Continue reading]]>

Before going into further details, let me first explain what ECCO is. It’s not your standard mathematical conference. It doubles as a summer school for undergrad and graduate students such as myself. As an example, this year we had the pleasure of learning from Günter Ziegler about polytopes, Vic Reiner about q-counting, Rekha Thomas about polynomial optimization and Lauren Williams about total positivity and cluster algebras. These difficult topics are broken down in such a way that even undergraduate students can follow along. These lectures are intermixed with presentations by graduate and postgraduate students and two plenary speakers. This year we had Sara Billey and Mauricio Velasco as our speakers and they were both amazing. Andrés R. Vindas Meléndez gives an exceptional recount of the conference on the AMS blogs network as does Viviane Pons here.

“But TCG,” you might be thinking, “every summer school conference has lectures and talks!” and you’d be right. So then why did I say going was one of the best things I ever did? Because there are a few things that ECCO does that make it special and blow most other schools out of the water.

**Community Agreement**

Firstly, there’s the community agreement. For those who have participated in conferences aimed at minorities, a community agreement, or a code of conduct, isn’t something new. It’s basically a mutual agreement between all participants. It’s an agreement that every person will be treated with respect and courtesy. This was the first time I saw this in a math conference which was a really nice addition. Adding this touch was the idea of Federico Ardila who is also the brain child of ECCO. He wanted to bring more diversity into mathematics and knew that in order to do that we must learn to respect one another.

Additionally, during the conference we spent a few minutes actually reading through the community agreement (found here) and talking to our peers about it. This was a nice touch in that it forced participants to actually be a part of the agreement instead of pushing it to the side and not actually abiding by its doctrine. This openness of dialogue allowed diversity to flourish at the conference and I will talk about this more later one.

**Problem Sessions**

Unlike most summer schools where you are taught things and have almost no time to digest the information, ECCO runs things a little differently. After each hour of lecture we had a 1.5 hour exercise/problem session. This allowed us to actually use the information we learned and to put into practice what the lecturers were talking about. Not only that, but the way the session was organized was well-done. We were all organized into groups of roughly 4-5 people in which there was always at least one postgraduate or professor and at least one undergraduate. The idea wasn’t just to practice what you just learned, but to help others understand the material as well. Although this process was slower, it allowed you to make sure that you actually knew the material before continuing instead of just rushing through a problem set without thinking. Sometimes it got the more experienced mathematician teaching the undergraduates about linear independence, followed by the professor themselves learning something new!

**Dancing**

One of the most noticeable differences at ECCO was what happened after the talks were done. In most conferences you end up going back to your room, grab something to eat, and probably work the rest of the night. Not in Colombia. In Colombia, after dinner the dancing begins, the part I had been looking forward to all day. It is the part where the students become the teachers and the professors become the students and the barriers that are typically seen between the two are obliterated. This is a nice touch because it allows students to see the professors as humans rather than as people who should be on a pedestal. It brings everyone down to the same level, usually taking away the fear from students that the professors are unapproachable.

**Math the Colombian way**

Although the previous differences were all readily noticeable, the biggest and the one that tied everything together was something more subtle. It was mentioned a couple times during the conference when Federico asked the conference what it means to do mathematics the Colombian way. To understand this, you have to actually be there, but I’ll try and break it down even though it’s hard to put into words. Math in Colombia is working together as a community. This is seen throughout the entire conference. In the community agreement everyone starts off right away agreeing to work together, no matter their differences. Then you have the problem sessions which brought the small groups into a tight-knit bond in which everyone helped everyone. You also had the nightly dancing sessions where whatever barriers that were left were shattered and everyone was included in the global dance nights. Everyone was made equal. No one was singled out to be better than anyone else. This made us all feel like a part of the same family. We were all one big community and it helped us stay comfortable and therefore kept us going strong throughout the conference. THIS is what we need more of in mathematics.

The single-handed biggest reason I loved this conference was the diversity and inclusiveness that was brought on by the community agreement and doing math the Colombian way. In particular, I want to mention three main things. The first two are things that don’t effect me individually but that I notice around me, the last had me crying mid-conference.

**Language**

A lot of conferences in math tend to be done exclusively in English. Even conferences held in France, which has a strong tradition of making sure their language isn’t consumed by another, tend to lean toward English. This is why it wasn’t very surprising to me to see that the lectures and most presentations were given in English. But the nice part about ECCO was that, unlike other conferences where participants would adamantly only speak English, at ECCO all the English speakers made a concerted effort to speak Spanish. Even those who spoke no Spanish in the beginning started speaking a few words by the end. Every effort was made to try and bring mathematics into Spanish. This is huge as typically mathematics is done in one of four languages: English, French, German or Russian (in alphabetic order). In order to do mathematics in Spanish, new terminology was needed to be invented and that’s great because it allows native Spanish speakers to be able to more readily and easily contribute to mathematics instead of having language be a barrier. It allowed a new venue for more Spanish speaking people to join the mathematics world, thus increasing the diversity within. This goal of bringing math to Spanish intrigued me as it made me wonder what math in Armenian would be like. It’s not something I’ve ever had to do, but as of right now there isn’t much math in Armenian. Armenia generally teaches math in Russian. So it gave me hope that one day I can do math in my native tongue, and that would be epic.

