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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.