## Research

My research area is commutative algebra, which concerns commutative rings and modules over them. Much of my work has involved studying these classical commutative algebra objects from a combinatorial perspective, which has been very fruitful. Combinatorial commutative algebra is a relatively new field, beginning in the 1970s with the work of Stanley and Hochster on simplicial homology and squarefree monomial ideals. The field has produced many notable areas of study, including the development of the theory of Stanley-Reisner rings and simplicial complexes, the connections between graphs and their edge ideals, and semigroup rings and toric varieties.

One fundamental problem in commutative algebra concerns the behavior of powers of an ideal *I* in a commutative Noetherian ring *R*. The Rees algebra is a classical commutative algebra construction whose properties give a lot of information about these powers, especially concerning their integral closures. When a second ideal is also considered, we would like to understand how powers of *I* and *J* relate to one another. My research concerns an analog of the Rees algebra called the intersection algebra, an object that came from commutative algebra with connections to algebraic geometry, combinatorics, number theory, and optimization. Much like the Rees algebra does for one ideal, the intersection algebra captures the information required to understand these relationships for two ideals. This algebra involves many of my mathematical interests, specifically combinatorial commutative algebra and designing and implementing computational algorithms, especially in Macaulay^{2.}

Studying this algebra has also lead to some other interesting examples of finitely generated algebras which have similar properties to the intersection algebra, which we are calling fan algebras. All of these examples come from certain semigroups in ^{2}. One of my current interests is in uniting these examples under one cohesive theory, and exploring their properties.

View Research Statement in its entirety as PDF

Publications:

On the Intersection Algebra of Principal Ideals *Sara Malec*

Intersection algebras for principal monomial ideals in polynomial rings *Florian Enescu, Sara Malec*

Intersection Algebras and Pointed Rational Cones *Sara Malec under the direction of Florian Enescu*

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