[This talk will be preceded by a special reception at 3:30pm honoring the two Michler fellow visiting us this semester: Anna Skripka and Emily Witt]
Introduced by Grothendieck, the notion of local cohomology is defined in a purely algebraic way. However, it also encodes fundamental geometric and topological data. For instance, local cohomology can help determine the number of equations needed to define a variety, or how the irreducible components of a ring's spectrum fit together topologically.
Unfortunately, local cohomology modules can be huge--i.e., they are typically not finitely generated--and the data they carry can be hard to access. It can even be difficult to determine whether a given local cohomology module is zero!
In this talk, aimed toward a general audience, we discuss methods developed to better understand local cohomology, which involve rings of differential operators, invariant theory, and graph theory. We also describe recently discovered connections between local cohomology and geometry/topology.