Fix a finite dimensional field extension $K/\mathbb{Q}$. It turns out that there is a canonical *ring* associated to $K$, which we’ll denote $O_K$, called the *ring of integers *of . Specifically, $O_K$ is defined to be the set of all elements of $K$ which are solutions to monic, integer polynomials. (As a sanity check, one can check the ring of algebraic integers of $\mathbb{Q}$ is $\mathbb{Z}$, which provides motivation for the term for the term “ring of integers”.) For example, the ring of integers of $\mathbb{Q}[\sqrt{-5}]$ is $\mathbb{Z}[\sqrt{-5}]$, but on the other hand, the ring of integers of $\mathbb{Q}[\sqrt{5}]$ is $\mathbb{Z}[\frac{1 + \sqrt{5}}{2}]$.

The next logical step is to ask what properties that rings of algebraic integers have. One might hope that the ring of algebraic integers is a unique factorization domain (UFD). However, in $\mathbb{Z}[\sqrt{-5}],$ we have that $2*3 = 6 = (1 + \sqrt{-5})(1 – \sqrt{-5})$, and it’s not too hard to show that the above equation gives two distinct factorizations of 6. However, one might notice that when passing to *ideals* in $\mathbb{Z}[\sqrt{-5}]$, then $(6)$ factors as the product of prime ideals

$(6) = (2, 1 + \sqrt{-5})(2, 1 – \sqrt{-5})(3, 1 + \sqrt{-5})(3, 1 – \sqrt{-5})$

and moreover, this factorization is unique. One can then go onto show that if $O_K$ is the ring of algebraic integers for some finite dimensional field extension $K/\mathbb{Q}$, then for any nonzero ideal $I \subset O_K, I$ can be factored *uniquely *as the product of prime ideals. This leads to the notion of the Dedekind domain, which generalizes this property.

Moreover, one can argue that one can make a *group* of these elements by including an extended notion of ideals known as *fractional ideals. *These are $O_K$-submodules $J \subset K$ for which there is an $r \in O_K$ with $rJ \subset O_K$. This is a group with a product operation similar to that of the rational numbers, so that $\frac{I_1}{J_1}*\frac{I_2}{J_2} := \frac{I_1I_2}{J_1J_2}$.

From this notion, one can define the *(ideal)* *class group* of the ring of algebraic integers $O_K$, defined to be the quotient of the above group by the group generated by all nonzero principal ideals. The class group tells us many facts about the associated field and its algebraic integers – it’s a good exercise to check that the ring of algebraic integers is a principal ideal domain if and only if its associated class group is trivial.

One of the first cool facts about this is that the class group is always a finite group! This also develops the subject of class field theory, the study of Galois extensions of $\mathbb{Q}$ whose Galois groups are abelian over $\mathbb{Q}$. This can be used to prove the Kronecker-Weber theorem, which says that for any abelian extension $K/\mathbb{Q}$ (i.e. any Galois extension $K/\mathbb{Q}$ for which $Gal(K/\mathbb{Q})$ is abelian), there is a cyclotomic field containing $K$. In short – the class group of a number field is a rich object worth studying!

]]>Jim Simons is an American mathematician, CEO, and philanthropist. Simons is the founder of Renaissance Technologies LLC, an investment management firm controlling over $60 billion, and of the Simons Foundation, a non-profit organization that funds education, outreach, and research in mathematics and the natural sciences.

Dr. Simons earned his PhD at the age of 23 from UC Berkeley under the guidance of Bertram Kostant. His thesis on holonomy systems broke ground in differential geometry and the Annals of Mathematics published a version of it [6]. At the age of 37, he won the American Mathematical Society’s Oswold Veblen Prize in Geometry. Simons most famous mathematical work is a 1980 paper coauthored with the celebrated mathematician S.S. Chern. Chern–Simons invariants are now fundamental in geometry, as well as in physics, and in number theory (see this recent quanta article on the work of Minhyong Kim).

Simons founded Renaissance Technologies LLC in 1982. The Medallion Fund, owned by Renaissance, is one of New York’s most successful hedge funds. They used *quantitative analysis *to make investment decisions long before it was the industry standard. Rather than hiring finance experts, Simons hired mathematicians, physicists, and cryptographers with no financial training. Their ability to reason to an abstract level made the Medallion Fund win out against companies operated by traditional Wall Street executives.

