But more importantly, what are the consequences for mathematics? ; ) While some are predicting that all of the at-home time we have in store will lead to another baby boom, knowing this audience I am going to predict… A MATH BOOM! Let’s face it, there is perhaps no sector of the population more suited to social distancing than mathematicians. Avoidance of public gathering is basically our lifeblood. There’s a reason some of our most renowned and productive research stations are isolated deep within the Black Forest, or in the middle of the Canadian Rockies. Unlike Ariel, we *don’t* want to be where the people are, because people have a well-known habit of messing with my concentration. This lockdown, we will be getting busy at home reading the papers in that special pile on our desk that only seems to grow in an ordinary semester, finishing proofs of tricky technical lemmas, and polishing up our pre-prints. I would bet on an observable uptick in arXiv uploads resulting from this whole situation, at least for us, the tribe of the portable research lab, pencil and notebook.

*Disclaimer #1:* Contrary to popular stereotype, mathematics is very much a social and collaborative activity, and there is no reason to believe that isolation is a beneficial precondition for its pursuit.

On the other hand, many of us are busy gearing up to face the bugbear we’ve steadfastly avoided for years — online teaching. After participating in a few trainings about (insert commercial video-conferencing service), I have to say it’s honestly better than I thought it would be, and even the dinosaurs of the department seem to be taking to it pretty well. It seems at least the basic task of communicating course material to students will be achieved at a reasonable level, though I’m still a bit skeptical of the substitute systems for evaluation and feedback on coursework. I’m concerned that the extra distance and technological hurdles will prevent students from taking the opportunities to talk one-on-one with their instructors, and also that instructors may fall into patterns that reduce their levels of availability. Then again, there may be some students that find the modes of electronic communication less intimidating than, say, showing up to office hours. Plus, blah blah millennials blah blah digital natives blah, right?

*Disclaimer #2:* I know there are many people out there (workers of many stations that do not have the luxury of working remotely, and healthcare workers in particular) for whom this pandemic is no frivolous affair. I wish them health and safety, and I hope my levity does not offend them.

While educators are making the online transition for our courses, it’s a little perplexing that more conferences aren’t doing the same. There are likely others, but I only know of one conference which was planned to be in-person and is making the switch to virtual (and it’s run by hardworking graduate students, on top of it all). I think this is great for at least three reasons. First, as early-career mathematicians, the ceasing of all conference activity for a long period could be injurious to our employment prospects, if indeed there are any jobs left when all the coronavirus dust has settled. Second, this will be a good excuse to have social contact, feel more normal, and less alone. Talk of virtually reviving the graduate student colloquium in my department has come up on these grounds. Third, thinking more long-term, I think something really cool could come out of this. The conference organizers are giving speakers the option to record presentations to upload, and such presentations, if well prepared, could be a real asset to people’s research profiles.

Think of having on your webpage not just a vague one or two paragraph description of your research plus links to papers, but also a 20-minute accessible video presentation on your work. This could take forms much more creative than your typical recording of a chalk talk at a conference, and really open up the world of math research — both amongst ourselves and to outsiders. I realize this is not a radically new idea: the promise of The Internet 2.0 was basically that we were all supposed to become content creators. As is often the case, academicians lag behind culture; but the video conference is a great excuse to think hard about how to produce good expository media about one’s own research.

I imagine this not as a way to supplant conference talks or research papers, but to supplement. The culture of academic publishing is such that research papers read dry and obscure to those outside the subfield, and (as a senior researcher and journal editor recently told me) there are (some) good reasons for this. But very often this style is a tool for exclusion. I am sure I am not alone in having had the experience of a professor handing me a paper or telling me to go read one as a way to make me buzz off. They don’t expect you to come back, and often they are right.

Lucid, informal, and big-picture explaining is usually reserved for talks, but on the one side, we can’t always make it to everyone else’s talks, and on the other (as Andrés points out), it’s a lot of work to put together a well timed and executed talk, only to do it once. It would be nice to be able to capture all of the thought and effort that goes in for posterity. I mean, your own work is the math that you should care the most about presenting — there’s no reason that the online calculus and algebra cartels should have a monopoly on well-produced, expository math videos.

*Disclaimer #3:* I have no idea how to produce any such media at present, but if I figure out anything cool, I will be happy to share. There was one tip in Mahrud’s recent post, and probably Mohamed Omar knows a thing or two as well.

Much is being made about how this is online education’s big opportunity, but this might also be a kick in the pants to the channels of research dissemination. I expect it’ll be rough at first, but when this dry-run is over, we may have some hard questions to face about how we’ve been doing things. At my university, (at time of writing) we are currently on pause to regroup with classes resuming on-line next week, and the virtual conference is next weekend, so I can’t yet really make any judgements on the tenability of these digital alternatives. In the meantime, I would be interested to hear how it’s going for graduate students in other places. Is your institution handling this pandemic very differently? Did anyone’s semester/quarter just get postponed or cancelled entirely? Is anybody being forced to move? What about plans to take quals/graduate/find a job? Are international students’ visas or applications being affected? Is your advisor taking advantage of the situation to *really* ghost you now?

I’m sure it’s still too early to really understand what this will all mean, so in the interim I wish everyone peace and safety, and that we can all treat one another with compassion. I hope to see you at the next virtual conference.

*Disclaimer*: The opinions expressed on this blog are the views of the writer(s) and do not necessarily reflect the views and opinions of the American Mathematical Society.

*Comments Guidelines*: The AMS encourages your comments, and hopes you will join the discussions. We review comments before they are posted, and those that are offensive, abusive, off-topic or promoting a commercial product, person or website will not be posted. Expressing disagreement is fine, but mutual respect is required.

2020 has been a complicated year so far, and things are only going to get more complicated as the COVID-19 pandemic. I’ve been thinking a lot about teaching recently, (as I’m the instructor for a class of undergrad learning assistants) and today I’m here to share my thoughts on how our teaching methods should change as we transition to remote teaching platforms. Later this week, I’ll also share my experiences with the Zoom platform – both as a speaker, and as an audience member.

First of all, no matter what technologies you or your university plan to use, **remember that not all students will have access to the internet**, or even internet with sufficient quality to participate in videoconferencing technology. It’s therefore imperative that you make your lesson as accessible as possible through as many forms of media as you can. This might include:

- posting your lecture notes online
- making lecture slides (and posting them online)
- recording your lecture

As a quick note on slides, I learned recently that** Google Slides offers a closed captioning service that works amazingly well** (much better than YouTube’s, for sure). So even if you use Beamer to make slides (as I do), I recommend porting them over to Google Slides just for the closed captioning ability. Here’s a quick picture of how to enable it:

Also, check to see what services/platforms/support your university is offering (for example, UT Austin is integrating Zoom, and I also learned that Hangouts Meet is offering free premium services as well).

Secondly, know that it’s difficult to suddenly switch to online/remote learning (especially in the middle of the semester!), and that the quality of education will suffer. But we can try to make the best of it that we can. In particular, **my recommendation is to focus on engaging your students**, rather than on trying to keep up with the pace of content in a normal semester.

It can be difficult to promote student engagement during a video lecture, but one thing you could try using are **survey/polling tools** such as Slido, Kahoot, etc. (UT Austin has a service called Instapoll). These tools can be used to engage students and to check for understanding on a basic level. However, I would recommend that you don’t assign grades/points to these assessments as again, not all students have the internet access/equipment to participate.

Other ways to engage students include using mediums outside of lecture, such as forum/discussion posts, email, etc. These are all additional means for students to ask questions and engage with the material at their own pace. You should encourage your students to use whichever means they prefer, and when possible, tailor your lectures to address the topics/questions that they bring up!

