known as the “Sophie Germain prize” Group 8 References Alkalay-Houlihan C., Sophie Germain and Special Cases of Fermat’s Last Theorem. http://www.math.mcgill.ca/darmon/courses/12-13/nt/projects/Colleen-Alkalay-Houlihan.p df​. Bertesteanu, L. (2019, October 23). 7 Famous Female Mathematicians and Their Legacies. Retrieved from ​https://letsgetsciencey.com/famous-female-mathematicians/ Bucciarelli, L. L., & Dworsky, N. (1980). ​Sophie Germain: An Essay in the History of the Theory of Elasticity​. D. Reidel Publishing Company. Case, B. A., & Leggett, A. M. (Eds.). (2005). ​Complexities: Women in Mathematics​. Princeton University Press. Dahan-Dalmédico, A. (1987). Mécanique et théorie des surfaces: les travaux de Sophie Germain. Historia Mathematica, 14​(4), 347-365.​ ​https://doi.org/10.1016/0315-0860(87)90066-8 Dahan-Dalmédico, A. (1991). Sophie Germain. ​Scientific American, 265​(6), 116-123. Del Centina, A. (2008). Unpublished manuscripts of Sophie Germain and a revaluation of her work on Fermat’s Last Theorem. Archive for History of Exact Sciences, 62(4), 349–392. https://doi.org/10.1007/s00407-007-0016-4 Fernandez Martinez, J. A. (1946). Sophie Germain. ​The Scientific Monthly, 63​(4), 257-260. Germain, S. (1833). ​Considérations générales sur l'état des sciences et des lettres aux différentes époques de leur culture​. Imprimerie de Lachevardière. Grabiner, J. V. (1982). Book Reviews [Review of the book ​Sophie Germain: An Essay in the History of the Theory of Elasticity,​ by L. L. Bucciarelli & N. Dworsky]. ​Isis, 73​(3), 448-449. Laubenbacher, R., & Pengelley, D. (2010). “Voici ce que j’ai trouvé:” Sophie Germain’s grand plan to prove Fermat’s Last Theorem. Historia Mathematica, 37(4), 641–692. https://doi.org/10.1016/j.hm.2009.12.002 Linke, I., & Hunsicker, E. (2018, March 7). FACES OF WOMEN IN MATHEMATICS. Retrieved from ​https://vimeo.com/259039018 List of women in mathematics. (2020, March 1). Retrieved from https://en.wikipedia.org/wiki/List_of_women_in_mathematics O'Connor , J. J., & Robertson , E. F. (1996, December). Marie-Sophie Germain. Retrieved from ​http://mathshistory.st-andrews.ac.uk/Biographies/Germain.html O'Connor , J. J., & Robertson , E. F. (2020, February). Female mathematicians. Retrieved from ​http://mathshistory.st-andrews.ac.uk/Indexes/Women.html Petrovich, V. C. (1999). Women and the Paris Academy of Sciences. ​Eighteenth-Century Studies, 32​(3), 383-390. Popova, M. (2017, February 22). How the French Mathematician Sophie Germain Paved the Way for Women in Science and Endeavored to Save Gauss's Life. Retrieved from https://www.brainpickings.org/2017/02/22/sophie-germain-gauss/ Riddle , L. (2001, July). Sophie Germain . Retrieved from https://www.agnesscott.edu/lriddle/women/germain.htm Riddle, L. (2009). Sophie Germain and Fermat’s LAst Theorem. Biographies of Women Mathematics. Retrieved from https://www.agnesscott.edu/lriddle/women/germain-FLT/SGandFLT.htm Tanzi, C. P. (1999). Sophie Germain’s Early Contribution to the Elasticity Theory. ​MRS Bulletin, 24​(11), 70-71.​ ​https://doi.org/10.1557/S0883769400053549 Group 4 - Hilbert’s Problem 1 David Hilbert, a German mathematician was born on January 23, 1862 in Königsberg and died on February 14, 1943 in Göttingen. During the second International Congress of Mathematicians in Paris in 1900, Hilbert gave a lecture, “Mathematical Problems” and introduced the list of Hilbert’s problems which consisted of 23 problems. These problems ranged in topic, including number theory, analysis, algebra and geometry. Until today, problems 3, 7, 10, 11, 14, 17, 19, 20, and 21 were solved. On the other hand, problems 8, 12, 13 and 16 were still unresolved, and problems 1, 2, 5, 6, 9, 15, 18, and 22 had achieved some partial acceptance, although there is no consensus whether the achieved results are indeed a solution. However, there are some of the problems, such as problems 4 and 23 are too vague to be said as solved. In 1920, Hilbert predicted that he would live to see the Riemann hypothesis solved and the solution for problem 7 would not be seen by anybody in the audience. However, in reality problem 7 was solved in 1934 (in Hilbert’s lifetime), and the Riemann hypothesis is still open until today. Let us go through some of the impact and significance of these problems. Problem 4 is about straight line as the shortest distance between two points. There is an axiom that if there are two triangles with two sides congruent and the angle between the sides congruent, then the triangles are also congruent implies the theorem of the shortest distance between two points. Hilbert is interested in the converse of the statement. This problem is more of a research rather than a problem involving geometry and other branches. It can be seen as a foundation of branches such as geometry. Mathematicians who studied this problem include Busemann and Pogorelov. Problem 10 asked for an algorithm to determine whether a given polynomial Diophantine equation with integer coefficients has an integer solution. Due to the work done for this problem, we can now code many different problems, including the Riemann Hypothesis, into polynomials. If the problem 10 was solvable, then we could solve all of these problems. This has shown decidable theories are complex and not practical. Even if there exists an algorithm, it might take too long to run. There are also other implications on the problem 10, for example it is now possible to prove a number is prime with a bounded number of operations. At the beginning of the 20th century, there was a little interest in the classical approach to calculus of variations. ​ th th th Hilbert attempted to draw people’s attention in calculus of variations by including the 19 ,​ 20 ,​ and 23 problems ​ ​ ​ in the list. Since 1900, there was a new interest arose in calculus of variations in the 20th century. Furthermore, in ​ 1900, Hilbert was the first one who gave a complete proof of the Dirichlet principle and consequently many different proofs were published during that time. Hilbert’s third problem can be simplified to the following: specify two tetrahedra of equal volume which are neither equidecomposable nor equicomplementable. Max Dehn (1878-1952) Born in Hamburg Germany, died in North Carolina USA. He was a student of David Hilbert known for proving the sum of angles of triangles is 180 degrees, and also the Dehn Invariant. He solved the third problem of Hilbert’s which was the first problem solved. Some Important Definitions: Equidecomposable is the ability to chop up one figure and build a new one out of its pieces ​ Two shapes are equicomplementable if congruent building blocks can be added to both of them to create two ​ equidecomposable supershapes Important things to note about Hilbert’s Third Problem: The progression of mathematical thinking that this question fostered is undeniable as seen through how this question led to the development and proofs of such things like the Dehn Invariant and Bricard’s condition. Although the problem was solved long ago it still has relevance today.