In 1994 Jim Simons and his wife Marilyn Simons established the Simons Foundation. The Simons Foundation funds exceptional work in mathematics, physics, the life sciences, and autism research. Additionally, it supports two independent online editorials: Quanta Magazine and Spectrum. Quanta Magazine reports advances in mathematics, physics, biology, and computer science. Spectrum provides news coverage of developments in autism research.

The Simons Foundation features high profile Simons Collaboration Grants. These projects consist in intensive collaborations among field leaders which last several years. Some ambitious ongoing collaborations are the Simons Collaboration on The Origins of Life and the Simons Collaboration on Homological Mirror Symmetry.

Dr. Simons has founded several non-profit organizations. He is a great benefactor of mathematical education. Math for America, founded in 2004, seeks in improve mathematical education in primary and secondary schools in the United States.

He helped establish the National Museum of Mathematics (NMM), located on 26th Street in Manhattan, just a few blocks from the Simons Foundation. The NMM is a hub of public mathematical outreach and caters largely to children, stoking their curiosity in mathematics. Simons was one of the first on the Forbes List to sign the giving pledge—a promise to donate most of one’s wealth to charity.

The Simons Center for Geometry and Physics (SCGP) is a research institute on the Campus of Stony Brook University. The SGCP emphasizes interdisciplinary research and collaboration among pure mathematicians and theoretical physicists–a motif which characterizes the past fifty or so years of geometry and topology. SCGP staff includes legends such as Sir Simon Donaldson, Kenji Fukaya, and Nikita Nekrasov.

Simons himself returned to research mathematics in 2007, coauthoring an important paper with Dennis Sullivan entitled “An Axiomatic Characterization of Differential Cohomology.” I can only hope to remain enamored of cohomological questions in my later life. As a young person working in mathematics, I find this totally uplifting.

The mathematical world is indebted to Professor Simons and his constant efforts to improve the quality of mathematics at all levels—from primary school education in developing countries to the grand vistas of 21st century symplectic geometry.

[1] Broad, W.J. G*iver, Does, Seeker, Ponderer. Billionaire Mathematician’s Life of Ferocious Curiosity. *New York Times, Web. 7 Jul. 2014.

[2] Kim, M. *Arithmetic Chern-Simons Invariant I.* https://arxiv.org/pdf/1510.05818.pdf 2016.

[3] Tindera, M. *How Hedge Fund Billionaire James Simons is Changing Math Education. *Forbes Magazine, Web. 21 Apr. 2017.

[4] Simons, J. and Chern, S.S. *Characteristic Forms and Geomtric Invariants. *99 (1):48-49. January 1974.

[5] Simons, J. and Sullivan, D. *An Axiomatic Characterization of Differential Cohomology. *https://arxiv.org/abs/math/0701077 2007.

[6] Simons, J. *On the transitivity of holonomy systems. *Ann. of Math. 76 (1962), 213-234.

[7] *About Jim and Marilyn Simons. *The Simons Effect, Stony Brook Foundation. Web. Accessed 10 Nov. 2017. https://www.stonybrook.edu/commcms/simonseffect/about/aboutsimons.html

However, through our training as mathematicians, are we asking ourselves: How are we contributing to this diversity? How do we create environments that embrace the identities of those who “do” mathematics? Are we making mathematics accessible and inclusive? These are questions that I ask myself, but would be interested in making them part of a larger narrative.

When I started thinking about these topics I found myself overwhelmed with the knowledge that this affects the lives of many on a daily basis. This is part of their personal story and in many ways an unavoidable part of their journey to become mathematicians. It affects my students who look into who does mathematics and may not see someone who they can relate to. It affects my peers and professors who may be the first “blank” or the only or the few “blank” in their classrooms, at a conference, or departments.

