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’ D​ isquisitiones 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: G​ ermain’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 xp​ ​ + y​p​ + zp​ ​ ≡ 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.​ h​ ttps://doi.org/10.1016/0315-0860(87)90066-8 Dahan-Dalmédico, A. (1991). Sophie Germain. S​ cientific 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). C​ onsidérations 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. E​ ighteenth-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. M​ RS 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. For example, there are more recent versions of the proof being made. The use of the Pearl Lemma, Cone Lemma and Bricard’s condition to solve the problem in a different way than Dehn,shows how every contribution made is extremely useful. Modern significance is also seen in how the solution of the problem has led to the development of other open questions. One question includes whether two polyhedra with the same Dehn invariant are necessarily equidecomposable, which developed after the converse was proven true. There is also the question of finding the minimum number of motions necessary for equidecomposability in four or more dimensions. This goes to show that solving a problem does not necessarily mean the end of the work that can be done for it. Group 4 - Hilbert’s Problem 2

References

Aigner, M., Ziegler Günter M., & Hofmann, K. H. (2014). Proofs from The Book [5th edition]. ​ Browder, F. (1976). Mathematical developments arising from Hilbert problems. Providence: ​ ​ American Mathematical Society. ISBN 0-8218-1428-1

Ciesielska, D., & Ciesielski, K. (2018). Equidecomposability of Polyhedra: A Solution of Hilbert’s Third Problem in Kraków before ICM 1900. The Mathematical Intelligencer, 40(2), 55–63. doi: ​ ​ ​ ​ 10.1007/s00283-017-9748-4

Hilbert, D. (1902). Mathematical Problems. Bulletin of the American Mathematical Society. 8(10): ​ ​ 437-479.

Joint review of The Honor Class: Hilbert’s problems and Their and Mathematical developments arising from Hilbert’s Problems. (2013). SIGACT News., 44(4), 18–24. https://doi.org/10.1145/2556663.2556665

Joyce, D. (n.d.). Mathematical Problems. Retrieved March 2, 2020, from https://mathcs.clarku.edu/~djoyce/hilbert/problems.html

K., L. (2015, February 24). Hilbert's Third Problem (A Story of Threes). Retrieved March 2, 2020, from https://mitadmissions.org/blogs/entry/hilberts-third-problem-a-story-of-threes/ ​ O'Connor, J. J., & Robertson, E. F. (n.d.). David Hilbert. Retrieved March 2, 2020, from http://mathshistory.st-andrews.ac.uk/Biographies/Hilbert.html

O'Connor, J. J., & Robertson, E. F. (n.d.). Max Wilhelm Dehn. Retrieved March 2, 2020, from http://mathshistory.st-andrews.ac.uk/Biographies/Dehn.html

Reid, C., & Weyl, H. (1970). Hilbert. Berlin, Germany: Springer-Verlag. ​ ​ Yandell, B. (2002). The Honors Class Hilbert’s Problems and Their Solvers. Natick, ​ ​ Massachusetts: A K Peters. ISBN 1-56881-141-1

Group 7: Fields Medal Overview

Medal ● Must be under 40 years old to qualify ● $15 000 cash prize with a gold medal ● Awarded every 4 years at International Congress of Mathematicians (ICM) with 2 - 4 winners ● Chosen by executive committee of International Mathematical Union (IMU).

John Charles Fields (1863-1932) ● Born: May 14, 1863 in Hamilton, Ontario. Died: August 9, 1932 ● 1864: Graduated with a BA in mathematics at the University of Toronto ● 1887: PhD thesis from John Hopkins University on Solutions by Definite Integrals of the Equation ​ ○ Then taught there for 2 years ● 1892-1900: Studied in Paris Gottingen and Berlin with mathematicians like Frobenius & Planck ● His strong support for research was influenced by Mittag-Leffler ● 1902: Appointed lecturer at U of T and stayed for the rest of his career. Eventually a full research professor ● Main research: Algebraic functions. ● 1906: Published book on Riemann-Roch Theorem and Weierstrass Gap Theorem

Events That Led to the Medal ● 1897: The series of ICM began. No congress was held 1914-1918 ​ ​ ● 1920: IMU was set up in Strasbourg (excluding Germany, Austria, Hungary, Bulgaria & Turkey) ● 1922: Fields suggested that the ICM be held in Toronto and was successful in gaining financial support ● 1931 (February): Fields had money left over after the 1924 ICM. He put forth a proposal letter to the IMU to a medal that would recognize mathematicians internationally.

