True to His Standards in Politics, Religion, and Math

 Augustin Louis Cauchy

Aug. 21, 1789-May 23, 1857

In Abstract Algebra we work with permutations a lot. What is a permutation you may ask? A permutation is essentially when you have a set of items and you change the order of the items, you then have created a permutation. In an earlier blog post we worked with permutations of a square. Here is a more specific definition of permutations from my text book, Contemporary Abstract Algebra by Joseph A. Gallian-

    "A permutation of a set $A$ is a function from $A$ to $A$ that is both one-to-one and onto. A permutation group of a set $A$ is a set of permutations of $A$ that forms a group under function composition."

Why am I interested in permutations? Well first off they are super cool, secondly, it is because this guy, Augustin Cauchy was interested in permutations too! Cauchy was one of the pioneers of analysis and the theory of permutation groups. He, like many mathematicians, added to several different areas of mathematics and physics including studying differential equations, infinite series and their convergence and divergence, determinants, calculus, probability, and optics.

Cauchy lived through the French Revaluation and when he was very young his parents feared for their safety and health as they had very little food and resources. However things got better as time passed. Laplace and Lagrange both visited Cauchy's home and encouraged his studies. In 1807 he graduated from the Ecole Polytechnique. He then attended an engineering school and eventually, in 1810, worked as an engineer for Napoleons' English invasion fleet at Cherbourg while continuing to do mathematical research on the side.

Around 1813 Laplace and Lagrange had convinced Cauchy to focus solely on mathematics. He came back to Paris and spent time teaching and doing research for years. Though Cauchy produced many important papers for mathematics, he had a way of offending his colleagues... he was very Catholic and came off as self-righteous and judgmental. Even poor, kind Mr. Abel was offended by this man! Abel exclaimed-"Cauchy is mad, and there is no way of being on good terms with him, although at present he is the only man who knows how mathematics should be treated. What he does is excellent, but very confused…"

In 1830 when Charles X was exiled and Louis-Philippe took the throne, Cauchy also went into exile since he refused to take an oath of allegiance. Because of this he lost all of his positions and chairs that he held in Paris. He began to teach in Turin and eventually attempted to teach the grandson of Charles X. This was an interesting experience I'm sure... Most articles reveal that the grandson was not the best of students and Cauchy was not the best of teachers, so the two made a less than stellar pair to say the least. 

Finally, in 1838 Cauchy returned to Paris and gained back his seat at the Academy, the Ecole Polytechnique, but was not allowed to teach as he still refused to take the oath of allegiance. By 1848 when Louis-Philippe was overthrown, Cauchy regained his chair at Sorbonne and held that position till he died. Some of his last words were, "Men pass away, but their deeds abide." Cauchy's works certainly abide today, and we are so thankful for that.


    Thanks bunches,

      -Haylee Jo Lau


Works Cited:

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A Revolution of Her Own

 Sophie Germain

Do you know how Archimedes died? He died by being speared to death by a Roman soldier. He had ignored the soldier because he was so intrigued by a geometric figure drawn in the sand. If you are like Sophie Germain, you too are wondering what was so interesting about a shape in the sand that you would lose your life over it? This was the story that first inspired Sophie to start studying mathematics.

Germain was born in Paris on April 1, 1776. Her life was surrounded by revolutions. The year she was born the American Revolution started and thirteen years later a revolution started in her own country. 

She was thirteen years old and stuck in her home since the revolts around their land made it unsafe to be out. There she started exploring her father's library which was where she found Archimedes story. She continued searching her father's books and began studying mathematics on her own, studying works by Newton and Euler in Latin and Greek. Germain had to study at night by candle light to hide from her parents.  Her family was middle class so her education had been limited and on top of that, social norms claimed that the studies Germain was carrying out were inappropriate for females. In an attempt to stop Germain before she got too carried away, her parents tried everything they could to stop her, but she fought back with just as much zeal till her parents decided that it could not be helped, their daughter was going to study mathematics one way or another.

In 1794, when  Sophie was about eighteen years old, the Ecole Polytechnique was founded in Paris. It was meant to train mathematicians and scientist for the country but did not accept women to study there. Despite that, Germain was able to get ahold of lecture notes and study the materials taught there. She eventually took the pseudonym M. LeBlanc and sent her work on analysis to a teacher whos lectures she was especially fond of, J.L. Lagrange.  Lagrange was so impressed, he insisted on meeting the author of the well done paper. When he found out that M. LeBlanc was really Sophie Germain, he looked past the fact that she was a female and decided to be her mentor, recognizing her capabilities in math. And just like that, with her new escort, she had been introduced into the formal society of mathematics.