Teaching in the time of the coronavirus will take more time and energy than we might be used to, but in times like these, students will remember and appreciate the compassion and effort we put in!

*Disclaimer*: The opinions expressed on this blog are the views of the writer(s) and do not necessarily reflect the views and opinions of the American Mathematical Society.

*Comments Guidelines*: The AMS encourages your comments, and hopes you will join the discussions. We review comments before they are posted, and those that are offensive, abusive, off-topic or promoting a commercial product, person or website will not be posted. Expressing disagreement is fine, but mutual respect is required.

Was that a bad way to start off an article concerning the late Katherine Johnson, the NASA legend whose persistence, precision, and proclivity for mathematics sent America to space in the 1960s?

Maybe it was. Many a person still asks me if I’ve seen the films and television series that try to capture the work of mathematicians: *Good Will Hunting, A Beautiful Mind, Hidden Figures, Interstellar, Numb3rs*. . . . The answer is usually no. I am maybe a little behind the times when it comes to math movies, but I am far less concerned with that, at the moment, than I am with my lack of knowledge of one of the most influential mathematicians in history. Before sitting down to write this post, I knew the following about Katherine Johnson: she was a woman, she was African-American, she worked at NASA, and she was virtually unrecognized until receiving the Presidential Medal of Freedom in 2015 and her story was told in the 2016 movie *Hidden Figures*. Oh, and of course the all-important fact that without her, sending American men to orbit the Earth and walk on the moon may have taken years longer than it actually did. Can I tell you about Hilbert? Yes. Can I tell you about Rudin, about Lebesgue, about Riemann? Of course. But could I tell you about this great woman who challenged the beliefs and the norm of her time? Until today, I am ashamed to admit that the answer to that question would be no.

I was raised with a conservative background, and surrounded by families who believed a woman’s primary purpose was to take care of a home. For years, I heard my parents praise the intelligence of my older brothers, especially the oldest in our family, who was overtly encouraged to take math courses none of the rest of us were, because he was apparently of mathematical mind. On the other hand, even though I usually did fairly well in my elementary math classes in high school, I was never told that I should consider the pursuit of a scientific career. When I mentioned one night at the dinner table that I wanted to major in math, I remember my father quizzically — if not astonishingly — asking me, “*You* like math?” Then came an obstacle I will never forget: two years of pouring as much undergraduate-level mathematics down my throat that I could to make it into a graduate program by fall 2018. In those two years, anxiety induced by my male classmates and, yes, sometimes my male professors, nearly crippled me (as a disclaimer, my male professors also encouraged me greatly: one of them became my mentor, and often went out of his way to encourage the women in the math program at my university). I thought that I was stupid and would never be as good as they were at mathematics. I can look back on that time now and see that I was clearly intelligent, possibly as intelligent as anyone in my school, perhaps just in a different way — a questioning, a persistent, a precise way.

And this is partly why my heart is drawn to the life of Katherine: she asked questions. She was precise and careful even when it hurt, even when the men around her weren’t used to being questioned by any woman, especially not by a black woman. She maintained her persistence through an era of cruel segregation, when she could only work in the same room as other black women, and still said that the level of segregation felt at NASA (then known as NACA) was not as bad as the level of segregation in the rest of her 1960s world. In spite of the lack of empathy of her surroundings, she was still brave. Still dedicated to the inexorable precision required by mathematics. Still humble. Still cognizant of the truth. Still aware of equality.

Forty years after her calculations allowed Neil Armstrong to walk on the moon and subsequently make it safely back to earth, I am thankful for Katherine’s legacy of leadership. Thankful for the role model she is for me. Thankful for the evidence that no matter a woman’s background, she can rise above the lies she may once have been told, accept herself and her intelligence, and make an impact. And lastly, thankful for the truth that Katherine stood for when she said, “I was no better than anyone else, but no one was any better than me.”

*Disclaimer*: The opinions expressed on this blog are the views of the writer(s) and do not necessarily reflect the views and opinions of the American Mathematical Society.

*Comments Guidelines*: The AMS encourages your comments, and hopes you will join the discussions. We review comments before they are posted, and those that are offensive, abusive, off-topic or promoting a commercial product, person or website will not be posted. Expressing disagreement is fine, but mutual respect is required.

I recently came back from a research visit to the Freie Universität in Berlin where I was visiting Professor Matthias Beck and the Discrete Geometry group there. I was invited to give a talk in their seminar and was also invited to give a talk in the Discrete and Convex Geometry Seminar at the Technische Universität. I want to take the time to reflect on giving talks and to think about things related to preparing and presenting talks. Note that these are just my thoughts and reflections; the things mentioned here may not apply to everyone or may not be the best approaches for different kinds of talks.

I have been in the audience for many talks in my time as a student. I have experienced some wonderful talks, a few more okay talks, and unfortunately a larger number of not-so-good talks. I believe that part of doing mathematics is also communicating it effectively (whatever that may mean). I was fortunate that during my time at the Freie Universität, Prof. Beck, graduate students, and post-docs started a “soft skills” seminar, and one of the seminar topics was on how to give a good talk. In the seminar we shared things that stood out to us at talks that we have sat through in the past. Some of these things were good and constructive, but many found “bad talks” more memorable. I’d venture to say that giving talks is quite challenging and is always an area where we can improve in.

Compiling some feedback I received about my talk and modifications/explanations from advice given in ([1], [2], [3], [4]), here is a list and some thoughts on things to do for a talk:

__Only give talks if you want to (or “have” to)__

- If you don’t have the desire to speak then you might have a negative attitude towards giving a talk and it will most likely lead to one of these “bad talks”
- If you “have to” present a talk (e.g., master’s defense, PhD defense, course requirement, etc.), know that life continues beyond these things. Believe in yourself, you got this.

__PLAN, PLAN, PLAN, PREPARE, PREPARE, PREPARE__

- I think most of the times I have felt that my talks have not gone well is partly due to my lack of preparation.
- You might feel better going into a talk if you feel prepared.
- When preparing your talks keep in mind that they may look different if you are giving a seminar talk, colloquium, expository talk, research-focused talk, board talk, slide talk, etc.
- Give a talk more than once, it will get better and better.
- Some people understand their own research better after preparing/giving a talk; this and potential feedback may help in writing up results.

__Know who your audience is and where you are talking__

- In preparing your talk, it is important to know who you are speaking to in order to prepare a talk that is an appropriate level.
- Will you be giving a board talk or slide talk? Does the room have boards accessible even if you are giving a slide talk?

__Be mindful of your audience/Aim to be inclusive__

- This can be interpreted in many ways and can be adapted at different moments.
- An example that comes to mind is when using colored chalk or markers; I had an audience member at a talk who had difficulty seeing (or could not see) a certain color on the board, I quickly changed colors.
- If a microphone is provided, please always use it. You never know who may have difficulties hearing.
- Thank your audience and the organizers of the seminar/conference and people that invited you. It is always nice to hear speakers are happy to be speaking with the audience.
- Another thing is not assume the gender or other identities of your audience.

__Do not overestimate the knowledge of your audience__

- When I gave my seminar talk, one of the audience members thanked me afterwards for defining a certain geometric object. My research is in geometric combinatorics, hers was in algebraic topology; while she uses tools from discrete geometry, not all the objects were common knowledge in her field.

__Tell a Story: Context, Motivation, Applications, and the Future__

- Your project/research tells a story. It is useful to keep that in mind and prepare your presentation as a guide to tell a story.
- What is the context of your talks? What is the aim of the project?
- Share how you became interested in the topic.
- Share how your work has applications to other fields of mathematics, society, science, etc.
- Share open problems or conjectures. This might get people invested in your work and you may find new collaborators!
- Every story has an ending. Aim to have a nice finish. This may include some of things mentioned above.

