Classification of singularities is an interesting problem in many areas of algebraic geometry, like the minimal model program. One classical approach is to assign to a singular subvariety a rational number, its log canonical threshold. For complex hypersurface singularities, this invariant has been refined by M. Saito to the minimal exponent. This invariant is related to Bernstein-Sato polynomials, Hodge ideals and higher Du Bois and higher rational singularities.

In joint work with Qianyu Chen, Mircea Mustață and Sebastián Olano, we defined the minimal exponent for LCI subvarieties of smooth complex varieties. We relate it to local cohomology, higher du Bois and higher rational singularities. In this talk, I will describe what was done in the hypersurface case, give our definition in the LCI case and explain the relation to local cohomology modules and the classification of singularities.