Fun Facts/Interesting Events ● There were no medals awarded between 1936 and 1950 due to the war ● First recipients (1936): Lars Ahlfors (Riemann surfaces) and Jesse Douglas (Plateau problem) ● Youngest recipient (1954): Jean-Pierre Serre (algebraic topology) at age 27 ● Only woman recipient (2014): Maryam Mirzakhani (Riemann surfaces)

Female Diversity ● Historically women were discouraged from education and mathematics due to society’s structure ● Women were not notable in math until the late 19th to early 20th century ● Some earlier notable women include ○ Emily du Chatelet (1706-1749): translated Newton’s Principia and made commentary ​ ​ ○ Mary Cartwright (1900-1998): 1st female elected Fellow of the Royal Society (1947) but was not given professorship ○ Anne Davis (present): first female elected professorship at Cambridge University (2002). ● 2016: 28.5% of all Mathematics and Statistics PhD’s in the U.S were awarded to women ● 1 out of 60 winners of the Fields medal were female

The “Curse” ● The theory that the young age of winners allows them to have nothing else to progress to afterwards ○ 25% less papers written per year after winning the medal (on average) ■ Post medal papers are cited less ● Grigori Perelman ○ Turned down the medal in 2006. Didn’t attend congress. ○ Russian mathematician who proved the Poincaré conjecture

References

(n.d.). Retrieved from http://mathshistory.st-andrews.ac.uk/Extras/Fields_letter.html

Barnes, M. E. (2007). John Charles Fields: A Sketch of His Life and Mathematics Work.

Chang, K. (2006, August 22). Highest Honor in Mathematics Is Refused. Retrieved from

https://www.nytimes.com/2006/08/22/science/22cnd-math.html

Chang, K. (2006, August 22). Highest Honor in Mathematics Is Refused. Retrieved from

https://www.nytimes.com/2006/08/22/science/22cnd-math.html

Fields, J. (1886). Symbolic Finite Solutions and Solutions by Definite Integrals of the Equation dny/dxn = xmy. American

Journal of Mathematics, 8(4), 367-388. doi:10.2307/2369393

Howison, S. (2014, August 13). A woman finally wins the Fields Medal after 50 years. Why did it take so long? | Sam

Howison. Retrieved from

https://www.theguardian.com/commentisfree/2014/aug/13/woman-wins-fields-medal-odds-maryam-mirzakhani

InternationalMathematicalUnion. (n.d.). Retrieved from

https://www.mathunion.org/imu-awards/fields-medal ​ James, Alex; Chisnall, Rose; Plank, Michael J. (2019): Supplementary material from "Gender and societies: a grassroots approach to women in science". The Royal Society. Retrieved from

https://doi.org/10.6084/m9.figshare.c.4638188.v2 ​ John Charles Fields. (n.d.). Retrieved from

http://mathshistory.st-andrews.ac.uk/Biographies/Fields.html

John Charles Fields (1863-1932). (2015, December 31). Retrieved from

http://www.fields.utoronto.ca/about/john-charles-fields

Kollar, J. (n.d.). IS THERE A CURSE OF THE FIELDS MEDAL?