Through the years Germain continued to do work with Lagrange and worked with Gauss (who also did not know she was a woman originally) and Jean-Baptiste- Joseph-Fourier and other mathematicians. And eventually, in 1816 on her third attempt, she won a prize from the French Academy of Sciences. She was the only one to have entered the contest originally.
With her paper Memoir on the Vibrations of Elastic Plates she not only won the prize but had been accepted to the Academy of Sciences. She would finally go and study mathematics as an equal collaborator and be able to refine her proofs in number theory.

Sophie Germain died on June 27, 1831 at the age of 55. She had been fighting breast cancer. Before she died, Gauss, an earlier mentor of hers, had petitioned the University of Gottingen to give her an honorary degree. However, she had passed away before they could get it to her.  Still, Germain was a revolutionary woman who worked hard to do what she loved despite the obstacles that loomed over her. And she made it! She is now a recognized and appreciated mathematician. Today some of her most important work is recognized in the theory of elasticity and in number theory.

Here is an example of some of the work she accomplished. If both p and $2p+1$ are prime, then $p$ is a Sophie Germain prime. The first few such primes are $2, 3, 5, 11, 23, 29, 41, 53, 83, 89, 113,$ and $131$. Around 1825 Sophie Germain proved that the first case of Fermat's last theorem is true for odd Germain primes. 

*My favorite quote by Sophie Germain, "Algebra is but written geometry and geometry is but figured algebra."


     Thanks bunches,
      -Haylee Jo Lau

Works Cited:
Contemporary Abstract Algebra, Joseph A. Gallian, Ninth Edition

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Graphs, Tables, and Groups- Oh My!

 Arthur Cayley

Last week we learned about Sylvester, this week let's look at one of his dearest friends who also had a liking of mathematics.

Sylvester commented, "Mr. Cayley, of whom it may be so truly said, whether the matter he takes in hand be great or small, "nihil tetigit quod non ornavit," which translates to "He touched nothing without embellishing it". Indeed Cayley had a knack for mathematics and many things. Born on August 16, 1821 in England, Cayley ended up writing enough papers on mathematics to create thirteen volumes, about 600 pages each, of just his work. He also had an interest in astronomy, mechanics, painting, traveling, architecture, and hiking. He taught at John Hopinks University for a bit and was a lawyer for fourteen years- the whole time writing papers for mathematics. Eventually he became a professor of mathematics at Cambridge University (also getting married the same year) until he died on January 26, 1895. There he not only taught mathematics and carried out research, but he successfully pushed for women to be accepted into Cambridge as well. 

Due to Cayley's amount of work mentioned above, it is not hard to believe that the man has contributed to many fields of mathematics- he helped come up with the theory of invariants, he worked with linear algebra, geometry, he invented Cayley graphs, Cayley tables, and worked quite a bit more with groups, the list could go on. For right now let's focus specifically on groups and Cayley tables as Abstract Algebra is all about groups. This will be fun!

Imagine you just got a brand new car (Santa was extra giving this year...).


It is in beautiful condition, but in awhile will need new tires. You buy tire 1, tire 2, tire 3, and tire 4. There is no specific order you have to put the tires on even though they are numbered. How many different combinations can you come up with? (*You should be able to put together 8 different combinations!)

While doing this, you may notice that you cannot move just one tire. If you want to switch one tire that means you are going to have to move at least one other tire. One other option is to move nothing and leave everything in its original place. 

Now that you have thought about this for a bit, let's look at something very similar with just a plain square. 


Here each vertex is labeled, we have a vertex set! Let's call our new set $V$ and let's say that $V =$ {$1, 2, 3, 4$}. We also have a set of edges, call this set $E$ and say our set $E =$ {{$1,2$}, {$2,3$}, {$3,4$}, {$4,1$}}. We now have a graph where $G = (V, E)$. How many different ways can we move these vertices? Well, squares are made from right angles and right angles are $90^{\circ}$right? So let's just rotate this square clockwise a couple times and see what we get.

      $R_{0}$ gives,              $R_{90}$ gives,             $R_{180}$ gives,              $R_{270}$ gives,                                                                                     
                                                    

      $H$ gives,                 $V$ gives,               $D$ gives,                $D'$ gives, 

                                                    
The top row is what we should get when shifting everything clockwise $90^{\circ}$ several times and includes what happens when we leave everything in its original spot, like what we mentioned earlier. The bottom row has first a horizontal flip, a vertical flip, a diagonal flip and then an off diagonal flip.
These eight different movements make up a specific group called $D_4$. And actually each movement that makes up $D_4$ is considered a function. So what happens when we combine two of these movements? Is that allowed? Well, when we think of these movements as functions we can then use function composition. Function composition works like so, let $f$ and $g$ be functions so then $f \circ g$ means "$g$ followed by $f$". Using our combination of movements let's apply this same idea. Let's take $HR_{90}$ which means "$R_{90}$ followed by $H$". This new combination of movements will give us $HR_{90} = D$. We can make more movements by combining our original eight from $D_4$! So then how many combinations of movements can we make? We can actually put all of this into a table called a Cayley table believe it or not, named after Arthur Cayley, the man who invented it.







