Equivariant Bordism Theory and Applications (23w5138)



Colleen Delaney (UC Berkeley)

Rolando Jiménez (UNAM)

(Purdue University)

(Universidad del Norte)


Imagine your favorite geometric object in nature. Chances are it exhibits pleasing symmetry. For example, a snowflake exhibits the same symmetries as a hexagon. Humans' innate attraction to symmetric geometric objects makes their mathematical study intrinsically appealing. However, there are an overwhelming number of ways in which geometric objects can exhibit symmetry, and so mathematicians have devised various approaches to understanding geometric symmetries. The main subject of this 5-day workshop is ``equivariant bordism theory," which is a mathematical framework that combines techniques from algebra with geometry in a way that makes it possible to sort through symmetric geometric objects systematically.

In addition to further developing equivariant bordism theory directly, the workshop participants will apply this theory to understand symmetries of high-dimensional objects in terms of lower-dimensional objects with similar symmetries, and, conversely, to understand when it is possible for an object with symmetry to be used to build a higher dimensional object with the same symmetry. The participants will also apply equivariant bordism theory to certain problems arising in quantum physics. Algebraic aspects of symmetry have been exploited to great effect in recent years in order to understand certain quantum mechanical systems called ``topological quantum field theories," and the workshop will push these applications further by coupling them to the geometric techniques enabled by equivariant bordism theory. The workshop activities have been carefully organized with the goal of nurturing new cosmopolitan research collaborations that unite experts and students from diverse backgrounds---both mathematical and personal.