National Mathematics Survey. (n.d.). Retrieved from https://math.mit.edu/wim/2019/03/10/national-mathematics-survey/

National Science Foundation - Where Discoveries Begin. (n.d.). Retrieved from

https://www.nsf.gov/statistics/women/

Riehm, C. (2002). The Early History of the Fields Medal. The Early History of the Fields Medal, Volume 49, Number 7. ​ ​ The Editors of Encyclopaedia Britannica. (2018, August 09). Fields Medal. Retrieved from

https://www.britannica.com/science/Fields-Medal

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https://www.open.edu/openlearn/science-maths-technology/mathematics-statistics/women-mathematics-the-history-be

hind-the-gender-gap GROUP 2- MARCH 3RD

Jesse Douglas lived from 1897-1965, in New York, USA. He developed a love for mathematics at a young age, starting in high school. Douglas went on to further develop this love and passion for the subject in both College (City College) and University. He was awarded a National Research Fellowship, and went to top Universities across the globe from 1926-1930. He conducted research in differential calculus, group theory and the calculus variations. In 1931 he published a detailed solution to the Plateau problem called Solution of the Problem of Plateau. For this work, he was awarded one of the first Fields Medals in 1936. The Plateau problem was concerned with how one determines the minimal surface for a given boundary. Douglas was determined to find an arbitrary parametric equation of a curve of the given contour to prove the existence of a minimal solution. He achieved this in his paper published in 1931, by minimizing his A-functional (unsure as to how he thought of this function). Douglas solved the Plateau problem in complete generality by seeking an arbitrary contour, unlike Tibor Radó who solved the Plateau problem by focusing on a single surface; a disc-like surface. Although Douglas too solved the Plateau problem for disc-like surfaces, he also studied the general case and considered multiple surfaces. These two mathematicians worked independently on the problem, yet reached similar conclusions despite their different approaches. Radó approached the problem using Koebe’s Theorem in regards to uniformity, while Douglas focused on minimizing his integral function. Integral equations were a highly active topic of research during this time. To this day, Douglas’ work remains important in the theory of minimal surfaces. His work in the Solution of the Problem of Plateau inspired and influenced other mathematicians such as Jürgen Jost and Richard Courant, who developed Douglas’ ideas on minimal surfaces. Lars Hörmander was a Swedish mathematician who was born in 1931 in Mjallby, Blekinge, Sweden. At the age of 15, Hörmander entered the senior secondary school where he could spend much of his time focusing on university- level mathematics. Hörmander entered the University of Lund in Sweden at age 17 and obtained his bachelor’s degree, master’s degree and Ph.D. After receiving his doctorate, Hörmander became a professor at the University of Stockholm in 1957. During the year of 1964 to 1968, he travelled to the United States and worked full time there as a researcher at the Institute of Advanced Study. He went back to the University of Lund as the Chair of mathematics in 1969 and retired in 1996. Hörmander devoted much time to the partial differential equation and was awarded the Fields Medal in 1962 because of his book Linear Partial Differential Operators, which was published in 1963. The book contains three parts; Part 1 introduces functional analysis, Part 2 and Part 3 discussed differential operators with constant and variable coefficients respectively. In Part 2 and 3, Hörmander used eight chapters in total to discuss how to solve different types of differential equations and also several specific problems such as the Cauchy problem and Elliptic boundary problems. Other than the Fields Medal, Hörmander was honoured with several other awards: the Wolf Prize in 1988 and the AMS Steele Prize in 2006. Reference: Courant, R. (1940). The Existence of Minimal Surfaces of given Topological Structure

under prescribed Boundary Conditions. Acta Math, 72, 51-98.

Douglas, J. (1931). Solution of the problem of Plateau. Transactions of the American

Mathematical Society. 33(1), 263–321. doi: 10.1090/s0002-9947-1931-1501590-9

Gray, J., & Micallef, M. (2013). The work of Jesse Douglas on Minimal Surface, 1–8.

Retrieved from https://arxiv.org/pdf/0710.5478.pdf

Hörmander Lars. (2013). Linear Partial Differential Operators. Berlin: Springer Berlin.

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andrews.ac.uk/Biographies/Douglas.html

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andrews.ac.uk/Extras/Hormander_books.html

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andrews.ac.uk/Biographies/Hormander.html

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Radó. Singapore: World Scientific.

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andrews.ac.uk/Honours/FieldsMedal.html