Then the question became, what do I do? I searched high and low for a magical answer and found that … it’s complicated. But, I think there are certainly small things we can do to open up the conversation as individuals and as a community. Here are a few things I’ve found useful, and I hope that this list helps others who wish to start discussing some of these questions:

**Create spaces for conversation.**- It may look like small coffee chats with peers, one or two conversations with faculty, or participating in conferences that facilitate these conversations. For example, the Society for Advancement of Chicanos/Hispanics and Native Americans in Science (SACNAS) is a great one! Others that come to mind are Latinxs in the Mathematical Sciences, Field of Dreams Conference, and Blackwell-Tapia Conference.
- As part of our AWM student chapter, we have created a Teaching and Diversity Seminar where we bring speakers to tackle some of these questions. We look in neighboring departments and we look for passionate individuals within our fields. One of our speakers this semester, Dr. Rochelle Gutierrez has a piece on this blog I encourage you to read. Other great speakers we’ve engaged with as part of our TA (teaching assistant) training include Aditya Adiredja and Esther Enright.

**Learn, keep learning, and challenge your assumptions.**Join a reading group, attend talks about these topics, or follow blogs by diverse mathematicians. Some of my favorites are Francis Su’s article, “Mathematics for Human Flourishing” and Piper Harron’s blog, The Liberated Mathematician. We may never have all the answers but we can become more aware that our roles as mathematicians extend beyond our discipline. Sometimes this looks like being mindful of our biases or empathizing with the experiences that are not similar to our own. It could mean being a voice and it could mean passing the mike to let other voices be heard.**Be honest, listen, and take care.**Embrace and share your stories as part of what makes you a mathematician. The challenges and triumphs. This is difficult if you find yourself in an environment that doesn’t embrace the identities you bring to your mathematics. Having open and honest conversations require listening to others with no judgment and accepting that their experiences have a place in our community. Some will be uplifting and others not so much. But it is important to create a space to share both. These conversations may be challenging so always seek to take care of yourself as well.

Mathematics is a beautiful field that blossoms with our own unique perspectives and experiences. Let’s work towards opening those conversations, let’s challenge our assumptions and foster the growth of a more diverse mathematical community … together.

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The idea that we can inculcate students into the practices of a discipline like mathematics relies heavily on theories that were developed in the early 1990s. In their 1991 book Situated Learning: Legitimate Peripheral Participation, Jean Lave and Eitenne Wenger propose an idea called legitimate peripheral participation. They suggest that learning is the process by which one comes to be a part of a community of practice; learning is a process of coming to negotiate the social meanings, moving one not towards central participation (since there is no definitive center) but rather to full participation. Legitimate peripheral participation, Lave and Wenger propose, is a theory of how people learn rather than a theory of how to teach. Schools teach many things other than just subject matter, and students participate in many different communities of practice, not just that of the mathematics classroom.

There is something qualitatively different about the mathematics in the mathematics classroom and the mathematics that professional mathematicians do. A professional mathematician works primarily to generate new knowledge rather than to merely learn past knowledge. Although students taught with discovery learning methods have occasionally discovered new theorems or invented new mathematical ideas, that is not the primary purpose of mathematics education, particularly at the K-16 level. A professional mathematician does not work in groups of three to four for fixed periods of time like the way a student might work in a Complex Instruction- or groupwork-based classroom; they have to learn how to structure their own time and to attend professional colloquia and conferences in order to present their ideas. Professional mathematicians also use very different tools than students; they would be more likely to be seen reading a professional journal than a textbook, and use more advanced specialized software such as GNU Octave or Sage instead of (say) a handheld graphing calculator.

The idea of legitimate peripheral participation, therefore, is tricky when you try to apply it to the K-16 classroom. We are not necessarily enculturating students into becoming mathematicians; our students come to us with many different goals. The classroom as a community of practice is a very different space in which full participation means to take part in an active role in classroom discourse, to make and defend ideas, and to begin to develop one’s own ideas while also mastering the fundamental concepts of mathematics. The norms we establish and the experiences that we give students in the classroom help to make them full members of the mathematics student community of practice. By instituting the Standards for Mathematical Practice in our classroom, we provide the foundation for students to begin as legitimate peripheral participants in their future studies and careers.

]]>These are the words I have seen five out of the six times I’ve opened an envelope after pouring my soul into studying for a prelim exam. That’s right – my prelim pass rate is 0.1667. That’s not even good in baseball, where the standard batting average is somewhere around a 0.300. Were I a baseball player, the coach would have benched me a long time ago (probably after my first three prelims, on which I went “oh for three”, as they say). The fact of the matter is that I’ve benched myself several times. Too many times. But I learned a lot sitting on that bench.