__Examples and Intuitive Definitions__

- Use helpful examples to illustrate definitions, special cases, or even proof ideas.
- Avoid technical definitions, this will more than likely lead to confusion. This happened during one of my talks; the definition was long and could been expressed more straightforward with an example.
- Try to have a unifying or guiding example. This is may be helpful and more friendly to the audience.
- Repeatedly remind the audience of unfamiliar definitions.
- PICTURES ARE YOUR FRIENDS!
- Use colors to highlight or underline key points.
- Use computations if they are illustrative of a main point.
- Realize that people in your audience who are interested in details can look at your papers/preprints. Examples > Technical Proofs

__Board Work and Punctuation/Symbols Matter__

- Try to write as neat as possible. For me, I typically write in cursive and so I try to write in print to have it be legible for others. Also, I try really hard to write as slow/effectively so that my writing does not tilt as I write across the board.
- Punctuation and symbols matter. There is a popular example of where commas are important: “Let’s eat grandma” and “Let’s eat, grandma.”
- Something similar arose during one of my talks. I hyphenated a mathematical term and that led to confusion because it has different meanings when you hyphenate. Be careful.

__Questions and how to handle them__

- Anticipate questions you may be asked.
- Encourage questions, this keeps people engaged and you can gauge whether people are following your talk.
- If there is a time constraint, you can also ask that people save their questions to the end.
- You might have an audience that does not have any questions during the talk, that is okay. You may want to ask the audience questions in return.
- When asked a question, acknowledge the person asking the question by walking in their direction and repeat the question for the rest of the audience who might have missed it.
- You might have an audience that actively (or excessively) asks questions. This shows interest, so don’t be deterred. At times this can be a bit challenging when questions are asked during talks. Feel free to answer them or save them for after the talk. You are in charge.
- Find a way to comfortably say “I don’t know the answer to your question.” Prof. Beck likes something along the lines of, “this is an interesting question, and I’d be happy to explore it—I don’t think I know the answer”.

__Time__

- Feel free to ask an organizer to let you know or give you a sign when you have a certain amount of time left.
- Know where the clock in the room is or have a watch.
- Try not to exceed your allotted time. If you exceed your time it shows that you may have been unprepared. Also, people might have to be somewhere right after your talk. Let’s aim to respect people’s time.

__Give credit where credit is due__

- Thank your co-authors/collaborators.
- Give references to other people’s work.
- Cite theorems, authors, and the year of the result. Mention references if possible. This also helps others seek out the relevant works if they are interested and it helps people put your work into context
- Give yourself credit. Tell people what work is yours! You’ve done great research, why not tell people it’s yours?

__BE YOUR AUTHENTIC SELF__

- Only YOU really know what this truly means!
- Don’t belittle your own work/results or downplay your knowledge. You are an expert! (this is something I struggle with all the time)
- Personalize your presentation. For some people this looks like showing humor, pictures, quotes, anecdotes, etc.
- Wear clothes and shoes you are comfortable with.
- Show enthusiasm for your work, in your own special way.
- People will reciprocate your energy.
- You are a rock star!

Further Reading – Other Resources for Presentations: Handouts and Links

__References:__

[1] Devadoss, Satyan L. “Planning Ahead for the Joint Meetings: Giving Good Talks.” *Notices of the AMS*, Vol. 66, no. 10, pp. 1647-1651.

[2] Gallian, Joseph A. “Advice on Giving a Good PowerPoint Presentation.” *Math Horizons*, vol. 13, no. 4, 2006, pp. 25–27.

[3] Kyra, Bryna. “Giving a Talk.” *Notices of the AMS*. Vol. 60, no. 2, pp. 242-244.

[4] McCarthy, John E. “How to Give a Good Colloquium” *Canadian Mathematical Society NOTES*, Vol. 31, no. 5, pp. 3-4.

*Disclaimer*: The opinions expressed on this blog are the views of the writer(s) and do not necessarily reflect the views and opinions of the American Mathematical Society.

*Comments Guidelines*: The AMS encourages your comments, and hopes you will join the discussions. We review comments before they are posted, and those that are offensive, abusive, off-topic or promoting a commercial product, person or website will not be posted. Expressing disagreement is fine, but mutual respect is required.

I didn’t go to the joint meetings (JMM) this year. This is despite the following good reasons I had to go:

- I’m in my fifth year, applying for jobs, and this is the time when you’re supposed to get out there and spread your name.
- I’ve been a few times before and actually kinda enjoy the spectacle of the “world’s largest gathering of mathematicians.”
- Flights to Denver were mega cheap, even as of like two weeks ago.

I even resisted some light pressure from peers and professors by staying put, and given the low airfare and reliable sources of on-campus support for academic travel, the trip would have cost me next-to-nothing. So why not?

A large chunk of the academic apparatus is set up to encourage you to travel. There are many grants available from many organizations to attend conferences, and travel support for attendees is one of the major budget items for many conference organizers. Many jobs include travel allowances, or require travel to present at conferences as part of the job description. Departments like mine seem to have sizable budgets for the express purpose of covering the travel expenses (and honorariums) of invited speakers who seem quite willing to travel many miles to spread their gospel. Travel is part of the job it seems – both a perk and a responsibility for the academic mathematician. Insofar as I can tell, there are a few source causes of the fact of academic travel, which I guess are obvious, but are worth recounting for what I want to say.

The first reason is personal. By disseminating your knowledge through the unique performative medium of a live-action talk, your work penetrates into mathematical culture and you become better-known to the community. You can also build your network by meeting folks with common interests in person, and perhaps sharing a drink or a bite. This can lead to collaboration, the production of new mathematics, and further opportunities to disseminate it, which I’m told also leads to jobs with greater prestige and pay. Briefly, geographic mobility begets social mobility.

The second reason is institutional. Imagine you already have a position of great prestige and pay. What cause do you have to get off your butt and go preach to the unwashed mathematical masses? Well, besides all of the personal incentives, your employer wants you to go out there because your renown is ultimately their renown. An institution accrues and maintains prestige by the the fact that its members are invited to speaking engagements, so they will want to make the mechanics of academic travel as easy as possible for you. The actual (as in non-rhetorical) you may have witnessed this system in action whenever a professor cancels class because they are out of town, or when you have been excused from your duties for same.

The last reason is similar, though more deeply cultural. Academia is replaying a decades-old fantasy which I think is common to many sectors of society: that the upper-classes are the jet-setters. Frequent travel is an emblem of status, and the other modes of academic life, namely those which demand contact with the immediate community, are subordinate to the higher purpose of missionary work. The work that requires travel, by its resource-intensive nature, must be limited to those of rarefied talent and ability. And while scarcity is the origin of this regard, in the present age of commodified luxury and full capitalization of earthly resources, it has become the norm – now you have to travel just to keep up with the Joneses. The gross domestic product thanks you.

A small perversion of this fantasy, it is no wonder that our community so celebrates the myth of Paul Erdös, the mathematician whose life was an amphetamine-fueled itinerant rampage of collaboration. From Erdös’ claim that mathematics was set back commensurately by his one-month abstinence from stimulants, one might also suppose that a refusal to travel could be injurious to mathematical progress. What self-respecting mathematician would abnegate their responsibility to speedily delivery the bounties of their enterprise by such refusal?

