# CMU Algebra and Combinatorics Seminar

## Spring 2019

### Meeting Times

Fridays, 11:15am–12:15pm, in Pearce 226.

### Schedule

 Date Speaker Affiliation Title 02/22 Steven Spallone Indian Institute of Science Education and Research Partition Arithmetic and Representation Theory 03/29 Eloísa Grifo University of Michigan Symbolic power and the containment problem 04/19 Bakul Sathaye Wake Forest University Boundaries of nonpositively curved manifolds and groups 04/26 Savannah Swiatlowski Central Michigan University Edge-Colored graphs and nilpotent Lie algebras

### Abstracts

Speaker: Steven Spallone
Title: Partition Arithmetic and Representation Theory
Abstract: There is a notion of dividing a partition by a natural number, with quotient and remainder. This theory can be used to solve various enumeration problems in the representation theory of $S_n$. For example, Macdonald used it to classify the irreducible representations of odd degree. We have used it to classify the irreducible representations with nontrivial determinant. Currently we are applying this theory to classify those with "spin structure". Asymptotically as $n \to \infty$, "$100\%$" of these representations have even degree, trivial determinant, and a spin structure. This talks will feature results from joint work with Amritanshu Prasad, Arvind Ayyer, and Jyotirmoy Ganguly.

Speaker: Eloísa Grifo
Title: Symbolic power and the containment problem
Abstract: Hilbert's Nullstellensatz gives a dictionary between algebra and geometry: solution sets to polynomial equations over the complex numbers (varieties) translate to (radical) ideals in polynomial rings. A classical theorem of Zariski and Nagata gives a deeper layer to this correspondence: polynomial functions that vanish up to a certain order along a variety correspond to a natural algebraic notion, called symbolic powers. Symbolic powers $I^{(n)}$ do not always coincide with the ordinary powers $I^n$, but it is natural to try to compare the two notions. One way to do this is via the Containment Problem, which asks when $I^{(a)} \subseteq I^b$. In this talk, we will introduce symbolic powers and the containment problem, and discuss some recent results.

Speaker: Bakul Sathaye
Title: Boundaries of nonpositively curved manifolds and groups
Abstract: The classical notion of curvature, i.e, sectional curvature, was developed by Riemann in the 19th century. Manifolds with nonpositive sectional curvature have been of interest due to the rich interplay between their geometric, topological and dynamical properties. In 1987, Gromov defined a notion of non-positive curvature for the larger class of geodesic metric spaces. I will define the boundary at infinity of these spaces and discuss how it can be used to construct manifolds that are nonpositively curved and yet do not have a Riemannian metric with non-positive sectional curvature. We will then further extend the notion of nonpositive curvature to groups. In the hyperbolic setting, groups with Menger curve boundary are known to be abundant. However, K. Ruane observed in early 2000s that it was not known whether non-hyperbolic groups have Menger curve boundary. I will end by giving the first examples of non-hyperbolic groups that have Menger curve boundary. Part of this is joint work with C. Hruska and M. Haulmark.

Speaker: Savannah Swiatlowski
Title: Edge-Colored graphs and nilpotent Lie algebras
Abstract: We consider an interesting family of edge-colored graphs. These are complete graphs with odd number of vertices and uniformly colored edges. We prove that the symmetry group of such a graph is a holomorph of the additive cyclic group $\mathbb Z_n$. We will also consider 2-step nilpotent Lie algebras associated to these graphs. We prove that the (Lie) automorphism group of the corresponding nilpotent Lie algebra contains the Dihedral group of order $2n$ as a subgroup.