My first year of graduate school was the first time that I truly grappled with math in a way that made me doubt my abilities, and failing both prelims I attempted at the end of that year only served to cement the feeling that I wasn’t cut out for grad school. I cried – a lot. I assumed my professors and colleagues saw me as a failure. But I pressed on through the next semester, and when the next round of prelims rolled around I got off the bench, grabbed my bat, and went to the plate again.

But I struck out again. And I didn’t understand. Because I had given up most of my Christmas break to study for this test. I had employed better strategies, I was more focused, and I had grown. But I saw that same four letter word in the envelope again: fail. And the worries that everyone at school viewed me as a failure came back again. I went home and I cried. Again.

Leading up to that second round of prelims, school had started once more, and suddenly I was a second-year, which meant that the new first-years started looking to me for advice. They started asking me questions as if they thought I knew something. Little did they know, I was still only marginally less confused than they were. And the dreaded question came: “which prelims have you passed?”

“Well… um… you see… about that…” What was I going to say? Would they take me seriously if I told them that I hadn’t passed any? Would they ever want my advice or help again? I decided to be honest, and at first it was hard. But gradually it became easier, and I wasn’t only honest about it, I was open about it. Slowly but surely I embraced the fact that I had failed and I got comfortable talking about it.

And, oh, how much freedom I felt. You see, when we hide our failures, when we keep them locked away from curious coworkers and friends and family, we must stay vigilant. We must put up a front so that the world thinks we are perfect and have it all together. We have to fight every day to maintain an unrealistic image. It’s exhausting and it only perpetuates the problem, as others begin to think that failure has its sights set on them and them alone.

But when we unlock the door and let even a small light in the room, it banishes the darkness. Sure, it hurts at first. But our eyes adjust and suddenly everything is clear. And when we share that with others, it gives them permission to fail. It gives them permission to let that guard down and be vulnerable and, dare I say it, human.

Now, failure isn’t as scary. Because I’ve seen it, I’ve felt it, I’ve experienced it, and I’ve come out on the other side. This doesn’t mean the fear doesn’t creep back in. It does. Frequently. In fact, I’m writing this as I decide which prelim to attempt this coming January, and I feel a keen sense of trepidation.

But we give power to failure when we don’t talk about it. We give it power when we hide it. So I write this to leverage the power for my good and the good of those around me. To remind myself why I don’t have to be afraid. To remind YOU why you don’t have to be afraid. Because you’re not alone in your failure, friend. And neither am I.

]]>Considering the intersection $\cap_x F_x$ we see that many of the sets could be skipped without altering the intersection.

Question: Is it possible to attain the same intersection by taking only countably-many of the subsets?

**Theorem:** There is a chain of subsets of the unit interval whose intersection does not equal the intersection of any countably-many of them. They may be chosen measurable.

In order to give a proof we first show a simple lemma.

**Lemma**: If $\{A_j\}_{j=1}^\infty $ is a chain then there exists a decreasing sequence out of its members

$$ G_1 \supset G_2 \supset \cdots $$

such that $\cap_j A_j = \cap_i G_i $ .

Proof: Take $G_1=A_1\ $, and let $j(1)=1\ $. We will move along the sequence $A_i$ and pick those that are needed in the intersection, which means those that are smaller. The details are as follows:

Let $j(2)$ be the first index bigger than $j(1)$ such that $A_{j(2)} \subsetneq G_{j(1)}$. If no such index exists then all the subsequent sets contain $A_1$, and so we can take the constant sequence $G_i=A_{j(1)}$ and it will satisfy the assertions in the claim.

Inductively, assuming that $G_1 \supset G_2 \supset \cdots \supset G_k$, and $j(1) < j(2) < \cdots < j(k)$ have been defined, we define $j(k+1)$ to be the least index after $j(k)$ such that $A_{j(k+1)} \subsetneq G_k=A_{j(k)}$, and $G_{k+1}=A_{j(k+1)}$. Again, if such an index does not exist then we could have the constant sequence $G_k$ satisfying the assertions of the claim.

If an infinite sequence $G_1 \supset G_2 \supset \cdots \ $ emerges eventually, then it is the sequence claimed, because each $A_i$ was given a chance at some point! (If, say, $A_{2017}$ is none of the $G_j$’s, then it means that it contained one of them.)

**Proof of the theorem:** Take $\mathfrak{C}$ to be the collection of all Lebesgue-measurable subsets of the unit interval with measure equal to 1 and order it with the inclusion relation “$\subset $”.

