5:00AM: Arf Rings and Strict Closure of SemigroupsClaudiu Raicu

3:45PM: On the settheoretic versions of conjectures of HoltzSturmfels and LandsbergWeymanLuke OedingA principal minor of a matrix is the determinant of a submatrix centered about the main diagonal. A basic linear algebra question asks to what extent is it possible to prescribe the principal minors of a matrix. We focus on the case of symmetric matrices. The equations the algebraic variety of principal minors of symmetric matrices tell when it is possible for a given vector of length 2^n to be the principal minors of a symmetric nbyn matrix. Holtz and Sturmfels found these equations in the cases n=3,4 and discovered an interesting connection to Cayley's hyperdeterminant. They conjectured that the hyperdeterminantal module generates the ideal of the variety. Starting with examples using basic linear algebra I will describe some of the questions that motivated the original question and show some of the beautiful symmetry and geometry of this variety. Next I will sketch a proof of the settheoretic version of the HoltzSturmfels conjecture. Finally I will point out an unexpected connection to the tangential variety of the Segre product of projective spaces, and sketch a proof of a settheoretic version of the LandsbergWeyman conjecture.