So here’s my real question. As highly educated people, we know that air travel is a particularly energy-intensive form of transportation. The emissions-per-passenger produced by a single transatlantic flight yields more CO2 than the average citizen of many countries produces in a year. Can we continue to justify our privilege of air travel for the sacred purpose of scientific progress when scientific progress also tells us that we, as a planet, cannot all afford to travel by air? Can we expect the peoples and nations of the world to take the scientific community seriously on climate change if we are not making strenuous efforts to reshape our own behaviors in accordance?

Don’t get me wrong: I love a good conference as much as the next person. I’ve had the good fortune of visiting places I would never have been able to afford or justify if not for academic travel. I’ve met wonderful people and been blessed to share a room or even a conversation with many mathematicians I greatly admire. I know there are experiences enabled by conference-going which have no substitute, and collaboration over video chat may never quite be the same as working at the same chalkboard. The expense of academic travel does bear value, yet I still don’t know if things have to be exactly the way they are.

It’s true, aviation only accounts for about two percent of all carbon emissions. But this is complicated by the fact that the particular type of high-altitude emissions from airplanes can be more dangerous in the short term. Also, in the US, two-thirds of air travel is accounted for by the twelve percent of the population that takes six or more round-trip flights per year — the “frequent flyers.” I’m certain many academics are among this class. Do we need to stop flying? Probably not entirely, but I feel some hypocrisy knowing that we would be in real trouble if everyone started flying as much as we do. I felt this sort of guilt before I learned the Swedes had a name for it: *flygskam*, or “flight shame.” As soon as I learned this, I felt the rush of relief that comes with learning there are other people out there like you, and that there’s a name for you, probably like how X-Men (I assume the term is gender-inclusive) feel when Dr. X taps them and gives them context and purpose. Needless to say, now I’m devoted to spreading awareness of the term.

In the interest of full disclosure, I should probably confess that I attended an AMS sectional meeting in Hawaii last year, and I have to say it was great. But I feel complicated about this privilege. This meeting was very well attended, and I’m sure organizers bank on the appeal of a meeting in Hawaii, but the decision to hold it there is demonstrably not good for the planet when compared to alternatives. As Denver is relatively centrally located, maybe JMM should be there every year? Or if we really want to go for it, we could campaign for the construction of a carbon-neutral/negative conference center at the geographic/population center of the US (near the Nebraska-Kansas border, or somewhere in central Missouri, or somewhere else depending on how you measure), with connecting high-speed rail, to be used for all national scientific conferences.

There are also advocates of the video-conferencing approach. We know it has limitations, but if university courses can be conducted online and at massive scale with the assurance of comparable student outcomes, I don’t see why a video conferencing solution couldn’t be appropriate for some purposes. I think part of the solution here could be purely technical. Humans have been organizing traditional conferences for decades so the mechanics are both familiar and highly-developed, while video conferencing is still (in my experience) often clumsy and frustrating. If someone would design a slick and reliable platform for organizing video conferences, I could see this becoming a thing. Imagine one portal with all the conference abstracts, schedule, relevant chatrooms, etc., and then you could easily enter and leave sessions at your leisure… say, if I don’t get a job due to my lack of conference attendance, maybe I could start this business…

One study found that CO2 emissions due to travel for the purpose of presenting scientific papers accounted for only 0.003 % of the annual total, somewhere between the transportation emissions of Geneva and Barcelona. This sounds maybe not that bad. But I think what sets the climate crisis apart from other challenges is that

it requires action on all fronts. We won’t achieve our goals on reducing carbon emissions by singling out individual sectors that need reform. We need to create a culture which considers the impact of all of our personal and professional activities on the environment, and as scientists, high priests of this secular era, we are responsible for leading the cultural shift. If we aren’t going to stop flying to conferences (and we aren’t, I guess) we need to start thinking of ways to offset this activity. We need climate-consciousness to be baked into the process of conference organizing. I don’t know of any math conferences that are explicitly trying to address their environmental impacts, but I would like to.

To be clear, I’m not calling for any sort of a heroic abandonment of all air travel by the scientific community or advocating the use of sanctimonious hashtags (see #istayontheground). I’m sure I will fly again for a conference, and probably even use a paper cup or two for coffee when I have forgotten my reusable mug. I just want to point out that the path of minimizing the consequences of our own actions is too tempting for a community that should be taking leadership, and that this path is made even easier by the fact that individualistic resource consumption and accumulation is still de rigeur in this country in general. Non-conformity might initially require a little bit of courage, but I think it’ll be a bit easier for the rest of society, and result in less political strife, if scientists act first.

*Disclaimer*: The opinions expressed on this blog are the views of the writer(s) and do not necessarily reflect the views and opinions of the American Mathematical Society.

*Comments Guidelines*: The AMS encourages your comments, and hopes you will join the discussions. We review comments before they are posted, and those that are offensive, abusive, off-topic or promoting a commercial product, person or website will not be posted. Expressing disagreement is fine, but mutual respect is required.

*We like to think that our life stories have happy endings, perhaps that we can carefully partition our lives into fourths of each year, and successfully say, “Well, after I learned this, my life was great!” But anyone who has lived life — so, I suppose, anyone reading this — knows that that is not what life is like. Life is a continuous (not discrete!) story with changing hurdles. The gist of this series called “Dear first year, this isn’t something you can plan for,” is that if anything has, grad school has shown me how much truth the quote “the best-laid plans of mice and men often go awry” holds. Every quarter of my first year had some unexpected obstacle or victory and sometimes both, and sometimes the victory turned into an obstacle. The following is the story of my third quarter as a math Ph.D. student at Oregon State University, along with some thoughts that stay with me from that time.*

I lived every term of my first year of grad school desperately hoping things would get easier. I still remember my first term as the most bitterly difficult of them all, but the truth is that each one of them — as my mentors warned me would happen — were approximately equally difficult (I recently thought maybe I should just turn this series into a memoir about the entirety of graduate school, since this past term, my fourth at Oregon State and the first of my second year, was busy as all heck and I felt like I’d stepped out of first year into a fire). I started spring quarter with the hopeful energy with which I had started every other quarter: with a determination to excel in my courses and return to my peak mental performance.

Winter quarter ended with the knowledge that my first attempt at Ph.D. qualifying exams was about ten days in the future. By this time, I understood that this attempt would be my practice run: I had been so completely overwhelmed during the past two terms that I hadn’t had the ability to study as much as is necessary for these exhausting tests, so I was intent on studying as much as I could throughout spring break and giving it an “honorable effort.”

The analysis exam went better than I thought it would, but that’s hardly surprising, since analysis is my field of study. Linear algebra, on the other hand, went absolutely terrible — I left the exam early, knowing without a doubt that I’d failed because I couldn’t get much more than one problem out of four solved. Two weeks later, I received the predictable news that I had failed both exams. In my head, I tried to tell myself I didn’t care, but failing those exams only made that little voice whisper more persistently, *You are a failure. You don’t deserve to be here. You don’t work hard enough. You didn’t deserve to be a Provost Scholar.*

Spring quarter, my course load was Real Analysis III (focused on general measure theory), Complex Analysis, and Partial Differential Equations III (largely bent toward applied mathematics; the last four weeks or so we discussed important topics in fluid mechanics). The beginning of the term, I was so excited about working really hard in complex analysis: we had reading assignments and problems to solve for each class day, as well as the typical set of four or five problems to hand in at the end of each week. Unfortunately, these daily assignments didn’t work out quite the way I expected. I hoped they’d be fairly simple: instead, they would often consume two or three hours of my time if I wanted to actually, really understand (they were graded on completion). As a result, I was left little time to work on the weekly homework, which was graded extremely carefully for correctness. The class I was excited to do well in quickly became the class for which I pulled multiple all-nighters, rarely managed to finish the homework, and was convinced I’d fail.

