Georgia Institute of Technology College of Sciences

We discuss the theory of symmetric Groebner bases, a concept allowing one to prove Noetherianity results for symmetric ideals in polynomial rings with an infinite number of variables. We also explain applications of these objects to other fields such as algebraic statistics, and we discuss some methods for computing with them on a computer. Some of this is joint work with Matthias Aschenbrener and Seth Sullivant.

The critical group of a graph G is an abelian group K(G) whose order is the number of spanning forests of G. As shown by Bacher, de la Harpe and Nagnibeda, the group K(G) has several equivalent presentations in terms of the lattices of integer cuts and flows on G. The motivation for this talk is to generalize this theory from graphs to CW-complexes, building on our earlier work on cellular spanning forests. A feature of the higher-dimensional case is the breaking of symmetry between cuts and flows. Accordingly, we introduce and study two invariants of X: the critical group K(X) and the cocritical group K^*(X), As in the graph case, these are defined in terms of combinatorial Laplacian operators, but they are no longer isomorphic; rather, the relationship between them is expressed in terms of short exact sequences involving torsion homology. In the special case that X is a graph, torsion vanishes and all group invariants are isomorphic, recovering the theorem of Bacher, de la Harpe and Nagnibeda. This is joint work with Art Duval (University of Texas, El Paso) and Caroline Klivans (Brown University).

In many applications in engineering and physics, one is interested in computing real solutions to systems of equations. This talk will explore numerical approaches for approximating solutions to systems of polynomial and polynomial-exponential equations. We will then discuss using certification methods based on Smale's alpha-theory to rigorously determine if the corresponding solutions are real. Examples from kinematics, electrical engineering, and string theory will be used to demonstrate the ideas.

Let I be an ideal in a polynomial ring R := F[x_n,...,x_1] Let A := R/I be the corresponding quotient ring, and let Q(A) be its field of fractions. The integral closure C(A, Q(A)) of A in Q(A) is a subring of the latter. But it is often given as a separate quotient ring, a presentation.Surprisingly, different computer algebra systems (Magma, Macaulay2, and Singular) choose to produce very different presentations. Some of these opt for presentations that have seductive forms, but miss the most important, namely a form that allows for determining when elements of Q(A) are in C(A,Q(A)). This is called membership and is directly related to determining isomorphism.