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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.

All are welcome!

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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.

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Singularities are measured in different ways in characteristic zero, positive characteristic, and mixed characteristic. However, the classes of singularities usually form analogous groups with similar properties, with an example of such a group being klt, strongly F-regular and BCM-regular.  In this talk we shall focus on a mixed characteristic counterpart of F-injective and Du Bois singularities.  We will see that the three have a unified description and satisfy many expected properties.

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In positive characteristic, the F-signature can be viewed as a quantitative measure of F-regularity – an important class of singularities central to the celebrated theory of tight closure pioneered by Hochster and Huneke, and closely related to
Kawamata Log Terminal (KLT) singularities via standard reduction techniques from characteristic zero. The definition can also be extended to divisor (or hypersurface) pairs, and the resulting F-signature functions given by scaling enjoy a number of nice properties such as convexity. In this talk, I will give an overview of some of the positive characteristic theory, and detail recent progress in developing an analogue in the mixed characteristic setting. Based in part on joint work with Hanlin Cai, Seungsu Lee, Linquan Ma, and Karl Schwede, this involves leveraging the perfectoidization functor of Bhatt-Scholze to define the perfectoid signature. In addition, I will mention joint work in progress with a subset of the same authors to extend this definition to pairs and show properties of the resulting perfectoid signature functions given by scaling.

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In this talk, I will discuss two methods: the 0th local cohomology and the bigraded algebra, that are used to extend the classical multiplicities such as the Hilbert-Samuel multiplicity, the Hilber-Kunz multiplicity, the mixed multiplicity from 0-dimensional ideals to general ideals.

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