We will reach a contradiction by assuming that the assertion of the theorem were false.

Take a chain $\mathfrak{F}=\{F_x\}_{x \in \Gamma}$ in $\mathfrak{C}$. There would be a countable subcollection $\{A_j\}_{j=1}^\infty $ giving the same intersection.

By the lemma, we can further restrict to a decreasing sequence $ G_1 \supset G_2 \supset \cdots ; G_j \in \{F_x\}_x$ and still have $ \cap_i G_i = \cap_j A_j = \cap_x F_x\ $.

It is a fact of measure theory that

$$\mu (\cap_i G_i ) = \lim_{j \rightarrow \infty} \mu (G_j) = \lim_{j \rightarrow \infty} 1 =1 \ .$$

Observe that $\cap_i G_i$ is in $\mathfrak{C}$, and thus a lower bound to the chain $\mathfrak{F}$.

What we have shown is that every chain in $\mathfrak{C}$ has a lower bound. By Zorn’s lemma, there exists a smallest element in $\mathfrak{C}$. However, for any element is $\mathfrak{C}$ removing a single point gives an even smaller set in $\mathfrak{C}$.

This contradiction shows that for some chains in $\mathfrak{C}$ the intersection cannot be replaced by a countable intersection.

Note: We can work with smaller $\mathfrak{C}$ such as the collection $\{[0,1]\backslash D \ \ | \ \ D \text{ is finite}\}$.

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**A great statement of purpose will make your application.** And while a not-so-great statement of purpose might not break your application, it would be a lost opportunity: the statement of purpose is your chance to convince the admissions committee that you are a good fit for the graduate program they oversee.

The trouble? Writing a convincing statement of purpose is tricky, and it comes naturally to almost no one. When I first tried to write mine, I spent a great deal of time staring at a blank page – writing a few words, only to delete them immediately.

The opening “when I was ten years old…” felt cliché. Plus, my interest in math was less a revelation than it was a snowball effect, made more difficult to ignore with each new, tantalizing piece of information I absorbed (the Fourier transform can decompose sound waves into their constituent parts?!). So pretending that my fifth-grade teacher or a childhood science fair was the singular impetus for my impending commitment to a lifelong career in mathematics seemed dishonest. On the other hand, starting the statement with some variation on “I am excited to apply for the PhD program in math at University X” felt too generic.

In the end, I decided to begin my statement of purpose with, well, a statement of purpose. Without preamble, I laid out my professional goals (they were specific, and somewhat unique) and explained how they had come to be. This opening allowed me to segue into my reasons for applying to each program, and from there, into my research background – two integral components to any statement of purpose.

I say this not to argue that this is the best or only way to structure a statement of purpose – it’s not – but to emphasize the following point: a great essay is always genuine, thoughtful, and specific. Providing unique details about your motivations (think: a story about a memorable encounter with mathematics, rather than a generic “I enjoy problem-solving”) will make for an honest, compelling essay. For more specific advice on crafting a statement of purpose, read this.

If time allows, share your essay with the professors who are writing your recommendation letters – it will allow them to write letters that reflect your strengths as relevant to the programs you’re applying to. And don’t forget to have friends and/or professors edit your essay.

If the prospect of crafting a statement of purpose is overwhelming, remember: at the end of the day, your goal in a grad school application is to communicate that you are prepared, both academically and personally, to do research in the program you’re applying to. That’s it. If you successfully communicate why you’re prepared for a research career in your statement of purpose, you’re well on your way to making a convincing argument for why you should be admitted. And once you are, remember:

**You are your own best advocate. **You may feel lucky to get into grad school when it happens – and you should! – but remember, too, that whichever graduate program you choose is lucky to have you. Advocate for yourself accordingly, and stick to your boundaries when it comes to work environment, hours, pay, health care, teaching load, and the like. While grad school requires a certain amount of sacrifice and compromise, on the whole, it should support, rather than hinder, your personal life – just like any other job.

Here’s a scenario I hear all the time: “My partner and I applied for all (or many) of the same grad schools, but we were accepted to different ones (on different sides of the country).” Sometimes, I’ll ask whether they communicated this fact to the relevant universities, and more often than not, I get a look of confusion in reply. If you’re accepted to a particular program and your partner isn’t, you can write a polite note to the department informing them of the situation – delicately, of course. Yes, you can – and should!