I could compare the three or so years prior to the beginning of May 2019 to being encapsulated in the deflector shields Droideka wear in *Star Wars*. Inside the deflector shield was math, math, and more math. Sure, being inside the deflector shield wasn’t a cakewalk, and the shield temporarily shut down in January 2018 when I learned that the only grandparent I had been very close to, my last surviving grandparent, had passed away. But it went back up again, shutting me inside with my math and not much else of the world.

Around the beginning of May, the deflector shield had sustained too much damage to protect me, and it burst. I started experiencing wretched allergies (did you know that Linn County, Oregon is the grass seed capital of the world?) and had to go to the health clinic three or four different times to try to (unsuccessfully) combat symptoms of allergies which left me with about 50% of my normal hearing. I randomly got heat exhaustion, even after drinking many fluids, after volunteering at a math outreach event in Eugene for the afternoon and then going on a bike ride in a Corvallis heat wave. I later learned that the random nausea was likely to be attributed to my new medication, the only side effect of which I had yet experienced was annoying itchiness on my extremities. Completely unexpectedly, I experienced a few weeks of relationship turmoil and confusion — the romance ended as suddenly and dramatically as it had begun, but the turmoil and confusion consumed my mind for months after. Then I received the news during one of my four recitations that my business calc class’s primary instructor was going on unforeseen leave; the next day I was asked to consider taking over a 100-person lecture, which I imminently decided to do, since it would give fellow grad students the opportunity to receive a teaching assistantship for the remainder of the academic year. I found a new apartment and my car got towed, and partly as a result of those two things experienced my first bout of the financial trouble grad students notoriously face.

All of this happened over the space of about two weeks. I said the deflector shield burst, didn’t I?

Having written almost three of these memoir-ish posts by now, one would think I would know how to end them. Do I discuss what I learned from that time, what I advise others to do? But I am a candid person; I have learned that honesty, even when it is brutal, is the best course of action; I have learned that lack of vulnerability is one of the tremendous weaknesses of humankind. So I’m afraid I will never be the one to wrap up a post like this with a nice little bow and make it pretty enough to put under a Christmas tree.

Perhaps what I think of when I think about the chaos and emotional toll spring term took on me, I am most thankful for the growth I see in myself — and not only the growth, but the evidence I gave myself that I am brave. It was not easy to know that I had to stop seeing someone I really enjoyed being around, but I did have to for the sake of multiple people involved, and so I did. It was not easy to keep showing up to teach a class where attendance on Fridays was around 15%, but I had to because I said I would, and I did. It was not easy to witness the subsequent strife in the mathematics department and feel that I was its cause, but I stood equally for both sides of the argument, knowing that I had made a decision and that I had to stand by it, so I did. It was not easy to read insulting student evaluations at the end of the term, knowing that I had poured so much time and energy into this body of students and into being as clear and precise as possible, but I knew I had to take their insults with a measure of salt, so I did. I did not know I could be strong, but I was, and I can be, and I am.

And you? You are strong too. Unfortunately, it is not the case that merely because we are human beings sequestered in learning the most beautiful field of study (I admit to being biased), we do not have to experience the pain of real life along with the pain of learning. I say this not to be pessimistic, but rather to tell you that I am aware that it is hard, and that showing up is hard, and that if you are showing up, you are standing strong in a hard battle. Don’t give up! I’m rooting for you.

*Disclaimer*: The opinions expressed on this blog are the views of the writer(s) and do not necessarily reflect the views and opinions of the American Mathematical Society.

*Comments Guidelines*: The AMS encourages your comments, and hopes you will join the discussions. We review comments before they are posted, and those that are offensive, abusive, off-topic or promoting a commercial product, person or website will not be posted. Expressing disagreement is fine, but mutual respect is required.

After a while I started a conversation with a couple of people about our jobs and what we enjoy about it. I told them about the research and teaching aspects of my graduate program. This semester I was a teaching assistant for a lower-division *Linear Algebra and Differential Equations* course. I find this course to be quite fun to teach, because I get to help students develop a geometric intuition for abstract mathematics and point to wonderful applications of that abstraction.

As it turned out, one of the people in our group was a graphics design student. He told me about a project involving linear algebra, and how he wished that he had taken more math courses. He also mentioned using the *Bézier curves* in his classes. I had never heard of that name, so I wrote a note to look into it later. This conversation reminded me of something I had read in Jordan Ellenberg pitch for Outward-Facing Mathematics:

“Those of us who teach spend a lot of hours talking about math in front of students who have been forced to be there. That makes it easy to forget that people out in the world generally admire math and are excited to learn about it, if we give them a way in!”

Back at home, I looked up *Bézier curves*, which lead me down a delightful rabbit hole of computer fonts and automobile design^{2}, and in the process I learned new math. In this post (and hopefully others) I am going to write about the wonderful mathematics that I learn inspired by people in other professions.

Geometry. Even the most basic illustrations involve lines and areas. To design the fancy $\LaTeX$ fonts used for mathematical symbols, for instance, each glyph is pieced together by many curves enclosing a shaded region.

This might sound like a trivial fact, like answering a toddler who asks “how are words written?” or at best something that typographers, not mathematicians, would find interesting. In that case, you might be surprised to hear that in the late 70’s and 80’s the AMS formed an advisory Standing Committee on Composition Technology^{3} and helped work on a then up-and-coming software by Donald Knuth called $\TeX$.

Stay with me and I will explain why I find this mathematically interesting.

Euclid postulated that given any two points, we can draw a straight line passing through them.

**Question:** in how many different ways can the statement above be generalized?

Here are a few I can think of:

- given three points, when can we draw a straight line passing through them? How about a circle or a conic section?
- given three points, what is the lowest degree polynomial $y=P(x)$ passing through them?
- given two lines in the space, when can we find a unique plane passing through them?

Each of these are interesting problems, typically studied by algebraic geometers. There are also others, but for now let’s consider the following:

- given two points and two lines passing through them, is there a cubic polynomial tangent to the given lines at the respective points?

This is referred to as spline interpolation.

The idea here boils down to finding a special basis $H_0(x)$, $H_1(x)$, $H_2(x)$, and $H_3(x)$ for the space of cubic polynomials in one variable so that given $y_0$,$y_1$,$m_0$, and $m_1$, we can quickly find the cubic polynomial we wanted by computing $P(x) = y_0 H_0(x)$ $+ y_1 H_1(x)$ $+ m_0 H_2(x)$ $+ m_1 H_3(x)$. Here is the idea:

Consider $P(x) = ax^3+bx^2+cx+d$, so $\tfrac{dP}{dx}(x) = 3ax^2+2bx+c$. Plugging in our initial conditions gives:

$$

\begin{aligned}

y_0 = P(0) &= d \\\

y_1 = P(1) &= a + b + c + d \\\

m_0 = \tfrac{dP}{dx}(0) &= c \\\

m_1 = \tfrac{dP}{dx}(1) &= 3a + 2b + c

\end{aligned}

$$

This is a system of linear equations:

$$

\begin{pmatrix} y_0 \\\ y_1 \\\ m_0 \\\ m_1 \end{pmatrix} =

\begin{pmatrix} P(0) \\\ P(1) \\\ P'(0) \\\ P'(1) \end{pmatrix} =

\begin{pmatrix}

0 & 0 & 0 & 1 \\\

1 & 1 & 1 & 1 \\\

0 & 0 & 1 & 0 \\\

3 & 2 & 1 & 0 \\\

\end{pmatrix}

\begin{pmatrix} a \\\ b \\\ c \\\ d \end{pmatrix}

$$

Since this matrix is invertible, we can find:

$$

\begin{pmatrix} a \\\ b \\\ c \\\ d \end{pmatrix} =

\begin{pmatrix}

0 & 0 & 0 & 1 \\\

1 & 1 & 1 & 1 \\\

0 & 0 & 1 & 0 \\\

3 & 2 & 1 & 0 \\\

\end{pmatrix}^{-1}

\begin{pmatrix} y_0 \\\ y_1 \\\ m_0 \\\ m_1 \end{pmatrix}

$$

Now we can go back to $P(x)$:

$$

\begin{aligned}

P(x)

