#cauchy-kovalevskaya theorem

[a-modernmajorgeneral]

Sofya Vasilyevna Kovalevskaya (born January 15, 1850, Moscow, Russia—died February 10, 1891, Stockholm, Sweden) was a mathematician and writer who made a valuable contribution to the theory of partial differential equations. She was the first woman in modern Europe to gain a doctorate in mathematics, the first to join the editorial board of a scientific journal, and the first to be appointed professor of mathematics.

In 1868 Kovalevskaya entered into a marriage of convenience with a young paleontologist, Vladimir Kovalevsky, in order to leave Russia and continue her studies. The pair traveled together to Austria and then to Germany, where in 1869 she studied at the University of Heidelberg under the mathematicians Leo Königsberger and Paul du Bois-Reymond and the physicist Hermann von Helmholtz. The following year she moved to Berlin, where, having been refused admission to the university on account of her gender, she studied privately with the mathematician Karl Weierstrass. In 1874 she presented three papers—on partial differential equations, on Saturn’s rings, and on elliptic integrals—to the University of Göttingen as her doctoral dissertation and was awarded the degree, summa cum laude, in absentia. Her paper on partial differential equations, the most important of the three papers, won her valuable recognition within the European mathematical community. It contains what is now commonly known as the Cauchy-Kovalevskaya theorem, which gives conditions for the existence of solutions to a certain class of partial differential equations. Having gained her degree, she returned to Russia, where her daughter was born in 1878. She separated permanently from her husband in 1881.

In 1883 Kovalevskaya accepted Magnus Mittag-Leffler’s invitation to become a lecturer in mathematics at the University of Stockholm. She was promoted to full professor in 1889. In 1884 she joined the editorial board of the mathematical journal Acta Mathematica, and in 1888 she became the first woman to be elected a corresponding member of the Russian Academy of Sciences. In 1888 she was awarded the Prix Bordin of the French Academy of Sciences for a paper on the rotation of a solid body around a fixed point.

Kovalevskaya also gained a reputation as a writer, an advocate of women’s rights, and a champion of radical political causes. She composed novels, plays, and essays, including the autobiographical Memories of Childhood (1890) and The Nihilist Woman (1892), a depiction of her life in Russia.

Memorial Museum-Estate of Sofia Kovalevskaya, Polibino, Pskov Oblast

[lthmath]

5th post on our Women in Mathematics Series

Sofia Vasilyevna Kovalevskaya was a Russian mathematician who made noteworthy contributions to analysis, partial differential equations and mechanics. She was the first major Russian female mathematician and a pioneer for women in mathematics around the world. She was the first woman appointed to a full professorship in Northern Europe and was also one of the first women to work for a scientific journal as an editor. 

Despite her obvious talent for mathematics, she could not complete her education in Russia. At that time, women there were not allowed to attend universities. In order to study abroad, she needed written permission from her father (or husband). Together with her husband, she emigrated from Russia in 1867. 

In October 1870, she moved to Berlin, where she took private lessons with Karl Weierstrass, as the university would not even allow her to audit classes. In 1874 she presented three papers—on partial differential equations, on the dynamics of Saturn's rings and on elliptic integrals—to the University of Göttingen as her doctoral dissertation. With the support of Weierstrass, this earned her a doctorate in mathematics summa cum laude, bypassing the usual required lectures and examinations. 

She thereby became the first woman in Europe to hold that degree. Her paper on partial differential equations contains what is now commonly known as the Cauchy–Kovalevskaya theorem, which gives conditions for the existence of solutions to a certain class of those equations.

[mathematicianadda]

Who was Sofya Kovalevskaya?

Some people, like Daniel Bernoulli, seem destined for a life of mathematics. Bernoulli, for example, was born to a wealthy family of mathematicians, and had easy access to the finest mathematics institutions in the world. Some people, however, became renowned mathematicians against the odds. Sofya Kovalevskaya was a mathematician who became well known for her work around the world, despite facing an uphill battle to study mathematics!

The Life of a Mathematician

Kovalevskaya was born in 1850 in Moscow. Her first exposure to mathematics came at a young age. Instead of using wallpaper, her parents covered the walls of her bedroom with sheets of calculus problems, which she said spurred an early interest in mathematics. However, after her childhood, paths to mathematics for women weren't easy - at the time, women were forbidden from attending universities in Russia, and the universities that accepted women were thousands of miles away. However, a friend of Sofia, a man named Vladimir Kovalevskij, pretended to be her husband so she could travel to Germany and study mathematics. 

Studying with Karl Weierstrass, a mathematician whose specialization was calculus, Sofia was able to graduate from the University of Heidelberg in 1874. However, owing to the discrimination against women at the time, she was barred from teaching at a university, despite being a doctor of mathematics. Thus, she spent the next few years away from mathematics, writing literature and helping Vladimir, who she decided to actually marry, with his business ventures. However, when Vladimir died, Sofia took up mathematics again.

