Friday, November 17, 2017 - 10:00 , Location: Skiles 114 , Timothy Duff , GA Tech , Organizer: Timothy Duff
Motivated by the general problem of polynomial system solving, we state and sketch a proof Kushnirenko's theorem. This is the simplest in a series of results which relate the number of solutions of a "generic" square polynomial system to an invariant of some associated convex bodies. For systems with certain structure (here, sparse coefficients), these refinements may provide less pessimistic estimates than the exponential bounds given by Bezout's theorem.
Friday, November 3, 2017 - 10:00 , Location: Skiles 114 , Jaewoo Jung , GA Tech , Organizer: Timothy Duff
We continue our discussion of free resolutions and Stanley-Reisner ideals. We introduce Hochster's formula and state results on the behavior of Betti tables under clique-sums.
Friday, October 27, 2017 - 10:00 , Location: Skiles 114 , Jaewoo Jung , GA Tech , Organizer: Timothy Duff
For any undirected graph, the Stanley-Reisner ideal is generated by monomials correspoding to the graph's "non-edges." It is of interest in algebraic geometry to study the free resolutions and Betti-tables of these ideals (viewed as modules in the natural way.) We consider the relationship between a graph and its induced Betti-table. As a first step, we look at how operations on graphs effect on the Betti-tables. In this talk, I will provide a basic introduction, state our result about clique sums of graphs (with proof), and discuss the next things to do.
Friday, October 20, 2017 - 10:00 , Location: Skiles 114 , Kisun Lee , Georgia Institute of Technology , Organizer: Timothy Duff
We will introduce a class of nonnegative real matrices which are called slack matrices. Slack matrices provide the distance from equality of a vertex and a facet. We go over concepts of polytopes and polyhedrons briefly, and define slack matrices using those objects. Also, we will give several necessary and sufficient conditions for slack matrices of polyhedrons. We will also restrict our conditions for slack matrices for polytopes. Finally, we introduce the polyhedral verification problem, and some combinatorial characterizations of slack matrices.
Friday, October 13, 2017 - 10:00 , Location: Skiles 114 , Libby Taylor , GA Tech , Organizer: Timothy Duff
We will give an overview of divisor theory on curves and give definitions of the Picard group and the Jacobian of a compact Riemann surface. We will use these notions to prove Plucker’s formula for the genus of a smooth projective curve. In addition, we will discuss the various ways of defining the Jacobian of a curve and why these definitions are equivalent. We will also give an extension of these notions to schemes, in which we define the Picard group of a scheme in terms of the group of invertible sheaves and in terms of sheaf cohomology.