In Abstract Algebra we can use Cayley Tables to see how groups interact with each other and to see if it could possibly be a group since there are specific criteria that need to be met for something to be a group.

And now you have explored a small part of Arthur Cayley's work with groups and gained a basic understanding of what we look at in Abstract Algebra. If you are still interested in groups and Cayley Tables, here is a link to a cite that goes a little more in depth on the subject.


    Thanks bunches,
      -Haylee Jo Lau
(P.S. Here is a cool video my Abstract Algebra teacher shared with our class to get us excited about groups, hope you enjoy!)




Works Cited:
Contemporary Abstract Algebra, Joseph A. Gallian, Ninth Edition
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Mathematics is the Music of Reason- J.J.S.

 James Joseph Sylvester


In my experience, many people picture mathematicians as cold, isolated, geniuses. Though Sylvester was an extremely capable man, he got things wrong and was far from cold and isolated. One student said about him,

    Sylvester’s methods! He had none. “Three lectures will be delivered on a New Universal Algebra,” he would say; then, “The course must be extended to twelve.” It did last all the rest of that year. The following year the course was to be Substitutions-Théorie, by Netto. We all got the text. He lectured about three times, following the text closely and stopping sharp at the end of the hour. Then he began to think about matrices again. “I must give one lecture a week on those,” he said. He could not confine himself to the hour, nor to the one lecture a week. Two weeks were passed, and Netto was forgotten entirely and never mentioned again. Statements like the following were not unfrequent in his lectures: “I haven’t proved this, but I am as sure as I can be of anything that it must be so. From this it will follow, etc.” At the next lecture it turned out that what he was so sure of was false. Never mind, he kept on forever guessing and trying, and presently a wonderful discovery followed, then another and another. Afterward he would go back and work it all over again, and surprise us with all sorts of side lights. He then made another leap in the dark, more treasures were discovered, and so on forever.
    -Ellery W. Davis

James Joseph Sylvester, was born on 3 September 1814 and grew up to become a bright, passionate man. He died on 15 March 1897, but not before leaving his mark on the world of mathematics. His major mathematical contributions were in  the theory of equations, matrix theory, determinant theory, and invariant theory (which was discovered with Cayley). Sylvester loved poetry and his proofs were often described as, "flowery and eloquent", and were said to match his sensitive and enthusiastic personality. 

We will often see terms that Sylvester coined such as matrix, invariant, discriminant, commutant, and more. There have also been ideas named after Sylvester such as Sylvester's determinant identity, the Sylvester-Gallai theorem, Sylvester Matrix, Sylvester's theorem and the list can go on. Taking a look at Sylvester's work, we can see his finger prints on Abstract Algebra today. And Sylvester's contributions to things like invariant theory  are a branch of Abstract Algebra. Another reminder that the math we study now was built from the shoulders of those who proceeded us.

In 1837 Sylvester earned second place in the Mathematical Tripos Examination but did not graduate with a degree as he was Jewish and refused to subscribe to the Thirty-Nine Articles of the Church of England, which was required to graduate at the time. However, Sylvester continued to spread his influence despite the roads that were closed to him because of his religious beliefs.

For three years Sylvester was the chair of natural philosophy at the University of London, then in 1841 he finally was able to receive his B.A. and  his M.A. from Trinity College, Dublin. By the time Sylvester was 27 he applied to be the chair of mathematics in the University of Virginia in Charlottesville in the United States. In 1843 he left the U.S. and returned to England. There he got into actuary work and decided to study law. This happened to be a very lucky choice on his part as it was what the mathematician Arthur Cayley was doing as well. They met, and though very different in many ways, became great friends who would end up doing lots of research together. Among Sylvester's many accomplishments in 1876 he was appointed to a prestigious position at the Johns Hopkins University, he founded the American Journal of Mathematics, and held a teaching position at Oxford from 1884 until he died.

What did I learn from studying Sylvester? It is okay to not get things right the first time... keep trying, you never know how much your contributions can benefit others.

Thanks bunches!
    -Haylee Jo Lau




Contemporary Abstract Algebra, Joseph A. Gallian, Ninth Edition













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