Grants and Contracts per year

## Grants and Contracts Details

### Description

Algebraic Geometry and Commutative Algebra have their common roots in the study of
solutions of polynomial equations. The set of common solutions is a geometric object such
as an ellipse and called a variety or a subscheme if one also takes the multiplicities of the
solutions into account. The polynomials in the equations generate an algebraic object, called
ideal. Liaison Theory addresses the question how one can decide if two geometric objects have
similar properties of one knows only their defining equations. Important contributions have
come both from the algebraic side and from the geometric side, and equally important aspects
are the theoretical results and the open questions on one hand, and the myriad ways that
liaison can contribute to seemingly unrelated topics on the other hand.
On the theoretical side, the known picture reveals a beautiful panorama of complete results
and highly suggestive partial results, begging for a resolution of the conjectures which, if true,
would make an incredibly elegant unifed whole. The new idea of enlarging the space in which
links take place has lead to further evidence that this is indeed possible. On the application
side, liaison theory has been used to answer question on Hilbert functions and study various
determinantal and monomial ideals.
Another focal point of my work has been the problem of developing new methods for deciding
wheather a graded algebra has the Weak Lefschetz Property. This property has its origins in
the Hard Lefschetz Theorem of topology. However, recent results show that it is related to
many other areas. For examples, recently new connections to combinatorics were established
because the presence of the Weak Lefschetz Property is intimately related to enumerations of
lozenge tilings and of families of non-intersecting lattice paths.

Status | Finished |
---|---|

Effective start/end date | 9/1/14 → 8/31/19 |

### Funding

- Simons Foundation

## Fingerprint

Explore the research topics touched on by this project. These labels are generated based on the underlying awards/grants. Together they form a unique fingerprint.

## Projects

- 1 Finished