Graduate school is a significant, long-term commitment that people undertake as fully-fledged adults, often with partners, dependents, and major life considerations (starting a family, caring for parents) in tow. In this respect, it is a far cry from undergrad, and should be approached accordingly. When making the decision about which programs to attend, remember that finding a supportive program and advisor, and communicating your personal and professional goals to them as appropriate, is key. Because ultimately, grad school should be a means to pursuing a fulfilling life and career.

For advice on how to survive – and thrive! – in grad school once you’re in, read my previous post here.

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You and nine other friends have been trapped by an evil hat-maker (who is a recurring character in these sorts of riddles). As part of his evil plan, the hat-maker has assigned each one of you a distinct hat color. These ten colors and their assignments are public knowledge, in the sense that they are known to both you and all of your friends. In order to test your affinity for your assigned color, the hat-maker has hidden ten hats of the prescribed colors in ten different boxes (one hat per box). The boxes are also colored with the ten different colors, but the hat contained inside a box may or may not correspond to the color of the box. Your group is now offered the chance to recover their hats, by participating in the following game.

One by one, each of you will be allowed to look inside up to five of the ten boxes. If you successfully find the box containing your hat, then the hat-maker will make a note of this and move on to the next person. (The hat itself is not yet removed from the box in this situation.) If all ten of you succeed, then (as a group) you win the game and are rewarded with the hats! However, if even one person fails to find their hat, then all of you are sentenced to bareheadedness. You are allowed to strategize with your friends before beginning the game, but no communication is allowed once the game starts. (Thus you cannot communicate what you find in your five boxes to your friends, and you can’t be sure which boxes your friends have looked inside unless you agree on this beforehand.) Note, however, that the boxes are inspected one after the other, so that an individual may decide which boxes to inspect based on the results of boxes that he or she has already inspected.

The simplest strategy is of course to simply have each member of the group inspect five random boxes. This obviously results in a $1/2^{10}$ chance of the group winning. Can you come up with a better strategy to stave off bareheadedness? What if instead of ten people (and hat colors), there are twenty people (and hat colors)? Can you come up with a strategy that gives a non-negligible chance of winning, even as the number of people grows to infinity?

This month’s puzzle was communicated by H. Dai.

]]>“The rapid advance of computers has helped dramatize this point, because computers and people are very different. For instance, when Appel and Haken completed a proof of the 4-color map theorem using a massive automatic computation, it evoked much controversy. I interpret the controversy as having little to do with doubt people had as to the veracity of the theorem or the correctness of the proof. Rather, it reflected a continuing desire for human understanding of a proof, in addition to knowledge that the theorem is true” – Bill Thurston (from [6])

A machine-checked proof is a proof written in a piece of software called a ‘proof assistant’ which ensures the proof complies with the ‘axioms of mathematics’ and the rules of logic. The question of the significance of computers in proving theorems can be polarizing (and for good reason). The quotations above represent some points of view relevant to this topic. In this post we will try answer three questions:

- What exactly is a computer-assisted proof?
- What are the advantages and the drawbacks of using computers to prove theorems?
- What should an interested person do to start learning to use proof assistants?

**The motivating problem**

Computer-assisted proof is a technique. Mathematicians care about new techniques when they solve some problem insoluble by old techniques. This is our motivating problem:

Above we see two solved problems. The first is a solved Rubik’s cube, while the second is Andrew Wiles’ proof of Fermat’s Last Theorem. A big difference between proving facts about Diophantine equations and solving Rubik’s cubes is that when a person solves a Rubik’s cube they know immediately it is solved, whereas Wiles’ proof (for example) took many months to properly referee.

As we progress in our mathematical education, our ability to check answers decreases. For example, my first lesson in mathematics was my mother teaching me to count. I was taught the numbers from one to ten *along with the fact that for small sums I could check my answer by counting with my fingers. *When one is taught to solve algebraic equations, one is also taught that answers can be checked by substitution. Checking Calculus is trickier, yet we can rely with good confidence on Wolfram Alpha. Upon reaching real analysis and abstract algebra, students check their work ultimately by handing it in and seeing if the professor buys their argument.

