Last summer, I had the amazing opportunity to participate in the Mathematical Association of America (MAA) MathFest. Each summer at MathFest one of the largest communities of mathematicians, students, and enthusiasts comes together to talk about the latest advances in mathematical research and education. One of my most memorable experiences was participating in the session * Great Talks for a General Audience: Coached Presentations by Graduate Students. *Here graduate students get coached by more experienced communicators on how to make their research accessible to a general audience, give a 20 minute presentation, and receive individual feedback by an undergraduate student and faculty member. Here I want to highlight some of the insight I gained through this experience.

**Ask yourself, who is your audience?** Are they kids, young adults, non-mathematicians, scientist, non-scientist, a combination? When addressing an audience you may not know its exact composition. I’ve always thought the best thing to do what to teach my audience my work. Make it accessible by treating it as a student-teacher interaction. However, in the limited time available may be impossible to achieve! Instead, **advertise your work. ** Talk about what drew you to your problem, what are the big ideas, why do you find it interesting?

**Less is more.** For anyone who has met me, they know when I am excited about an idea I tend to ramble. I used to think my presentations needed to stand on their own. Every detail I thought was important should be precisely stated in writing. Truth be told, audiences have a limited attention span and a presentation is a tool to highlight ideas. Your use of slides or board space is what brings your talk to life! Make it your story.

**Avoid jargon. **When you spend a lot of time talking to specialist in your field, it is hard to imagine changing how you talk about your work. However, using layman language is necessary to make your ideas engaging outside of your field. You can use a picture to build ideas. Be generous with using more simple examples that motivate your result. I am sure at many stages of your academic moment you’ve experience that ‘A-ha!’ moment. Give that gift you your audience.

**If you’ve found yourself eager to learn more about communicating mathematics, I encourage you to apply to MathFest (deadline April 30) this summer! Below are additional resources:**

- AMS Communicating Mathematics in the Media: A Guide
- MAA Audience Awareness
- MAA Mathematical Communication
- Steven Strogatz, “Writing about Math for the Perplexed and the Traumatized”
- Follow other math communicators on social media, some examples on Twitter include: Tai-Danae Bradley (@mathma), Dr. Eugenia Cheng (@DrEugeniaCheng), Steven Strogatz (@stevenstrogatz) and many more!

In this chapter, Rands, McDonald, and Clapp take up Casid’s ideas and apply them to classrooms. They argue that classrooms are arranged so that “students will enter the classroom on time, take their seats, remain stationary throughout the class so the knowledge imagined to reside in the instructor can follow the sight lines from teacher to student in a timely fashion into the minds of the students, and finally leave class” (p. 153). In my math classroom, the chairs default to facing the smartboard, with an assumption that there is one teacher who stands at the front of the classroom, lecturing, and that only that one person writes on the board. In fact, the smartboard sometimes gets confused if two people are holding a pen at a time, so it is challenging for two people to be working problems on the board at the same time. I have to intentionally ask students to move into groups; although they will sit near their group, they do not sit in groups by default.

Likewise, class occurs within a fixed range of time; if students arrive early for class, they have to remain in the hall; if they stay late in class, another class of students will enter. Students from two separate classes are not supposed to interact; even if students enter for the next class, they do not ask the previous class what was covered. The digital smartboard automatically erases at the end of class, meaning that the possibility that the next class will get to see and discuss what was on the previous class’s whiteboard is foreclosed. There is a clearly designated time for mathematics and aside from homework, the rest of the hours of the day are not for mathematics. Once class is dismissed, although students sometimes come up to ask me questions (rarely about mathematics, usually more about their grades or their worries about the exams), they do not continue to discuss mathematics with each other; the time for discussing mathematics is over, delimited into this narrow, confined space.

Rands et al. proposes that instructors should challenge norms in the classroom around time, proposing that classes could meet at unusual times, such as 6am or 11pm, on different times each day or for different lengths of time each class. Classes could also meet in different spaces; they propose that teachers “go on a field trip, conduct class at a coffee shop or a park, or hold class outside” (p.164). I have asked why we do not have evening or early morning classes, and the response is that they are not as popular anymore now that online classes are available and that classes are only to be scheduled at times that fit student preferences.

I am thinking, too, about ways in which I could change the way time is organized within in my classroom. One thing I have been doing is to forego the traditional break in favor of a more open classroom, where students can get up and attend to personal matters as they arise, particularly when students are working in groups. Another strategy I am trying is being available by email on evenings and weekends when students are actually working on homework, rather than only checking email during office hours.

If we want to transform our teaching, thinking about how we (together with our students) use time and space both inside and outside of the classroom is essential. In future posts, I will discuss additional articles that I have consulted about time and space in the context of education.

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$$ \forall x, \ \ \forall y \ \ \ d(f(x),f(y)) \leq L d(x,y) \ .$$

The definition makes sense as long as a distance is defined on the spaces. This makes Lipschitz maps highly versatile. Almost every space you deal with daily is either a priori a metric space (a set plus a distance function) or can be made one by endowing it with a distance. (Check for instance the word metric on a group that builds metric spaces out of groups, opening the doors to the beautiful topic of Geometric Group Theory.)

We will, nonetheless, limit ourselves to studying (some of) the properties of Lipschitz maps between Euclidean spaces. Properties?! Look at the definition again, what else could we expect of a Lipschitz map? What else could that condition possibly impose on $f$?

