The AMS Mathematics Research Communities (MRC) is an NSF-funded program to help graduate students and postdocs jumpstart their careers. Every summer the AMS runs three sessions, each on a specific subject. Budding researchers in that area work intensively with each other, alongside expert advisors, on open problems. Many of the 2016 participants already published their MRC results. The AMS offers some further funding for collaborations that began at an MRC and helps cover costs of travel to the annual Joint Mathematics Meetings (JMM), where the MRC topics have special sessions. Select senior participants become organizers for the JMM special session, providing a further organizational experience opportunity. During the program, mentors also offer workshops on professional development to assist attendees in navigating the space of job applications and grant proposals. Women and members of underrepresented minority groups are especially encouraged to apply.

In the summer of 2017, the AMS ran sessions entitled *Homotopy Type Theory*, *Beyond Planarity: Crossing Numbers of Graphs*, and *Dynamical Systems: Smooth, Symbolic, and Measurable* at the gorgeous Snowbird Ski Resort in Utah. I participated in the Homotopy Type Theory (HoTT) MRC. Prior to arriving, I was assigned a five-person research group and two advisors. The program began on Monday morning with opening talks and concluded with a banquet on Friday night. Over these five days we spent twenty-four total hours in intensive group work and sat in six hours of lecture. On Wednesday, as tradition dictates, we took the afternoon off to hike and explore the mountainous landscape.

After dinner on Monday we had a poster session. For this hour, some participants displayed posters describing their personal research and we bothered them with stupid questions. On Wednesday evening Dr. Emily Riehl ran an informal professional development discussion where we asked about job searches and the research process. On Thursday night, the organizers orchestrated a formal such panel and addressed questions that participants submitted by email in advance. Some questions answered: What should a research statement contain? What constitutes a good cover letter? How do you choose research problems? How do you apply for NSF grants? What are non-academic options for math PhDs? How do you balance becoming a mathematician with interests like starting a family? How do you justify your work to prospective employers? They gave detailed practical advice.

What I am most pleased with is, however, a subtler happening. Homotopy type theory is an interdisciplinary topic. The field germinated when Voevodseky, Awodey, and Warren observed that the structure of certain functional programming languages made the data types act like topological spaces [4,5]. Following a special year (2012–2013) at Princeton’s Institute for Advanced Study, it was officially known that algebraic topology could indeed be done in such a language (specifically, in a “dependent type theory”) [3]. So, both mathematicians and computer scientists attended the MRC. By Friday, everybody was on the same page despite severe background differences.

It is a pleasure to watch researchers working to overcome this language obstacle. Often, their attempts are successful—in fact, many of them are amazing. I am most excited to see the long-term collaborations ignited by this MRC and hope they will lead to a more efficient dictionary between pure mathematics and theoretical computer science.

On Friday evening, a representative from each group gave a seven minute report on the progress made and on their plans for future research. We thanked the organizers and made plans to contact each other.

A final word: Because attendees work on specific problems, the scope of an MRC session must be narrow. So, it is likely that one’s niche field will not have a session during a given summer. But, it is a good opportunity and the wise graduate student should annually check the Mathematics Research Communities’ webpage to see if a program suitable to them is offered.

**Acknowledgements **I thank the HoTT MRC organizers, J. Daniel Christensen (University of Western Ontario), Chris Kapulkin (University of Western Ontario), Daniel R. Licata (Wesleyan University), Emily Riehl (Johns Hopkins University), and Michael Shulman (University of San Diego) for making the experience worth writing about.

References

[1] Allyn Jackson, *Building a Research Career: Mathematics Research Communities, *The Notices, 2008.

[2] The American Mathematical Society, *About the Mathematics Research Communities Program, *http://www.ams.org/programs/research-communities/mrc/. Accessed 2017.

[3] *Homotopy type theory: Univalent foundations of mathematics*. The Univalent Foundations Program, Institute for Advanced Study, 2013.

[4] Steve Awodey and Michael A. Warren, *Homotopy theoretic models of identity types*, Math. Proc. Cambridge Philos. Soc. 146 (2009), no. 1, 45–55.

[5] Vladimir Voevodsky, *A very short note on homotopy λ-calculus*, notes from seminars given at Stanford University, 2006.