**What is a proof assistant?**

A computer proof assistant allows for more systematic checking of mathematical arguments. A user writes their proof up in a *semi-formal language* (meaning not as formal as formal logic and not as informal as ordinary mathematics). The proof assistant checks the proof against some foundation of mathematics. Normally when we think of foundations, we think of set theory. Yet for technical reasons proof assistants are implemented with ‘type theoretic’ foundations. Type theory is another foundation of mathematics which was actually proposed by Russell and Whitehead around the same time that others proposed ZFC set theory. Although there are philosophical implications of varying foundations across pieces of mathematics, for us, this is a moot point and we will say no more on the matter.

Below is a picture of a correct proof and an incorrect proof in the proof assistant Lean.

The proof on top is clearly wrong because of the error message showing up in red. The proof on the bottom is quickly seen to be correct since no error occurs. Just like with a Rubik’s cube, it is immediately clear if a proof is correct or not.

This seems a bit complicated. Is it ever necessary? It can be. For example, Hales’ proof of the Kepler conjecture was too complicated to be checked by journal referees. That proof was eventually verified with a proof assistant called HOL-light. The project to formally verify Kepler’s conjecture was called the Flyspeck project. Flyspeck took several years to complete and 5000 processor hours on the Microsoft Azure Cloud [5]. Some people hoped for a less computer-heavy proof so that mathematicians could read the proof.

Georges Gonthier along with his colleagues at Microsoft Research produced the first formally verified proof of the four color theorem [2]. This is different than the original computer-based proof of the four color theorem, which was essentially a standard mathematical argument that involved an absolutely massive computer calculation. Gonthier’s work certified that the algorithm purported to proof the four color theorem actually did what we believed it did.

Gonthier’s team also formally proved the Feit-Thompson odd order theorem, a cornerstone of the classification of finite simple groups, using the proof assistant Coq [3]. The original Feit-Thompson paper was 255 pages. Other high-profile projects include formal proofs of the prime number theorem, Gödel’s incompleteness theorems, and the central limit theorem.

**How to get started?**

These tools are not used exclusively for massive proofs that take years. There exist formal libraries containing theorems and definitions in real analysis, general topology, representation theory, and abstract algebra. Proof assistants are also used in industry to verify software and algorithms. This is quite powerful. As soon as one can state “this program P has no bugs” in a mathematically rigorous way, one can try to prove it. And with a formal proof software developers can be sure their programs are error-free.

A number of proof assistants are available. They are all free.

Isabelle-HOL is a proof assistant created by Lawrence Paulson. It is based on higher-order logic. Isabelle has massive libraries already in place as well as some of the most powerful automation available. Here, automation means the proof assistant can find short proofs for you. HOL-Light is a similar program, with a smaller kernel, written by John Harrison.

Coq and Lean are all based on dependent type theory. They were developed by teams led, respectively, by Thierry Coquand and Leo de Moura. That means data types can depend on other data types–we want that. For example, we’d like a type Fin$(k)$ whose inhabitants are finite sets of cardinality $k$, for arbitrary $k$. But these proof assistants, despite being newer, have weaker automation because it is, for technical reasons, harder to implement automation in dependent type theories (at least at the moment).

Online manuals exist for all the proof assistants mentioned above. Lean 2 has an interactive web tutorial. The current version of Lean is Lean 3, but the author found this tutorial a good way to get a flavor for proof assistants in general.

**Problems and outlook**

Proof assistants are not yet at the point where they can reasonably be used by working mathematicians. Hales did use HOL-light for his work on the Kepler problem, but this project was not the sort of thing mathematicians would do unless they absolutely had to. The current libraries are not large enough to include everyday arguments about everyday theorems. For example, we may go to prove a standard result about compact Lie groups only to discover the Haar theorem (proving the existence of Haar measures) is not in our library. This theorem is quoted all the time but its proof is lengthy and with present technology we should expect a formal proof of the Haar theorem to take a few years.

A more fundamental objection says that mathematics is, to use Thurston’s language, ultimately about the *human understanding* of mathematical objects and that proofs are there only secondarily to prevent our understanding from wandering off. In the words of the great combinatoricist G.-C. Rota, “saying a mathematician ‘proves theorems’ is like saying an author ‘writes words’.” It would follow that algorithmic ‘proof search’ type arguments are undesirable.

