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A numerical semigroup is a subset of the natural numbers that is closed under addition. There is a family of polyhedral cones \(C_m\), called Kunz cones, for which each numerical semigroup with smallest positive element \(m\) corresponds to an integer point in \(C_m\). It has been shown that if two numerical semigroups correspond to points in the same face of \(C_m\), they share many important properties. In this way, the faces of the Kunz cones naturally partition the set of all numerical semigroups into "cells" within which any two numerical semigroups have similar algebraic structure. In this talk, we survey what is known about the face structure of Kunz cones, and how studying Kunz cones can inform the classification of numerical semigroups. No familiarity with numerical semigroups or polyhedral geometry will be assumed for this talk.

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Over a polynomial ring, an ideal \(I\) is said to be linearly presented if its first syzygy module is generated by linear relations. More generally, the resolution of \(I\) is linear for \(p\) steps if this linearity continues through the first \(p\) steps. In this talk, we explore some fundamental properties of ideals with partially linear minimal free resolution, such as asymptotic behavior of powers, Castelnuovo-Mumford regularity, and lower bounds on the number of minimal generators. In particular, we prove the Eisenbud-Ulrich conjecture on powers of linearly presented ideals and establish a linear effective bound toward the more general Eisenbud-Huneke-Ulrich conjecture.

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