&= \begin{pmatrix} x^3 \\\ x^2 \\\ x \\\ 1 \end{pmatrix}^T

\begin{pmatrix} a \\\ b \\\ c \\\ d \end{pmatrix} \\\

&= \begin{pmatrix} x^3 \\\ x^2 \\\ x \\\ 1 \end{pmatrix}^T

\begin{pmatrix}

0 & 0 & 0 & 1 \\\

1 & 1 & 1 & 1 \\\

0 & 0 & 1 & 0 \\\

3 & 2 & 1 & 0 \\\

\end{pmatrix}^{-1}

\begin{pmatrix} y_0 \\\ y_1 \\\ m_0 \\\ m_1 \end{pmatrix} \\\

&= \begin{pmatrix} x^3 \\\ x^2 \\\ x \\\ 1 \end{pmatrix}^T

\begin{pmatrix}

2 & -2 & 1 & 1 \\\

-3 & 3 & -2 & -1 \\\

0 & 0 & 1 & 0 \\\

1 & 0 & 0 & 0 \\\

\end{pmatrix}

\begin{pmatrix} y_0 \\\ y_1 \\\ m_0 \\\ m_1 \end{pmatrix} \\\

&= \begin{pmatrix} 2x^3-3x^2+1 \\\ -2x^3+3x^2 \\\ x^3-2x^2+x \\\ x^3-x^2 \end{pmatrix}^T

\begin{pmatrix} y_0 \\\ y_1 \\\ m_0 \\\ m_1 \end{pmatrix} \\\

&= \begin{pmatrix} H_0(x) \\\ H_1(x) \\\ H_2(x) \\\ H_3(x) \end{pmatrix}^T

\begin{pmatrix} y_0 \\\ y_1 \\\ m_0 \\\ m_1 \end{pmatrix} \\\

&= y_0 H_0(x) + y_1 H_1(x) + m_0 H_2(x) + m_1 H_3(x)

\end{aligned}

$$

These polynomials are the cubic Hermite splines.

Interpolating piece-wise cubic curves certainly is not the end of the story. As studied by algebraic geometers, multivariate spline theory and geometric modeling of curves and especially algebraic surfaces of higher degree is an active area of research. Even more, we can consider complex analytic functions and arrive at periodic designs. All of that also for another time.

From here it is a short walk to define Bézier curves. To keep this post short I will point to two references instead^{5}, but if you are interested to explore, there are many connections to numerical analysis and even a constructive proof of the Stone–Weierstrass approximation theorem.

If you are interested in the computations, you can experiment with cubic and quadratic Bézier curves or learn more about modern fonts and create one or make an animation instead.

To close, I want to give a nod to David Austin’s essay in which he suggests that “the singular value decomposition should be a central part of an undergraduate mathematics major’s linear algebra curriculum.” The book used in the course I am teaching stops at simple applications to physics, but perhaps introducing applications from other areas like computer science or the arts, even for students taking lower-division mathematics courses, would encourage people (including mathematicians) to view the subject in a more approachable light.

Notes and footnotes:

- The animations above are created by the Manim engine, as seen in 3Blue1Brown videos. I am confident that somewhere deep inside the Python libraries used in Manim, there are Bézier curves smoothing the transitions.

- In case anyone found my costume scary, I was ready to recite the quote by Gauss that “mathematics is the queen of sciences,” so technically I was wearing a princess costume! ︎
- If you haven’t heard yet, the new Tesla Cybertruck has decided to buck the trend of using smooth surfaces for vehicles in favor of low-poly designs. Oh well. ︎
- I couldn’t find much on this committee beyond an invitation by them to join the $\TeX$ user’s group in Knuth’s 1979 book “$\TeX$ and METAFONT,” published by the AMS. ︎
- These polynomials are not the same as Hermite polynomials
*a la*the quantum harmonic oscillator, but they’re both named after the same Charles Hermite. ︎ - Bill Casselman’s feature column From Bézier to Bernstein and these slides from a Computer Graphics and Imaging course at UC Berkeley are good places to read more. ︎

*Disclaimer*: The opinions expressed on this blog are the views of the writer(s) and do not necessarily reflect the views and opinions of the American Mathematical Society.

*Comments Guidelines*: The AMS encourages your comments, and hopes you will join the discussions. We review comments before they are posted, and those that are offensive, abusive, off-topic or promoting a commercial product, person or website will not be posted. Expressing disagreement is fine, but mutual respect is required.

Before I dive into the gist of Professor Thompson’s argument, I think it is important to reiterate why diversity matters in mathematics. Here, I risk making Professor Thompson into a strawman; she’s not asserting diversity in mathematics is unwelcome, just that diversity statements should be removed from hiring. But humor me so I can climb on this soapbox.

First, creating a more equitable society and correcting past injustices that have disadvantaged underrepresented minorities is the most obvious reason diversity should matter. Of course, I have heard the rebuttal, “Yes, but why is it a responsibility of mathematicians to facilitate this change in our field?” Well, mathematicians have the power to enact substantial change by incorporating diversity initiatives into hiring, extracurricular programs, and candid reassessments of the academic climate. Sure, mathematicians may not cause systemic change in society at large, but undoubtedly academics have the power to influence climate and advocate for their values at their own universities.

More selfishly, collaborative environments benefit from diversity. A Tufts study on collaboration efforts of mock jurors found that diverse groups “deliberated longer, raised more facts about the case, and conducted broader deliberations” (6). While the Tufts study focused only on racial diversity, it is emblematic of a larger trend in social psychology which has demonstrated positive effects of diversity in a variety of collaborative environments (7). Crucially, mathematics is more collaborative now than it has ever been and, unfortunately, is not much more diverse (5). Through this lens, if we care about the advancement of our field, we should value diversity for its practical use in addition to its moral imperative.

Now, to the substance of Professor Thompson’s argument:

One of Thompson’s major planks is that a diversity statement is “a political test with teeth.” Thompson likens diversity statements to McCarthy-era loyalty oaths (back in the 1950s, the UC system forced faculty to sign pledges that they were loyal to America and not the Communist party, infamously firing those who refused to comply). Gently put, this is an odd comparison. Even if we accept Thompson’s claim that diversity statements are “political,” they hardly seem comparable to McCarthy-era extremism with respect to harm and disruptiveness. People didn’t sign the loyalty oath, likely because it aimed to exclude, isolate, or punish individuals for their political beliefs. A diversity statement’s entire purpose is to include historically excluded, silenced, or isolated minorities and allow them space in the academic community.

Moreover, is assessing whether job candidates treat people as individuals really a political statement as Thompson asserts? A person’s background influences the way they interact with most things–the classroom is no exception. Consider a student who can’t afford school supplies. Likely, that student will encounter challenges many others won’t: working a job outside class, distracting financial concerns, or even how to take notes each day. I’m not arguing that the instructor should give preferential treatment to this student; just that an inclusive instructor should strive to work with each student to help them realize their academic goals, being sensitive to the backgrounds different students bring into the classroom.

