UC Berkeley

Commutative Algebra and Algebraic Geometry Seminar

February 24, 2015

939 Evans Hall

3:45PM: Exponential Varieties

Bernd Sturmfels

Exponential varieties arise from exponential families in statistics. These real algebraic varieties have strong positivity and convexity properties, generalizing those of toric varieties and their moment maps. Another special class, including Gaussian graphical models, are varieties of inverses of symmetric matrices satisfying linear constraints. We present a general theory of exponential varieties, with focus on those defined by hyperbolic polynomials. This material appeared in a recent paper with Mateusz Michalek, Caroline Uhler, and Piotr Zwiernik.

5:00PM: Monotonicity of Hilbert functions

Justin Chen

One of the most natural questions that can be asked of a Hilbert function is whether or not it is (always, weakly) monotonic, i.e. never has a strict decrease. A long-standing conjecture in this area is that the Hilbert function of any one-dimensional complete intersection is always monotonic. I will survey various results, ranging from classical to fairly recent, relating to when the Hilbert function of a 1-dim Cohen-Macaulay ring or module is or is not monotonic.

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