One of the most surprising facts about Lipschitz maps is the following:

**Radamacher’s Theorem:** If $f: \mathbb{R}^n \longrightarrow \mathbb{R}^m$ is Lipschitz, then it is differentiable a.e. (i.e is differentiable everywhere on $\mathbb{R}^n$ except maybe on a set of Lebesgue measure zero.)

Not surprised? Let’s look at the simplest case where $f: \mathbb{R}^1 \longrightarrow \mathbb{R}^1$ and all we know about $f$ is this:

$$ \forall x, \ \ \forall y \ \ \ |f(y) – f(x)| \leq L |y-x| \ .$$

Let’s divide to get an equivalent statement

$$ \forall x, \ \ \forall y\ (\neq x) \ \ \ -L \leq \frac{f(y) – f(x)}{y-x} \leq L \ ,$$

which means that all difference quotients are bounded. Radamacher’s theorem asserts that this boundedness (alone) implies the existence, for a.e. $x$, of the limit

$$ \lim _{y \rightarrow x} \frac{f(y) – f(x)}{y-x} \ \ \cdot $$

Not at all obvious!

For a proof of Radamacher’s theorem see Geometric Analysis [1] notes of Piotr Hajlasz.

Another nice property of Lipschitz maps is their extendibility:

**Theorem (McShane): **Assume $A$ is a subset of a metric space $X$ — so, it is a metric space itself — and $f:A \longrightarrow Y$ is Lipschitz. Then there exists a map $F: X \longrightarrow Y$ such that $F(a)=f(a)$ for all $a \in A$, $F$ is Lipschitz with the Lipschitz constant (the $L$ in the definition) the same as that of $f$.

This property shows that the differentiability result above holds if $\mathbb{R}^n$ is replaced with an open $\Omega \subseteq \mathbb{R}^n$.

By Radamacher’s theorem, a Lipschitz map $f: \mathbb{R}^n \supseteq \Omega \longrightarrow \mathbb{R}^n$ has derivative a.e on $\Omega$. Thus, its Jacobian is defined a.e. which leads us to another theorem.

**Theorem (Change of Variables) : **Assume $ \Omega \subseteq \mathbb{R}^n$ is open, $\phi : \Omega \longrightarrow \mathbb{R}^n$ is Lipschitz, and 1-t0-1. Then for any measurable set $E \subseteq \Omega$ and any measurable function $u: \mathbb{R}^n \longrightarrow \mathbb{R}$

$$ \int_E (u \circ \phi)(x) \ |J_{\phi}(x)| \ dx= \int_{\phi (E)} u(y) \ dy \ , $$

provided that $u\geq 0$ or one of the integrands is integrable. Measurability of the integrands is part of the assertion of the theorem as is the fact that the integrability of one of the integrands implies that of the other.

There are many versions of this theorem and many different proofs in different generalities. A nice and complete proof (of a more general case) can be found in [2].

In the realm of integration, the **area formula** and **co-area formula** hold for Lipschitz maps. (Theorem 3.23 in [1])

Note how much easier it is to show that a map is Lipschitz than is to verify for instance its differentiability and (local) integrability of the derivative. Some form of integrability/boundedness is required of the derivative of a function to satisfy the change of variables formula. This observation tells us that Lipscitz maps are in fact “nicer” than differentiable maps. Their derivatives are smoother and tamer, in some sense. The following theorem is a testimony to this notion.

**Theorem (Federer): **Let $f: \mathbb{R}^n \supset \Omega \longrightarrow \mathbb{R}$ be Lipschitz. For any $\epsilon > 0$ there exists a $g \in C^1(\mathbb{R}^n)$ such that

$$ \mathcal{L}^n (\{ x \in \Omega: f(x) \neq g(x) \}) \ < \ \epsilon \ .$$

Observe that $C^1$ maps are locally Lipschitz by the Mean Value Theorem because on compact sets the derivative is bounded. So, the closest cousin to Lipschitz functions are $C^1$ functions. In fact, many proofs about Lipschitz maps can be reduced to verifications for $C^1$ cases.

Moving to the next property on our list, **the Luzin N property:** A Lipschitz map from $\mathbb{R}^n$ to $\mathbb{R}^n$ that takes measure zero sets to measure zero sets. This interestingly implies that any measurable set is mapped to a measurable set. Of course, for a differentiable map (not $C^1$) this does not have to hold — another reason why Lipschitz maps are more than being a.e. differentiable.

Since the definition of a Lipschitz map is through the distance function, of course Hausdorff measures must come into the picture somewhere:

**Theorem: **If $f:X \longrightarrow Y$ is Lipschitz between metric spaces with Lipschitz bound $L$, then for any $E \subseteq X$, and any $ s $,

$$ \mathcal{H}^s (f(E)) \leq L^s \mathcal{H}^s (E) \ ,$$

which in particular gives the Luzin N property above.

Finally, I finish with a fact that my advisor (Piotr Hajlasz) and I verified recently:

**Theorem: **Let $f:\mathbb{R}^m \longrightarrow \mathbb{R}^n$ be Lipschitz. If the derivative of $f$ at a (single) point has rank $k$, then $Df$ will have rank at least $k$ on a set of positive $m$-measure.

Again, note that the property is an elementary fact if the map is $C^1$.

The V-shaped $y=|x|$ is not that broken after all!

If you know of some other facts about Lipschitz maps that you find cool, please write in the comments!

References:

[1] Geometric Analysis (notes) Piotr Hajlasz, University of Pittsburgh.

[2] P. Hajlasz, Change of variables formula under minimal assumptions. *Colloq. Math. *64 (1993), 93–101.

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!

]]>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.

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