The abstract world of mathematics reveals many hidden realities of the physical world. Rodrigo De Pool Alcantara, a postdoctoral researcher at the University of Notre Dame in the Department of Mathematics, acknowledges and applies this idea to the concept of symmetry.
Working in the labs of Andrew Putman, professor of topology, and Nick Salter, assistant professor, De Pool researches group theory and topology. To him, understanding symmetry offers a gateway to uncovering the unseen patterns that govern mathematical and physical systems alike. His work delves deeply into the ways in which shapes, objects, and equations can transform yet remain fundamentally the same.
De Pool significantly focuses on group theory, the field of mathematics that studies symmetries. To visualize this, think of a snowflake. You can rotate a snowflake and still see the same picture. In mathematics, this is made precise with the notion of a group–a structure that records all possible transformations which preserve the essence of an object. This idea of a group appears in many areas of science as well, like particle physics, crystallography, and classical mechanics. Group theory helps to discover the properties of the system which are not apparent at first sight.
De Pool studies two main families of groups: mapping class groups and braid groups. Each provides a unique perspective on how complex systems behave when undergoing transformation.
Mapping class groups capture the symmetry of surfaces, which can be stretched, twisted, or flipped, but not torn. For instance, an equation with two (complex) variables defines a surface within four-dimensional space. De Pool investigates the symmetries of such surfaces, exploring how they can be deformed in continuous yet distinct ways. These transformations collectively form the mapping class group. His work examines how these groups relate to each other through structures known as homomorphisms, which are functions that preserve the essential group properties.
These investigations are known as rigidity problems in group theory. “The rigidity conjecture for mapping class groups provides a geometrical description for the interactions between these groups” says De Pool. His work attempts to give these ideas a concrete description.
The other family of groups that De Pool studies are known as braid groups. These groups can be visualized as intertwining strands, with each unique pattern of intertwining representing an element of the braid group. Braid groups appear in many areas of science, but he studies them purely from the mathematician’s point of view. As with mapping class groups, he solves rigidity problems for braid groups to determine how they interact with each other, seeing what properties are preserved and what properties must change.
Early on as a student, De Pool was interested in the power of abstractly studying these symmetry groups. He has specifically taken a liking to mapping class groups and braid groups because “they live in between group theory and topology, where one can use geometric intuition to study groups,” says De Pool.
Born and raised in Venezuela, De Pool moved to Valencia, Spain to finish high school. He then did his undergraduate schooling at the Autonomous University of Madrid, completing a double major in mathematics and computer science. Furthering his education in Madrid, he received a master's degree in mathematics from the Autonomous University of Madrid and received his doctorate from the Institute of Mathematical Sciences (ICMAT).
When looking for postdoctoral programs in mathematics, De Pool looked mainly to places that did research close to his. He came to admire Notre Dame for its large math research group and prestigious researchers, especially in geometry and topology–the areas of his interest. In his free time outside of his research, De Pool enjoys running and Latin dancing, including salsa, bachata, and merengue. He is excited to further his career at Notre Dame and looks forward to getting to know the community.