Mathematicians help train the next generation of positive thinkers

A trio of researchers paves the way for future progress in an up-and-coming field that unites geometry and number theory in powerful new ways.

three people sitting t a table inside a library writing with a pen and paper
Mathematicians (from left) Julia Hartmann, Valentijn Karemaker, and Renee Bell are co-organizers of the conference “Explicit Methods in Arithmetic Geometry in Characteristic p,” a meeting that provides early career researchers with a place to address tractable mathematical problems. 

As postdocs across the region embark on summer research projects and internships, Penn mathematicians Renee Bell, Julia Hartmann, and Valentijn Karemaker are using their time between semesters to address challenges in a mathematics field called arithmetic geometry, and to help early-career researchers gain skills important to their success as the next generation of mathematicians. 

Unlike fields like biology or engineering, where an aspiring academic fresh off of earning a Ph.D. joins a lab, mathematicians must establish their research agendas quicker and more independently. “Postdocs do have a faculty mentor formally, but they are usually not part of a bigger project,” says Hartmann. 

To assist these peridoctoral researchers—those currently working on a Ph.D. or who have recently graduated—the Penn trio, with Padmavathi Srinivasan from Georgia Tech and Isabel Vogt from MIT, organized a summer conference. Participants will focus on problems in arithmetic geometry, a relatively young field that applies techniques from algebraic geometry to solve problems in number theory. 

Seemingly simple questions in number theory such as “Can the sum of two cubes be a cube?” are, in reality, very difficult to prove mathematically. One example is Fermat’s Last Theorem, a number theory problem solved using techniques from geometry more than 350 years after its proposal. 

Conference participants, all mathematicians who are two years away from earning a Ph.D. or up to three years into a postdoc, who work in fields like algebra, geometry, and number theory, will attend lectures on specific techniques followed by break-out sessions aimed at combining their expertise to work on challenging math problems. “Mathematics is such an old field that pretty much all problems that can be solved by a single person have been solved, so you have to work together,” explains Bell. 

Problems will focus on fields with characteristic p (for “prime” numbers), which, because of their finite nature, are more capable of being solved using computational approaches. The organizers hope using this approach, geared toward problems that researchers can make headway on within a week, will serve as the starting point for future projects and kickstart collaborations that could address more challenging problems and conjectures. 

“[Arithmetic geometry] is a field that’s not very old yet, less than a century old, when compared to some other branches of mathematics, like number theory which is thousands of years old. There are more and more people, especially young people, who are now working on this intersection, and that’s something we all appreciated and recognized,” says Karemaker. 

Researchers at the conference will also learn key skills for their careers like writing mathematical research papers, establishing a research agenda, managing large collaborations, and writing grant applications. Senior advisors and mentors will be on hand to talk to discuss their own technical challenges and math problems, as well as to provide general career guidance. 

Hartmann describes her own transition from Ph.D. student to independent researcher as challenging, and she emphasizes how crucial these conferences are for early career mathematicians. What makes this one even more unique, she says, is that Bell and Karemaker are themselves both young mathematicians, with Karemaker having earned her Ph.D. in 2016 and Bell in 2018 “It’s much better to have younger people there who know what the difficulties are,” she says. 

karemaker, hartmann, and bell posing in a library
Karemaker, Hartmann, and Bell all know the importance of summer math programs and supportive mentors as inspiration for their own career paths. “It’s a really good opportunity to meet a whole lot of new people who are all extremely friendly and supportive, and I think that atmosphere is something that we’re definitely aspiring to,” Karemaker says about the upcoming conference. 

The conference is being organized entirely by women. They worked hard to ensure 50 percent female representation for the conference as a whole. “We’re excited for the way that will contribute to the atmosphere,” says Bell. “Hopefully people get used to the feeling of there being lots of women around, and we will help these collaborations propagate.”

Throughout their careers, Bell, Hartmann, and Karemaker have experienced firsthand the importance of summer math programs and supportive mentors as inspiration. 

Karemaker became interested in math as a career after participating in summer camps that opened her eyes to the engaging and abstract world of mathematics. “It’s really solving puzzles with other people, which is how I feel doing research is these days,” she says. 

Bell says that she didn’t know that mathematics could be a career before her undergraduate studies, but thanks to encouraging professors and summer research programs was able to see and experience different types of math that she found incredibly interesting. She’s also one of the first people from her high school to earn a Ph.D.

As an elementary school student, Hartmann found herself interested in a number of subjects, but focused on math after scoring well on a regional competition and as a result, became more involved with math through summer programs. After speaking with a Ph.D. student, she spent time exploring university-level courses and found that math suited her best. 

Regardless of the conference’s technical outcomes, Hartmann is encouraged to see the next generation of mathematicians becoming engaged and collaborative early. “It’s good to find something that then will evolve into some bigger project that you can work on,” she says. “Any publication is a good publication, but it’s more important for them to create the network.” 

Renee Bell is a Hans Rademacher Instructor of Mathematics in the Department of Mathematics in the School of Arts and Sciences at the University of Pennsylvania.

Julia Hartmann is a professor in the Department of Mathematics in the School of Arts and Sciences at the University of Pennsylvania.

Valentijn Karemaker was a postdoctoral fellow in the Department of Mathematics in the School of Arts and Sciences at the University of Pennsylvania. She will begin working as a tenure-track assistant professor at the Mathematical Institute of Utrecht University in July.