**People of Color + Women**

It might not surprise you that a conference in Colombia would have a high number of Colombians, but it was refreshing to see a large number of non-white people doing mathematics. It is well-known that mathematics is traditionally done by white male heterosexuals, so to see so many non-white people doing some amazing mathematics brought a smile to my face. But what was even better was seeing my friends who grew up in predominantly white areas have an extra hop in their step, a glow on their face, a wider smile than normal, when looking around and seeing people who look like them doing some amazing mathematics. It drove them to do better and to recommit to doing math. It made them feel a part of the community. This is how you get more diversity into math.

Not only that but the number of women at the conference was also fairly high (for a math conference). Although I didn’t count the number of women, but it felt as though roughly 30% of the people in attendance were women. Not only that, but half (!!) of the speakers (half the lecturers, half the plenary speakers, half the presentations) were women! This is something you rarely see. Most mathematics is dominated by males (as an example, in my combinatorics department we have 7 males and 0 females) and so it’s refreshing to see a concentrated push to get more women into mathematics. I’m hoping in future years to see the number of women participants to be 50% as well and want to work with others to try and make that a reality.

**LGBTQIPA+**

The last topic I’ll talk about, and for me the biggest, is the LGTBQIPA+ community in mathematics. In recent years I’ve noticed something strange around me. Everywhere I looked, I felt alone. I felt like the only gay person doing mathematics. This is a weird feeling for me because I have always had queer friends everywhere I went. But in academia the LGTBQIPA+ community is few and far between. My friends point out there are a lot of queer people, but every name they give is a PhD student and every one of these students eventually leave academia. Sure there was Alan Turing, but he killed himself due to how horrible the English made his life because of his homosexuality. And yes, there’s an entire group (consisting of 10 people) called Spectra in the AMS, but none of them are people I know and they all worked in different areas. Also, that brings the grand total of queer mathematicians to 10 (not counting Turing since he’s dead) out of how many thousands of mathematicians? So I’ve started to feel more and more lonely in my math community. It felt like I didn’t belong. All my friends are amazing and they always try to make me feel welcome, but it’s just not the same.

Therefore it came as a big surprise to me when I counted at least 5 gay people at ECCO. My biggest shock, and the thing that has changed everything in my life, was a professor, and one of the lecturers of the conference: Günter Ziegler.

First a little background. Ziegler for me had always just been a name on multiple books and articles. I had put him in the category of “exceptional geniuses I will never meet”. So to have met him at the conference was already an honour and a privilege. The way he presents mathematical material is flawless. Not only that, but he recently was elected as president of the Free University of Berlin (one of the biggest universities in Germany). But for me, meeting him for his achievements wasn’t the big part. The biggest part came when one of my friends pointed out that Günter was also an out gay man! You can read a letter he wrote to Turing here where he describes life as a gay mathematician in Germany. The second I found about this article I hid myself in my hotel room and I read it. I actually couldn’t believe my eyes. Here, a man I considered a genius, a man I considered to be one of the best mathematicians of all time, a man who was now at the head of one of the best universities in the world, was just like me. The amount of emotion that flooded my system at that time will stay off the record, but it suffices to say that for the first time in 3+ years, I didn’t feel alone.

Not only that, but having a community agreement gave me the confidence in order to do something I had never done before. I invited my math friends at ECCO to a gay club. The awesome part was that people actually joined me! We ended up having 7 people in total go to the club. Sure, more than half of us weren’t gay, and sure, the gay club wasn’t that amazing, but just being out, in a gay club, with mathematicians… was amazing. I felt I belonged.

**In Closing**

So in the end I felt like ECCO was the best conference I had ever been to. Having a community agreement allowed everyone to be open about themselves allowing queer people to be out. This in turn allowed me to feel a part of a community that for the past 3 years I had been struggling to feel a part of. I finally feel like I belong in mathematics and that I’m not alone. This is why ECCO is one of the best conferences ever. Doing math the Colombian way opens doors to allow people on the sidelines like myself to feel welcome. It allows diversity to flourish and creates an open community of all sorts of humans. It’s the type of conference style I’d like to see replicated far and wide. We should all be doing mathematics the Colombian way.

