Unraveling fractals with Shiping Cao, our PIMS-Simons PDF at the University of Washington.
By Lisa Sammoh, Communications and Events Assistant.
Tell us about your academic journey.
By chance, I delved into the realm of fractals during my bachelor’s degree in physics at Nanjing University (China). Guided by Professor Hua Qiu, I explored energy forms on the Sierpinski gasket, igniting my passion for mathematical structures. Seeking further guidance, I reached out to Professor Robert S. Strichartz (Bob) at Cornell University and joined as a Ph.D. student. Under Bob’s mentorship, I contributed to the existence of Dirichlet forms on Julia sets, which was published in Advances in Math. Despite limited meetings due to Bob’s declining health, his guidance and Hua’s collaboration propelled our exploration of Dirichlet forms on fractals with irregular geometry.
Simultaneously, Professor Laurent Saloff-Coste deepened my understanding of heat kernel estimates and became my advisor after Bob’s passing. For my postdoctoral position, I approached Professor Zhen-Qing Chen, renowned for connecting Dirichlet forms and Markov processes. Working under Zhen-Qing’s guidance, I unintentionally resolved the longstanding question regarding the convergence of effective resistance in pre-Sierpinski carpets, while investigating the homogenization of random environments on fractals.
What field are you in and what is your current research on?
My primary research focuses on Dirichlet forms applied to fractal spaces. Dirichlet forms are generalizations of divergence forms on Euclidean domains, representing bilinear forms with the Markov property on metric measure spaces. The semigroup of a regular Dirichlet form generates a Hunt process. By employing this analytical approach, we can construct and analyze stochastic processes on fractals.
During my Ph.D., I successfully constructed local regular self-similar Dirichlet forms on specific Sierpinski carpets with reduced local symmetry. This construction utilized the strong recurrence of Markov processes in low dimensional spaces, and I aim to extend this understanding to higher dimensions in future work.
Additionally, I am collaborating with Zhen-Qing on investigating the homogenization of random environments on self-similar sets. This involves assigning random conductance weights to each copy of the fractal or the approximating domain, with the goal of comprehending the scaling limit.
What does research and life balance mean to you?
I prefer taking one or two days off during the weekend given the absence of any urgent tasks. I find great joy in spending quality time with my friends, often through shared dinners. During my Ph.D. studies, I developed a fondness for baking cakes and sharing them with friends.
Also, near Seattle, there are abundant natural landscapes to explore, such as Mount Rainier, Olympic Park, and North Cascade. Summer provides an excellent opportunity to appreciate nature alongside friends. Taking a brief hiatus after a few days of work rejuvenates the mind, and I have occasionally discovered new ideas during these trips.
What are you looking forward to learning/ seeing about Washington and your postdoctoral position at the university?
At UW, there is a thriving probability group that is exciting to be a part of as I delve into new topics in probability. Furthermore, I want to enrich my teaching skills. Currently, I am instructing a linear algebra course and actively seeking to understand and address the needs of my students. Building connections with a few students towards the end of the quarter was a delightful experience too.
Shiping will be speaking at the PIMS Emergent Research Seminar Series, on May 24, 2023, at 9:30 AM Pacific. Details on his talk and poster, Convergence of resistances on generalized Sierpinski carpets, can be found here.