Yet there is no reason why one cannot both understand mathematical objects and use computer proof assistants. Machine checking is not synonymous with proof search. Presently, we have human journal referees check detailed technical arguments. If it were possible to use a computer to check these, we would lose nothing and gain more accountability.

Steven Wolfram has recently gotten interested in formal proof. People from Wolfram have worked on getting Mathematica and formal proof to coalesce [1]. The relevant linguists, computer scientists, and mathematicians are constantly considering ways to get the computer code to look more like ordinary mathematics. An ultimate goal is streamlining the refereeing process.

All journals require us to write papers in LaTeX–this was not always so. Perhaps in the future, some journals will require proofs written in a quasi-formal language for computer checking. Then, journal referees could focus more energy on the clarity and the overall presentation of mathematical articles–thus *furthering human understanding of mathematics.*

**Acknowledgements. **I am most indebted to Jeremy Avigad at Carnegie Mellon University and Tom Hales at the University of Pittsburgh for teaching me what I know about proof assistants. I thank also Emily Riehl at Johns Hopkins University for introducing me to Bill Thurston’s brilliant article “On Proof and Progress in Mathematics”, which I referenced several times in this post. And lastly, I dedicate this note to the late Vladimir Voevodsky. I never met him personally but his influence on my many teachers, his papers, and his recorded lectures were highlights of my undergraduate education.

**References**

[1] Geuvers, H., England, M., Hasan, O., Rabe, F., Teschke, O. Intelligent Computer Mathematics, 10th International Conference, CICM 2017, Edinburgh, UK, July 17-21, 2017, Proceedings.

[2] Gonthier, G.: Formal proof—the Four Color Theorem. Notices of the AMS 55(11), 1382–1393 (2008)

[3] Georges Gonthier, Andrea Asperti, Jeremy Avigad, Yves Bertot, Cyril Cohen, Fran¸cois Garillot, Stéphane Le Roux, Assia Mahboubi, Russell O’Connor, Sidi Ould Biha, Ioana Pasca, Laurence Rideau, Alexey Solovyev, Enrico Tassi, and Laurent Théry, A machine-checked proof of the odd order theorem, Interactive Theorem Proving – 4th International Conference, ITP 2013, Rennes, France, July 22-26, 2013. Proceedings (Sandrine Blazy, Christine Paulin-Mohring, and David Pichardie, eds.), Lecture Notes in Computer Science, vol. 7998, Springer, 2013, pp. 163–179.

[4] Hales, T., & Hales, T. C. (2012). *Dense sphere packings: a blueprint for formal proofs* (Vol. 400). Cambridge University Press.

[5] Hales, T.C. Introduction to the Flyspeck project. In Thierry Coquand, Henri Lombardi, and Marie-Franc¸oise Roy, editors, Mathematics, Algorithms, Proofs, number 05021 in Dagstuhl Seminar Proceedings, Dagstuhl, Germany, 2006. Internationales Begegnungs- und Forschungszentrum f¨ur Informatik (IBFI), Schloss Dagstuhl, Germany. http://drops.dagstuhl.de/opus/volltexte/2006/432.

[6] Thurston, W.: 1994, ‘On Proof and Progress in Mathematics’, *Bulletin of the American Mathematical Society* **30**(2), 161–177.

The spotlight this month discusses an article entitled, “Gerrymandering, Sandwiches and Topology” written by Pablo Soberón. This article is particularly interesting as just last week the Supreme Court heard a case regarding gerrymandering in Wisconsin. Recall that gerrymandering refers the process of drawing congressional districts in such a way that the “dominant” political party will win in a majority of the districts. There are many mathematical articles written about gerrymandering that discuss different ways in which mathematical formulas can be used to create a “fair” political map. The article in the Notices has a different take and shows that there are also mathematical theorems that can be used to create an “unfair” map that fits all criteria that are in place to help avoid gerrymandering. The article discusses three well known algebraic topology results including the so-called ham sandwich theorem. The author presents proofs of these results in a very accessible manner so even a graduate student with little background in algebraic topology can understand. In addition, the author shows how, when using these three theorems, we can still create a division of a state that the dominant political party will win. We encourage you to take a moment out of the busy time of the semester to read this article, or if not this one find an article that strikes your fancy. Until next time, enjoy your semester and tune in in a couple of weeks for the next AMS Notices Spotlight.

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