*Juan Lozano is a second-year math student at The University of Texas at Austin who participates in the Directed Reading Program, an experiential learning program that matches undergraduate and graduate students in mathematics for projects that deepen students' understanding of high-level mathematical concepts. UT Austin's is the nation's largest and most diverse Directed Reading Program in the country.*

I was there to deliver the first math talk of my college career. Three months before, I knew nothing about the topic and now my job was to tell my audience of fellow undergraduate math students everything they needed to know about geometric measure theory—in 15 minutes. Would they be engaged? Would they get it?

The end-of-semester symposium is one of the key aspects of the Directed Reading Program (DRP), an extracurricular activity at UT Austin that pairs about 100 undergraduate students a year with graduate student mentors to explore math topics outside of their traditional coursework. The DRP gives undergraduates a rare taste of life as a graduate student, exposes them to topics off the beaten path and builds community.

Along with my mentor, Darren King, I figured out which parts of my project I should work into my talk and how to present them. Almost as important as deciding what to leave in were the aspects I decided to leave out of my presentation, because wading too deeply into technical details would risk losing my audience. My goal was to give my talk in a way that showed the geometric motivations of the subject enough so that the audience would understand and believe the results I presented without the whole technical proof.

When I reached the end of my talk, I got a flurry of interested questions about a particularly astonishing way in which measure theory allows you to extend the notion of a "dimension"—engagement that was both rewarding and validating.

In addition to teaching students how to communicate clearly and concisely, the symposium gives audiences a chance to broaden their mathematical horizons.

"Younger students are challenged by material ranging from just out of reach to very far ahead [to] nearly incomprehensible," says Jacob Caudell, another undergraduate who participated with me in the program in the fall of 2016. "And older students get to see treatments of material that they might have never seen or haven't seen in a while."

**Expanding Horizons**

Andrew Blumberg, an associate professor of mathematics, founded the on-campus program in 2013 with the help of a National Science Foundation grant and additional funding from the Department of Mathematics, modeling it after one at his alma matter, the University of Chicago. Some of his friends started the first program in Chicago when he was a graduate student.

"The program was fantastic, and I felt like UT Austin should have one," says Blumberg.

Quite a few top-tier universities offer Directed Reading Programs, but UT Austin's program has grown to be the largest of its kind in the country. About 100 undergraduate students participate at UT Austin (compared to the second largest at the University of Chicago with an average of 44 students a year). The program also has become more diverse than others like it around the country: about 1 in 3 undergraduate participants are women and more than 1 in 5 come from racial and ethnic populations that have been traditionally underrepresented in math.

Now even the College of Natural Sciences' signature undergraduate research program, the Freshman Research Initiative, has incorporated the Directed Reading Program into the curriculum. The FRI's one mathematics-based research stream, called Geometry, Symmetry, Groups and Fields or the Symmetry stream, involves DRP activities both in the summer and during the fall semester of stream participants' sophomore year. Like their counterparts elsewhere in Natural Sciences, they work alongside more seasoned mentors in their discipline to explore solutions to real problems in the field.

**A Taste of Graduate Student Life**

The experience of an undergraduate in mathematics is often very different from that of a graduate student or a professional mathematician. Most undergraduates go through a series of courses with rigidly defined syllabi and clearly defined problems for homework or exams that help them understand math broadly. Meanwhile, graduate students must learn more independently about a certain subject area. Guided by their advisor, they grapple in very different ways with specialized mathematical problems that come up in mathematics' academic literature, but not typically in the undergraduate curriculum. Thus, for students considering graduate school, it's hard to gauge what life will be like as a graduate student.

Lisa Piccirillo, one of the lead organizers of the DRP and a graduate mentor in the program, points out that DRP "is one of the [few] experiences you can have as an undergrad that's indicative of what it is like to be a graduate student."

For undergraduates, the experience of independent reading in mathematics can be daunting at first, especially if, as Caudell described, one goes into the program with "limited exposure to math outside of a neatly designed curriculum." But for him, and many other participants in the program, having the guidance of a mentor was instrumental in successfully learning how to read the literature independently.

Moreover, that process of understanding what graduate school is like can be instrumental in helping DRP participants make informed decisions about their future. Both Caudell and Gregory Lyons, an undergraduate who participated in the program for two semesters, came into DRP planning to go to graduate school in mathematics. Getting the experience of being in the DRP helped reaffirm their conviction. Meanwhile, Arlenne Gonzalez, a third-year physics and math double-major who participated in the DRP, found that after doing a DRP project on the mathematical aspects of quantum mechanics, the mathematics of physics was less intimidating. She decided that in the future she wanted to focus on experimental physics.

More generally, the DRP provides a window into the mathematics that happens outside of the curriculum, and in a different style and pace, and therefore helps shape undergraduates' view of the potential areas of research and applications they might pursue in the future. For example, before starting my own work in the DRP, I was interested mostly in topology and geometry; but working with my mentor, I got to explore a topic in analysis, something I didn't see strongly overlapping with my main interests. Working in an area of analysis called geometric measure theory, though, I got to see a lot of the geometric intuition that goes into the workings of analysis, an aspect not usually taught in the undergraduate curriculum. This has broadened my view of mathematics, also making me better at understanding both analytic arguments and geometric arguments throughout mathematics.

**Building Community**

Although UT Austin's Department of Mathematics is ranked 14th among graduate programs in the U.S. and is also one of the top research departments in the country, undergraduates can feel like they're playing in a very different league from the rest of the department. Piccirillo says the DRP "helps build a mentoring community between the undergrads and the grad students, where the undergrads can say, 'Hey do you have five minutes, I need help figuring this out.'" It's also good experience for graduate students who may supervise students in the future as professional mathematicians.

The DRP is also enriching the interactions between the Math Department and other departments in the College of Natural Sciences. The program has many students who work in areas at the intersection of mathematics and other fields of science, especially physics, economics and computer science. Many DRP projects are integral to the understanding of more advanced sciences, such as physics undergraduate John Goldak's project in Symplectic Geometry, which he says is akin to "setting the mathematical table that allows us to do the physics that we observe everyday."

"People seem to genuinely care about [physics students'] experience while learning mathematics," says Arlenne Gonzalez, also a physics major, who enjoyed having the opportunity to learn from math experts.

Maxwell Stolarski, a graduate student mentor, was surprised by the number of interdisciplinary projects: "Students give presentations on topics … with applications to computer science, financial math, options pricing … Their collective knowledge of the many applications of various mathematical topics leads to many interesting projects."

In fact, many students come to the DRP proposing projects in applied math, such as mathematical biology or mathematics applied to computer science, according to Picarillo. She and the other organizers try to pair them with appropriate mentors, but sometimes struggle to find graduate students in the Department of Mathematics with the appropriate applied knowledge. Because of this, the program sometimes recruits mentors from applied math areas in other departments, such as computational linguistics graduate students in the Department of Linguistics.

The DRP helps make math more approachable, with the one-on-one relationship between mentor and mentee, and helps curious students explore math outside of the standard curriculum, all while giving mentor and mentee valuable experiences for their future careers and new opportunities to forge community.

*If you are an undergraduate student in the College of Natural Sciences who would like to participate in the program, or a graduate student interested in being a DRP mentor, email the organizers at **drp@math.utexas.edu*

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