# The pursuit of discovering new things: Meet Cintia Pacchiano, our PIMS PDF at the University of Calgary.

By Lisa Sammoh, Communications and Events Assistant.

Cintia Pacchiano is one of the 2022 PIMS Postdoctoral Fellowship (PDF) cohort, and is currently at the University of Calgary. Her research area is in mathematical analysis, and more precisely, the theoretical aspects of nonlinear partial different equations (PDEs). She completed her Ph.D. at the Aalto University School of Science in Finland, as part of the Nonlinear PDE (NPDE) research group under the supervision of Prof. Juha Kinnunen. During her PhD, Cintia discovered the analysis on general metric measure spaces. She also obtained her Master’s degree from the University of Bonn, and a Bachelor’s in Mathematics from Universidad Nacional Autónoma de México (UNAM). We connected with Cintia to learn more about her academic career, and how the experience of moving to Calgary, Alberta has been like for her so far.

**Tell us about yourself and the academic work you do.**

My enthusiasm for mathematics first grew after taking an Analysis course in my undergraduate degree, and I felt it deepened the concepts I had already learned in Calculus and Linear Algebra by capturing their essence and extending them to other spaces different from the Euclidean one. From this, I learned that we could in fact prove the existence of solutions to differential equations. All of this would not have been possible without the encouragement of my professor, Mónica Clapp, whose supervision and guidance led me into the more analytical area of mathematics, leading up to my bachelor’s thesis on the “Brezis-Nirenberg’s Problem”. The work confirmed my preference for studying Partial Differential Equations and Variational Methods. While in Germany for my M.S. in Mathematics, I worked on the Boltzmann Equation, particularly on “Hilbert expansions in Kinetic Equations”, under the supervision of Prof. J.L.L. Velázquez.

I then joined Aalto University in Finland as a PhD candidate and was part of the Nonlinear PDF research group. There, I discovered the analysis on geometric metric measure spaces, and more specifically on two research themes forming a common thread: the Total Variation Flow (TVF) and Quasiminimizers of a (p,q)-Dirichlet integral. My group and I worked on several papers during this time period — in the papers “Variational solutions to the total variation flow on metric measure spaces” ( Nonlinear Analysis; 220) and “Existence of parabolic minimizers to the total variation flow on metric measure spaces” (manuscripta mathematica; 170), in general terms, we first defined variational solutions to the total variation flow (TVF) in metric measure spaces. We establish their existence and, using energy estimates and the properties of the underlying metric, we give necessary and sufficient conditions for a variational solution to be continuous at a given point. As far as we know, this is the first time that existence and regularity questions are discussed for parabolic problems with linear growth on metric measure spaces.

My research area is in mathematical analysis, and more precisely, theoretical aspects of nonlinear partial differential equations (PDEs). In the past years, I have focused on the existence and regularity properties of functions related to the calculus of variations on metric measure spaces that support a weak Poincaré inequality and doubling measure. I have concentrated especially on variational solutions to the total variation flow, and quasiminimizers to the (p,q)-Dirichlet integral. The main interest of this work is to extend some classical results of the calculus of variations to metric measure spaces. Variational methods appeared as an answer to the problem of finding minima of functionals. It is about giving a necessary and sufficient condition for the existence of the minimum, as well as conditions that allow its calculation and algorithms that let us compute it. Variational calculus is intimately linked with the theory of partial differential equations, since the conditions for existence of a solution to the minimization problem normally depend on the fact that said solution satisfies a certain differential equation.

The subject of analysis, more specifically, first-order calculus, in metric measure spaces provides a unifying framework for ideas and questions from many different fields of mathematics. Analysis on metric spaces is nowadays an active and independent field, bringing together researchers from different parts of the mathematical spectrum. It has applications to areas as diverse as geometric group theory, nonlinear PDEs, and even theoretical computer science. This can offer us a better understanding of the phenomena and also lead to new results, even in the classical Euclidean case.

**How did you come across the PIMS position? What are some experiences you have had since beginning as a PDF fellow at the University of Calgary?**

I came across the PIMS PDF position afterward, and was lucky to have had a chance to speak to Profs. Cristian Rios and Tracey Balehowsky (UCalgary) before applying. I was happy I had found problems and applications to work on that resonated so well with my work and that we were both interested in. Not only that, it was exciting to know there was a plethora of questions still that we could ask and learn new things from. My scientific goals are to do high impact research, publish the results in top journals, and to participate in international collaboration related to PDEs, variational methods and potential theory. I have found that research allows you to pursue your interests, to learn something new, broaden your mind, and challenge yourself in new ways. The work is hard and the standards are rigorous, but I find it to be a rewarding and enjoyable experience.

Since starting, and apart from our ongoing research, I am also teaching a course this semester — Math 277, Multivariable Calculus for Engineers and Scientists. I am also leading a small group of graduate students as part of the PIMS First-Year Interest Group (FYIG) program, where I present them with topics related to Variational Methods and Partial Differential Equations. I think having had the chance to also previously teach in two countries (Finland and Mexico) with different educational systems over the course of my academic career has been a fortunate experience as well. My long-term professional goal is to teach and research. I deeply enjoy being a teacher, and I believe that to become a good one, you have to keep learning. To be able to inspire and support others as my professors have with me is also important. Students in general need to be challenged alongside offering support. We should dare to let them think on their own.

**Outside of work, what has life been like in Calgary?**

I always try to make space for my hobbies. I enjoy running, painting, knitting, and playing the piano. I also enjoy taking pictures; mostly of animals, as I have always loved them. It’s a pastime I have come to take quite seriously since the pandemic began. Otherwise, if I am not training for a marathon, then my weekends are usually for long distance running. Afterwards, I will lay on my couch to watch a series or listen to some Bob Dylan music. I also recently went to Banff (Alberta) for the first time and that was very beautiful. Living in a new country gives me the opportunity to explore and discover new things every day.

*Cintia will be speaking at the PIMS Emergent Research Seminar Series, on February 22, 2023, at 9:30 AM Pacific. Details on her talk and poster, ***Total Variation Flow on metric measure spaces**, *can be found **here**.*