]]>Ledermann and Weir take a slightly unique approach in the theory of groups. Their text is slightly difficult to follow in a lot of places as they tend to group things together in non-standard ways. The biggest difference I found with his use of symbols is when talking about homomorphisms. In particular most texts will look at a homomorphism: \(\theta : G \rightarrow G^{\prime}\) and when going from \(G\) to \(G^{\prime}\) they apply a function such as \(\theta (g)\); Ledermann and Weird on the other hand choose to use the notation of \(g\theta\). Although this format might be more concise, it can at times be a little confusing as to what each symbol is supposed to stand for. My assumption is that he’s trying to show that a homomorphism is similar to group actions with the action being the map \(\theta\), but for an introductory piece, it can be particularly confusing.

The biggest part I did enjoy about this book was the list of topics. Not only did he cover the standard topics of groups, subgroups, cosets, normal groups, conjugacy classes, etc. he also threw in some topics you don’t normally see in an introductory course such as double cosets, free groups, the derived series, soluble and nilpotent groups. He tends to go through topics fairly quickly so it can be a little difficult to follow at times, but it’s not that bad to follow and can generally be followed on your own. Another part about this book I thoroughly enjoyed was that every exercise had a solution in the back of the book. Exercises are essential when learning any new topic and when you are trying to teach yourself, having questions without answers can be a little difficult at times because you can never be sure you’re actually doing it correctly.

All in all I think this book is a really good book for introductory group theory. If you’re willing to invest a little bit of time understanding the notation, you’ll learn more out of this book than most other books on group theory.

]]>I think Enderton does a really nice job introducing logic to those who have never studied it before. Although not always the easiest to follow Enderton lays out all the necessary topics in a nice organised fashion so it’s fairly simple to follow everything. I found his explanation of the pumping lemma to be lacking which made it difficult to follow, but his proof of the compactness theorem more than makes up for it. He does a good introduction of not only proposition and predicate logic, but also goes into second-order logic as well and tackles it the same way as his previous material so it’s easy to follow it all. Although this wasn’t my favourite logic book, it is a nice book to peruse.

]]>This book was definitely one of the best books I’ve seen for introducing computation theory. Michael Sipser does an amazing job introducing not only Turing Machines, but also different types of machines such as RAM and Finite Automata. He not only gives a good intuitive description and explanation of different machines, but he also does a really good job explaining different languages. Although some of his proofs can be a little difficult to follow, they are all understandable to anyone who is coming at the topic for the first time. He also does a good job explaining the different issues that are relevant in computation theory. unlike most books that I’ve seen so far, he gives a very detailed proof of the Cook-Levin Theorem (That SAT is a NP-Complete language) and also gives multiple examples of proving NP-Completeness which is nice as that seems to be one of the more confusing parts of computation theory for most students. He also gives answers to many of his examples which makes it so that it’s easy to follow on your own and learn everything without needed secondary help.

]]>This week we’re doing another book on Logic! Mendelson does an ok job with his book in introductory logic. He sets up a nice introduction to logic concepts, but then fails to deliver much needed exposition. Although the book is very descriptive and helps to get your hands very dirty in logic, it can sometimes be a little to follow what Mendelson is trying to prove and whether a certain problem is an exercise or a proof. Although at times it’s a little hard to follow this text does give an amazing introduction to the topic of logic. It doesn’t assume any presupposed knowledge and actually goes over almost every topic that a new student would be expected to know in the topic. Not only that, but he also breaks down Number Theory to prove Gödel’s incompleteness theorem and also does a fairly good job of showing the different axiomatizations of set theory. Not only that, but his introduction to second-order logic and modal theory are also fairly nicely laid out. I chose not to look at his computability section as it seemed that it would be lacking since it only seemed to barely touch on the subject and there are tons of books out there that go more in depth on the topic and do a phenomenal job. I’d definitely recommend this book with a secondary companion text to help guide you along if you have trouble understanding what Mendelson is trying to state.

]]>Armstrong does an amazing job with his introduction to group theory. Unlike most texts he uses a geometric approach for a lot of his work in groups. With that he uses dihedral groups a lot in his exposition and is always going back to it (and symmetric groups) for examples. I thought it to be refreshing to see group theory presented from an alternative angle. Although he doesn’t provide answers for his exercises, all of the exercises are answerable using the text provided. Some of them were difficult enough to pose some fun challenges while reading the book. Although the book is not the best introductory text I’ve found out there, I thought Armstrong’s introduction of finitely generated abelian groups was very well done. It definitely was better than most of the ones I had seen out there and definitely worth reading more into. The final theorem he put in his book was the Nielsen-Schreier Theorem which states:

Every subgroup of a free group is free

Which is a super fun concept and is not fully introduced in a lot of the texts that I’ve read in the group theory world. If you want a text that does group theory from the slight angle of geometry I definitely recommend this book.