In 1880, she wrote a paper on Abelian integrals that was well received. Following this, she wrote three papers on the refraction of light. This led to her appointment as a temporary professor at the University of Stockholm. Five years later, in 1889, she was granted the title of full professor, the first woman to earn the title in the 19th century at a European university (two women had been professors in the previous century). However, her life was cut tragically short, as she died two years later at the age of 41.

Kovalevskaya's Accomplishments

In addition to being a pioneer for women in the field of mathematics, Kovalevskaya made several important contributions to mathematics, many of which are still being studied today! Though she only wrote a few papers on mathematics, they contained several groundbreaking discoveries.

Her graduate school work on partial differential equations led to the creation of the Cauchy-Kovalevskaya Theorem. This theorem described a set of solutions for a set of differential equations.

Arguably Kovalevskaya's most notable contribution to mathematics was the idea of a Kovalevskaya Top. The 'top' isn't something that can be spun on a table, but rather deals with mathematical questions about rotation. Previously, Euler and Lagrange had studied the motion of rotating objects. However, Euler had only examined objects based on the center of gravity, while Lagrange had only looked at symmetrical objects. Kovalevskaya examined objects that weren't symmetrical, which led to the idea of the Kovalevskaya Top! This discovery won Kovalevskaya the Prix Bordin from the French Academy of Science, one of the world's most prestigious institutions at the time!

from The Center of Math Blog https://ift.tt/37HcdSn from Blogger https://ift.tt/37KBlYc

[365daysoflesbians]

JANUARY 15: Sofia Kovalevskaya (1850-1891)

Once upon a time long ago, in a place far away from here (in high school), I was a total science nerd and was planning on becoming an astrophysicist. That obviously didn’t happen, and I could go on and on about the place granted to women in the field of sciences, and especially in theoretical sciences such as math, but it boils down to: it’s not that they don’t exist, it’s that their stories are dismissed or erased. But this trend is changing (go see Hidden Figures if you haven’t already), and we’re bringing our own contribution today by telling you about about Sofia Kovalevskaya, Russian math whiz from the 19th century.

Sofia Kovalevskaya had a short but remarkable life as she overcame the prejudices of her time regarding women’s education. She had a lifelong interest in women’s rights – and her sister was the feminist and revolutionary socialist Anne Jaclard. Sofia developed very early on a flair for mathematics: the story goes that when she was 11, she had the walls of her nursery papered with pages of Ostrogradski's lecture notes on differential and integral analysis (you can’t really go much more full-on nerd than that). In her autobiography, she wrote of this precocious introduction to math:

The meaning of these concepts I naturally could not yet grasp, but they acted on my imagination, instilling in me a reverence for mathematics as an exalted and mysterious science which opens up to its initiates a new world of wonders, inaccessible to ordinary mortals.

She had plans to continue studying math at university, but there was one problem: in Russia at the time, young women could neither attend university, nor travel without permission from their father or husband. So she entered a marriage of convenience with a young paleontology student (Vladimir Kovalevsky, who would notably go on to work with Darwin). In 1869, they arrived in Heidelberg, Germany, but another problem came up there: women weren’t allowed to enroll in courses (sounds familiar?). Sofia was having none of that, so she lobbied the university’s authorities and eventually was granted permission to attend courses in math & physics, albeit unofficially.

The rest of her academic career was in the same spirit: her genius was recognized by many, she had papers published in prestigious journals, and she became the first woman in Europe to earn a doctorate – in math, summa cum laude, from the University of Göttingen. Bypassing the usual required lectures and exams, she presented three papers which were deemed worthy of a doctorate: one on partial differential equations, which led to the Cauchy–Kovalevskaya theorem, one on the dynamics of Saturn's rings, and one on elliptic integrals. On the other hand, even with such a degree, it took her a long time to find employment, and her gender often worked as an obstacle in her career. In 1883, however, she became a privat-docent at Stockholm University and from there on, worked her way up the ranks, until 1889 when she was appointed a professor (she was the 3rd woman to hold a chair at a European university). During her time in Sweden, Sofia won several important prizes for her work, including the Prix Bordin, and in 1889, thanks to the influence of fellow Russian mathematician Chebyshev, she was elected a corresponding member of the Russian Imperial Academy of Sciences.

Also while in Sweden, Sofia met the Swedish author Anne Charlotte Edgren-Leffler. While each was married and/or had strong relationships with men, the two shared a (very) close friendship (gal pals!) until Sofia’s death. They notably wrote plays in collaboration (since Sofia also had literary interests) and Anne Charlotte wrote an actual biography of Sofia in 1892, as a complement to Sofia’s autobiography, Nihilist Girl (1890). ‘Literary partnership’ – so that’s how they used to call it back in the day.  