Studies also support the notion that individual identity influences performance in the classroom. For instance, two different studies (one conducted in Florida, one in Tennessee) found that having a teacher of the same race contributed positively to academic success (1, 2). Other studies reiterate that representation matters and that even math classrooms aren’t immune from the effect one’s background brings. For instance, a University of Massachusetts Amherst study found that “increasing the visibility of female scientists, engineers and mathematicians […] profoundly benefits [young women’s] self perception in STEM” (8).

While I agree with Thompson that treating people as individuals is an assertion of how society “ought to be organized,” I believe characterizing this sentiment as “political” misconstrues the meaning by associating it with partisan politics.

All of this is to say: it’s not political to treat people as individuals. It’s human and it’s logical.

Then, Professor Thompson criticizes the fact that “the diversity ‘score’ is becoming central in the hiring process.” Thompson’s language implies that other factors like caliber of research take a backseat to diversity which, when looking at the faculty of any R1 University, seems misleading. The New York Times rebutts this point best:

“The ethos [of mathematics] is characterized as meritocracy [and] is often wielded as a seemingly unassailable excuse for screening out promising minority job candidates who lack a name-brand alma mater or an illustrious mentor. Hiring committees that reflect the mostly white and Asian makeup of most math departments say they are compelled to “choose the ‘best’” […] even though there’s no guideline about what ‘best’ is.”

To paraphrase, hiring committees are just like the rest of us: subject to implicit bias. Certainly, the diversity statement plays a crucial role in patching “the leaky pipeline.”

Moreover, the diversity statement also communicates to underrepresented minorities that a math program cares about creating an inclusive research community. The University of Michigan recently conducted a study on academic attrition and found that, for underrepresented minorities, academic climate was a major factor in their decision to leave (4). In other words, stressing a department’s belief in the value of diversity helps positively shape department norms and combat attrition. The same Times article wrote about Edray Goins, a black mathematician who left a “better” position in a hostile academic environment for a department which emphasized inclusivity (3). Fortunately, Professor Goins chose to remain in academia, but his story is the exception, not the rule.

To her credit, Thompson ends by asserting that “mathematics must be open and welcoming to everyone, to those who have traditionally been excluded, and to those holding unpopular viewpoints.” Unfortunately, the substance of her previous argument makes these words feel empty.

If we truly care about increasing diverse representation in mathematics, we should pursue every available avenue. Diversity statements are only one piece of the puzzle, but they are important nonetheless.

1. Long Run Impacts of Same Race Teachers: https://www.nber.org/papers/w25254

2. Representation in the classroom: the effect of own race teachers on student achievement: https://www.sciencedirect.com/science/article/abs/pii/S0272775715000084

3. For a black mathematician, what it’s like to be the only one: https://www.nytimes.com/2019/02/18/us/edray-goins-black-mathematicians.html

4. Exit Interview Study: Executive Summary, University of Michigan: https://advance.umich.edu/wp-content/uploads/2018/09/UM-Exit-Interview-Study-2014-Report-executive-final.pdf

5. Women, Minorities, and Persons with Disabilities in Science and Engineering: https://www.nsf.gov/statistics/2017/nsf17310/digest/fod-women/mathematics-and-statistics.cfm

6. Racial diversity improves group decision making in unexpected ways: https://www.sciencedaily.com/releases/2006/04/060410162259.htm

7. More sources on the value of diversity in performance: https://blog.capterra.com/7-studies-that-prove-the-value-of-diversity-in-the-workplace/

8. Female Scientists Act like a Social Vaccine to Protect Young Women’s Interest and Motivation in the Sciences, UMass Amherst Study Shows: https://www.umass.edu/newsoffice/article/female-scientists-act-social-vaccine-protect-young-women%E2%80%99s-interest-and-motivation-sciences

*Disclaimer*: The opinions expressed on this blog are the views of the writer(s) and do not necessarily reflect the views and opinions of the American Mathematical Society.

*Comments Guidelines*: The AMS encourages your comments, and hopes you will join the discussions. We review comments before they are posted, and those that are offensive, abusive, off-topic or promoting a commercial product, person or website will not be posted. Expressing disagreement is fine, but mutual respect is required.

There is so much that is peculiar, irregular, silly, or downright twisted in mathematical verbiage that, certainly, we could all benefit from some soul-searching on the language of our culture. Some of mathematics usage is confusing (e. g. overuse of “normal” and “regular”) and some irritating (personal peeve: persistent classroom use of “guy” to refer to mathematical expressions – I know anthropomorphization makes things friendly and all, but I’m not sure that thinking of all mathematical objects as “guys” is good for our ongoing gender problem). And then there are other things that just floored me the first time I heard them (um, “clopen,” anyone?), not to mention our obsession/affliction with eponymy and its discontents. There is a dissertation in linguistic anthropology waiting to be written on mathematical usage, and perhaps several that already have been.

It would be all well and good to litigate the social and political aspects of mathematical speech, but who really has the time?^{ 1 }This is a graduate student blog, and, you know, life is already hard enough, so we must have some recreation. Proposed solution: **the mathematical crossword puzzle**, or more accurately, crossword puzzle with a strong mathematical bias – a venue to examine and lightheartedly ponder our field’s history, culture, language and content without needing to delve into heated public debate. On the other hand, maybe the chance for debate is sort of the point.^{ 2 } Entertainment which is presented as critical thinking and that leads to higher-level critical thinking is a high kind of art.

I suppose, based on my own experience, that many crossword solvers will relate to the experience of hating puzzle-makers for clues that make no sense, are elitist, presume familiarity with arcane or dated bits of culture, etc. To draw a parallel, I submit that this is exactly the sort of experience many students are having in math classes, at any level. That you are the kind of person that is willing to put up with being treated with such pomposity and contempt, until you are suddenly on the other side of this diode-like arrangement, is something one might infer from the fact that you are in mathematics graduate school and reading a math blog to boot, which is to say: I bet the intersection of math-o-philes and cruciverbalists is not so small.

But! We must do better than our teachers by seeking to not alienate, condescend, and exclude, and in order to get there first we must try. As a long-time-solver-first-time-constructor, let me say the following:

- Constructing is hard! Harder that you might think, harder than I thought at least. The junky, off-putting clues you find in crosswords are much more likely due to (i) the jams a maker finds them- self in while constructing and (ii) laziness at dealing with these jams, than they are to any kind of elitism or snootery.
- Regarding the handling of such jams, no matter how hard you try you are still a victim of your own biases. There is perhaps no way around this, at least not on an individual level. A diversity of backgrounds among puzzle-makers and solvers (draw the mathematical analogy) will lead to a richer and less homogenized and consistently frustrating experience. This is the general nature of the criticisms levelled at the New York Times editorialship by Rex Parker et al., and it leads to a big and important conversation on power, privilege, who’s being represented and who’s being excluded. I have made a best effort at inclusiveness in the theme and content of this puzzle, which I’m sure is still abjectly deficient in some respects.
- This puzzle has a few more black squares than is typical/admissible for your average newspaper puzzle. Here’s my accounting for this: many puzzles are built around a “theme,” a collection of clues that are linked by some common feature. Clues in this set are called “themers.” I tried to cram too many themers into this one, and in order to cope with the resulting jams, I had to black some stuff out.
- I couldn’t (and this is maybe a relief for solvers) find a way to reasonably make all of the clues mathematics related. So some are intersectional, and some are out of left-field. I learned a lot of trivia while making this, and my hope is that you might learn some too.

I hope you enjoy! If you are moved to create your own math-puzzles, I am also sharing the kinda janky LaTeX file I used to make this one, in case it helps.