]]>Edward John Lemmon was a logician whose main area of expertise was modal logic. His book ‘Beginning Logic‘ is likely one of the better ones out there for those just getting into logic in order to understand the different rules in modal logic. Lemmon begins with an introduction on logic and why it is necessary to talk about the subject. After giving a very well laid out overview of why logic is required he begins by talking about the rules of logic and why each one is necessary. Although the book on it’s own is likely difficult to understand, it is a good reference book to see why certain rules are the way they are. (By rules I refer to the rules of derivation from one set of formulae to another). Not only does he give a detailed explanation of each rule he also goes into the concepts of completion and why each set of rules are complete in their respective areas.

He covers 2 separate areas of logic: propositional and predicate. Propositional logic is logic that only uses ‘operators’ such as \(\rightarrow , \land , \lor , —\). (Here he uses \(—\) instead of the now traditional \(\neg\)). Propositional logic is also sometimes referred to as sentential logic or propositional calculus. In predicate logic Lemmon adds the symbol \(\exists (x)\) for ‘there exists an x’, and \((x)\) for ‘for all x’. This format is not necessarily traditional, but Lemmon is working before standards were fully developed. His syntax can be slightly difficult to follow, but after working on it, it is not the worst syntax out there.

Who this book is for: This book should be used in conjunction with some other books and is good for someone who is looking for an english language description of logic and the different rules associated with proving results.

]]>Continue reading]]>

If you look at the primes, they seem to go on forever! And thanks to amazing mathematicians from the past we know that they do! But HOW do we know that they actually do? How can you be sure there is a number after 97 that is prime? Some will just point to their computer and say, “Computers can prove that!” and they would be partially right. Computers have helped us find extraordinarily HUGE prime numbers. For example, remember those special prime numbers called Mersenne primes which are primes of the form \(2^{p} – 1\) where p is also prime? Well, a worldwide collaboration was started in order to find the largest prime known to man and they called themselves GIMPS.

The organization is called GIMPS or Great Internet Mersenne Prime Search. You can find their website here: [GIMPS] Quite recently they found the largest prime ever known to man. What is the prime number you may be wondering? It is: \(2^{57,885,161}-1\). You might be thinking that this looks like a small number, but it really isn’t. This prime is 17,425,170 digits long. That is HUGE! It is so big that if you were to print out the number on paper (without commas) it would be 4,283 pages long!

It took years for the computers to calculate that prime number, so as we questioned earlier, how do we know if there is a prime number greater than this or not? It is so large we can barely use computers to go any higher, and we can forget about writing the number down (you try writing out 4,283 pages of numbers!). So then how can we KNOW if there are infinite number of primes or not? Well this is exactly what a guy named Euclid was pondering over 2,000 years ago. Euclid was able to prove that there are an infinite number of primes! So no matter how big of a prime number we find, there will always be a bigger one. So cool! But how did Euclid come up with this?

Below is a rough transcript of how Euclid created his proof, but before we go over the proof here a lemma that Euclid uses in order to prove it that you should know:

Lemma 1: Every number greater than 1 can be broken down into multiples of prime numbers. e.g: \(12=2 \cdot 2 \cdot 3, 4=2 \cdot 2, 3=3, 5=5\). Notice that prime numbers just equal themselves, and composite numbers are multiples of primes!

Euclid’s Proof: Suppose that there are only a finite number of primes. This means that there is a number x such that any number greater than x is not prime. All these numbers greater than x thus must be composite and by the lemma we just discussed must be a combination of prime numbers. So if we take every prime number less than x and multiply them together we get a number n. Now if we add 1 to n we get a prime number! (Notice that since 2 is a prime, thus multiplying all the primes together would get us an even number, which would automatically not be prime. So we add 1 to make it an odd number.) Why/how is n+1 prime?! Because no matter which prime number we select and we divide n + 1 by it we will get a remainder of 1. And since we have used every prime number less than x, and every number between x and n are composite, there are no other prime numbers that (n+1) can divide into. So (n+1) must be prime! But, we had stated earlier that x is the greatest prime and since (n+1)>x we see that our earlier thought was wrong (that there are a finite number of primes) and so there must be an infinite number of primes!

[Note: This type of proof is called proof by contradiction. I’ll eventually lay the groundwork for how this works, but for now you can look it up in google if you’d like.]

Wow, that was complex, but so rewarding! We can now for sure state that there are an infinite number of primes! Fun part is, that this was only 1 of many proofs that there are infinite primes! Any questions? Leave a comment!

Questionably yours,

The Cali Garmo