Her legacy subsists in many ways. In the scientific world, the Association for Women in Mathematics launched Sonya Kovalevsky High School Mathematics Day, a program that funds workshops across the US to encourage girl to explore math. The AWM also sponsors the annual Sonya Kovalevsky Lecture, which highlights significant contributions of women in the fields of applied or computational mathematics. Every two years, the Alexander Von Humboldt Foundation (Germany) bestows the Sofia Kovalevskaya Award to promising young researchers. And Sofia even has a lunar crater and an asteroid named in her honor (wlw really are out of this world).

[kskskskye]

Sofia Kovalevskaya is the kind of person we, in modern times, would call a "feminist icon." She never let others' treatment of women get her down, from the time her father prevented her from studying math, to when she was studying and even when she was working. She was able to achieve great feats at a time when women, especially those in Russia, were severely disadvantaged academically.

Her contributions to mathematics through the Cauchy-Kovalevskaya theorem and elliptic integrals are also crucial to our present understanding of differential equations and calculus as a whole. Even with her challenges, she still made great strides in the field.

Let her be an inspiration to many aspiring mathematicians worldwide.

Let this alternative assessment be a testament to a great mathematician, a great student, a great daughter, and a great woman.

hello lovelies.

there is a good chance that this will be my last update.

i've been struggling with pneumonia recently, and now i've also contracted influenza. i'm worried my time might come soon, and i won't be able to provide you guys with a proper sendoff.

so... thank all of you. really. for being there for my highs and my lows! you were the best crowd to talk to :)

this might be the last time i say it, but i truly wish all of you are safe, well, and warm. always. may your passion for math be everlasting, even when the world tries its hardest to tear you down.

love, sofia <3

[lthmath]

Sofia Vasilyevna Kovalevskaya

Sofia Vasilyevna Kovalevskaya was a Russian mathematician who made noteworthy contributions to analysis, partial differential equations and mechanics. She was the first major Russian female mathematician and a pioneer for women in mathematics around the world. She was the first woman appointed to a full professorship in Northern Europe and was also one of the first women to work for a scientific journal as an editor.

Despite her obvious talent for mathematics, she could not complete her education in Russia. At that time, women there were not allowed to attend universities. In order to study abroad, she needed written permission from her father (or husband). Together with her husband, she emigrated from Russia in 1867.

In Octoer 1870, she moved to Berlin, where she took private lessons with Karl Weierstrass, as the university would not even allow her to audit classes. In 1874 she presented three papers—on partial differential equations, on the dynamics of Saturn's rings and on elliptic integrals—to the University of Göttingen as her doctoral dissertation. With the support of Weierstrass, this earned her a doctorate in mathematics summa cum laude, bypassing the usual required lectures and examinations.

She thereby became the first woman in Europe to hold that degree. Her paper on partial differential equations contains what is now commonly known as the Cauchy–Kovalevskaya theorem, which gives conditions for the existence of solutions to a certain class of those equations.

30/01/1884 - Sofia Kovalevskaya gives her first university lecture. This was the first regular lecture by a woman at a research institution in any field in modern times.

[lthmath]

Sofia Kovalevskaya

She was the first major Russian female mathematician, responsible for important original contributions to analysis, differential equations and mechanics, and the first woman appointed to a full professorship in Northern Europe. She was also one of the first women to work for a scientific journal as an editor.

Sophia went to Berlin to study with the world-renowned mathematician, Karl Theodore Weierstrass. Weierstrass was reluctant to take Kovalevsky as a student and gave her a test with difficult math problems before agreeing to work with her. He was so impressed by her performance on his test that he tried to get her admitted to the university. He was not successful, so Weierstrass tutored her privately for the next four years.

1874 - Sofia Kovalevskaya presented three papers: on partial differential equations, on the dynamics of Saturn's rings and on elliptic integrals—to the University of Göttingen as her doctoral dissertation. Her paper on partial differential equations contains what is now commonly known as the Cauchy–Kovalevskaya theorem. This theorem is the main local existence and uniqueness theorem for analytic partial differential equations associated with Cauchy initial value problems. [ If you want to find out more about this theorem and you are in a uni - type mode, check this:https://goo.gl/SDnt24 ]

1883 a colleague offered her a teaching position at Stockholm University. After only a year, she was appointed to full professor and published her research on light refraction.

30/01/1884 - she gives her first university lecture. This was the first regular lecture by a woman at a research institution in any field in modern times.  In 1885 Kovalevsky was appointed chair of mechanics.

While searching for more information about her, I discovered that she actually wrote 2 books, one of them is about her childhood. Professor Kovalevskaya was already an international celebrity, and partly for the wrong reasons: less as the distinguished mathematician she actually was than as a "mathematical lady"-a bizarre but fascinating phenomenon. Her book was an immediate success. She had written it in Russian, but its first publication was a translation into Swedish, the language of her adopted homeland, where it appeared thinly disguised as a novel under the title "From Russian Life: the Rajevski Sisters".

Book Recommendation: "A Russian Childhood" by Sofya Kovalevskaya.