^{ 1 }Maybe the same people that spend time making crosswords, ahem.

^{ 2 }Regular NYT puzzle solvers may know the boisterous commentary of Rex Parker and others in the puzzle blogosphere.

*Disclaimer*: The opinions expressed on this blog are the views of the writer(s) and do not necessarily reflect the views and opinions of the American Mathematical Society.

*Comments Guidelines*: The AMS encourages your comments, and hopes you will join the discussions. We review comments before they are posted, and those that are offensive, abusive, off-topic or promoting a commercial product, person or website will not be posted. Expressing disagreement is fine, but mutual respect is required.

The Society for Advancement of Chicanos/Hispanics and Native Americans in Science (SACNAS) is a society that aims to further the success of Hispanic and Native American students in obtaining advanced degrees, careers, leadership positions, and equality in STEM. SACNAS was founded in 1973 by underrepresented scientists to address the representation of Chicanos/Hispanics and Native Americans in STEM. Diverse voices can expand scientific and mathematical knowledge as well as bring creative solutions to scientific problems. This is one of SACNAS’s motivations for building an inclusive, innovative, and powerful national network of scientists, which now includes over 6,000 society members, over 115 student and professional chapters, and over 20,000 supporters of SACNAS throughout the USA. Contrary to the name, the society is welcoming of people from all backgrounds, identities, fields of study, and professions. SACNAS is the largest multicultural STEM diversity organization in the US.

SACNAS has programs and events that train and support the diverse STEM talent that is found in this country. This is done in partnership with the student and professional chapters, the leadership programs, Native American programs, regional meetings, and policy and advocacy initiatives. SACNAS also hosts THE National Diversity in STEM Conference. This year’s 2019 SACNAS National Conference in Honolulu, Hawai’i brought in over 5,000 participants! Next year the 2020 SACNAS National Conference is in Long Beach, California!

Mathematicians and mathematics have always been a strong part of SACNAS. In fact some of the founders of SACNAS include mathematicians, such as Dr. Richard Tapia (Rice University) and Dr. William Vélez (University of Arizona). I am fortunate to have met these two great mathematicians, who at different times in my academic journey have shared their wisdom and thoughtful advice.

My first SACNAS conference was in 2011 in San José, California. I was a second-year undergraduate student attending his first scientific conference. I was eager to learn and excited for all the opportunities that would be presented at this conference, but I did not know what to expect. Fortunately, I found a community of mathematicians who share similar goals for diversifying mathematics and who genuinely care in supporting the success of students. I trace my interest in combinatorics to the 2011 SACNAS National Conference, where I had the opportunity to attended the NSF Mathematics Institutes’ Modern Math Workshop. That year’s keynote lecture on “Counting Lattice Points in Polytopes” was presented by Dr. Federico Ardila (San Francisco State University). As an example of the power of networking, community, and mathematics at SACNAS, four years later Federico became one of my master’s thesis co-advisors. More than that, I found an unconditional mentor, friend, and research collaborator and I owe part of this to SACNAS for providing a space for a student like me to grow academically and professionally.

The Modern Math Workshop is a two-day workshop that takes place in conjunction with the national meeting of the SACNAS conference and showcases the contemporary research happening at NSF-funded mathematical sciences institutes around the country. It became a collaboration with SACNAS in 2006 and has been jointly organized by the Mathematical Sciences Institutes since 2008. Since 2011 this event has been funded by the NSF through the Mathematical Sciences Institute Diversity Initiative. The workshop is a mix of activities including research expositions aimed at graduate students and researchers, mini-courses aimed at undergraduates, a keynote lecture by a distinguished scientist, and a reception where participants can learn more information about the Mathematical Sciences Institutes.

In addition to the Modern Math Workshop, there are scientific symposia organized by mathematicians, there are oral graduate presentations, and both graduate and undergraduate poster presentations.

I do not know if it was because we were in the beautiful city of Honolulu, that the sky was much bluer and the ocean water much clearer, but there was certainly an extra revitalizing energy present at this year’s SACNAS conference. Below are some of the mathematical events that went on (and that I participated in) at this year’s SACNAS conference. I am sure there were more that I missed out on.

This year’s Modern Math Workshop was organized by the Mathematical Sciences Research Institute (MSRI) There were two mini-courses aimed at undergraduate students. One was lead by Dr. Wilfrid Gangbo (UCLA) and Dr. Anastasia Chavez (UC Davis). The workshop also included research talks aimed at graduate students and faculty and were delivered by representative mathematicians from each of the NSF Math Institutes. Additionally, there was a panel which addressed topics such as: imposter syndrome, how to choose a graduate program, how to stay motivated, how to choose a mathematical field, etc. Below are some of the speakers and panelists.

- Katherine Breen (Institute of Pure and Applied Mathematics (IPAM))
- Xinyi Li (SAMSI – Statistical and Applied Mathematical Sciences Institute)
- Gabriel Martins (California State University, Sacramento)
- Robin Neumayer (Northwestern University)
- Marilyn Vazquez (Mathematical Biosciences Institute (Ohio State University); Institute for Computational and Experimental Research in Mathematics (ICERM))

I was able to sit in Dr. Anastasia Chavez’s mini-courses, which was “An introduction to matroid theory.” My discrete mathematical mind was very happy to hear and learn from my friend on a topic that is incredibly interesting. You can find her slides here.

Apart from the Modern Math Workshop there were three great events/experiences that I would like to share with you all.

- Dr. Rebecca Garcia (Sam Houston State University) and Dr. Kamuela Yong (University of Hawai’i – West O‘ahu) organized the very first “Pacific Islanders in Mathematics” session. This was a historic event (the organizers are writing a more detailed article to be shared with the public) and it featured amazing speakers including:
- Kyle Dahlin (Purdue University): Avian Malaria & Hawaiian Honeycreepers – Modeling of the Effectiveness of Vector Control
- Dr. Marissa Loving (Georgia Tech): Determining Metrics using the Lengths of Curves
- Ashlee Kalauli (UC Santa Barbara): Solving the Word Problem for Artin Groups
- Dr. Efren Ruiz (University of Hawai’i – Hilo): Rings Associated to Directed Graphs

- Dr. Pamela Harris and I co-organized, “Latinxs Count!”, an algebraic and geometric combinatorics research talk session at SACNAS. It featured a talk by me and three amazing speakers :
- Andrés R. Vindas Meléndez (University of Kentucky): An Invitation to Ehrhart Theory
- Laura Escobar (Washington University in St. Louis): Polytopes and Algebraic Geometry
- Ryan Moruzzi, Jr. (Ithaca College): Exploring Bases of Modules using Partition overlaid Patterns
- Rosa Orellana (Dartmouth): The Combinatorics of Multiset Tableaux

- Dr. Pamela Harris was also one of the featured speakers at the SACNAS National Conference. Her featured talk titled, “DREAMing,” shared her story as DREAMer and her mathematical journey into research and mentoring.

I am blessed to have such a supportive mathematics/SACNISTA familia. To end the blog post, I want to share something I mentioned at the conference. I overheard several people say that the math they do is not useful, but I want to challenge each of us to think more about the meaningfulness of our mathematics. Sure, my math may not be applicable (at least right now) to anything “useful”, but it is meaningful to me. It has given me a career path, it has allowed me to make wonderful friends and connections, and I get to share the beauty and meaning of it with people all over the world. But, that’s a whole other topic for a blog post (too deep for this blog post), so I hope that you got a glimpse of the mathematical events that I experienced at this year’s SACNAS National Conference! I look forward to seeing and meeting some of you at the 2020 SACNAS National Conference in Long Beach, CA!

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