Abstract: In this talk, we discuss the relationship between the Castelnuovo–Mumford regularity, which measures algebraic complexity, and the geometric configurations of zero-dimensional schemes. If Γ is a zero-dimensional scheme, then its Castelnuovo–Mumford regularity can be understood as the number 1+d, where d is the smallest degree for which the restrictions of homogeneous polynomials of degree d are sufficient to express all regular functions on Γ. If the scheme is reduced, this is equivalent to determining the degree of interpolating polynomials. 

As our main result, we describe the geometric configuration of Γ when its regularity is close to the known upper bound \ceiling((d-n-1)/t(Γ))+2, where t(Γ) denotes the smallest integer t such that Γ admits a (t+2)-secant t(Γ)-dimensional space. This is joint work with Euisung Park.

Recently, progress has been made in the minimal model program (MMP) for foliated varieties. A natural question arises: do foliations with some fixed invariants form a bounded family? In the first part, I will begin with an introduction on the foliations and MMP. In the latter part, I will talk on the boundedness results on foliations, including foliated surfaces of general type and Fano toric foliations.