# Interview with Professor Heekyoung Hahn

#### "Math is not something to be afraid of. Rather, it is something that you should enjoy playing with."

Professor Heekyoung Hahn was interviewed in Girls' Angle Bulletin's most recent issue.  The magazine is published six times a year by Girls’ Angle: A Math Club for Girls, to communicate with its members and to share ideas and information about mathematics.  The article is reprinted below from the Girls’ Angle Bulletin, Volume 11, Number 5.

##### An Interview with Heekyoung Hahn

Heekyoung Hahn is an Assistant Research Professor in the Department of Mathematics at Duke University. She received her doctoral degree in mathematics from the University of Illinois at Urbana-Champaign under the supervision of Bruce Berndt.

Ken:  What’s an early memory of something that excited you to mathematics?
Heekyoung:  From my big brother’s friend (I had a big family, I am the youngest one) I heard that mathematicians can “count” all the natural numbers (yes, infinitely many numbers). Even more surprising to me, the number of all the natural numbers is equal to that of the integers. I was about a 4th or 5th grader at that time. At that time I knew, for sure, what the natural numbers and the integers were. I thought, if one could “count” them (which was crazy), the number of all the integers should be twice that of the natural numbers + 1 (because you have the positive numbers, negative numbers and zero). I really wanted to know if this is the case and wanted to study math more.

Ken: Could you please describe the journey you traveled to become a mathematician?

Ken:  Yours is a remarkable story! What do you enjoy about being a mathematician? What is your life like as a mathematician?
Heekyoung:  Many subjects, including most of science, can describe how things work (based on experimental tests, observations), but math is the one that can explain “why” things work in the way they do. Unlike much of humanity, we seek the simplest and easiest way to explain this “why”, permanently. Clean and simple. I like this. In math, we can even discuss infinite things, unbounded stacks of principles, and we often find the untouchable perfect truth behind them. This fascinates me. There has never been a subject that has beaten math, and there never will be one. In my classroom, I always told my students that if God gave me a second chance at life, then I would certainly choose again to be a mathematician.

Ken:  Does math come easily to you, or is it something you have to work hard at?
Heekyoung:  I would say both. After reading articles or hearing lectures, I usually have some idea or some thoughts on a particular question. But those ideas always require very hard work on my part until all of my misunderstandings and confusions are cleared up. Only then will I have good questions to ask. It is like patching up the puzzle. One misstep could cause complete nonsense. Sometimes, mathematicians have to fight the fear that what we are working on will completely blow up and not work.

Ken:  Your research bridges many fields - algebra, geometry, analysis, number theory… Could you please explain some of the “big” questions that you are interested in and why you find these questions compelling?
Heekyoung:  The Langlands functoriality conjecture is one of the deepest conjectures in number theory unifying the concepts in representation theory, algebraic geometry and the theory of automorphic forms. My current research is motivated by this conjecture. What is fascinating to me in my recent research is that I isolated a concrete question in algebraic group theory from this big conjecture. Moreover, the methods that I used to answer this question boiled down to representation theory and to computing concrete combinatorics problems. I like that because they are quite elementary.

Ken:  What is one of your favorite results that you proved? How did you prove it?
Heekyoung:  One of the main tools to study the relative Langlands functoriality conjecture of Sakellaridis and Venkatesh is via the theory of (relative) trace formulas. There has been so much great work done in this very important area. I am kind of new to this, though I was able to prove a simple twisted version of the relative trace formula over special subgroups. I am very happy that I was able to contribute to this area, although it may be only a little contribution. The proof comes down to integrating a kernel function over special subgroups; the key idea was to choose the subgroups in an interesting way so that one could mimic the twisted trace formula in this setting.

Ken:  When you get stuck on a problem, what do you do to try to get unstuck?
Heekyoung:  STOP thinking about the question for a few days! It is very important for me to undo my obsession on that problem. Do non-math activities, like hiking or taking kids to the children’s museum, and so on. When I come back, I restart the problem again from the very beginning. I revisit each step again to see if I am missing something. Sometimes, at each step, I ask myself, “What if this step were not true at all?” That way, I can confirm that I am at least walking along a right path.

Ken:  You run a math program for high school girls called SWiM, which stands for Summer Workshop in Math. What inspired you to create this program? What is the program like?
Heekyoung:  I have always been interested in promoting math in general. For example, at Duke, I have founded and organized the PLUM lecture series (Public Lectures Unveiling Math) aimed at a general audience and focused on promoting mathematics by presenting inspiring stories about mathematics. Math is not something to be afraid of. Rather, it is something that you should enjoy playing with. It is important for us to think like this, and therefore we should inspire our kids to think this way. Parents’ influence on how their kids think about math and how to enjoy it are crucial. One of the challenges with the SWiM program that Ingrid and I were trying to overcome is to make it enjoyable as well as challenging ("Ingrid" is Professor Ingrid Daubechies - there is an interview with Professor Daubechies in this Bulletin, Volume 1, Number 6, and Volume 2, Numbers 1-4). Just because something is difficult does not mean we can’t enjoy it. Math might be difficult, but that is okay, since we are capable of overcoming it. There is no reason to dumb Math down to make it look easy. In fact, many participants of the SWiM program get excited when they are challenged. All SWiM participants attend two math courses, do afternoon group work, and attend SWiM lectures given by local professors at Duke, University of North Carolina, and North Carolina State University, as well as go on field trips. Participants in the program learn not only about exciting mathematics, but also about discovering new arguments, and explaining insights to their peers. The participants give a group presentation at the end of the program. All the presentations are broadcast in live stream video on the internet. Participants will also attend useful panel discussions as well as participate in social activities. For more information about the program, search for “Duke SWiM math” on the internet or visit the web page:  Duke SWiM 2019.

Ken:  What advice do you have for a teen who aspires to become a mathematician? What should she study? How should she spend her “math time”?
Heekyoung:  Simply enjoy it. Personally, I found it a lot of fun to figure out how certain formulas are formed and why they should be true. To every single formula and a term, ask yourself why it has to be that way, or try to explain to your friends how it works, or perhaps think together to see if there is any room to negotiate, I mean, to make it better or different. Ken: What do you like doing when you’re not doing math? Heekyoung: Hiking. I love hiking. Walking around the trails in the woods and mountains makes me calm and helps me sort out my tangled thoughts and concerns.

Ken: Thank you for this interview!

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Reprinted from the Girls’ Angle Bulletin, Volume 11, Number 5.  The Girls’ Angle Bulletin is a bimonthly publication of Girls’ Angle.  Electronic Version: ISSN 2151-5743  Print Version: ISSN 2151-5700 www.girlsangle.org  © Copyright 2018 by Girls’ Angle, Inc. All Rights Reserved.