For the first four years of my sentence, I worked to learn mathematics in total isolation. It was my safe, lonely hiding place away from the nightmare in my head and all around me, where I could vent all my frustrations and hopes into something meaningful, tangible... something I could cling to during the most chaotic period in my life. Mathematics gave me this beautiful oasis that I could escape to mentally, providing reprieve from that miserable, suffocating shadow that was cast over my life. But the pain I felt was woven into the fabric of my mathematical world. Although I knew it was unhealthy, the pain felt necessary for what I was doing. Math called to me in my darkest, most painful moment, and recognizing that, I continued to draw from that pain to fuel my studies. In that way, I justified my unwillingness to let go of it, holding my wounds open rather than letting them heal. But beginning a little over four years into my sentence, something started to change.

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When I first learned of the Prison Mathematics Project through a 2018 story in Math Horizons, I felt completely detached from the rest of the world. Despite having corresponded with a few professors, I still felt an immeasurable distance between myself and the world of "true mathematicians." I had studied and learned a lot of mathematics, I had begun to dip my toes into a specific field of research, and I had started to internalize the belief that if I kept on this trajectory, I could make a real impact in math. But I also had major reservations about whether I'd ever be accepted or taken seriously. It wasn't that I needed acceptance; I love math for its own sake and always will, whether I have people with whom to share that love or not. But I recognized that mathematics at its best is a community. People build off the work of other people, have conversations, and collaborate. Perhaps I could achieve my goals on my own, and indeed that twisted, rebellious flame inside me that was already fueling my work seemed to grow slightly brighter at the thought of being shunned or rejected. But deep down, I knew that to be my best and to truly reach my potential, I'd need other people. I do not doubt that throughout my career I will encounter people who will learn of my background and view me as an outsider, or worse. But the PMP has shown me that countless people believe I deserve kindness, acceptance, and a mathematical community that is free from judgment. Their existence alone is proof of that fact, and they have gone above and beyond to verify it to me.

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The people I spoke to within the PMP also challenged me. I am highly internally motivated, so it was a real surprise to be pushed. Yet that was exactly what I needed in the moment. I explained my passion for mathematics and for my specific field, describing the problems I was working on in detail, and that I hoped to continue honing my skills in the areas needed in my field, like the deeper applications of complex and microlocal analysis. I thought perhaps the PMP could help me locate appropriate textbooks or papers (which is difficult to do on my own, having no direct internet access), or answer this or that technical question as they arise by connecting me with a specialist in the area. Instead, I was strongly encouraged by several people to reach out directly to the top researchers in my field.

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I was hesitant. Mathematics and, in particular,the theory of scattering resonances had become my whole world. As such, I had essentially deified the top researchers in my field. My biggest concern was reaching out too early, before I had something tangible to share. I recalled the advice I had received about a year earlier: "It is not enough to read and understand mathematics... You must be able to do something of your own with it." I had some ideas, but no real results to show for my research, and I didn't want to waste the time of the people who revealed this world to me. I thought I should just keep working until I had some decent results or a draft of a paper to share with someone.

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I meditated, as I often do when faced with a decision. I liken meditation to listening to music. I let the thoughts play in my head like songs in a playlist, not trying to control their direction anymore than one would try to control the direction of a song playing through headphones. I just listen, trying to feel the music of my mind. In the same way that hearing the right song at the right time adds that extra dimension of beauty to life for that moment, when the right thought passes through my mind, I know—intuitively—that it has some deeper purpose for being there. I believe, through many experiences, that if you listen to the music closely enough, things reveal themselves this way.

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In this particular meditation, the music told me to write the person whose work was inspiring my first attempts at research: Professor Tanya Christiansen. I had no way to know it at the time, but this decision would end up being one of the most important in my life, as it led to the mentorship I needed to grow not only as a mathematician, but as a person.

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At the time I first reached out to Professor Christiansen, I had read every paper of hers I could find. I had fallen in love with the mathematical theory of resonances, and her work played a huge role in that. I already mentioned one of her results in my last post, namely a proof that there are potentials that produce no resonances, which struck all the most beautiful mathematical chords in my mind; it is such an unexpected result, given that all potentials in dimension one have infinitely many resonances, and it really showed me the depth and wonder of this field. But it was a different idea of hers that inspired my first ever attempts at research, and it was this idea that primarily occupied my first letter to her.

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I was inspired by her approach to a certain open problem in our field, namely, the existence of resonances and asymptotic growth of the resonance counting function. Her approach was to prove that there are many potentials with resonance counting function that achieve the optimal order of growth by combining a specific example of Zworski with methods from the theory of several complex variables. Taking my own spin on the idea, I wanted to use this same example, but to relate it to perturbations of the potential by comparing scattering determinants and trying to obtain some uniform resonance stability results. I wrote to her, introduced myself, and asked her what she thought about my idea. Though I felt confident in my abilities and that I was making the right choice in writing her, I still thought it was highly unlikely that she would reply. It just didn't seem possible that someone so successful in this field would respond to some vague ideas from someone in my position...

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When I first received her response, I actually thought somehow this must be from a different Tanya Christiansen. My mind couldn't grasp the reality that she wrote back to me! Yet there it was, thoughtful, considering my questions and asking some of her own... offering bits of advice, taking my ideas seriously. The fact that she took the time to respond to me was extremely validating and encouraging. With every fiber of my being engaged, I hung on every word she wrote and couldn't wait to start looking into the questions she asked me.

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There are certain untethered events in life that transcend the moment in which they happen, like that initial meditation early in my sentence when mathematics first called to me, or the moment I learned that my first paper was published, cementing an achievement I'd poured my soul into to obtain. The energy of those moments is so strong that it connects all past and all future versions of myself to it; I can return to those moments whenever I want and feel them as though they were happening all over again. Reading that letter from Professor Christiansen is one of those moments, and it marks the beginning of another massive change in my life that wouldn't have been possible without her.

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Over the course of our correspondence, she not only showed me the ropes of the mathematical research process, teaching me more about math in the first year than I had taught myself during the previous 5 years combined. She also built my confidence and convinced me that I belong in mathematics and in our field. It can only be the result of genuine thoughtfulness and expertise to repeatedly ask me questions at just the right level of difficulty, nudging me through sticking points but pushing me to work things out on my own whenever possible. I worked on the things she suggested, thought about her ideas, always trying to build from her suggestions as a starting point, adding something more each time. I took notes not only on what she was asking me, but also on the specific way she asked it and why. "Can you quantify that statement?" "Can you prove this by a different method?" I'd learn that these kinds of questions lead to more results and better proofs. She gave me a glimpse into her own research process, and I absorbed it like a sponge.

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It was the mentorship I needed to really understand what mathematical research was all about, but it was so much more than that. I found myself reflecting daily on the juxtaposition of our two worlds: mine, dark and isolated, and hers full of the mathematical achievement I longed for. I couldn't help but be inspired by the kind of person she must be to offer me her time and expertise freely. And like the sun rising slowly but surely over the horizon, her wisdom and the respect she showed me during our collaboration swept that darkness out of my life, replacing it with something much stronger.

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Gratitude. Every single day, I wake up grateful for what she did for me. What probably seems like a simple thing to most people—guiding me in the research process and taking the time to teach me about our field—meant more tome in those dark days than I could ever explain, and it has continued to carry me through the toughest period in my life. It also gave me a new goal, one beyond mathematical achievement alone: to rid my world of darkness, to allow myself to heal, and to be a source of positivity and inspiration in this world, as she has been for me.

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Header photo by ArtemSapegin on Unsplash.

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