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Lovász-Saks-Schrijver ideals, LSS ideals for short, are a family of ideals associated to graphs that were introduced in the context of orthogonal representations of graphs and studied for the first time in 1989 by Lovász, Saks and Schrijver. The study of such ideals lie in the intersection between algebraic geometry, commutative algebra and combinatorics as some geometric and algebraic properties can be exhibited from combinatorial invariants of the graph and viceversa. Our goal is to study this relationship, focusing on some algebraic properties of LSS ideals such as defining a unique factorization domain.

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Koszul algebras appear in many areas of algebra, geometry and topology. Two of their
remarkable properties are that the minimal free resolution of the ground field k can be explicitly
described as a generalized Koszul complex, and there is an elegant formula relating the Hilbert
function of the algebra and the Poincare series of k. Consider the binomial edge ideal J(G) associated
to a finite simple graph G. Such an ideal is always generated by quadrics. It has a quadratic Groebner
basis if and only if the graph G is closed. There are many other cases when J(G) is Koszul. The study
of Koszul binomial edge ideals was initiated by V. Ene, J. Herzog, and T. Hibi in 2014. We characterize
the Koszul binomial edge ideals by a simple combinatorial property of G. This is joint work with 
A. LaClair, M. Mastroeni, and J. McCullough.